# Transformation matrix for an expanding space

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## Summary:

How should I write the transformation matrix for an expanding space?

## Main Question or Discussion Point

Hello. I am confused with this matter that how should we write the transformation matrix for an expanding space. consider a spacetime that is expading with a constant rate of a. now normally we scale the coordinates for expansion which makes the transformation matrix like this:

\begin{pmatrix} -1 & 0 & 0 & 0\\ 0 & a & 0 & 0 \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & a\end{pmatrix}

but there's one thing I can't figure out. If we take one point in this space and attribute a frame of reference to it, this tranformation matrix will scale it up to the rate of expansion, yet that frame also moves and I think it should also translate while scaling. because when expanding, distance between points increase, and increasing distance means the origin of the frame also changes because it is farther from any point before. which means the frame actually moved. so does that mean matrix would be like this?

\begin{pmatrix} -1 & 0 & 0 & a\\ 0 & a & 0 & a \\ 0 & 0 & a & a \\ 0 & 0 & 0 & a\end{pmatrix}

PeroK
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Summary:: How should I write the transformation matrix for an expanding space?

Hello. I am confused with this matter that how should we write the transformation matrix for an expanding space. consider a spacetime that is expading with a constant rate of a. now normally we scale the coordinates for expansion which makes the transformation matrix like this:

\begin{pmatrix} -1 & 0 & 0 & 0\\ 0 & a & 0 & 0 \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & a\end{pmatrix}

but there's one thing I can't figure out. If we take one point in this space and attribute a frame of reference to it, this tranformation matrix will scale it up to the rate of expansion, yet that frame also moves and I think it should also translate while scaling. because when expanding, distance between points increase, and increasing distance means the origin of the frame also changes because it is farther from any point before. which means the frame actually moved. so does that mean matrix would be like this?

\begin{pmatrix} -1 & 0 & 0 & a\\ 0 & a & 0 & a \\ 0 & 0 & a & a \\ 0 & 0 & 0 & a\end{pmatrix}
I don't see why you need to move the origin, as it were. In any case, a translation doesn't change the metric. I'm not sure what your second matrix might mean. It's not even symmetric.

when we have the scale factor we can compare the previous frame with the new one but this only shows how a body ( a point, an event) comoves with the space. what if I want to erase the effect of the expansion of this space. I mean look at this gif please:

you can see the stars are comoving with the grid space. transformation matrix with a scale factor only gives me the stars at any given moment but w.r.t the grid space which is still expanding. I mean that the scale factor doesn't neutralize the expansion, it only gives us a way to find new coordinates after expansion. what I want is a transformation that gives me the space without the expansion. when you scale it up you only changed the distances. but the space itself isn't fixed. I want the transformation matrix for neutralizing expansion and getting a fixed space . this image has the luxury of a fixed absolute frame. if I have a star that is moving with the rate of expansion and toward the center, that star is fixed and is not comoving. now that star feels the change of space under its feet right? the star can move toward the center and without a need for the absolute frame, measures the expansion and the rate of change in space. how can I write the transformation matrix for this? a fixed star the sees the space moving beneath its feet. how would the star measure the change?

PeroK
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I'm sorry, I'm not sure I understand what you are trying to do.

PeroK
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Do your matrices in post #1 represent coordinate transformations or metrics? If the former, why is there a $-1$ in the first entry?

scaling the frames doesn't neutralize the expansion. it gives us the new coordinate. I want to know how to transform the expanding space into a fixed one by neutralizing the effect of expansion. If that star cancels the expansion (which is by magnitude=rate of expansion and direction=toward center) it is stationary but still it can feel the space moving around it. that shows that the star itself is now fixed in the absolute frame. being stationary gives it an advantage point to talk about the expanding space while being stationary itself. it can explain the expansion of the space without being affected by it. how can I show this point of view?
(There's a big chance I wrecked the matrices)

Last edited:
PeroK
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The general equations are what I need. if I have a sphere that is expanding with the rate of expansion, the changes that it feels in spacetime are equivalent to the changes of spacetime in that region of space. now I need geometrical equations that could explain the changes. in two dimensional case with a center (which is the origin or absolute frame) I can say the changes of the spacetime (or equivalently changes of the frame in that space) is $vt$ which is the speed of the frame times elapsed time (which is arbitrary since $v$ is constant). now this is in two dimensions and with a center, but if I choose a space that is homogeneous and isotropic, that space doesn't have a center or boundaries and so, it needs to be a circle (for 2-D) and a sphere for 3-D. how do I say in mathematical form how the sphere changes in space?
I think there is a confusion of too many ideas in there. Expressions like "changes of the frame in that space" really don't make any sense to me.

I think you are talking about changes of coordinates in an expanding universe? But, you posted under General Maths, I think.

I suggest trying to clarify your question and posting under "General Relativity".
thank you for your suggestion but it really isn't about general relativity itself. that expression that you quote is this: If $r^2=v(x^2+y^2)$ is an expanding circle which is to be considered the space that we are considering ( limited part of the Euclidean space) and $M: (x,y)$ is an arbitrary point in that space and $M$ is moving toward the center of the circle with the speed $v$, and $N: (x_{1},y_{1})$ is another arbitrary point in that circle, then $M$ will observe $N$ moving away from it with the speed of $v$ which is what the center of the circle will observe too since it's stationary and thus $M$ is stationary too. now $M$ will observe the whole circle and the points in it (except origin of course) moving away from it with $v$ and for $t=t_{2} - t_{1}$ the changes in space for the point $M$ will be $vt$. now let's look from $M$'s point of view. it can see both these views: either it's stationary and the space is moving around it or it can see that it's moving itself and the space is fixed. both these views are equivalent (Principle of Relativity). now until this point everything is easy. what I don't know and want to know is that how to generalize this equivalence to a minkowski space that is homogeneous and isotropic which would be without any center and origin thus making being stationary literally impossible. but with considering an expanding sphere (as a substitute for $M$) in minkowski space this generalization could be achieved. what I can't figure out is the math.

note: now that I look at it I can see one thing might come to your mind which is the scale factor in FLRW metric. look, for the simplification of the situation I considered $N$ to be seemed like it is moving with $v$ in the $M$'s view. I do understand how the scale factor works. If you can't neglect the scale factor in this scenario then consider $N$ to be infinitesimally distant from $M$. then the scale factor is not needed anymore.

PeterDonis
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Moderator's note: Moved to the cosmology forum.

PeterDonis
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a spacetime that is expading with a constant rate of a
I don't know what you mean by this. I suspect you are confusing "rate of expansion" with "scale factor"; see below.

normally we scale the coordinates for expansion which makes the transformation matrix like this:
$$\begin{pmatrix} -1 & 0 & 0 & 0\\ 0 & a & 0 & 0 \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & a\end{pmatrix}$$
This matrix is not a transformation matrix; it's the metric tensor written as a matrix, and $a$ is the scale factor.

The rest of what you are saying makes no sense to me, but I suspect it's all wrong anyway because of the basic mistake above.

I don't know what you mean by this. I suspect you are confusing "rate of expansion" with "scale factor"; see below.

This matrix is not a transformation matrix; it's the metric tensor written as a matrix, and $a$ is the scale factor.

The rest of what you are saying makes no sense to me, but I suspect it's all wrong anyway because of the basic mistake above.
If you read the last message it explains the matter.

thank you for your suggestion but it really isn't about general relativity itself. that expression that you quote is this: If $r^2=v(x^2+y^2)$ is an expanding circle which is to be considered the space that we are considering ( limited part of the Euclidean space) and $M: (x,y)$ is an arbitrary point in that space and $M$ is moving toward the center of the circle with the speed $v$, and $N: (x_{1},y_{1})$ is another arbitrary point in that circle, then $M$ will observe $N$ moving away from it with the speed of $v$ which is what the center of the circle will observe too since it's stationary and thus $M$ is stationary too. now $M$ will observe the whole circle and the points in it (except origin of course) moving away from it with $v$ and for $t=t_{2} - t_{1}$ the changes in space for the point $M$ will be $vt$. now let's look from $M$'s point of view. it can see both these views: either it's stationary and the space is moving around it or it can see that it's moving itself and the space is fixed. both these views are equivalent (Principle of Relativity). now until this point everything is easy. what I don't know and want to know is that how to generalize this equivalence to a minkowski space that is homogeneous and isotropic which would be without any center and origin thus making being stationary literally impossible. but with considering an expanding sphere (as a substitute for $M$) in minkowski space this generalization could be achieved. what I can't figure out is the math.

note: now that I look at it I can see one thing might come to your mind which is the scale factor in FLRW metric. look, for the simplification of the situation I considered $N$ to be seemed like it is moving with $v$ in the $M$'s view. I do understand how the scale factor works. If you can't neglect the scale factor in this scenario then consider $N$ to be infinitesimally distant from $M$. then the scale factor is not needed anymore.

PeterDonis
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If you read the last message it explains the matter
Sorry, it still doesn't make any sense to me. It still looks to me like you are making a fundamental mistake at the very start: what you are calling a "transformation matrix" is actually the metric. You need to address that first before anything else.

until this point everything is easy. what I don't know and want to know is that how to generalize this equivalence to a minkowski space that is homogeneous and isotropic which would be without any center and origin thus making being stationary literally impossible. but with considering an expanding sphere (as a substitute for M) in minkowski space this generalization could be achieved. what I can't figure out is the math.
I'm not at all sure I follow but do you mean the math for a expanding sphere in time in Minkowski space? Like frontwaves of light observed from different points? That's basically SR's math then.

Sorry, it still doesn't make any sense to me. It still looks to me like you are making a fundamental mistake at the very start: what you are calling a "transformation matrix" is actually the metric. You need to address that first before anything else.
Yes. you are right. that is entirely wrong. this is actually because i'm confused with handful of things. so please forgive me for misguiding you in the start. also I think the equation for expanding circle is wrong. I think it should be this: $v^2t^2=x^2+y^2$. I'm sorry.

I'm not at all sure I follow but do you mean the math for a expanding sphere in time in Minkowski space? Like frontwaves of light observed from different points? That's basically SR's math then.
No. If you want to know which point of view to consider I have to say I'm looking only for the sphere's point of view only. nothing else. because when the sphere is expanding in Minkowski space with the constant rate of expansion, that sphere can see itself stationary while the rest of the world are changing because of expansion. or it can consider the world to be stationary and it's moving. this creates a relationship between the sphere and space which is unique and it can give us better perspective on the expansion of space itself.

I'm really sorry for screwing up the matrix and other stuff.

No problem. You do know Minkowski space doesn't expand, do you?

PeterDonis
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Yes. you are right. that is entirely wrong.
In which case you need to go back to the beginning and fix that error, before even trying to do anything else.

also I think the equation for expanding circle is wrong
I don't know if it's even wrong at this point, because I don't see how it's relevant. You need to go back and fix your original error first.

when the sphere is expanding in Minkowski space with the constant rate of expansion
The model you are referring to here is sometimes called the "Milne universe" or the "empty universe", and is only valid if there is no matter, no energy, no radiation, no dark energy, nothing in the universe. So it's not relevant to the universe we actually live in.

that sphere can see itself stationary while the rest of the world are changing because of expansion
No, it can't. The different parts of the "sphere" are moving relative to each other, and the sphere as a whole is the "world"--there is no "rest of the world" other than the sphere.

You seem to have some fundamental misconceptions about the models you are trying to use. What references have you read on this subject?

No, it can't. The different parts of the "sphere" are moving relative to each other, and the sphere as a whole is the "world"--there is no "rest of the world" other than the sphere.

You seem to have some fundamental misconceptions about the models you are trying to use. What references have you read on this subject?
Apparently the wrong ones. what should I read to get this right?

note: I didn't understand what you meant by "the sphere as a whole is the world". the sphere is just in a part of space and it's expanding with the rate of expansion so I thought if that point in two dimensions with moving with the rate of expansion and toward center could be stationary, this one can too. canceling the expansion by magnitude (rate of expansion) and direction ( all directions). of course this doesn't contradict with principle of relativity since its statement about being stationary is only valid locally. Now I thought if that point could observe the changes in its own space by being stationary, this one can too.

PeroK
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Apparently the wrong ones. what should I read to get this right?

note: I didn't understand what you meant by "the sphere as a whole is the world". the sphere is just in a part of space and it's expanding with the rate of expansion so I thought if that point in two dimensions with moving with the rate of expansion and toward center could be stationary, this one can too. canceling the expansion by magnitude (rate of expansion) and direction ( all directions). of course this doesn't contradict with principle of relativity since its statement about being stationary is only valid locally. Now I thought if that point could observe the changes in its own space by being stationary, this one can too.
Here's an Insight into the cosmological expansion:

https://www.physicsforums.com/insights/inflationary-misconceptions-basics-cosmological-horizons/

Just could read that and see what you make of it.

for example in this picture what I'm saying would be: (the picture shows how the distance between A and B is constant after expansion)

the B moves toward A so it can still have the same distance but w.r.t to previous grids. I mean that I want to take one grid in one moment and consider it my time of reference. and then B will always set its constant distance to A w.r.t the reference grid in the reference time. now everything that B considers as its own motion is actually the expansion's. which allows it to have the space that is actually fixed.

PeroK
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If we are talking about an expanding homogeneous and isotropic universe, then the expansion looks the same from every point. The points on a sphere centered on a given point have nothing in common except that they are the same distance from that given point. Those points don't have a special relationship beyond that. The universe from any one of those points looks the same as it does from the point at the center.

If you want to have a sphere centered on a given point "contract" to cancel out the expansion, then that doesn't achieve anything. Viewed from the centre of the sphere, the points are indeed not receding. But, viewed from outside the sphere, the sphere and all the points in it are still receding. It's simply that the sphere retains its size, like an set of gravitational bound objects would.

If you model the entire universe as a 3-sphere embedded in four spatial dimensions, then the sphere is all there is. There is no center - the embedding serves as a way of describing the large-scale geometry of a closed universe.

If we are talking about an expanding homogeneous and isotropic universe, then the expansion looks the same from every point. The points on a sphere centered on a given point have nothing in common except that they are the same distance from that given point. Those points don't have a special relationship beyond that. The universe from any one of those points looks the same as it does from the point at the center.

If you want to have a sphere centered on a given point "contract" to cancel out the expansion, then that doesn't achieve anything. Viewed from the centre of the sphere, the points are indeed not receding. But, viewed from outside the sphere, the sphere and all the points in it are still receding. It's simply that the sphere retains its size, like an set of gravitational bound objects would.

If you model the entire universe as a 3-sphere embedded in four spatial dimensions, then the sphere is all there is. There is no center - the embedding serves as a way of describing the large-scale geometry of a closed universe.

As I said sphere's statement about being stationary is only valid locally. meaning that you're right and I agree with you. it's just supposed to do what I said in #18.

PeroK
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As I said sphere's statement about being stationary is only valid locally. meaning that you're right and I agree with you. it's just supposed to do what I said in #18.
A sphere on a cosmological scale is not a local object. A small patch on the sphere might be considered locally. But, two distant points on the sphere, as far as they (or any one else) are concerned, are just two distant points receding from each other with the Hubble flow.

A sphere on a cosmological scale is not a local object. A small patch on the sphere might be considered locally. But, two distant points on the sphere, as far as they (or any one else) are concerned, are just two distant points receding from each other with the Hubble flow.
I'm feeling there's a problem with the sphere. I don't know what you mean by a sphere but I mean just a simple geometrical 3-D shape. its radius is definitely not at Mpc level. consider its radius one meter or even less. does that clarify anything or did I understand what you said entirely wrong?

PeroK
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I'm feeling there's a problem with the sphere. I don't know what you mean by a sphere but I mean just a simple geometrical 3-D shape. its radius is definitely not at Mpc level. consider its radius one meter or even less. does that clarify anything or did I understand what you said entirely wrong?
What does a sphere 1m in diameter have to do with cosmology or universe expansion? Or Minkowski space?

What does a sphere 1m in diameter have to do with cosmology or universe expansion? Or Minkowski space?
Expansion is homogeneous. it occurs everywhere. so I'm just looking for understanding what this expansion really is. not figuring out how to convert coordinates in the Hubble flow. we have comoving coordinates for that. even at this length expansion occurs, even if it's about $10^{-16}$ meter per second for a meter. Although its effective role appears in long distances I'm trying to work with the expansion not practical situations.

PeroK
Expansion is homogeneous. it occurs everywhere. so I'm just looking for understanding what this expansion really is. not figuring out how to convert coordinates in the Hubble flow. we have comoving coordinates for that. even at this length expansion occurs, even if it's about $10^{-16}$ meter per second for a meter. Although its effective role appears in long distances I'm trying to work with the expansion not practical situations.