# Using graphing functions to explain expansion and universal time dilation?

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I usually don't frequent physics, being a chemistry student in college, as I have never been a fan of it, but I do have a rather unusual question to ask. Could we represent the expansion of the universe after the Big Bang with a graphing function, such as x=y10? With expansion being represented by the distance between the upper and lower arms of the function? I'm asking this question because I've been thinking (A dangerous pastime for me) that since time is essentially the non-co-occurrence for lack of a better word, of two events, could we not represent our current position on the universal timeline with line such as y=0? With the rate of expansion slowly diminishing over time as the slope of x=y10 gradually becomes less and less? If so, this also begs the question, is time slowing down because expansion of the universe is slowing down because time can be represented as the differential increase of distance between y=0 and x=y10? Please point out where I'm wrong, I do not have any more than introductory college experience so please help me out with this.

Edited by DanTrentfield
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The expansion appears to be accelerating rather than slowing.

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The expansion appears to be accelerating rather than slowing.

Excuse me, you're right...... I confused my functions, let me correct that before I look like an idiot....

I also changed the X exponent to 10 because it's not only a more realistic representation of what I was trying to convey as it shows a sudden expansion and then gradual further expansion which is gradually diminishing, but it complies with Einstein's belief in 10 dimensions if I am not mistaken? My knowledge in physics is extremely limited as previously stated so please forgive me if I'm wrong about that.

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There is no universal time dilation. Time dilation requires a gravitational gradient. In Cosmology the background field is homogeneous and isotropic which is no preferred direction or location. Essentially this describes a uniform mass distribution.

In the FLRW metric This fluid distribution is defined by

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

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

a is the scale factor which correlates expansion

$Proper distance =\frac{\stackrel{.}{a}(t)}{a}$

$H(t)=\frac{\stackrel{.}{a}(t)}{a(t)}$

, s,k is the curvature constant, Omega the density.

You can see the similarities with SR in this (I copied this from my previous posts below)

Lorentz transformation.
First two postulates.
1) the results of movement in different frames must be identical
2) light travels by a constant speed c in a vacuum in all frames.
Consider 2 linear axes x (moving with constant velocity and \acute{x} (at rest) with x moving in constant velocity v in the positive \acute{x} direction.
Time increments measured as a coordinate as dt and d\acute{t} using two identical clocks. Neither dt,d\acute{t} or dx,d\acute{x} are invariant. They do not obey postulate 1.
A linear transformation between primed and unprimed coordinates above
in space time ds between two events is ( this below doesnt have curvature as SR assumes Euclidean)
$ds^2=c^2t^2=c^2dt-dx^2=c^2\acute{t}^2-d\acute{x}^2$
Invoking speed of light postulate 2.
$d\acute{x}=\gamma(dx-vdt), cd\acute{t}=\gamma cdt-\frac{dx}{c}$
Where$\gamma=\frac{1}{\sqrt{1-(\frac{v}{c})^2}}$
Time dilation
dt=proper time ds=line element
since$d\acute{t}^2=dt^2$ is invariant.
an observer at rest records consecutive clock ticks seperated by space time interval dt=d\acute{t} she receives clock ticks from the x direction separated by the time interval dt and the space interval dx=vdt.
$dt=d\acute{t}^2=\sqrt{dt^2-\frac{dx^2}{c^2}}=\sqrt{1-(\frac{v}{c})^2}dt$
so the two inertial coordinate systems are related by the lorentz transformation
$dt=\frac{d\acute{t}}{\sqrt{1-(\frac{v}{c})^2}}=\gamma d\acute{t}$

Here is relativity of simultaneaty coordinate transformation in Lorentz.

$\acute{t}=\frac{t-vx/c^2}{\sqrt{1-v^2/c^2}}$
$\acute{x}=\frac{x-vt}{\sqrt{1-v^2/c^2}}$
$\acute{y}=y$
$\acute{z}=z$

as far as graphing expansion the calculator in my signature has graphing functions, that allow you to graph several key regions such as the cosmological event horizon, particle horizon etc

http://www.einsteins-theory-of-relativity-4engineers.com/LightCone7/LightCone.html

there is an advanced user section that contains the formulas used in the calculator

Edited by Mordred
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Excuse me, you're right...... I confused my functions, let me correct that before I look like an idiot....

I also changed the X exponent to 10 because it's not only a more realistic representation of what I was trying to convey as it shows a sudden expansion and then gradual further expansion which is gradually diminishing, but it complies with Einstein's belief in 10 dimensions if I am not mistaken? My knowledge in physics is extremely limited as previously stated so please forgive me if I'm wrong about that.

To be clear, I mean the actual cosmic expansion that we observe out in the universe appears to be accelerating rather than slowing down.

If I have my timeline correct, the very, very beginning of the universe saw an extremely brief period of very rapid expansion before the rate settled down a bit, but that rate of expansion has since been increasing rather than continuing to slow down.

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Yeahhh..... Hmm..... how do I put this? I was so far off on my basic representation of a universal timeline and explanation for the existence of time that I basically put aside the famous quote "Genius is 1% inspiration and 99% perspiration" In this case it was 99% inspiration and 1% perspiration, because this was inspired by working out a graphing equation for my intro college calc class. Well at least one positive thing can be learned from this: I don't know jack about physics.

There is no universal time dilation. Time dilation requires a gravitational gradient. In Cosmology the background field is homogeneous and isotropic which is no preferred direction or location. Essentially this describes a uniform mass distribution.

In the FLRW metric This fluid distribution is defined by

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

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

a is the scale factor which correlates expansion

$Proper distance =\frac{\stackrel{.}{a}(t)}{a}$

$H(t)=\frac{\stackrel{.}{a}(t)}{a(t)}$

, s,k is the curvature constant, Omega the density.

You can see the similarities with SR in this (I copied this from my previous posts below)

Lorentz transformation.
First two postulates.
1) the results of movement in different frames must be identical
2) light travels by a constant speed c in a vacuum in all frames.
Consider 2 linear axes x (moving with constant velocity and \acute{x} (at rest) with x moving in constant velocity v in the positive \acute{x} direction.
Time increments measured as a coordinate as dt and d\acute{t} using two identical clocks. Neither dt,d\acute{t} or dx,d\acute{x} are invariant. They do not obey postulate 1.
A linear transformation between primed and unprimed coordinates above
in space time ds between two events is ( this below doesnt have curvature as SR assumes Euclidean)
$ds^2=c^2t^2=c^2dt-dx^2=c^2\acute{t}^2-d\acute{x}^2$
Invoking speed of light postulate 2.
$d\acute{x}=\gamma(dx-vdt), cd\acute{t}=\gamma cdt-\frac{dx}{c}$
Where$\gamma=\frac{1}{\sqrt{1-(\frac{v}{c})^2}}$
Time dilation
dt=proper time ds=line element
since$d\acute{t}^2=dt^2$ is invariant.
an observer at rest records consecutive clock ticks seperated by space time interval dt=d\acute{t} she receives clock ticks from the x direction separated by the time interval dt and the space interval dx=vdt.
$dt=d\acute{t}^2=\sqrt{dt^2-\frac{dx^2}{c^2}}=\sqrt{1-(\frac{v}{c})^2}dt$
so the two inertial coordinate systems are related by the lorentz transformation
$dt=\frac{d\acute{t}}{\sqrt{1-(\frac{v}{c})^2}}=\gamma d\acute{t}$

Here is relativity of simultaneaty coordinate transformation in Lorentz.

$\acute{t}=\frac{t-vx/c^2}{\sqrt{1-v^2/c^2}}$
$\acute{x}=\frac{x-vt}{\sqrt{1-v^2/c^2}}$
$\acute{y}=y$
$\acute{z}=z$

as far as graphing expansion the calculator in my signature has graphing functions, that allow you to graph several key regions such as the cosmological event horizon, particle horizon etc

http://www.einsteins-theory-of-relativity-4engineers.com/LightCone7/LightCone.html

there is an advanced user section that contains the formulas used in the calculator

Also Mordred, thank you for the brilliant explanation on how I was wrong, I learned a lot from it, and am continuing to learn from it, I had no idea that time dilation was dependent on gravitational gradients, I thought it had to do solely with velocity relative to another object, this has definitely sparked my interest.

Edited by DanTrentfield
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Inertia vs gravity through the principle of equivalence gives the same results. However expansion isn't inertia based though it seems to be. The distances increase but no galaxy gains inertia due to expansion

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