# Temporal Uniformity

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Hi Michel,

I can render some different views of the graph for you. But I will try and explain what you are seeing.

1.) Mathematica renders the graphs inside a box. I can remove this box if you like. It is only there to show the axes and provides a ruler.

2.) I'm sorry about that effect. I simply set the view such that the axis lined up that way. It removed a lot of clutter while keeping certain lines that I wanted to display.

3.) The green and aqua rings represent the expansion of the sphere which are growing at a constant velocity. Those rings also can be viewed as temporally displaced views. The blue strips, which represent the clocks, encompasses those triangles structures you are talking about. I can remove all of the reference points around the trianglular paths if you like. They are there to show you how everything stays aligned with the sphere. If you look at the triangle structure in the blue strips, they are getting wider and wider for the images that show acceleration.

The acceleration you see in the above graphs does not mean that the 4D sphere is expanding at an accelerated rate. The sphere is expanding at the speed of light. It is the clocks, that are traversing across the surface of this expanding sphere, that are accelerating as shown in most of the images from the previous post.

I also forgot to mention that I plot the up and down path of photons for each clock. That is why you see the diamond shape paths inside the blue strips.

Edited by Daedalus

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Hi Michel,

I can render some different views of the graph for you. But I will try and explain what you are seeing.

1.) Mathematica renders the graphs inside a box. I can remove this box if you like. It is only there to show the axes and provides a ruler.

2.) I'm sorry about that effect. I simply set the view such that the axis lined up that way. It removed a lot of clutter while keeping certain lines that I wanted to display.

3.) The green and aqua rings represent the expansion of the sphere which are growing at a constant velocity. Those rings also can be viewed as temporally displaced views. The blue strips, which represent the clocks, encompasses those triangles structures you are talking about. I can remove all of the reference points around the trianglular paths if you like. They are there to show you how everything stays aligned with the sphere. If you look at the triangle structure in the blue strips, they are getting wider and wider for the images that show acceleration.

The acceleration you see in the above graphs does not mean that the 4D sphere is expanding at an accelerated rate. The sphere is expanding at the speed of light. It is the clocks, that are traversing across the surface of this expanding sphere, that are accelerating as shown in most of the images from the previous post.

I also forgot to mention that I plot the up and down path of photons for each clock. That is why you see the diamond shape paths inside the blue strips.

But, if the expanding sphere represents a system of reference (is this the case?), the clocks in your graphs should expand at the same rate with the sphere, and not remain the same width as you are showing. As much as I can understand, your graphs show a clock reducing in size as time passes by.

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But, if the expanding sphere represents a system of reference (is this the case?), the clocks in your graphs should expand at the same rate with the sphere, and not remain the same width as you are showing. As much as I can understand, your graphs show a clock reducing in size as time passes by.

I don't think that it would be proper to think of this as a scaling system. The clocks remain the same size as the sphere is getting larger. You could think of the clocks as getting smaller, but there is no need for both to scale in size. If the sphere and the clocks both expanded, then the space surrounding the clocks would never eventually flatten out.

Time Dilation

Before we continue this discussion it is important to note that I have choosen natural units for the speed of light in the equations from the previous posts such that $c=1$. This is a result of parameterizing the equations such that the photon traverses the length, $L$, in the interval $u=0$ to $u=1$. I'm sorry I didn't mention this in the previous posts. I've just had other things on my mind lately. I realized that I omitted the fact that I had used natural units in my graphs and in deriving the time dilation equations when I reviewed my notes. This means that all positions, velocities, and accelerations are proportional to $c$. We can rewrite the equations such that $c=1$, and undo this naturalization once we have derived our equations.

We will also use the arc length integral to derive the length of the path and ultimately the time dilation equation:

$\int_{0}^{1}\sqrt{w'(u)^2+x'(u)^2+y'(u)^2}\, du$

where

$w(u)=\left (W_{\gamma}\right ) \, \cos(\Delta \theta_{x}) \, \cos(\Delta \theta_{y})\, -\, \left (Y_{\gamma}\right ) \, \cos(\Delta \theta_{x}) \, \sin(\Delta \theta_{y}) \, \cos(\Delta \theta_{w})\, +\, \left (Y_{\gamma}\right ) \, \sin(\Delta \theta_{x}) \, \sin(\Delta \theta_{w})$

$x(u)=\left (W_{\gamma}\right ) \, \sin(\Delta \theta_{x}) \, \cos(\Delta \theta_{y})\, -\, \left (Y_{\gamma}\right ) \, \sin(\Delta \theta_{x}) \, \sin(\Delta \theta_{y}) \, \cos(\Delta \theta_{w})\, -\, \left (Y_{\gamma}\right ) \, \cos(\Delta \theta_{x}) \, \sin(\Delta \theta_{w})$

$y(u)=\left (Y_{\gamma}\right ) \, \cos(\Delta \theta_{y}) \, \cos(\Delta \theta_{w})\, +\, \left (W_{\gamma}\right ) \, \sin(\Delta \theta_{y})$

We will use the simplifed version of the above equation and we will not consider acceleration as previously stated:

$w(u)=\left (\Delta W\right ) \cos\left (\Delta \theta_{x}\right)\, \cos\left (\Delta \theta_{y} \pm \Delta \theta_{\gamma}\right)$

$x(u)=\left (\Delta W\right ) \sin\left (\Delta \theta_{x}\right)\, \cos\left (\Delta \theta_{y} \pm \Delta \theta_{\gamma}\right)$

$y(u)=\left (\Delta W\right ) \sin\left (\Delta \theta_{y} \pm \Delta \theta_{\gamma}\right)$

Time Dilation - W Axis

This post is only going to look at deriving time dilation for motion along the W axis. We will derive time dilation along both the W and X axis in the next post. Since we are not going to be changing our X position, our equations simplify to the following (Note: $c=1$ so it has been removed due to using natural units, also we are not considering acceleration):

$w(u)=\Delta \tau\left (\mathit{Vl}_w \, \left (n+u\right ) + \mathit{Wl}_0\right ) \cos \left (\pm \frac{2\, u-1}{2\, \mathit{Vl}_w \, \left (n+u\right ) + 2\, \mathit{Wl}_0}\right )$

$x(u)=0$

$y(u)=\Delta \tau\left (\mathit{Vl}_w \, \left (n+u\right ) + \mathit{Wl}_0\right ) \sin \left (\pm \frac{2\, u-1}{2\, \mathit{Vl}_w \, \left (n+u\right ) + 2\, \mathit{Wl}_0}\right )$

Now that we have the equation in the form that defines motion with uniform velocity along the W axis, we can simplify the terms inside the square root of the arc length integral (Note: I have factored the result and arranged the terms so that we can easily derive time dilation in the following steps):

$w'(u)^2+x'(u)^2+y'(u)^2=\Delta \tau^2 \left ( \frac{\left(\mathit{Vl}_w \left(2\, n+1\right ) + 2\, \mathit{Wl}_0\right)^2}{\left(2\, \mathit{Vl}_w \left(n+u\right ) + 2\, \mathit{Wl}_0\right)^2}+\mathit{Vl}_w^2\right )$

Next we will undo the natural units for the speed of light:

$\Delta \tau^2 \left ( \frac{\left(\frac{\mathit{Vl}_w}{c} \left(2\, n+1\right ) + 2\, \frac{\mathit{Wl}_0}{c}\right)^2}{\left(2\, \frac{\mathit{Vl}_w}{c} \left(n+u\right ) + 2\, \frac{\mathit{Wl}_0}{c}\right)^2}+\left(\frac{\mathit{Vl}_w}{c}\right)^2\right )=\Delta \tau^2 \left ( \frac{\left(\mathit{Vl}_w \left(2\, n+1\right ) + 2\, \mathit{Wl}_0\right)^2}{\left(2\, \mathit{Vl}_w \left(n+u\right ) + 2\, \mathit{Wl}_0\right)^2}+\left(\frac{\mathit{Vl}_w}{c}\right)^2\right )$

Finally, we can substitute this result into the arc length integral and factor out $\Delta \tau$:

$\Delta \tau \int_{0}^{1}\sqrt{ \frac{\left(\mathit{Vl}_w \left(2\, n+1\right ) + 2\, \mathit{Wl}_0\right)^2}{\left(2\, \mathit{Vl}_w \left(n+u\right ) + 2\, \mathit{Wl}_0\right)^2}+\left(\frac{\mathit{Vl}_w}{c}\right)^2}\, du$

Now that we have the length of our path, we can easily derive the time dilation equation:

$\Delta \tau \left(\int_{0}^{1}\sqrt{ \frac{\left(\mathit{Vl}_w \left(2\, n+1\right ) + 2\, \mathit{Wl}_0\right)^2}{\left(2\, \mathit{Vl}_w \left(n+u\right ) + 2\, \mathit{Wl}_0\right)^2}-\left(\frac{\mathit{Vl}_w}{c}\right)^2}\, du \right)^{-1}$

Because the curvature of a sphere diminishes as it gets larger we must include $\mathit{Wl}_0$ as it does affect the radius of the sphere. We can also show that this time dilation equation for an expanding sphere approaches those in SR as time approaches infinity. This can be done by evaluating the limit as our integer multiple of $\Delta \tau$, or the variable $n$, approaches infinity:

$\lim_{n \to \infty}\Delta \tau \left(\int_{0}^{1}\sqrt{ \frac{\left(\mathit{Vl}_w \left(2\, n+1\right ) + 2\, \mathit{Wl}_0\right)^2}{\left(2\, \mathit{Vl}_w \left(n+u\right ) + 2\, \mathit{Wl}_0\right)^2}-\left(\frac{\mathit{Vl}_w}{c}\right)^2}\, du \right)^{-1}=\Delta \tau \left(\sqrt{ 1-\left(\frac{\mathit{Vl}_w}{c}\right)^2}\right)^{-1}$

That concludes tonights post. I have some serious family matters to attend to, so I will not be able to conclude the discussion on time dilation until next week.

Edited by Daedalus
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(...) Or I understand nothing.

That's it.

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(...) Or I understand nothing. That's it.

To try and explain it better Michel, it is not the galaxies, stars, or planets that are expanding. Rather, it is the space between the galaxies that is expanding. The mathematics is based on a four dimensional sphere that is growing in size which represents the space that is expanding. The clocks are not expanding. Their size remains constant.

We can break the four dimensional sphere into two, three dimensional, spheres that combine to represent the "bubble" of space that we exist in. The surface of one sphere represents the XY plane and the other is the YZ plane. Because the overall four dimensional sphere is expanding, these three dimensional spheres are also expanding. The effect is such that we perceive an expanding space with no center for the expansion. If we could somehow see four dimensional space, we would see that there is a center which is located at time zero.

I have created the following image to demonstrate the concept:

The image decomposes the four dimensional sphere into two, three dimensional spheres labeled, WXY and WYZ. The middle, XYZ, sphere is a result of combining the XY surface of the WXY sphere with the YZ surface of the WYZ sphere. The smaller circles on both, WXY and WYZ, spheres demonstrate the portion of the two spheres which combine to make our three dimensional "bubble". Please note that this image is not drawn to scale as it would be impossible to accurately show the relationships.

Also, I have not labeled the W axis as it could lead to confusion because it is very difficult to render four dimensional space. One might misinterpret the W axis as being perpendicular to itself when combining the WXY and WYZ spheres when in fact it is the same axis for both spheres. The image is intended to show how we lose the W axis as a result of how we perceive a four dimensional sphere that is expanding at the speed of light.

I hope this clarifies any confusion with this system and to how the mathematics was derived.

Edited by Daedalus
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Time Dilation - W and X axes

Now we will look at time dilation for the W and X axes. The process is the same as we have worked out previously:

$w(u)=\Delta \tau\left (\mathit{Vl}_w \, \left (n+u\right ) + \mathit{Wl}_0\right ) \cos \left (\pm \frac{2\, u-1}{2\, \mathit{Vl}_w \, \left (n+u\right ) + 2\, \mathit{Wl}_0}\right )\cos \left (\frac{\mathit{Vl}_x \, \left (n+u\right )+ \mathit{Xl}_0}{\mathit{Vl}_w \, \left (n+u\right ) + \mathit{Wl}_0}\right)$

$x(u)=\Delta \tau\left (\mathit{Vl}_w \, \left (n+u\right ) + \mathit{Wl}_0\right ) \cos \left (\pm \frac{2\, u-1}{2\, \mathit{Vl}_w \, \left (n+u\right ) + 2\, \mathit{Wl}_0}\right )\sin \left (\frac{\mathit{Vl}_x \, \left (n+u\right )+ \mathit{Xl}_0}{\mathit{Vl}_w \, \left (n+u\right ) + \mathit{Wl}_0}\right)$

$y(u)=\Delta \tau\left (\mathit{Vl}_w \, \left (n+u\right ) + \mathit{Wl}_0\right ) \sin \left (\pm \frac{2\, u-1}{2\, \mathit{Vl}_w \, \left (n+u\right ) + 2\, \mathit{Wl}_0}\right )$

Now that we have the equation in the form that defines motion with uniform velocity along the W and X axes, we can simplify the terms inside the square root of the arc length integral:

$w'(u)^2+x'(u)^2+y'(u)^2=\Delta \tau^2 \left ( \frac{\left(\mathit{Vl}_w \left(2\, n+1\right ) + 2\, \mathit{Wl}_0\right)^2}{\left(2\, \mathit{Vl}_w \left(n+u\right ) + 2\, \mathit{Wl}_0\right)^2}+\frac{\left (\mathit{Vl}_x \, \mathit{Wl}_0 - \mathit{Vl}_w \, \mathit{Xl}_0\right )^2}{\left (2\, \mathit{Vl}_w \left(n+u\right ) + 2\, \mathit{Wl}_0\right )^2}\cos^2 \left(\frac{2\, u-1}{2\, \mathit{Vl}_w \, \left (n+u\right ) + 2\, \mathit{Wl}_0}\right)+\mathit{Vl}_w^2\right )$

Next we will undo the natural units for the speed of light (Note: the equations are getting too big for LaTeX to display the image. We will define smaller equations so that we can put everything together):

$\alpha^2=\frac{\left(\frac{\mathit{Vl}_w}{c} \left(2\, n+1\right ) + 2\, \frac{\mathit{Wl}_0}{c}\right)^2}{\left(2\, \frac{\mathit{Vl}_w}{c} \left(n+u\right ) + 2\, \frac{\mathit{Wl}_0}{c}\right)^2}=\frac{\left(\mathit{Vl}_w \left(2\, n+1\right ) + 2\, \mathit{Wl}_0\right)^2}{\left(2\, \mathit{Vl}_w \left(n+u\right ) + 2\, \mathit{Wl}_0\right)^2}$

$\beta_{x}^2=\frac{\left (\mathit{Vl}_x \, \mathit{Wl}_0 - \mathit{Vl}_w \, \mathit{Xl}_0\right )^2}{\left (2\, \mathit{Vl}_w \left(n+u\right ) + 2\, \mathit{Wl}_0\right )^2}\cos^2 \left(\frac{2\, u-1}{2\, \mathit{Vl}_w \, \left (n+u\right ) + 2\, \mathit{Wl}_0}\right)=\frac{\left (\mathit{Vl}_x \, \mathit{Wl}_0 - \mathit{Vl}_w \, \mathit{Xl}_0\right )^2}{c^2 \left (2\, \mathit{Vl}_w \left(n+u\right ) + 2\, \mathit{Wl}_0\right )^2}\cos^2 \left(\frac{c \left (2\, u-1\right )}{2\, \mathit{Vl}_w \, \left (n+u\right ) + 2\, \mathit{Wl}_0}\right)$

$\beta_{w}^2=\left (\mathit{Vl}_w\right )^2=\left (\frac{\mathit{Vl}_w}{c}\right )^2$

Putting everything together we get:

$w'(u)^2+x'(u)^2+y'(u)^2=\Delta \tau^2 \left (\alpha^2+\beta_{x}^2+\beta_{w}^2\right )$

Expanding this out gives us the following (I hit the max image size on this one):

$w'(u)^2+x'(u)^2+y'(u)^2=\Delta \tau^2 \left ( \frac{\left(\mathit{Vl}_w \left(2\, n+1\right ) + 2\, \mathit{Wl}_0\right)^2}{\left(2\, \mathit{Vl}_w \left(n+u\right ) + 2\, \mathit{Wl}_0\right)^2}+\frac{\left (\mathit{Vl}_x \, \mathit{Wl}_0 - \mathit{Vl}_w \, \mathit{Xl}_0\right )^2}{c^2 \left (2\, \mathit{Vl}_w \left(n+u\right ) + 2\, \mathit{Wl}_0\right )^2}\cos^2 \left(\frac{c \left (2\, u-1\right )}{2\, \mathit{Vl}_w \, \left (n+u\right ) + 2\, \mathit{Wl}_0}\right)+\frac{\mathit{Vl}_w^2}{c^2}\right )$

Finally, we can substitute this result into the arc length integral and factor out $\Delta \tau$ (This is where the LaTeX image size is too big. We'll have to use the defined equations.):

$\Delta \tau \int_{0}^{1}\sqrt{\alpha^2+\beta_{x}^2+\beta_{w}^2}\, du$

Now that we have the length of our path, we can derive the time dilation equation:

$\Delta \tau \int_{0}^{1}\sqrt{\alpha^2-\beta_{x}^2-\beta_{w}^2}\, du$

I realized that I made a mistake when discussing the limits of these equations as the variable, $n$, approaches infinity. To explain this better, I will demonstrate the limits with the variable $n$ and with the variable $\mathit{Wl}_0$. We need to use $\mathit{Wl}_0$ as our limiting variable because it is this variable that defines the initial radius of the sphere. Basically, we didn't need the variable $n$ because it was only used to align the cycles of the light clock which can also be done with $\mathit{Wl}_0$. Thus, the variable $n$ and $\mathit{Wl}_0$ are practically the same except the variable, $n$, works in conjunction with the parameter, $u$.

Limits using $n$ :

$\lim_{n \to \infty} \frac{\left(\mathit{Vl}_w \left(2\, n+1\right ) + 2\, \mathit{Wl}_0\right)^2}{\left(2\, \mathit{Vl}_w \left(n+u\right ) + 2\, \mathit{Wl}_0\right)^2}=1$

$\lim_{n \to \infty} \frac{\left (\mathit{Vl}_x \, \mathit{Wl}_0 - \mathit{Vl}_w \, \mathit{Xl}_0\right )^2}{c^2 \left (2\, \mathit{Vl}_w \left(n+u\right ) + 2\, \mathit{Wl}_0\right )^2}\cos^2 \left(\frac{c \left (2\, u-1\right )}{2\, \mathit{Vl}_w \, \left (n+u\right ) + 2\, \mathit{Wl}_0}\right)=0?$

$\lim_{n \to \infty} \left (\frac{\mathit{Vl}_w}{c}\right )^2=\left (\frac{\mathit{Vl}_w}{c}\right )^2$

Limits using $\mathit{Wl}_0$ :

$\lim_{\mathit{Wl}_0 \to \infty} \frac{\left(\mathit{Vl}_w \left(2\, n+1\right ) + 2\, \mathit{Wl}_0\right)^2}{\left(2\, \mathit{Vl}_w \left(n+u\right ) + 2\, \mathit{Wl}_0\right)^2}=1$

$\lim_{\mathit{Wl}_0 \to \infty} \frac{\left (\mathit{Vl}_x \, \mathit{Wl}_0 - \mathit{Vl}_w \, \mathit{Xl}_0\right )^2}{c^2 \left (2\, \mathit{Vl}_w \left(n+u\right ) + 2\, \mathit{Wl}_0\right )^2}\cos^2 \left(\frac{c \left (2\, u-1\right )}{2\, \mathit{Vl}_w \, \left (n+u\right ) + 2\, \mathit{Wl}_0}\right)=\left (\frac{\mathit{Vl}_x}{c}\right )^2$

$\lim_{\mathit{Wl}_0 \to \infty} \left (\frac{\mathit{Vl}_w}{c}\right )^2=\left (\frac{\mathit{Vl}_w}{c}\right )^2$

We can see that by using $\mathit{Wl}_0$, we do not affect the equation for time dilation for the W axis in the previous post. I made this mistake because the correct variable to use was not easily apparent. This also brings us to the point that it is extremely important to check and recheck ones work often to make sure that you have derived the correct calculations : ) All things aside, we can now show the limit of this time dilation equation for the W and X axes:

$\lim_{\mathit{Wl}_0 \to \infty}\Delta \tau \left (\int_{0}^{1}\sqrt{\alpha^2-\beta_{x}^2-\beta_{w}^2}\right )^{-1}=\Delta \tau \left(\sqrt{ 1 - \left (\frac{\mathit{Vl}_x}{c}\right)^2-\left(\frac{\mathit{Vl}_w}{c}\right)^2}\right)^{-1}$

We could derive time dilation for the Y axis, but the equations would come out the same as we have for the X axis. This is because all we have to do is rotate the path using the initial angles we have defined in the previous post. I do have the equations that consider acceleration. However, I will save that for a later post. Also, we can see that the limits of both time dilation equations are exactly the same as the ones I originally posted on page 1 post# 7:

$\Delta t_{r} \sqrt{1-\frac{V_{w}^{2}}{c^{2}}-\frac{V_{r}^{2}}{c^{2}}}=\Delta t_{n} \sqrt{1-\frac{V_{w}^{2}}{c^{2}}}$

Next, we will begin discussion on the spiraling singularity. This singularity is not a gravitational singularity. It is a mathematical singularity. But, we will discuss this in the next post : )

I almost forgot that we need to derive the time dilation equations for the mechanical clock / rotating mechanism. We will discuss this after we take a look at the singularity. The reason for this is because I have only been able to work out the time dilation equation for the W axis. I will need to obtain a newer version of Mathematica (the one that uses the GPU to crunch) to solve the equation for the W and X axes because my computer takes forever to solve the equations and factor the results.

Edited by Daedalus
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• 1 month later...

I know it has been a while since I last made a post, but I would like to continue and discuss the spiraling singularity and how I have interpreted this mathematical result of my equations. Before we get into the details, I would like to recap on what has brought us here and why I believe this mathematical framework gives us some insights into the nature of the universe.

My original goal for Temporal Uniformity was to explain the phenomena of time by stating a hypothesis that dark matter is temporally displaced matter. This requires us to view the temporal dimension no differently than we do any other spatial dimension. This brings into question why we do not perceive this dimension to have any physical length, but experience a forward direction through time. This was derived by applying the mathematics of special relativity in such a way that when we solved for the speed along the W axis, we obtained the speed of light regardless of our motions through the other three observable spatial dimensions (I will re-derive the equation for the benefit of newcomers):

The concept:

Special relativity takes into account two main frames of reference. One is the rest frame where the observer is at rest and the other is a frame of reference in motion relative to the observer. Temporal uniformity takes into account an additional frame of reference called the Big Bang frame of reference. The Big Bang FoR has its origin located at the center of a four dimensional sphere that is expanding. This expanding sphere represents the Big Bang and we denote the subscript, $b$, to identify this FoR. The rest frame is called the time-normal FoR and is denoted with the subscript, $n$. This leaves us with the FoR that is in motion relative to the time-normal FoR. We call this frame of reference the time-relative FoR and denote the subscript, $r$, to identify it.

Both the time-normal and time-relative FoRs share the same magnitude of their space-time vectors from the Big Bang FoR's origin. In other words... from the Big Bang FoR we can say that the space-time vector with the tail located at the origin and the head located at the position of any observable body is equal in magnitude and therefore must be temporally aligned such that these bodies exist at the same point in time (i.e. everything you can see and touch exists at the same point in time).

The time-normal FoR moves away from the Big Bang FoR's origin only along the W axis. But, the time-relative FoR not only moves away from the origin along the W axis, but also moves away from the time-normal FoR along dimensions perpendicular to the W axis. The key component to this view is that an observer located at the origin of the Big Bang FoR can rotate their FoR such that the time-relative FoR can become the time-normal FoR and vice-versa. This results in both, time-normal and time-relative, FoRs moving away from the Big Bang FoR's origin at the same rate as derived below:

Time as defined by SR for an observer positioned at the origin of the Big Bang FoR:

$\Delta t_{b} = \frac{2L}{c}$ where $L=|AB|$

Time as defined by SR for a body in the time-normal FoR being observed by an observer positioned at the origin of the Big Bang FoR (or vice-versa):

$\Delta t_{n} = \frac{2\, N}{c}$ where $N=\sqrt{\left (\frac{V_{w} \times \Delta t_{n}}{2}\right )^{2}+L^{2}}$

$V_{w}$ is the velocity through the fourth dimension and $L=|AB|$

Multiply both sides by $c$ and square the results:

$c^2 \, \Delta t_{n}^2 = 4 \, \left( \frac{V_{w}^2 \, \Delta t_{n}^2}{4} + L^2\right )=V_{w}^2 \, \Delta t_{n}^2 + 4\, L^2$

Subtract $V_{w}^2 \, \Delta t_{n}^2$ from both sides:

$c^2 \, \Delta t_{n}^2 - V_{w}^2 \, \Delta t_{n}^2 = 4\, L^2$

Factor out $\Delta t_{n}^2$ from the left side:

$\Delta t_{n}^2\, \left(c^2 - V_{w}^2\right ) = 4\, L^2$

Factor out $c^2$ from $c^2 - V_{w}^2$ on the left side:

$\Delta t_{n}^2\, c^2\, \left(1 - \frac{V_{w}^2}{c^2}\right ) = 4\, L^2$

Divide both sides by $c^2\, \left(1 - V_{w}^2 / c^2\right )$:

$\Delta t_{n}^2 = \frac{4\, L^2}{c^2\, \left(1 -\frac{V_{w}^2}{c^2}\right )}$

Take the square root of both sides:

$\Delta t_{n} = \frac{2\, L}{c\, \sqrt{1 - \frac{V_{w}^2}{c^2}}}$

Substitute $\Delta t_{b}$ in place of $2\, L / c$:

$\Delta t_{n} = \frac{\Delta t_{b}}{\sqrt{1 - \frac{V_{w}^2}{c^2}}}$

Time as defined by SR for a second body in the time-relative FoR being observed by an observer positioned at the origin of the Big Bang FoR without rotating the FoR (or vice-versa):

$\Delta t_{r} = \frac{2\, R}{c}$ where $R=\sqrt{\left (\frac{V_{w} \times \Delta t_{r}}{2}\right )^{2}+\left (\frac{V_{r} \times \Delta t_{r}}{2}\right )^{2}+L^{2}}$

$V_{w}$ is the velocity through the fourth dimension, $V_{r}$ is the velocity relative to the first body in the time-normal FoR, and $L=|AB|$

Performing the same steps as we did above yields:

$\Delta t_{r} = \frac{\Delta t_{b}}{\sqrt{1 - \frac{V_{w}^2}{c^2} - \frac{V_{r}^2}{c^2}}}$

Relating time dilation for both, time-normal and time-relative, FoRs through $\Delta t_{b}$:

$\Delta t_{b} = \Delta t_{n} \, \sqrt{1 - \frac{V_{w}^2}{c^2}}$

$\Delta t_{b} = \Delta t_{r} \, \sqrt{1 - \frac{V_{w}^2}{c^2} - \frac{V_{r}^2}{c^2}}$

This give us the following relationship:

$\Delta t_{r} \sqrt{1-\frac{V_{w}^{2}}{c^{2}}-\frac{V_{r}^{2}}{c^{2}}}=\Delta t_{n} \sqrt{1-\frac{V_{w}^{2}}{c^{2}}}$

We need to solve for $\Delta t_{r}$. First we'll divide $\sqrt{1-V_{w}^{2}/c^2}$ by both sides:

$\Delta t_{r} \frac{\sqrt{\left(1-\frac{V_{w}^{2}}{c^{2}}\right)-\frac{V_{r}^{2}}{c^{2}}}}{\sqrt{\left(1-\frac{V_{w}^{2}}{c^{2}}\right)}}=\Delta t_{n}$

Simplify the fraction by moving everything inside the square root and dividing out the terms in parentheses:

$\Delta t_{r} \sqrt{1-\frac{\frac{V_{r}^{2}}{c^{2}}}{1-\frac{V_{w}^{2}}{c^{2}}}} = \Delta t_{r} \sqrt{1-\frac{\frac{V_{r}^{2}}{c^{2}}}{\frac{c^{2}-V_{w}^{2}}{c^{2}}}} = \Delta t_{r} \sqrt{1-\frac{V_{r}^{2}}{c^{2}-V_{w}^{2}}} = \Delta t_{n}$

Solve for $\Delta t_{r}$ by dividing the square root and its contents by both sides:

$\Delta t_{r} = \frac{\Delta t_{n}}{\sqrt{1-\frac{V_{r}^{2}}{c^{2}-V_{w}^{2}}}}$

Interpreting the result:

This result makes sense because all observable bodies have a relative temporal velocity of zero, $V_{w}=0$, with all other observable bodies. In other words, we can see these objects and they do not disappear or appear at our point in time. This suggests that they are moving at the same rate through the temporal dimension as we are. This in turn yields the standard time dilation equation:

$\Delta t_{r} = \frac{\Delta t_{n}}{\sqrt{1-\frac{V_{r}^{2}}{c^{2}-0^{2}}}}=\frac{\Delta t_{n}}{\sqrt{1-\frac{V_{r}^{2}}{c^{2}}}}$

However, we can determine our temporal velocity by solving for $V_{w}$:

$V_{w}=\frac{\sqrt{c^{2}(\Delta t_{n}^{2}-\Delta t_{r}^{2})+\Delta t_{r}^{2}\times V_{r}^{2}}}{\sqrt{\Delta t_{n}^{2}-\Delta t_{r}^{2}}}=\sqrt{c^{2}+\frac{\Delta t_{r}^{2}\times V_{r}^{2}}{\Delta t_{n}^{2}-\Delta t_{r}^{2}}}$

Since all observers will place themselves in the time-normal frame of reference, or the rest frame, they will have a zero relative velocity and their temporal velocity is equal to the speed of light, $V_{w}=c$:

$V_{w}=\sqrt{c^{2}+\frac{\Delta t_{r}^{2}\times 0^{2}}{\Delta t_{n}^{2}-\Delta t_{r}^{2}}}=\sqrt{c^{2}}=c$

Because we move through the temporal dimension, W, at the speed of light, this fourth spatial dimension has collapsed according to length contraction. This is a valid statement if we consider the Big Bang FoR and vice-versa. This results with us perceiving three observable dimensions of space and one spatial dimension that becomes our temporal dimension, time:

$L'=L\, \sqrt{1-\frac{V_{w}^{2}}{c^{2}}}=L\, \sqrt{1-\frac{c^{2}}{c^{2}}}=0$

This explains why we experience a forward motion through time but do not perceive a length along the temporal axis. I also explain this with a little more detail in the following section about the theory behind the math.

The theory behind the math:

The consequence of this result is that we are located on a layer of a four dimensional sphere, or 3-sphere, that is expanding at the speed of light:

The mathematics in the previous posts demonstrate that as this sphere gets larger, the mathematics approach the equations found in special relativity due to the diminishing curvature of this expanding 3-sphere:

$w'(u)^2+x'(u)^2+y'(u)^2=\Delta \tau^2 \left ( \frac{\left(\mathit{Vl}_w \left(2\, n+1\right ) + 2\, \mathit{Wl}_0\right)^2}{\left(2\, \mathit{Vl}_w \left(n+u\right ) + 2\, \mathit{Wl}_0\right)^2}+\frac{\left (\mathit{Vl}_x \, \mathit{Wl}_0 - \mathit{Vl}_w \, \mathit{Xl}_0\right )^2}{c^2 \left (2\, \mathit{Vl}_w \left(n+u\right ) + 2\, \mathit{Wl}_0\right )^2}\cos^2 \left(\frac{c \left (2\, u-1\right )}{2\, \mathit{Vl}_w \, \left (n+u\right ) + 2\, \mathit{Wl}_0}\right)+\frac{\mathit{Vl}_w^2}{c^2}\right )$

Taking the limit as $\mathit{Wl}_0$ approaches infinity:

$\lim_{\mathit{Wl}_0 \to \infty} \frac{\left(\mathit{Vl}_w \left(2\, n+1\right ) + 2\, \mathit{Wl}_0\right)^2}{\left(2\, \mathit{Vl}_w \left(n+u\right ) + 2\, \mathit{Wl}_0\right)^2}=1$

$\lim_{\mathit{Wl}_0 \to \infty} \frac{\left (\mathit{Vl}_x \, \mathit{Wl}_0 - \mathit{Vl}_w \, \mathit{Xl}_0\right )^2}{c^2 \left (2\, \mathit{Vl}_w \left(n+u\right ) + 2\, \mathit{Wl}_0\right )^2}\cos^2 \left(\frac{c \left (2\, u-1\right )}{2\, \mathit{Vl}_w \, \left (n+u\right ) + 2\, \mathit{Wl}_0}\right)=\left (\frac{\mathit{Vl}_x}{c}\right )^2$

$\lim_{\mathit{Wl}_0 \to \infty} \left (\frac{\mathit{Vl}_w}{c}\right )^2=\left (\frac{\mathit{Vl}_w}{c}\right )^2$

Time dilation - X and W axes:

$\Delta \tau \left(\sqrt{ 1 - \left (\frac{\mathit{Vl}_x}{c}\right)^2-\left(\frac{\mathit{Vl}_w}{c}\right)^2}\right)^{-1}$

Note: Because we are talking about an expanding sphere, the W axis can be considered parallel to the local direction of the expansion when choosing a frame of reference which labels the temporal dimension as the W axis.

We can hypothesize that dark matter fills the layers of this 3-sphere much like the layers of an onion, which would be nothing more than temporally displaced mass-energy. Because all mass-energy would be expanding with this 3-sphere, everything would be moving locally in the same direction and expanding outward at the speed of light. This means that the light from temporally displaced mass-energy will never be able to reach us and vice-versa. This explains why we don't see dark matter or any length along the W axis. This can be demonstrated if we visualize two photons travelling in the same direction, but offset from each other along the W axis. The photon at the end will not be able to catch up with the photon ahead of it. The photon at the front will not be able to change direction, and therefore will never share or intersect the same location with the photon behind it. However, this temporally displaced mass-energy would still be affected by gravitational fields from all spatial dimensions. This would explain why gravitational effects of dark matter can be seen as a halo around a galaxy and possibly explains why gravity seems to be weaker than the other forces of nature due to it propagating through all four dimensions of space. It would also explain why galaxies can rotate as fast as they do without flying apart as well as provide mass-energy located at our point in time with a scaffold to form around.

If we look at any type of sphere, we can clearly form rays which propagate from the center of the sphere outward at any angle we wish to specify. Even though these rays would be moving away from the center of this sphere at an equal rate, they would have a different relative velocity from each other across the surface of this sphere in accordance to the angle they are propagating away from the center. This concept, along with considering localized effects of four-dimensional space, explains why the spaces between galaxies are expanding and why we do not see a center for this expansion. This brings us to an explanation for dark energy as discussed below.

Massive bodies cannot go faster than the speed of light. Because the above mathematics shows us that everything is moving at the speed of light along the temporal axis, temporally displaced mass-energy cannot accelerate us toward the "future" because it would theoretically take an infinite amount of energy to do so. The inverse of this statement is that it would take an infinite amount of energy to decelerate a massive body that is moving at the speed of light. Therefore, temporally displaced mass-energy cannot pull us towards the "past". The only thing temporally displaced mass-energy could do is produce gravitational effects at our point in time and pull us along dimensions that are perpendicular to the temporal axis. Dark energy, which accelerates the expansion of space, can be visualized as temporally displaced mass-energy closer to the outer edge of this 3-sphere that is pulling us along with its expansion. Not only could there be more mass-energy closer to the edge of this expanding sphere, but it would also have had more time to expand resulting in much larger spaces between galaxies. This can be visualized by examining the cross section of the 3-sphere as shown in the above graphic.

Not only can temporally displaced mass-energy explain dark matter, and possibly dark energy, but it can also explain dark flow if we consider that there is more mass-energy closer to the outer edge of this 3-sphere and that it has grouped into larger super clusters of galaxies than what we observe at our point along the temporal axis because it has had more time to do so:

Dark flow is an astrophysical term describing a peculiar velocity of galaxy clusters. The actual measured velocity is the sum of the velocity predicted by Hubble's Law plus a small and unexplained (or dark) velocity flowing in a common direction.

For the same reasons as above, we can also explain the Great Attractor:

The Great Attractor is a gravity anomaly in intergalactic space within the range of the Centaurus Supercluster that reveals the existence of a localised concentration of mass equivalent to tens of thousands of Milky Ways, observable by its effect on the motion of galaxies and their associated clusters over a region hundreds of millions of light years across.

The reason why there would be a higher concentration of mass-energy toward the outer edge of this 3-sphere is due to the size of layers that radiated outward first. If we visualize a cross-section of a sphere that is composed of layers, we can see that the outer layers would have more mass-energy than the inner layers. Mass-energy in the outer layers would have been expanding longer than the inner layers. This implies that super clusters in the outer layers are composed of more galaxies than the inner layers resulting in a higher density for these outer layers. Also, the spaces between the galaxies in the outer layers would have had more time to expand and therefore would be larger than the spaces between the galaxies located in the inner layers. These spaces and super clusters can be seen as structures in the image of the WMAP for our point in time located along the W axis. I hypothesize that all layers along the W axis would have similiar composition:

The mathematics I have derived in the previous posts can explain other cosmic phenomena such as the smoothness of the CMBR, or cosmic microwave background radiation. This has to do with the spiraling singularity as shown below:

We normally think that everything expands away from each other much like what happens during an explosion. This causes a problem for what we see with the CMBR in how it can have such a uniform temperature. How is it possible for the entire CMBR to have this uniformly distributed temperature when one side of the expanding universe hasn't had enough time to communicate its temperature with the other side? However, the mathematics that I have derived tells a very different story:

By analyzing a single photon path we can see that it actually forms a type of hyperbolic spiral. This means that the energy of the Big Bang did not just expand outward in all direction instantly, but converged to its present state as can be seen in the first image of the spiral that combines multiple photon paths. The above graph clearly shows that the photon path becomes incredibly dense towards time zero of the expansion of the 3-sphere, or Big Bang. We can also see the symmetry of two photon paths moving in opposite directions as shown below:

By zooming out, we can see how these paths converge to straight lines as time approaches infinity:

But we must zoom in on the spiral so that we can see how mass-energy could have communicated its temperature before expanding outward:

Zooming in closer:

The above image clearly illustrates how our mass-energy came into contact with mass-energy from the other side of the universe before expanding outward and converging to the expansion we see today, solving the horizon problem. The mathematics I have derived, along with the evidence provided by dark matter, dark energy, and dark flow, suggests that the idea of temporally displaced mass-energy is plausible.

The big question now is what happens with inflation? Does this model allow for inflation or is it no longer valid within the context of temporal uniformity? I call on the experts to shed light on these questions to either validate or invalidate some of the speculative and hypothetical claims made within this post. That concludes tonight's posting. I hope you have enjoyed reading this and I look forward to any debate that may arise.

Edited by Daedalus
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Next week we will discuss the equation for the path of rotating body moving outward along the W axis and derive time dilation accordingly. As promised, I will also post the interpolation equation that allows us to specify locations with respect to intervals that can be sampled at any given time. This is different from Newton's interpolation formula which requires each location to be sampled within equal intervals of time.

Newton's interpolation formula:

$\sum_{i=0}^{n-1} \left [ \sum_{j=0}^{i} \left [ \left ( -1 \right )^{j} \binom{i}{i-j} S_j \right ] \times \binom{i-t-1}{i} \right ]$

The variable $n$ is the number of measurements or specified locations, $S_j$ is the location specified by the $j$ index, the $i$ index works in conjunction with the $j$ index, and $t$ is time.

Edited by Daedalus
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• 1 month later...

Thoughts on

Temporal Uniformity

When is now ?

This is a awkward question

Common sense is indeed violated but going against common sense is not applicable to uncommonly experienced situations such as relative motion near the speed of light.

Fortunately common sense dose tell us that a characteristic speed of vacuum waves cannot possibly depend on who’s vacuum they are propagating in.

Events can always be described without coordinates but special relativity deals specifically with how to use coordinates in space witch exhibits non intuitive behaviour when comparing lengths and time among different observers

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• 10 months later...

When we make a measurement of a physical phenomena, we can only do so by taking advantage of an attribute inherent in the phenomena itself. For instance, we use a rod to measure length. The rod has markings that allow us to measure lengths that are within its boundaries. We use a voltmeter to measure electrical potential differences between two points. The voltmeter uses attributes inherent in electricity and magnetism to make its measurements. This leads us to measurements of time as provided by a clock. All clocks use oscillations of known intervals to measure time. This can be demonstrated in the fact that a sun-dial uses the oscillations as provided by the rotation cycle of the Earth, grandfather clocks use pendulums, and light clocks bounce light between two reflective surfaces. There are many more mechanisms that clocks use to measure time, such as oscillations as provided by crystals, but the point is that they are all based on mechanisms that oscillate. Oscillation in itself is motion and, more specifically, it is motion that repeats with a specified frequency. It is because time can only be measured with motion that motion must be inherent in time. The same statements are true for motion through any other spatial dimension. We can use oscillation to measure our speed along the x axis the same way a speedometer uses the oscillations of the vehicle's wheels to measure the speed of the vehicle. But, if the vehicle always had a constant velocity along the x axis, then we could also measure physical processes in the vehicle in relation to the vehicle's x position. However, we have the freedom to change our speed and direction along the x axis, making this pseudo-temporal dimension unordered. The physical nature of time seems to be one where motion is the only attribute inherent in the phenomena and it is restricted to moving forward along the temporal dimension.

I'm a little thrown by your use of the word "oscillation" and the word" speed". Units of time arbitrary progress one at a time to the next number according to whatever unit we use, time doesn't oscillate any more then the dimension of "width" oscillates. Which brings me to my next point, which is that not all oscillation is the intuitive physical motion you'd expect. We have no physical knowledge of a photon before we measure it, it's oscillation is only modeled in terms of it's probability or the uncertainty of it's field, a photon by the dimensional analysis of it's mathematics does not have physical components which oscillate with any sort of velocity like rippling a piece of paper, and neither does time. Things can oscillate, but the oscillation doesn't have to take place with physical dimensions, other properties of particles can oscillate as well, properties that are not as physical.

If time is motion no different than motion in other spatial dimensions, then how can we expect to time travel to a past that only exists as a memory or to a future that will exist as a memory? We would have moved away from this point in time and would have not made it to a point in the future. If time travel was possible, then I would have to currently exist at all points in time from birth to death. This is because if I could time travel, then it would not be possible for me to take the mass-energy of the entire universe with me and arrange it the way it was back then or will be in the future. Plus, this behaviour is not true for any other spatial dimension. I may have existed at location zero on the x axis, but I am no longer at that position. I have moved on and when I go to visit location zero on the x axis, I find that it is not that same as before. There is always something different such as new cars in the parking lot. Furthermore, If all things exist currently at all points in time, then wouldn't the past and future attract gravitationally? Time may not flow linearly but surely we only exist at a given point in time and not simultaneously at all points. From this view I suggest that time travel to a memory of the past or to a memory that will exist in the future is impossible. However, traveling to a point in time that is parallel to ours may reveal a new view of our universe. These temporally displaced universes would move through time at the same rate as we do, except they are ahead or behind us along the temporal dimension. We could fast forward or rewind time in all instances and show that each universe would have its own unique history, etc... But I have not taken GR into account, so I am looking for people to disprove this view or support it and perhaps we all can learn something to the nature of time.

Contituting time as motion doesn't make sense, relative physical motion is modeled as a change in position over time, so you'd have a change in the position of time over time, when time doesn't have any spacial position to begin with. Also, with quantum statistics takes care of all this "memory" and determinism business. Essentially, no observed result that occurs can be though of being based on previous results, which means you can't mathematically have the future be determined in any way, and it more or less supports that there's no memory of the past since the direct results of the locations of particles were not based on those previous results. Instead, you have the probability of particles, and the probability clouds of those particles can change or transform different spacial coordinates, but where you actually see something end up is still randomness.

I can see that you think of time as a coordinate that, when it changes, other matter changes in response to it, which is true in a way, but there's still properties of time that make it different than other dimensions and make it so distinguishable that we give it it's own name, so that it doesn't change exactly like other dimensions. So when you mention that "all things exist currently in all points in time", or base assertions on it, I just think of the quantum statistics. With all this in mind, I think you need to revise the theory and distinguish between boundaries of physical dimensions and non-physical ones.

Edited by EquisDeXD
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I'm a little thrown by your use of the word "oscillation" and the word" speed". Units of time arbitrary progress one at a time to the next number according to whatever unit we use, time doesn't oscillate any more then the dimension of "width" oscillates. Which brings me to my next point, which is that not all oscillation is the intuitive physical motion you'd expect. We have no physical knowledge of a photon before we measure it, it's oscillation is only modeled in terms of it's probability or the uncertainty of it's field, a photon by the dimensional analysis of it's mathematics does not have physical components which oscillate with any sort of velocity like rippling a piece of paper, and neither does time. Things can oscillate, but the oscillation doesn't have to take place with physical dimensions, other properties of particles can oscillate as well, properties that are not as physical.

The point of the paragraph you quoted is to demonstrate the general workings of a clock. It does not in any way attempt to describe QM effects. Sure, you can attempt to bring QM into the discussion. However, you cannot truly be certain whether these oscillations, as you say, are not as physical. A counter-example to your argument can be seen in string theory.

String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything (TOE), a self-contained mathematical model that describes all fundamental forces and forms of matter. String theory posits that the elementary particles (i.e., electrons and quarks) within an atom are not 0-dimensional objects, but rather 1-dimensional oscillating lines ("strings").

You even state that "time doesn't oscillate any more then the dimension of width oscillates". However, you failed to recognize that units of width do oscillate in their own way. I can create a wheel that marks out one meter per cycle of the wheel. Thus each unit of width that I measure represents a cycle. I can add up the number of cycles the wheel has made and arrive at a measurement of width. So, unless you have a working theory of everything and can be so bold as to rule out physical motion at the quantum level, let's keep QM out of the discussion.

Contituting time as motion doesn't make sense, relative physical motion is modeled as a change in position over time, so you'd have a change in the position of time over time, when time doesn't have any spacial position to begin with.

Here you are compounding this notion of time with how you currently understand it. There is no change in position of time over time. That makes no sense at all. However, you have to ask yourself a more fundamental question to the nature of time. One that brings into question whether or not measurements of time are needed at all in order to describe a physical system. The answer to the question is that time, as is currently measured, is not needed because we can use motion instead.

If I have a light clock, I can either count the number of round trips (cycles) the pulse of light has made or I can measure the distance the pulse of light has traversed each cycle. Either way, both measurements can be used to quantify motion. One method quantifies motion to the count of cycles a clock makes and the other quantifies motion to the distance traversed by the oscillating mechanism the clock uses.

You may argue that QM doesn't operate in such a classically defined way. However, much like in GR, if you cannot tell the difference between being in a room that is accelerated versus one that is in a gravitational field, then how can you tell the difference between clocks that use classically defined motion versus the few that are based on principles of QM. Both types of clocks simply count the cycles of their oscillating mechanism. The difference is that if you base time on the cycles of the oscillating mechanism, then you must account for the time dilation of clocks in relative motion to each other. However, if you base time on the distance traversed by such mechanism, then time dilation is automatically included. This is evident in that the path of the oscillating mechanism for the clock in motion relative to the observer is elongated such that the clock in relative motion will appear to run slower than the observer's clock. This is demonstrated in the following image where the observer's clock is to the left and the clock in motion relative to the observer is to the right.

As you can see, the path of the oscillating mechanism for the clock that is in relative motion to the observer is elongated. The point is that if we can use motion instead of counting cycles to define equations that describe the motion of an object, then time itself must be based on motion. The result is that the differences between the temporal and spatial dimensions can be resolved within the framework of relativity by using nothing more than motion itself.

I can see that you think of time as a coordinate that, when it changes, other matter changes in response to it, which is true in a way, but there's still properties of time that make it different than other dimensions and make it so distinguishable that we give it it's own name, so that it doesn't change exactly like other dimensions. So when you mention that "all things exist currently in all points in time", or base assertions on it, I just think of the quantum statistics. With all this in mind, I think you need to revise the theory and distinguish between boundaries of physical dimensions and non-physical ones.

In order to understand the full implications of the theory, you must read beyond the first post in the thread. I have provided a framework that resolves the differences between the temporal and spatial dimensions. Of course, there is plenty of speculation on my part. The mathematical model I provided is missing several key elements. However, I do support my argument throughout the thread and continue to develop the model. I have even created a new model since my last posting that takes into account motion across expanding space. You are at liberty to disagree with me, and I do encourage criticism. The only thing I ask is to be thorough in your objections, which requires understanding the model itself.

Edited by Daedalus
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...

You are at liberty to disagree with me, and I do encourage criticism. The only thing I ask is to be thorough in your objections, which requires understanding the model itself.

Sorry Daedalus - he isn't at liberty to disagree with you, we banned him yesterday. He is/was a sophisticated troll and you probably would have ended up pulling your own hair out trying to get him to see reason.

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• 2 years later...

Is time a dimension without any past or present or future? Does one move through it at the rate of one second per second?

That is to say the same rate as the second hand ticking on a clock is also the same rate one experiences time in real time?

Is there a reason why the three arrows of time [ biological / thermodynamic / cosmological ] move in only the one direction?

Why is the Second Law Of Thermodynamics the one major one that is actually non reversible from a temporal perspective?

Is time what space expands into? What happens to time once it passes the event horizon? Does it speed up or slow down?

Or stay just the same? How is it possible for a photon not to experience time but travel through it? Has time always existed?

There has been a lot of talk about the nature of time in the forums here lately. So, I'm going to try and answer these questions here in my thread on temporal uniformity. Before I can do this, I think it's important to bring everyone up to speed on how time is defined within the framework of temporal uniformity. I'm going to start from the beginning and make a series of falsifiable statements that I believe encompass the very nature of time itself. However, I have to apologize up front for such a long post, but it's necessary to introduce simple building blocks and go from there.

What is Time?

In order to understand the nature of time, we need to understand what it does and how it behaves. This requires making valid observations about time and its properties, and then construct a definition for time that encompasses these observations. So, let's list all of the observations we can make about time that we know are true.

• We experience the passage of time as a duration, which is relative to the observer's frame of reference. Therefore, time is observed to speed up or slow down relative to various frames of reference.
• Although the passage of time is relative to an observer, the passage of time is continuous. e.g. we don't observe objects disappearing and reappearing at different points in time because their clocks speed up or slow down.
• We observe a clear direction through time such that we exist in the present moving towards the future leaving the past behind.
• Although we experience a direction through time, we cannot see the dimension of time.
• When plotting the path matter and energy takes through space, an extra variable is required to order events such as changes in position. In physics, we define this extra variable as time and we use clocks to measure it.
• Because it is mathematically impossible for anything to move the space with an infinite speed, the act of changing position through space requires a duration of time. This is what it all boils down to in the end. This entire post is meant to emphasize this observation which demonstrates that time is a mathematical consequence of traversing distances through space at finite speeds. This will be the theme for my arguments regarding the nature of time.

I really can't think of anything else that we can say about time that is an actual observation, but there are a few preconceptions that we can list that fall right in line with surreptitious57's questions. These preconceptions usually revolve around time travel; the idea of being able to physically move through space-time in such a way that allows matter and energy to travel to the past.

• Time might exist physically as a temporal dimension where matter and energy continues to exists along this dimension arranged the way it was in the past and will be arranged in the future.
• Time might exist as some physical dimension that everything moves through, but only in one direction such that we could travel to the past if we could reverse our direction through time.
• Time might exist according to the many worlds interpretation of quantum mechanics and there could be an infinitude of parallel universes that are defined by every possible event that can happen.
• If time travel is possible, then temporal paradoxes are possible, and you could go back in time and kill your grandfather before your father was born. The paradox that I find most interesting is the one Michio Kaku mentioned about traveling to the past and killing yourself right before you time traveled.

Temporal Uniformity attempts to explain our observations of time and reconcile our preconceptions using a consistent mathematical definition for what time actually is. In doing so, we can explain all of our observations of time while demonstrating that time travel, as understood as the act of traveling to the past and interacting with matter and energy as it was arranged back then, is impossible. This frees us from temporal paradoxes, defines parallel universes which allow dark matter, dark energy, and dark flow to exist, and provides us with answers to some of surreptitious57's questions. We'll begin by discussing space and energy.

Space and Energy

In temporal uniformity, space is defined as a collection of points where each point is separated by some distance from each other and are organized along dimensions. Basically, it's just a metric space that is occupied by energy in all of its various forms. As will be demonstrated, there is no need to introduce some temporal dimension. However, we still need a way to order events, but we don't need time in order to do so. After all, there are really only two things the make up the universe; space and the various forms of energy that occupies it.

I think most people have this misconception that there exists some fabric of space that can expand or shrink. I could be wrong, but it seems more likely that what we perceive as expanding or contracting space is actually the result of how various fields that are generated by matter and energy affect the motion of matter and energy that traverses such fields. Furthermore, things like "quantum foam that is theorized to be the 'fabric' of the Universe, but cannot be observed yet because it is too small" and virtual particles that pop in and out of existence are the result of how energy behaves at different points in space and at different scales. If we existed as charged particles moving through a magnetic field, then the space around us would be defined by the shape of the magnetic field, which is definitely non-Euclidean. Of course, gravitational fields are very different than magnetic fields. However, both fields are generated by the energy contained within matter and both fields affect how energy moves through space. The important thing about these fields is that they change the metric of space for the energy that exists in it, which in turn affects how energy moves through it.

Regardless if space actually bends or not, once you define a metric space and introduce energy, there is only one thing that is physically possible; energy can now change position through space. After all, the only thing energy can do with respect to space is change position through it. Granted, energy can interact with itself and other forms of energy in several different ways, but it can only do so if it is allowed to change position through space and come into contact with other forms of energy in the environment. If energy could not change position through space, then no interactions could take place, and the universe would be forever static. Light wouldn't be able to traverse the vast distances between galaxies, and it would be impossible to perceive the universe because electrical signals in our brains wouldn't be able to traverse the neural pathways. The universe as we experience it can only exist when energy is allowed to change position.

Now, I need to introduce a point of clarity here for the sake of our experts, moderators, and everyone else. There has been much debate regarding motion when it comes to quantum mechanics (QM) where motion is not as easily defined. Most people assume that motion is simply the act of a solid body changing position through space such as a baseball. However, subatomic particles such as electrons do not behave as if they were tiny baseballs orbiting a tightly packed nucleus of protons and neutrons that are also like tiny sized baseballs. So, this idea of motion simply doesn't suffice to explain what is observed.

In order to resolve this discrepancy, we are forced to define motion simply as the act of energy changing position through space. After all, doesn't this encompass the very essence of what is observed at any scale? For example, when we look at standing waves or just waves in general, we see motion. However, it is not the same type of motion exhibited by a baseball.

Figure 1 - Wave motion.

Are the particles of the wave moving up and down creating the appearance that the wave is moving to the right, or are the particles actually moving to the right? How can we define the position of the wave? It's not like we can define a center point that allows us to track how the wave changes position like the baseball. Regardless of how complicated the motion is, we still know that the structure of the wave is in motion and its energy is changing position through space.

So far, we have defined the universe as being comprised of only two things; space and the energy that occupies it. Time has not been included in our definition for space because, as it will be shown later, it exists as a property of space and not as a separate phenomenon. We will define what time is but, for or now, let's recap the finer points discussed thus far.

• The universe only consists of space that is occupied by energy in its many forms.
• Motion is defined as the act of energy changing position through space.

If you agree with these statements, then the discussion can move forward. Now that we have defined space and energy, we need to discuss how we measure and quantify it.

Measurements

Whenever we take measurements, we can only do so by exploiting an attribute inherent to whatever it is we are measuring. For instance, space is composed of points that can be organized along spatial dimensions where each point is separated by some distance. One of the defining attributes inherent to space is distance separating the points within it. Without distance separating the points, there wouldn't be any space. So, the only way we can directly or explicitly measure distance through space is by using a predefined unit of distance. The unknown distance through space is quantified in multiples of this predefined unit of distance. Direction through space is also quantified using predefined units of distance along the dimensions of space defined by a coordinate system. Therefore, distance is an attribute that defines what space is and we can only directly measure space using a unit of distance. Just look at how we measure length, direction, areas, and volumes. All of these things that define what space is, are the result of measurements of distance.

The same is true for anything else we measure. We are forced to use mass to measure mass. Given a balance / scale and several unit masses, we can measure the unknown mass of some object. If we are measuring probabilities, then we are forced to take measurements of the number of times something happened in a particular way. No matter what it is we are measuring, we always have to use the thing we are measuring to take the measurement. Otherwise, it would be impossible to measure. The fact of this statement can be seen in our basic units of measurement such as feet, meters, grams, kilograms, etc... Each one defines a predefined unit of the thing we are measuring.

Now, you might argue that you can use measurements of time and speed to derive the value for the measurement of distance an object has traversed. However, such a value is implicit because it relies on measurements that are directly / explicitly taken of other physical phenomena that are plugged into an equation provided by a theory of ones choosing that defines how one or more of these explicit measurements relate to the value of the unknown quantity. The derived value of the measurement was not directly observed and relies on the accuracy and precision of the theory that defines such relationship. This provides further evidence that we can only use the physical phenomena itself to explicitly measure it. Let's recap.

• In order to explicitly take a measurement of physical phenomena, you have to use the phenomena itself to take the measurement.
• Values derived for measurements using equations defined by physical theories are implicit because the measurements were not directly observed and rely on the correctness of the theories that provided the equations.

Since we are forced to use the phenomena itself in order to measure it, we simply have to ask ourselves how we measure time. Understanding the mechanics that drives our measurements of time should reveal the very nature of time itself. Such an understanding should resolve temporal paradoxes, define the driving mechanism for change, describe relativistic affects such as time dilation, and explain the arrow of time.

Measuring Time with Motion

Why is it important to define and understand how we take measurements? How can we even begin to answer questions about time if we don't even understand the underlying nature inherent to time itself? It all comes back to the statement that In order to measure any physical phenomena, we are forced to use an attribute inherent to the phenomena itself. When we examine how we measure time, we find that we always use motion to take the measurement. A sun dial works because the Earth rotates, grandfather clocks works due to the swing of their pendulum, spring watches use a spring to turn gears, digital watches use electricity to cause crystals to oscillate, light clocks reflects photons between two plates, and atomic clocks measure quantum mechanical properties derived from the motion of the energy contained within atoms (I will address atomic clocks later in this post). The mechanism that measures time in all of these devices does so by some form of energy changing position through space.

If we can agree that we use motion as defined by energy changing position through space to measure time, then motion must be inherent to time. If motion is inherent to time, then time is purely spatial and is governed by distance, which makes sense because the only thing that exists in the universe is energy and space. Because it is mathematically impossible for energy to traverse distances through space with an infinite speed, the passage of time must occur. So, our experience of the passage of time is a mathematical consequence of motion being restricted to finite speeds and nothing more. This is no different than how objects seem smaller at greater distances. Such things are mathematical consequences of distance. Therefore, in temporal uniformity, time is defined as a mathematical consequence of energy changing position through space with finite speeds.

However, we measure motion as a change in space over a change in time. How can time be a consequence of motion if our definition for motion is based on units of distance and time? This contradiction can be resolved by analyzing how we use motion to measure motion. Because energy has to change position through space for change in the environment to occur, when we use a clock to measure the rate of change of a physical property, we are actually using the motion of the mechanism in the clock to measure the motion of the energy causing the physical property to change. In essence, we are using motion to measure motion no different than how we use distance to measure distance.

Besides, a change in position $\Delta\,x$ does not require measurements of time. The change in position is purely a spatial property which relates back to measuring distances. We introduced the concept of measuring time because energy can change position at different rates. However, we don't need measurements of time to measure these different rates at which energy can change position through space. All we need are measurements of the distance traversed for some unit of motion compared to the distance traversed by some form of energy. If this is true, then we should be able to remove the time variable $t$ from every equation in physics, and replace it with a measurement of distance. We can demonstrate this concept using a light clock.

Figure 2 - The light clock to the left is at rest and the clock to the right is moving relative to the observer's coordinate system.

The light clock to the left is stationary in the coordinate system or frame of reference (FoR) of the observer, and the clock to the right is in motion relative to the observer's FoR. For now, let's examine the light clock to the left that is stationary. There are only two ways to use the clock. We can either count the number of times photons have traversed the distance between the two reflective plates and define that measurement as a unit of time, or we can measure the distance the photons traversed between the plates. Both measurements are equally valid and allows us to quantify motion and order events.

$t = d_c$

where $t$ equals a unit of time and $d_c$ is the distance traversed by the clock mechanism. Therefore, a unit of time is nothing more than a normalization of a unit of distance. If we choose to use measurements of distance, then we can define speed as the change in distance traversed by energy divided by the change in distance traversed by the clock mechanism.

$\text{speed} = \frac{\Delta\,d_e}{\Delta\,t} \ \ \text{or} \ \ \frac{\Delta\,d_e}{\Delta\,d_c}$

where $\Delta\,d_e$ is the change in distance traversed by energy when the change in distance traversed by the clock mechanism equals $\Delta\,d_c$. If the speed is constant, we could multiply the total distance our clock mechanism has traversed by our newly defined speed, and we can derive the distance the energy traversed through space without having to use units of time.

$d = \frac{\Delta\,d_e}{\Delta\,d_c} \times d_c$

where $d$ is the calculated distance. Again, this is no different than measuring distance with a ruler. We are simply quantifying the distance traversed by energy in multiples of the unit distance traversed by the clock mechanism. So instead of measuring motion in units of distance per units of time, we are comparing distance to distance, which adheres to the rule that we have to use the phenomena itself to take measurements. The standard equation for motion using only measurements of distance is defined no differently than when we use values of time. So, we can completely rewrite every equation in physics that uses measurements of time to use measurements of distance instead.

$d = \frac{1}{2} \left(\frac{\Delta\,d_e}{\Delta\,t^2}\right) t^2+\left(\frac{\Delta\,d_e}{\Delta\,t}\right) t + d_0$

$d = \frac{1}{2} \left(\frac{\Delta\,d_e}{\Delta\,d_c^2}\right) d_c^2+\left(\frac{\Delta\,d_e}{\Delta\,d_c}\right) d_c + d_0$

As a result, measurements of time are the only units in physics that can be replaced by units of distance. Such a contradiction is a violation of dimensional analysis, which provides further evidence that when we use time to measure rates of change in physical properties, we are actually using motion to measure the motion of energy changing position. Again, let's recap.

• There are only two measurements a clock can make; a measure of the number of times the clock mechanism has completed a cycle, or the distance the clock mechanism traversed throughout the cycle.
• What we experience as the passage of time is the mathematical result of energy being restricted to finite speeds. Since it is mathematically impossible for energy to traverse space with an infinite speed, the passage of time must occur.
• For any equation in physics, we can replace the time variable $t$ using measurements of distance.
• Units of time become normalizations for units of distance and are interchangeable.

Although we can demonstrate that we use motion to measure time, there exists a different view regarding time that is based on change. That it is change that drives the mechanism of time. So, let's now discuss how change is propagated throughout space.

Change in a System

Newton said it best in his laws for motion. For every action there is an equal and opposite reaction, and an object will maintain a constant velocity unless acted upon by an outside force. Although these statements apply to the classical world of physics, they also apply to the QM world as well when we consider that the only thing energy can do within space is change position. If energy was not allowed to change position, then how could it interact with other energy in the environment? How could the window break if the ball did not change position through space, and how could we observe the ball breaking the window if the energy of the light emitted from these object is also not allowed to change position through space? The only observation we can make is that change in a system can only occur when energy is allowed to change position through space and interact with other energy in the environment. Thus, the very definition of interaction is energy changing position.

Because energy can change position through space, an organized collection of particles will become disorganized if the particles are allowed to move around freely. However, although physics allows it, we never see the disorganized particles reverse their motions and reorganize. So, we observe what we call the arrow of time. However, entropy and the arrow of time can be explained by energy changing position. Let's take our collection of particles and place them at the origin point of a coordinate system. Each particle in the collection has a defined position in that coordinate system, and the amount of disorder in the collection increases as the particles move farther away from their original positions. So, we could define the measure of entropy for the collection of particles as the total sum of each particles change in distance from its original position in the collection. If all the particles were at their original position, then measure of entropy would be zero. As entropy increases, the particles move farther away from their original position. However, this doesn't explain the arrow of time. To understand why smoke doesn't go back into the cigarette or why an ice berg doesn't jump out of the water and reform the glacier, we only have to examine the physics of a single particle.

In order to set the particle in motion, we have to apply some amount of energy $E_1$ to it. Now that the particle is changing position, the disorder or entropy of the collection of particles is increasing. To stop entropy from increasing, we have to apply the same amount of energy $E_1$ to bring the particle to rest. However, the entropy of the collection of particles is still higher than it was before we applied energy to the particle. In order to reverse entropy and restore the collection of particles to its original form, we have to apply some energy $E_2$ to the particle to reverse its motion. Then, we have to apply the same amount of energy $E_2$ to bring the particle to rest at its original location restoring the collection of particles to their original state. Therefore, it takes more energy to restore the collection of particles to their original state than it does to increase its entropy.

The amount of energy needed to set a single particle in motion and increase the entropy of the collection of particles is

$\text{increase entropy} \rightarrow E_1$

The amount of energy needed to reverse the motion of the particle and lower the entropy of the collection of particles is

$\text{reverse entropy} \rightarrow E_1 + 2 E_2$

Because the amount of energy needed to increase the entropy of a system is less than the amount of energy needed to restore the system to its original state, the collection of particles will continue to become disordered and the arrow of time is observed. So, although physics allows for these particles to reorganize themselves, it takes more energy to restore a collection of particles to their original form than it does to cause them to become disordered. This is why the arrow of time seems to move in only one direction,

• Entropy and the arrow of time are caused by energy changing position through space.
• It takes more energy to organize a collection of particles than it does to cause them to be disordered.
• The only thing various forms of energy can do as a result of interacting with other forms of energy or itself is to change position through space such that change in the environment can only occur when energy is allowed to change position through space.

So, even entropy and the arrow of time can be explained by energy changing position, but we still need to consider atomic clocks. After all, they are based on QM where the motion of energy is not so easily defined.

QM and Motion

I believe the biggest confusion regarding QM for most people is what we are actually measuring when we apply energy to a quantum mechanical system. Because we cannot physically see these subatomic particles such as electrons, we cannot directly measure how energy within the system is changing position. Instead, we attempt to detect their position in space by using some form of energy to interact with these particles or fields. Then, we determine the position where they occurred along with how many times we were able to detect the particle at that position. This allows us to build a probability space of where the particle occurs and the probability that you will find it at any given position in that space. The following image demonstrates these atomic orbitals, which "can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus."

Figure 3 - False-color density images of some hydrogen-like atomic orbitals (f orbitals and higher are not shown). (Wikipedia atomic orbitals)

Measurements in QM such as those taken for quantum numbers or quantum fields are completely different than measuring how the energy contained within these particles or fields is changing position through space as defined in the classical sense. QM deals with aspects of energy that can't be explained using classical mechanics.

A classical field is a function defined over some region of space and time.%5B3%5D Two physical phenomena which are described by classical fields are Newtonian gravitation, described by Newtonian gravitational field g(x, t), and classical electromagnetism, described by the electric and magnetic fields E(x, t) and B(x, t). Because such fields can in principle take on distinct values at each point in space, they are said to have infinite degrees of freedom.%5B3%5D

Classical field theory does not, however, account for the quantum-mechanical aspects of such physical phenomena. For instance, it is known from quantum mechanics that certain aspects of electromagnetism involve discrete particles—photons—rather than continuous fields. The business of quantum field theory is to write down a field that is, like a classical field, a function defined over space and time, but which also accommodates the observations of quantum mechanics. This is a quantum field.

However, QM behaviors occur within space and must be caused by energy changing position and interacting with energy in the QM system / environment. As defined earlier, the only thing energy can do within space is change position. Granted, energy takes on many forms where each form has special properties that cause different forces to be generated that in turn affect the motion of energy in the QM system but, despite the cause that set it in motion, the only thing energy can do as far as space is concerned is move through it, which allows energy to interact with the environment.

Even if we never learn how energy within these particles or waves is changing position through space, how could change in an atom or particle occur if energy wasn't allowed to change position and interact with other energy within the system? Just because we don't know or can explain how this energy changes position doesn't mean that motion does not occur within QM systems. Furthermore, force is propagated in QM through force carriers. If the energy contained within these particles was not allowed to change position through space, then the universe would not be able to exist much less use them to take measurements of QM systems.

An argument was made that we cool the atoms in atomic clocks to near absolute zero as to eliminate motion from the QM system, and yes this is somewhat true. However, it is not an honest argument. We don't super cool the atoms to eliminate the motion of the energy in the atoms. We super cool the atoms to eliminate the motions of any energy within the environment that would interfere with the motion of the energy contained in our atoms. Then, when we use energy to interact with these atoms to create atomic clocks, we can measure the QM affects that result from the energy changing position within the atoms themselves. We may never know how the energy within atoms is actually changing position through space in a classical sense but, because energy has to change position through space to interact with other energy in the system, we observe QM properties that are a result of such motions. Our entire existence occurs because energy changes position through space.

• Measurements of quantum numbers, fields, and other QM effects are not the same as measurements of energy changing position through space.
• QM effects arise when energy changes position through space.
• Energy contained within atomic clocks must change position through space for the clock to work.
• Because we can't measure how energy in atomic clocks change position through space, we have to count the number of times some quantum number completes a cycle.

I realize that my argument for QM might be considered weak. However, if we agree that the universe is comprised of space and the energy that occupies it, then the only way change can propagate is by energy changing position through space, which includes QM systems.

I would also like to add that even though quantum teleportation seems to allow information to traverse distances through space instantaneously, energy must still be allowed to change position through space for it to work.

Quantum teleportation is a process by which quantum information (e.g. the exact state of an atom or photon) can be transmitted (exactly, in principle) from one location to another, with the help of classical communication and previously shared quantum entanglement between the sending and receiving location.

I figured this additional statement was required to make because I can see someone arguing this point.

Temporal Uniformity

So far, we have explained a few of our observations of time:

• We experience the passage of time as a duration, which is relative to the observer's frame of reference. The passage of time occurs because energy cannot move through space with an infinite speed.
• We observe a clear direction through time such that we exist in the present moving towards the future leaving the past behind. It takes more energy to restore a collection of particles to their original arrangement than it does to cause them to become disordered.
• When plotting the path matter and energy takes through space, an extra variable is required to order events such as changes in position. Instead of measurements of time, we can now use measurements of distance when plotting that path energy takes through space and to order events.

We still need to explain why time can speed up or slow down, why we don't see a dimension of time, and how we can resolve temporal paradoxes by removing the temporal dimension of time altogether. In order to do this, we have to discuss how four spatial dimensions can be used to describe how we experience time including relativistic affects such as time dilation and length contraction.

Temporal uniformity states that the universe consists of four dimensions of space where the energy contained within this space is moving away from a single point at the speed of light where the big bang event occurred. Once the energy from the big bang cooled enough to form matter, this matter became trapped in a manifold of space defined by the boundary of a 3-sphere or four-dimensional sphere that is radially expanding at the speed of light. The reason why matter and energy is bound to a 3-sphere is because the energy that formed this matter started its journey moving away from a four-dimensional point in space at the speed of light where the big bang event occurred. Because matter and energy is moving at the speed of light along a radial vector of a 3-sphere, we can only observe light that moves in the three remaining dimensions that are perpendicular to these radial vectors. So, this three-dimensional boundary of this four-dimensional sphere becomes the space that we and everything that is observable exists; essentially our view of the universe. Figure 4 illustrates a two-dimensional cross-section of a four-dimensional sphere with the observer $O_b$ located at the coordinate where the big bang event occurred and the observers $O_n$ and $O_r$ located on a boundary defined by the four-dimensional sphere.

Figure 4 - A two-dimensional cross-section of a 3-sphere with three observers defined at various locations.

This view of space and the energy that occupies it gives us a natural explanation for why we observe the space in between galaxies expanding. It also explains why space is expanding everywhere with no observable center to the expansion. We simply can't see the center to this expansion because it exists in four-dimensional space.

However, it's not space or time that is moving or expanding. Instead, it's the energy radiating away from a four-dimensional point at the speed of light that is causing the observers $O_r$ and $O_n$ to move away from each other with a velocity $V_r$. Although this provides us with an explanation for how space can expand, it doesn't explain the acceleration that is observed, or provide us with an explanation for gravity. However, the affect of the expansion for this 3-sphere is numerically identical to the "bug on a band" problem. The following is from a post I made in the thread, "Is Krauss looking at this right?".

This [ant on a rubber rope] problem has a bearing on the question of whether light from distant galaxies can ever reach us if the universe is expanding.[2] If the universe is expanding uniformly, this means that galaxies that are far enough away from us will have an apparent relative motion greater than the speed of light. This does not violate the relativistic constraint of not travelling faster than the speed of light, because the galaxy is not "travelling" as such—it is the space between us and the galaxy which is expanding and making new distance. The question is whether light leaving such a distant galaxy can ever reach us, given that the galaxy appears to be receding at a speed greater than the speed of light.

"Two views of an isometric embedding of part of the visible universe over most of its history, showing how a light ray (red line) can travel an effective distance of 28 billion light years (orange line) in just 13 billion years of cosmological time. Click the images to zoom. (Wikipedia)"

The expansion of space is often illustrated with conceptual models which show only the size of space at a particular time, leaving the dimension of time implicit.

In the "ant on a rubber rope model" one imagines an ant (idealized as pointlike) crawling at a constant speed on a perfectly elastic rope which is constantly stretching. If we stretch the rope in accordance with the ΛCDM scale factor and think of the ant's speed as the speed of light, then this analogy is numerically accurate—the ant's position over time will match the path of the red line on the embedding diagram above.

The point I was making in the quote is that if space is the rope and it stretches in accordance with the ΛCDM scale factor and we think of the ant's speed as the speed of light, then this analogy is numerically accurate to the observed expansion of space. Therefore, the concept that we are bound to a 3-sphere that is expanding at the speed of light through four-dimensional space fits within our current framework of physics.

So, in order to explain dark matter, dark energy, and dark flow we have to expand upon the idea of being bound to a 3-sphere in four-dimensional space, which we will do later in the section on Space, Gravity, and Time Dilation. The reason why it is important to introduce this into the discussion is because it provides us with a mechanism that explains why we can't perceive four-dimensional space directly and allows us to explain gravity and gravitational time dilation. As explained above, since we are bound to a 3-sphere that is expanding radially at the speed of light, we can only see light that moves along dimensions of space that are perpendicular to radial vectors that comprise the 3-sphere. Thus, we cannot physically see this four-dimensional space, and can only observe how it affects energy as it moves through it.

Furthermore, if we consider that the energy of the big bang event didn't radiate away from this four-dimensional origin point all at once, there would exist layers of 3-spheres each expanding radially at the speed of light that contain their own matter and energy. Each layer would be a separate universe altogether, but each layer would still exist in the same four-dimensional space and potentially interact with each other through gravitational fields. Because each layer has a radial velocity that is zero relative to the radial velocity of other layers, matter and energy within these other layers could still affect each other gravitationally. However, because each layer has a radial velocity equal to the speed of light relative to the four-dimensional point in space where the big bang event occurred, the gravitational affects of each layer could only affect matter and energy along dimensions that are perpendicular to radial vectors that comprise the 3-spheres. Because matter and energy within layers along the outermost edge of the expansion of the big bang are moving away from each other with a greater velocity $V_r$ than the inner layers, these outer layers would pull the matter and energy within the inner layers along with it. This not only explains dark matter, but also provides a mechanism for dark energy, which would also be observed as dark flow. Figure 5 illustrates these layers as three-dimensional spheres.

Figure 5 - A three-dimensional representation of the universe demonstrating various layers defined by the surface of the spheres that contain individual local views of space.

Figure 6 illustrates how these 3-spheres are comprised of matter and energy, and how dark matter and dark energy can be represented as different 3-sphere layers.

Figure 6 - A two-dimensional cross-section of four-dimensional space showing how dark matter existing at different layers or 3-spheres can give rise to dark energy and dark flow.

Some of you might disagree with this view space and time because we are taught that anything with mass cannot move at the speed of light. However, when we use special relativity to derive our four-velocity through space-time for an object at rest, we find that the magnitude of this four-velocity is equal to the speed of light.

The magnitude of the four-velocity of a massive particle is c. Let us for simplicity just consider special relativity.

So where is the proper time. As such we are discussing physical massive particles. You can write this as

where , "classical 3-velocity".

The magnitude is given by

depending on your conventions. Either way, the magnitude is given by c.

Then passing to the rest frame of the particle we see that

(in the rest frame)

Now that we have defined space as being four-dimensional, and that matter and energy moves through this space at the speed of light restricted to the boundary of 3-spheres, we can explain why time dilates. This has nothing to do with some actual physical dimension of time. Instead, it relates back to how energy can only traverse distances through space at finite speeds. If the metric of space changes how distances are traversed, then naturally clocks will measure time differently based on an observer's FoR.

Space, Gravity, and Time Dilation

If we consider that time is just a property of space that is a consequence of energy moving with finite speeds, then it becomes easy to explain time dilation.

Figure 7 - The light clock to the left is at rest and the clock to the right is moving relative to the observer's coordinate system (figure 2 revisited).

If we examine the light clock on the right that is in motion relative to the observer's FoR, we will notice that the path the photons have to traverse between the two plates has been elongated. Because the speed of light is the same for all observers, the observer at rest relative to the clock in motion will notice that the clock ticks more slowly. The tick of the clock doesn't change because it moves through time slower than the observer. The tick of the clock changed because the metric of space has changed causing distances to be affected. The path that the photons have to traverse has been elongated, which affects how the energy of the photons changes position through space. It's as simple as that. There is no special temporal dimension where the clock is moving slower than the observer.

In order to understand gravity, we have to understand that what we observe in the universe is actually a three-dimensional projection of matter within four-dimensional space. Because energy is moving away from a four-dimensional point in space at the speed of light in all directions (where the big bang event occurred), we lose the ability to perceive light along our local radial vector of space. So, if the universe was three-dimensional, and we were bound to the surface of a three-dimensional sphere that was radially expanding at the speed of light, we would only perceive the universe as having two-dimensions as defined by the surface of the sphere. However, the universe would still consist of three dimensions of space. As previously discussed, when we apply this concept to four-dimensional space, we end up with three observable dimensions of space.

Now, as objects gain mass they slow down as a result of conservation of momentum. This is also applies to motion through four-dimensional space. However, objects that gain mass don't seem to slow down along the radial vector of expansion and fall outside of the boundary defined by the 3-sphere and completely disappear from our local view of the universe. You can think of this no different than cars moving down the highway at different speeds. Eventually, cars that are moving slower down the highway will disappear from the view of cars moving faster. So, we are forced to expand upon the concept of being bound to a 3-sphere that is expanding radially at the speed of light because just defining a boundary that arises as a consequence of the speed of light isn't enough to explain how conservation of momentum plays its part and why objects with mass simply don't vanish from this boundary.

To explain why matter and energy remains bound to these 3-spheres, we have to speculate that these layers that were formed by matter and energy moving away from a four-dimensional point in space at the speed of light where the big bang event occurred is more than just a result of speed. Instead, if we consider that most of the energy from the big bang event is in the form of gravitational waves, then we can consider that this boundary defined by 3-spheres are actually gravitational waves that formed when the big bang event occurred. Therefore, not only do these 3-spheres arise from matter and energy moving at the speed of light away from a four-dimensional point in space, but that most of the energy contained within these layers is in the form of four-dimensional, spherical, gravitational waves.

If we can consider this speculation, then we can explain why objects with mass don't disappear from our own boundary defined by the 3-sphere, and we arrive at a natural explanation for gravity and gravitational time dilation. First of all, due to the law of the conservation of momentum, as objects gain or lose mass its speed will change along all dimensions of space. This includes the higher four-dimensional space that, as explained previously, we can't physically see. However, if these layers defined by the 3-spheres are also gravitational then, as objects gain mass or energy, they will attempt to slow down along their local radial vector. However, instead of completely moving outside of the boundary defined by the 3-sphere, they can only move so far until this extra mass / energy is countered by the gravitational field that is defined by the 3-sphere itself. As the objects gain mass, they can only move so far until the field counters this motion and continues to carry this matter and energy along with the expansion. So, instead of a smooth boundary as defined by the 3-sphere, it's more like the surface of a golf ball where the dimples represent the displacement of objects with mass relative to the smooth boundary of the 3-sphere itself.

Figure 8 - Gravity as demonstrated from a two-dimensional space that exists in a three-dimensional universe.

As figure 8 illustrates, if we existed on the two-dimensional surface of a three-dimensional sphere that is expanding radially at the speed of light, then objects with mass would create an impression in the higher three-dimensional space the comprises the universe. This causes an elongation of distances when observed from the higher, three-dimensional space as can be seen in figure 8. However, we cannot observe this higher dimension of space because we are moving at the speed of light along our radial vector from the center of the sphere to the point we are located. The means that we can only see light that is perpendicular to this radial vector. Although the true distance through the higher dimension of space is the length of the green curve from the points C to D, we'd only perceive the distance through space as the orange line from the points A to B. To us, it would appear flat.

This distortion of distance in higher dimensional space, is why gravity and gravitational time dilation exists. As an observer moves along the green path starting from point C moving towards the massive object in between C and D, for every unit of distance the observer traverses in the two-dimensional space, the distance traversed in the higher three-dimensional space increases. Although we can't see this higher dimension of space, the observer would still be traversing greater and greater distances for each unit of distance traveled in the two-dimensional space. This is why the observer experiences the force of gravity. The distance in the higher dimension of space is actually increasing for each unit of distance traversed within the two-dimensional manifold, and it is this effect that causes clocks to tick more slowly in gravitational fields. No different than distances being elongated for the light clock that is in motion relative to the observer, the distances as observed in the higher dimensional space is being elongated by the gravitational field and causing the clock to tick more slowly.

Again, none of this has to do with the existence of some physical dimension of time that we call the temporal dimension. Instead, time dilation, gravity, and gravitational time dilation can all be explained by changes in the metric of space that causes distance to contract or expand, which in turn affects the distance the mechanism in the clock has to traverse in order to measure time. It's as simple as that, and we don't have to invoke some temporal dimension to measure time and order events.

The Nature of Time

When we apply this concept to four-dimensional space, we can explain why we perceive the universe as having three spatial dimensions while experiencing time. However, we do not need to invoke some physical temporal dimension where energy exists as it was in the past or as it will in the future. When we consider that time is a mathematical consequence of energy traversing distances through space at finite speeds, then the only way one can "time travel" is to physically arrange all of the matter and energy to the way it was in the past or how it will be arranged in the future, which is physically impossible.

When our equations, such as those in general relativity, give us a negative change in our time variable $t$, we don't have to interpret this as a way to travel back in time. The view as provided by temporal uniformity suggest that instead of traversing some temporal dimension, that we are really traversing the higher, four-dimensional space that comprises the the universe such that we end up at a point in four-dimensional space outside of the boundary as defined by the four-dimensional sphere or 3-sphere that comprises our local three-dimensional view of the universe. In the case of wormholes, we'd just end up in a different three-dimensional space defined by a completely different 3-sphere. The matter and energy located in our manifold of space would still be expanding outward with the rest of the layers and you'd simply end up in a different layer.

So, using the framework of temporal uniformity, we can now answer some of surreptitious57's questions.

Is time a dimension without any past or present or future? Does one move through it at the rate of one second per second?

That is to say the same rate as the second hand ticking on a clock is also the same rate one experiences time in real time?

Time is a mathematical consequence of matter and energy being restricted to changing position through space at finite speeds. It's simply impossible for anything to traverse space with an infinite speed, which is required in order to change position without any duration of time to be experienced. We perceive the universe because electrical signals in our brains traverse the distances that separate our neurons. If these electrical signals could traverse the neural pathways with infinite speed, then our brains could process every single piece of information received from the environment and we could have every single thought instantaneously. If the electrical signals in our brains traversed the distances along our neural pathways with infinite speed, then they could traverse these pathways an infinite number of times before any amount of time would pass. The result of this would be as though time stopped.

So, at least within temporal uniformity, time is not a dimension at all and, as demonstrated earlier, we don't even need measurements of time in order to formulate the laws of physics or order events. Because points in space exist regardless if time is defined or not, the energy that occupies space can change position within it. The result is the occurrence of motion, and it is the motion of energy through space that causes change in the environment to occur. Therefore, matter and energy can only move through space because time does not exist as a physical dimension all by itself or even as a part of space. Time is nothing more than a mathematical consequence of distance.

Is there a reason why the three arrows of time [ biological / thermodynamic / cosmological ] move in only the one direction?

As explained in the section, Change in a System, the arrow of time occurs because it takes more energy to restore a collection of particles to a particular arrangement than it does to cause the collection of particles to become disordered.

The amount of energy required for entropy to increase is

$\text{increase entropy} \rightarrow E_1$

where the amount of energy required to reverse entropy is

$\text{decrease entropy} \rightarrow E_1 + 2 E_2$

Is time what space expands into?

In the section, Temporal Uniformity, we demonstrate how space can expand naturally without having to invoke some fabric of space that is expanding itself. The effect is best illustrated in the following image.

What happens to time once it passes the event horizon? Does it speed up or slow down?

Or stay just the same?

Although we didn't discuss black holes, we did define how gravitational fields affect the metric of space causing distances to expand or contract. If distances become elongated, then the mechanism inside a clock that uses motion to measure time will tick more slowly. Time dilation occurs because the metric of space changes how distances are defined and, even if we can't perceive this contraction or expansion of space as is the case for gravitational fields, the effect arises because distances are affected. Because relativity theory has shown that the speed of light is the same for all FoRs when measured using the same units, then energy itself and not just clocks is affected by this contraction or expansion of space. So, not only do clocks tick more slowly, but energy itself has to traverse distance for change in the environment to occur. So, observers / people will also age more slowly when placed within gravitational fields relative to observers / people outside of theses fields.

However, just because physical processes are affected by the elongation or contraction of distances within or outside gravitational fields doesn't mean that these processes are literally moving slower or faster through time. Sure, people will age differently when observed from varying FoRs, but this effect is purely spatial. That is why my theory is called temporal uniformity. Although we can experience time differently, we still remain temporally uniform as energy changes position through space. In other words, we don't observe object disappearing and reappearing because clocks slow down or speed up.

How is it possible for a photon not to experience time but travel through it? Has time always existed?

From the FoR of a photon, if such a thing could be said, time could not be measured at all. Take a look at the light clock to the right that is in motion relative to the observer.

As the speed of the clock approaches the speed of light relative to the observer, the path that the light has to traverse increases more and more. If the clock was allowed to travel at the speed of light, then the path the photon's have to traverse between the reflective plates becomes stretched to infinity. Therefore, it would be impossible for such a clock to measure time.

As for the question, "Has time always existed?", I believe the answer depend on if space has always existed. Although my reply to this is pure speculation, I believe that space and energy has always existed because energy cannot be created or destroyed. So, I believe that space had to exist in order for energy to fill it. If we consider the universe as described by temporal uniformity, not only can we define the big bang resulting in multiple layers of 3-spheres that represent separate three-dimensional universes, but also multiple big bang events that can occur throughout this physical four-dimensional space.

So, not only have we defined time within the framework of temporal uniformity beyond the simple definition of that which clocks measure, but we also have derived a definition that encompasses all of our observation of time while demonstrating that our preconceptions regarding time travel are wrong. Sure, there are some speculation on my part, but these speculations fit within our current framework of physics. To demonstrate this I'll will re-post the mathematics I derived that uses the equations of special relativity for time dilation to derive the notion that we traverse four-dimensional space at the speed of light.

One way to view time as motion through the fourth dimension would be as follows:

We will begin at a position located at time zero at the center of the big bang. This frame of reference will remain at time zero at the center of the big bang, observing the expansion of the universe from this view. Let there be a light clock located at this frame of reference and let us synchronize this light clock in accordance to the definitions in special relativity. We shall denote the time it takes for light to travel from point A to point B and back to point A in this frame of reference as $\Delta t_{b}$. The subscript $b$ denotes the big bang reference. According to SR the light clock in this frame of reference would measure time according to:

$\Delta t_{b} = \frac{2L}{c}$ where $L=|AB|$

If time was nothing more than motion, then the observer in this reference frame would infer motion through time by observing the expansion of the big bang. So their light clock would still be able to bounce light between two reflective surfaces because, even though they are not moving with the expansion of the big bang, motion would still be possible.

Now let us consider the motion of an identical light clock that is radiating outward from the big bang, through the fourth dimension, relative to the big bang frame of reference. We shall denote the time observed by this clock as $\Delta t_{w}$. The subscript $w$ denotes the spatial axis of time. The use of $w$ to label the time axis is to avoid confusion with the time variable $t$. This frame of reference is called the time-normal frame as it defines a frame of reference that is only moving through the fourth dimension, time, away from the big bang. This is normally called the lab frame of reference. It is important to note that in all frames of reference, an observer will see their light clock no differently than the observer at the big bang frame of reference. The equations of relativity predict how one observer will view the other and vice-versa. This means that the observer in the big bang frame of reference will not see the path of the light from the light clock in motion relative to their position as a straight up and down path. The same is true for the observer in motion as they will see the light from the light clock positioned at the big bang frame of reference the same way. The path that is observed is that of a triangle instead of one that is straight up and down. Because Einstein proved the constancy of the speed of light, the difference in time measured as observed from each frame of reference is:

$\Delta t_{n} = \frac{2N}{c}$ where $N=\sqrt{(\frac{V_{w} \times \Delta t_{n}}{2})^{2}+L^{2}}$

$V_{w}$ is the velocity through the fourth dimension and $L=|AB|$

Finally, we shall place another identical light clock that is in relative motion with the clock in the time-normal frame of reference, except these two clocks share the same position, speed, and direction through the temporal dimension. Therefore, the only relative motion between them is through spatial dimensions that are perpendicular to their forward motion through time. This means that both clocks have a relative velocity of zero along the temporal dimension and a non-zero relative velocity along the $x, y$ and $z$ axis of space. We shall denote time in this frame of reference as $\Delta t_{r}$. The subscript $r$ denotes the time-relative frame of reference. We shall derive the equation for time dilation for this frame of reference in respect to the big bang frame of reference. This allows us to derive the following relationship:

$\Delta t_{r} = \frac{2R}{c}$ where $R=\sqrt{(\frac{V_{w} \times \Delta t_{r}}{2})^{2}+(\frac{V_{r} \times \Delta t_{r}}{2})^{2}+L^{2}}$

$V_{w}$ is the velocity through the fourth dimension, $V_{r}$ is the velocity through all other spatial dimensions, and $L=|AB|$

Now that we have derived the equations for both frames of reference that are moving away from the big bang, we can relate them back to the big bang frame of reference according to the following:

Solving for $\Delta t_{n}$ in the time-normal frame of reference we get:

$\Delta t_{n} = \frac{2L/c}{\sqrt{1-\frac{V_{w}^{2}}{c^{2}}}}=\frac{\Delta t_{b}}{\sqrt{1-\frac{V_{w}^{2}}{c^{2}}}}$

Solving for $\Delta t_{r}$ in the time-relative frame of reference we get:

$\Delta t_{r} = \frac{2L/c}{\sqrt{1-\frac{V_{w}^{2}}{c^{2}}-\frac{V_{r}^{2}}{c^{2}}}}=\frac{\Delta t_{b}}{\sqrt{1-\frac{V_{w}^{2}}{c^{2}}-\frac{V_{r}^{2}}{c^{2}}}}$

We can see that from the above relationships that the time-normal frame of reference is related to the time-relative frame of reference as follows:

$\Delta t_{r} \sqrt{1-\frac{V_{w}^{2}}{c^{2}}-\frac{V_{r}^{2}}{c^{2}}}=\Delta t_{n} \sqrt{1-\frac{V_{w}^{2}}{c^{2}}}$

This allows us to derive the following relationship between the time-normal frame of reference and the time-relative frame of reference:

$\Delta t_{r} =\Delta t_{n} \frac{\sqrt{1-\frac{V_{w}^{2}}{c^{2}}}}{\sqrt{1-\frac{V_{w}^{2}}{c^{2}}-\frac{V_{r}^{2}}{c^{2}}}}=\frac{\Delta t_{n}}{\sqrt{1-\frac{V_{r}^{2}}{c^{2}-V_{w}^{2}}}}$

This result makes sense because everything has a relative temporal velocity of zero:

$\Delta t_{r}=\frac{\Delta t_{n}}{\sqrt{1-\frac{V_{r}^{2}}{c^{2}-0^{2}}}}=\frac{\Delta t_{n}}{\sqrt{1-\frac{V_{r}^{2}}{c^{2}}}}$

But, $V_{w}$ becomes apparent from the big bang frame of reference such that:

$V_{w}=\frac{\sqrt{c^{2}(\Delta t_{n}^{2}-\Delta t_{r}^{2})+\Delta t_{r}^{2}\times V_{r}^{2}}}{\sqrt{\Delta t_{n}^{2}-\Delta t_{r}^{2}}}=\sqrt{c^{2}+\frac{\Delta t_{r}^{2}\times V_{r}^{2}}{\Delta t_{n}^{2}-\Delta t_{r}^{2}}}$

Interpreting this result seems to reveal that we move at a velocity other than the speed of light through the dimension of time. This is due to the relative velocity $V_{r}$ through the other spatial dimensions. However, when we place ourselves in the big bang frame of reference, we realize that $V_{r}=0$. This is because all observers will place themselves in the time-normal frame of reference with all other bodies being time-relative in respect to their position. Therefore, all observers have zero relative velocity.

$V_{w}=\sqrt{c^{2}+\frac{\Delta t_{r}^{2}\times 0^{2}}{\Delta t_{n}^{2}-\Delta t_{r}^{2}}}=\sqrt{c^{2}}=c$

So if we can deduce through mathematics that time is motion through the fourth dimension at the speed of light, shouldn't my notions about time travel be true in regards to the impossibility of traveling to a time that exists as a memory of the past or will exist as a memory in the future?

Thank you for reading this post. I hope that I have clarified, at least within the framework of temporal uniformity, what time actually is. In doing so, we have resolved temporal paradoxes by showing that time travel is impossible, provided explanations for the multi-verse and how dark matter, dark energy, and dark flow exists within our four-dimensional universe, and have defined a consistent definition to explain time.

Edited by Daedalus
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Nice try Michael,

In the rest frame $\tau=t$ $x^0=ct$

for the rest frame $U^0= dx^0/d\tau =cdt/d\tau=c\gamma$

Cancelling out the c's

$dt/d\tau=\gamma$

Because we are in the rest frame $\gamma=1$ and $\tau=t$

You are saying the speed of time is one second per second.

a sec per sec is not a rate.

Typically $dt/d\tau$ is considered the rate of time dilation.

Edited by david345
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Well this being my first time looking at this thread thus far I'm impressed by your rigor. Unlike most speculation models your backing up yours with mathematics.

I still have to look over what you have thus far in greater detail however I see a couple of points and potentially counter arguments against your model. Looking over this this far it looks like your using primarily SR as opposed to GR. I can see some inherent problems with this but not in something previously mentioned throughout this thread.

A common mistake in many speculations is that there is a belief that expansion is based primarily upon distance measurements as its primary source of evidence. This is far from the truth.

In fact an even larger body of evidence is thermodynamics.

Rather than go through an entire course in Cosmology applications of thermodynamic laws I'll jump to some primary relations.

$E=(\rho c^2+p)R^3$

Expansion is adiabatic if there is no net flow or outflow of energy. So that

$\frac{de}{dt}=\frac{d}{dt}(\rho c^2+p)R^3=0$

Let p be proportional to $\rho c^2$

$p=w\rho c^2$

This link gives the appropriate equations of state

Without going through all the steps expansion is accurately described via the FLRW metric acceleration equation which can correlate energy density/pressure and temperature relations.

The acceleration equation is

$\frac{\ddot{a}}{a}=-\frac{4\pi G\rho}{3c^2}(\rho c^2+3p)$

$H^2=\frac{\dot{a}}{a}=\frac{8\pi G\rho}{3c^2}-\frac{kc^2p}{R_c^2a^2}$

where k is the curvature constant.

now the curvature constant can have three main configurations 1,0-1.

You've probably know about the stress energy tensor but this set of relation is handy to know.

$T^{\mu\nu}=(\rho+p)U^{\mu}U^{\nu}+p\eta^{\mu\nu}$

Which correlate the stress energy tensor to energy density/pressure in Minkowskii metric form.

If you look through the universe geometry formula below I go into basics on the FLRW metric and how the curvature constant affects light paths.

The FLRW metric to distance formula is.

$d{s^2}=-{c^2}d{t^2}+a{t^2}d{r^2}+{S,k}{r^2}d\Omega^2$

$S\kappa r= \begin{cases} R sin r/R & k=+1\\ r &k=0\\ R sin r/R &k=-1 \end {cases}$

For example when k=0 light rays remain parallel, they Will either converge or defract depending if the curvature constant is positive or negative.

http://cosmology101.wikidot.com/redshift-and-expansion

http://cosmology101.wikidot.com/universe-geometry

http://tangentspace.info/docs/horizon.pdf:Inflation and the Cosmological Horizon by Brian Powell

http://arxiv.org/abs/1304.4446:"What we have leaned from Observational Cosmology." -A handy write up on observational cosmology in accordance with the LambdaCDM model.

http://arxiv.org/abs/astro-ph/0310808:"Expanding Confusion: common misconceptions of cosmological horizons and the superluminal expansion of the Universe" Lineweaver and Davies

http://www.mso.anu.edu.au/~charley/papers/LineweaverDavisSciAm.pdf:"Misconceptions about the Big bang" also Lineweaver and Davies

The above links are basic articles on common misconceptions.

These three will get you started on the ideal gas laws applications involved. The last will help you with your GR, and incorporates GR to the FLRW metric via the Einstein field equations and thermodynamic laws.

http://arxiv.org/pdf/hep-ph/0004188v1.pdf:"ASTROPHYSICS AND COSMOLOGY"- A compilation of cosmology by Juan Garcıa-Bellido

http://arxiv.org/abs/astro-ph/0409426An overview of Cosmology Julien Lesgourgues

http://arxiv.org/pdf/hep-th/0503203.pdf"Particle Physics and Inflationary Cosmology" by Andrei Linde

http://www.wiese.itp.unibe.ch/lectures/universe.pdf:"Particle Physics of the Early universe" by Uwe-Jens Wiese Thermodynamics, Big bang Nucleosynthesis

http://www.blau.itp.unibe.ch/newlecturesGR.pdf"Lecture Notes on General Relativity" Matthias Blau

the main concern is if you want your model seriously considered you will have to correlate the ideal gas law aspects to the EFE and FLRW metric to explain the thermodynamic history of our universe via your model

Including Big bang nucleosynthesis and the corresponding particle species % found in the CMB.

Anyways it's late at my locale, I will provide some arguments on galaxy rotation curves in terms of an ideal gas isothermal sphere and the NFW profile. As well as touch on the Integrated and non integrated Sachs Wolfe effect. For dark matter I will probably add some details on gravitational lensing which Will correlate its gravitational influence.

Dark flow itself is a locale group interaction its not applied the the entire observable universe. Our local group is moving to a sulercluster called the great attractor. Accronym name.

The main point is as far as I can tell your model is based mainly upon SR and how light can cause influence upon observable influences. Unfortunately Cosmology doesn't based its model strictly upon light. It's intensely includes particle physics and thermodynamic laws as well.

For example the temperature of the universe at any time by the inverse of the scale factor a(t).

I would also look into cosmological redshift and luminosity relations

Edited by Mordred
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Well this is the first time I have seen this thread and I am looking forward to reading all the posts carefully.

It is a long time since I was motivated to get my boots out of the mud and read detailed thoughts about cosmology.

+1

There is much to consider here and the quantity accounts for why we don't see posts from you more often but this caught my eye

What is Time?

In order to understand the nature of time, we need to understand what it does and how it behaves. This requires making valid observations about time and its properties, and then construct a definition for time that encompasses these observations. So, let's list all of the observations we can make about time that we know are true.

• We experience the passage of time as a duration, which is relative to the observer's frame of reference. Therefore, time is observed to speed up or slow down relative to various frames of reference.
• Although the passage of time is relative to an observer, the passage of time is continuous. e.g. we don't observe objects disappearing and reappearing at different points in time because their clocks speed up or slow down.
• We observe a clear direction through time such that we exist in the present moving towards the future leaving the past behind.
• Although we experience a direction through time, we cannot see the dimension of time.
• When plotting the path matter and energy takes through space, an extra variable is required to order events such as changes in position. In physics, we define this extra variable as time and we use clocks to measure it.
• Because it is mathematically impossible for anything to move the space with an infinite speed, the act of changing position through space requires a duration of time. This is what it all boils down to in the end. This entire post is meant to emphasize this observation which demonstrates that time is a mathematical consequence of traversing distances through space at finite speeds. This will be the theme for my arguments regarding the nature of time.

I really can't think of anything else that we can say about time that is an actual observation, but there are a few preconceptions that we can list that fall right in line with surreptitious57's questions. These preconceptions usually revolve around time travel; the idea of being able to physically move through space-time in such a way that allows matter and energy to travel to the past.

• Time might exist physically as a temporal dimension where matter and energy continues to exists along this dimension arranged the way it was in the past and will be arranged in the future.
• Time might exist as some physical dimension that everything moves through, but only in one direction such that we could travel to the past if we could reverse our direction through time.
• Time might exist according to the many worlds interpretation of quantum mechanics and there could be an infinitude of parallel universes that are defined by every possible event that can happen.
• If time travel is possible, then temporal paradoxes are possible, and you could go back in time and kill your grandfather before your father was born. The paradox that I find most interesting is the one Michio Kaku mentioned about traveling to the past and killing yourself right before you time traveled.

Temporal Uniformity attempts to explain our observations of time and reconcile our preconceptions using a consistent mathematical definition for what time actually is. In doing so, we can explain all of our observations of time while demonstrating that time travel, as understood as the act of traveling to the past and interacting with matter and energy as it was arranged back then, is impossible. This frees us from temporal paradoxes, defines parallel universes which allow dark matter, dark energy, and dark flow to exist, and provides us with answers to some of surreptitious57's questions. We'll begin by discussing space and energy.

Several of your bullet points boil down to saying that we can observe effects in the material universe that require another generalised axis, dimension or formalised degree of freedom to explain and write equations for.

I agree with this completely and have made this point before, although my example (nuclear disintegration/radioactivity) does not require continuity but relates the phenomena directly to the counting numbers.

Edited by studiot
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Well this being my first time looking at this thread thus far I'm impressed by your rigor. Unlike most speculation models your backing up yours with mathematics.

I still have to look over what you have thus far in greater detail however I see a couple of points and potentially counter arguments against your model.

Well this is the first time I have seen this thread and I am looking forward to reading all the posts carefully.

It is a long time since I was motivated to get my boots out of the mud and read detailed thoughts about cosmology.

+1

I appreciate both of you taking the time to familiarize yourselves with my theory of temporal uniformity. I look forward to discussing the theory with both of you and anyone else that wants to join in the conversation. Please note, that I posted my theory when I first joined SFN. Back then I was definitely a green horn when it comes to discussing science and, even though I have extended my knowledge of physics, I still do not have all of the knowledge needed to fully argue all of the finer points needed to construct a full scientific theory. However, I have been working rigorously to remedy my ignorance.

With that being said, I am very talented when it comes to mathematics and I love solving mathematical problems and discovering new mathematical relationships (new to me anyway ). The following links are to various posts here at SFN that I have made regarding mathematical relationships that I've discovered:

and mathematical challenges that I have posted (most of them are my own design):

I mention this because as you review the entire thread, you will notice that I have matured in my ability to make a valid argument. However, the mathematical models that I have proposed within this thread are flawed, and I fully realize this. I'm still teaching myself Tensor Calculus so that I can properly apply my theory and incorporate general relativity. As you have no doubt noticed, most of the mathematics presented thus far are based solely on special relativity. So, you could state that what I have constructed is actually a special theory of temporal uniformity. Of course, I have no problem collaborating with others that can help me incorporate general relativity into temporal uniformity, but I have no problem attempting this on my own. After all, learning new mathematics and physical concepts is the fun part. Plus, I love making mathematical models whether they are accurate or not. It's still good practice.

With that in mind, I look forward to debating temporal uniformity and I hope that I can continue to refine and mold my theory into an actual scientific theory that can be considered to have merit. Of course, the only way I can do this is by exposing what I have thus far and debating the concept with my peers. Again, thank you for taking interest in my work.

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Energy is not an object, it is a property of objects. An electron has energy, or an electron-nucleus system has energy. It is not a "thing" unto itself. You can talk about energy transfer, but energy motion.

And until you can give an equation describing it, you can't infer motion in QM. (I mean, you can, and obviously do, but it's not rigorous) What's the equation of motion of an electron in an atom? Of its spin orientation?

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Well I for one give you props for recognizing this example of model building has flaws, and not pushing " This is how it is". As such it's an excellent example of model building the right way albeit improvements to conform to the physics and ideal gas laws are needed. Wish other posters had the same diligence.

If you take the time to look over the materials I provided above you will notice one of the links is a full (but older textbook). The Mathius Blau article is roughly 990 pages covering GR, with the last chapters covering Cosmology.

Later when I get a chance I'll post more training material for you, with GR and thermodynamic applications. In particular the Einstein field equations.

If you can afford and isn't adverse to buying textbooks. For GR I would recommend buying General relativity by Wald

http://www.amazon.ca/General-Relativity-Robert-M-Wald/dp/0226870332

+1

Edited by Mordred
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Mordred

one of the links is a full (but older textbook).

As you know I am a muddy boots, dirty hands technologist.

As such I have always felt GR to be inferring too much from too little.

So I was suprised to learn that over the years there have been several theories of general relativity, with different terms and constants in the equations changing as new material has arisen.

So I recommend you be very aware of which version you incorporate material from.

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Noted the textbook I was referring to is the particle physics by Liddle. It's primarily based on SO(5) rather than SO(10). The author took the time to cover the now improved physics understanding in the opening notes.

It still has tons of useful information provided one remembers our understanding has improved since its writing.

Edited by Mordred
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Well I for one give you props for recognizing this example of model building has flaws, and not pushing " This is how it is". As such it's an excellent example of model building the right way albeit improvements to conform to the physics and ideal gas laws are needed. Wish other posters had the same diligence.

If you take the time to look over the materials I provided above you will notice one of the links is a full (but older textbook). The Mathius Blau article is roughly 990 pages covering GR, with the last chapters covering Cosmology.

Later when I get a chance I'll post more training material for you, with GR and thermodynamic applications. In particular the Einstein field equations.

If you can afford and isn't adverse to buying textbooks. For GR I would recommend buying General relativity by Wald

http://www.amazon.ca/General-Relativity-Robert-M-Wald/dp/0226870332

+1

As you know I am a muddy boots, dirty hands technologist.

As such I have always felt GR to be inferring too much from too little.

So I was suprised to learn that over the years there have been several theories of general relativity, with different terms and constants in the equations changing as new material has arisen.

So I recommend you be very aware of which version you incorporate material from.

Thank you Mordred and Studiot! I really do appreciate the help, and I will study the material and begin researching those topics as soon as I can.

Energy is not an object, it is a property of objects. An electron has energy, or an electron-nucleus system has energy. It is not a "thing" unto itself. You can talk about energy transfer, but energy motion.

And until you can give an equation describing it, you can't infer motion in QM. (I mean, you can, and obviously do, but it's not rigorous) What's the equation of motion of an electron in an atom? Of its spin orientation?

Swansont, I'm not sure if any answer that I reply with will be satisfactory. Right now I'm at work programming a new module for the company. So, I am unable to give a lengthy reply at the moment. Once I get home from work, I will see what I can come up with for a reply.

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I didn't supply a reference, but the book I drew from is

http://www-astro.physics.ox.ac.uk/~pgf/Pedro_Ferreira/The_Perfect_Theory.html

Please note this is a great source of understanding and further reading but it is not technical enough for your purposes.

Edited by studiot
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With your strong interest in mathematics here is two articles covering the mathematics involved in particle physics and how it ties into GR in the second article. First article is primarily differential geometry coverage.

http://arxiv.org/abs/0810.3328A Simple Introduction to Particle Physics

http://arxiv.org/abs/0908.1395part 2

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Again, thank you Mordred! I really do appreciate it

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