# Galaxy rotation rates explained without Dark Matter

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f) If you give me a standard GR Tensor expressing the distribution of Gravitational Potential, then this is the same as the density of the energy field in my interpretation - no change required.

I see absolutely no reason to accept that. It is up to you to prove, in appropriate mathematical detail, that the Einstein Field Equations (which describe the geometry of space time) can also be used to describe the density and flow of an energy field.

Edited by Strange
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(g) Showed that an observer will always measure light's speed as constant ONLY if space's geometry is fixed, but filled with a field that determines light's speed and the rate of Time.

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(g) Showed that an observer will always measure light's speed as constant ONLY if space's geometry is fixed, but filled with a field that determines light's speed and the rate of Time.

Nope.

However, this does mean that you are rejecting GR in its entirety. So you have quite a lot of work ahead of you to come up with a credible theory.

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But there is already such a field. It's called gravity. Also how do you explain the frame-dragging

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(g) Showed that an observer will always measure light's speed as constant ONLY if space's geometry is fixed, but filled with a field that determines light's speed and the rate of Time.

Claimed ≠ showed

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e) Not no knowledge, but am not too familiar with the maths to show it in Tensor form

Okay some other form then - but I am not too familiar with the twistor or spinor fomulations.

Is it time to end this madness?

Edited by ajb
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But there is already such a field. It's called gravity. Also how do you explain the frame-dragging

Excellent. The Schwartzchild metric is a static solution. The Kerr metric which includes frame dragging is not.

Tensors are independent of coordinate choices. (Metric choice)

(g) Showed that an observer will always measure light's speed as constant ONLY if space's geometry is fixed, but filled with a field that determines light's speed and the rate of Time.

Incorrect thats a conclusion drawn from not fully understanding of how the static solution of the Schwartzchild solution works to include particle movement.

That is defined by the particles geodesic equations.

Edited by Mordred
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e) Not no knowledge, but am not too familiar with the maths to show it in Tensor form

Is it even possible to know anything about GR without knowing tensors?

f) If you give me a standard GR Tensor expressing the distribution of Gravitational Potential, then this is the same as the density of the energy field in my interpretation - no change required.

The gravitational potential is a very Newtonian notion. In general one thinks of the metric as the 'potential'- not that this is really a careful statement. What you can do is look at the Newtonian limit and then your recover a potential and the Laplace equation.

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Declan my advise is don't draw conclusions until you have time to study the material properly.

You probably don't want this thread to get locked.

Take some time from this thread to work out the mathematical details. If you have questions on GR don't hesitate to start a new thread in mainstream (as your questioning mainstream physics.)

You can always come back to this thread when your better prepared and armed.

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Mordred, what do you think about GR without tensors?

To me this seems quite unthinkable. Or really, we would be replacing tensors with other equivalent objects that will be less clear. For example, we could write out everything in GR in terms of scalar functions, but this would look a mess. You could use twistors and some people do, but still these are quite nice geometrically - they are still elements of a vector space and so on.

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Take this thought experiment. Remove every star and blackhole. Have a homogeneous and isotropic fluid.

Gravity still works.

Mordred, what do you think about GR without tensors?

To me this seems quite unthinkable. Or really, we would be replacing tensors with other equivalent objects that will be less clear. For example, we could write out everything in GR in terms of scalar functions, but this would look a mess. You could use twistors and some people do, but still these are quite nice geometrically - they are still elements of a vector space and so on.

I tried that once. I stopped after 11 pages of partial derivatives. (Most of them factored out) I never did complete all the partial derivitaves in the Einstein field equations.

Most cases I work on is homogeneous and Isotropic fluids which the FLRW metric greatly simplifies.

Twistors use of symmetry relations is a handy tool. One that does greatly simplify a lot of complexity. I ran into a professor who specialized in twister theory once. He ran me through some of the basics behind it.

I honestly wish I had more time with him

Edited by Mordred
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I ran into a professor who specialized in twister theory once.

I have met Penrose a couple of times, but never actually spoken to him!

Anyway, because to me 99.99% of physics is geometry, the use of tensor, tensor-like objects (connections, densities etc) and sections of fibered manifolds and fibre bundles seems natural and unavoidable. I am not sure how one would go about classical field theory without these tools.

Edited by ajb
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For Declan.

$\frac{dx^\alpha}{dy^{\mu}}=\frac{dx^\beta}{dy^{\nu}}=\begin{pmatrix}\frac{dx^0}{dy^0}&\frac{dx^1}{dy^0}&\frac{dx^2}{dy^0}&\frac{dx^3}{dy^0}\\\frac{dx^0}{dy^1}&\frac{dx^1}{dy^1}&\frac{dx^2}{dy^1}&\frac{dx^3}{dy^1}\\\frac{dx^0}{dy^2}&\frac{dx^1}{dy^2}&\frac{dx^2}{dy^2}&\frac{dx^3}{dy^2}\\\frac{dx^0}{dy^3}&\frac{dx^1}{dy^3}&\frac{dx^2}{dy^3}&\frac{dx^3}{dy^3}\end{pmatrix}$

This is just the Einstein metric tensor. Each 4×4 matrix has the number of degrees of freedom.

I have met Penrose a couple of times, but never actually spoken to him!

Anyway, because to me 99.99% of physics is geometry, the use of tensor, tensor-like objects (connections, densities etc) and sections of fibered manifolds and fibre bundles seems natural and unavoidable. I am not sure how one would go about classical field theory without these tools.

You can't Not even a supercomputer can do so. Without tools such as tensors, twistors, guage groups etc.

Even N body codes rely on those tools

Lol a good example is the three body problem using Keplers laws. Now imagine an entire galaxy.

Particle fields are far more complex with their added degrees of freedom.

Edited by Mordred
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I don't think people generally know how geometric just standard classical mechanics is - in either the Hamiltonian or Lagrangian formalism. Again, tensor calculus plays a central role. So when people say that they don't need tensors in physics I am really sceptical. Of course, it maybe the case that they are using tensors and not understanding that they are; say they have fixed some global coordinates or are only working on a given coordinate patch.

Edited by ajb
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I don't really have a problem with describing GR in terms of a space-time density.

If we consider the typical 2D reduction ( in order to better visualize it) of the depressed rubber sheet with a mass on it, we can get the exact same equations of motion by using a variable spacing of the co-ordinates, i.e. a density.

And I've had this discussion with you before AJB, where I spoke of GR as a geometric theory, but you corrected me by saying that it is a field theory where the geometry is the field.

So Declan's interpretation, while not mainstream, certainly doesn't change GR.

But that's not the OP, is it ?

What I really have a problem with is how this geometric field ( essentially a co-ordinate system ) can flow, disappear ( or sink ) into a Black Hole, and, in order to satisfy continuity, must be sourced elsewhere.

And that idea is central to his claim to explain galactic rotation without dark matter.

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Thats true. As one that has tried it back when I first started studying Cosmology. I usually laugh my head off on such a claim.

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I don't really have a problem with describing GR in terms of a space-time density.

Great, so you can formulate this 'space-time density' and show how it is equivalent to GR?

If we consider the typical 2D reduction ( in order to better visualize it) of the depressed rubber sheet with a mass on it, we can get the exact same equations of motion by using a variable spacing of the co-ordinates, i.e. a density.

This is an analogy for GR. And I am not sure I follow what you are saying - variable spacing of coordinates?

And I've had this discussion with you before AJB, where I spoke of GR as a geometric theory, but you corrected me by saying that it is a field theory where the geometry is the field.

All classical field theories are geometric. The usual way of understanding the gravitational field is as a metric on space-time; but there are other ways of understanding this.

So Declan's interpretation, while not mainstream, certainly doesn't change GR.

But that's not the OP, is it ?

We don't see how his interpretation really relates to the mathematics.

What I really have a problem with is how this geometric field ( essentially a co-ordinate system ) can flow, disappear ( or sink ) into a Black Hole, and, in order to satisfy continuity, must be sourced elsewhere.

And that idea is central to his claim to explain galactic rotation without dark matter.

We now don't think that he is describing 'flow of coordinates', but dynamics of some other field that he does not really specify.

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Think of a sheet of graph paper, AJB.

Changing the spacing of co-ordinates is a way of describing intrinsic curvature.

And one could then make the argument ( admittedly with a lot of hand waving ) that changing the spacing of co-ordinates is akin to changing the density of space-time.

You're right, extremely 'messy', but it might work.

The dynamics ( and nature ) of this field, which I wrongfully assumed was simply co-ordinate in nature, are what's central to his idea.

And I don't believe that has any validity.

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It's a little reminiscent of a GRIN lens. (GRadient INdex). Instead of a curved surface causing refraction, the index of the material changes in the radial direction, perpendicular to the light (cyl. coordinates. n varies with r, light propagates along z) even though the surfaces are flat.

But nothing is flowing — it's static. If you have flow, you have a sink and need a source.

(Plus, if there's an actual medium, you have the whole aether thing going on, and you can test for that.)

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This publication though extremely complex (field theory usually is) has probably one of the best coverage of fields in general.

Chapter 1 alone will take considerable math skills and study. The article also covers twistors, spinors etc.

"Feilds"

https://arxiv.org/abs/hep-th/9912205#

Over the years I've always kept this article handy. Its been a lifesaver when studying new metrics.

Another handy tool provided you can buy textbooks is

"One hundred Roads to Reality" by Sir Roger Penrose. For a non model specific coverage on various applications his 1000 page plus book is incredibly enjoyable.

He drops a considerable amount of humour into his descriptives. In particular at his own models. ( try not to laugh too hard at his zig zag model)

@Declan chapter one of fields will of particular use in clearing up some of your misunderstandings.

Edited by Mordred
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To ajb:

I didn't say we don't need Tensors, I said it is possible to understand Relativity without getting involved with the Tensor maths.

I can explain exactly how Time Dilation, Length Contraction and Mass Increase works without involving any Tensors.

It is also to understand qualitatively how GR works without knowing how to do the Tensor maths.

The Tensor maths is a neat mathematical tool for doing exact calculations on the non-linear equations governing the motions of test particles within space-time - agreed.

To Strange:

Accepting that the speed of light is not constant is not a complete rejection of GR - not at all - even though the constancy of the speed of light was Einstein's first assumption that led him to Relativity, the whole theory remains essentially intact by allowing light's speed to change (along with the rate of time) and have a space-filling field with variable density.

To MigL:

It's not the coordinate system that disappears - it is the energy field that fills the coordinate system that disappears. And when we say 'disappears' it is in the same sense as normal matter disappearing into a black hole (where is the source of normal matter in the Universe). Both the matter and the energy field (which are essentially the same thing anyhow) become part of the black hole.

The black hole may eventually evaporate (via hawking radiation) and release the matter/energy back into the Universe anyhow - so nothing lost forever.

To Mordred:

I wouldn't mind leaving the thread open and try to understand GR Tensor maths a bit better so that we can have a more meaningful discussion on this (if I can find the time to understand it).

I still think it should be possible to move past the sticking point of Tensor maths and discuss the idea whilst acknowledging that this is an area that needs to be worked on.

To ajb:

In above comment: "it is also to understand" should read "it is also possible to understand".

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Accepting that the speed of light is not constant is not a complete rejection of GR - not at all - even though the constancy of the speed of light was Einstein's first assumption that led him to Relativity, the whole theory remains essentially intact by allowing light's speed to change (along with the rate of time) and have a space-filling field with variable density.

Of course, you have a proof of this?

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Here we go again...

To Swansont:

How can we test for an Aether?

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I'm placing these into a single post as I may use this again.

In the presence of matter or when matter is not too distant physical distsnces between two points change. For example an approximately static distribution of matter in region D. Can be replaced by tve equivalent mass

$M=\int_Dd^3x\rho(\overrightarrow{x})$ concentrated at a point $\overrightarrow{x}_0=M^{-1}\int_Dd^3x\overrightarrow{x}\rho(\overrightarrow{x})$

Which we can choose to be at the origin

$\overrightarrow{x}=\overrightarrow{0}$

Sources outside region D the following Newton potential at $\overrightarrow{x}$

$\phi_N(\overrightarrow{x})=-G_N\frac{M}{r}$

Where $G_n=6.673*10^{-11}m^3/KG s^2$ and $r\equiv||\overrightarrow{x}||$

According to Einsteins theory the physical distance of objects in the gravitational field of this mass distribution is described by the line element.

$ds^2=c^2(1+\frac{2\phi_N}{c^2})-\frac{dr^2}{1+2\phi_N/c^2}-r^2d\Omega^2$

Where $d\Omega^2=d\theta^2+sin^2(\theta)d\varphi^2$ denotes the volume element of a 2d sphere

$\theta\in(0,\pi)$ and $\varphi\in(0,\pi)$ are the two angles fully covering the sphere.

The general relativistic form is.

$ds^2=g_{\mu\nu}(x)dx^\mu x^\nu$

By comparing the last two equations we can find the static mass distribution in spherical coordinates.

$(r,\theta\varphi)$

$G_{\mu\nu}=\begin{pmatrix}1+2\phi_N/c^2&0&0&0\\0&-(1+2\phi_N/c^2)^{-1}&0&0\\0&0&-r^2&0\\0&0&0&-r^2sin^2(\theta)\end{pmatrix}$

Now that we have defined our static multi particle field.

Our next step is to define the geodesic to include the principle of equivalence. Followed by General Covariance.

Edited by Mordred
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In my comment I was talking about qualitative understanding (i.e. understanding the principles, rather than the numbers).

However, now may be a good time to help me understand the maths that you are familiar with.

I sort of understand the example you have given, correct me if I am wrong:

- you are taking a density of matter in a region of space D and integrating it over that area to get a total mass M, which is at the Origin.

- Then you are showing the change in position of a test particle due to this mass M? Or is it indicating how far away the two particles are (in a variable geometry space)?

A couple of questions:

(1) What is the N subscript on the Potential?

(2) Why is there no time variable?

(3) I have seen the Omega symbol elsewhere - it expands to another formula doesn't it? Can you explain the Omega to me?

And also the ds^2, is that the double differential of distance (i.e. the acceleration of the test particle)?

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