Speed of light on cosmological scales

[Grinkle]

TL;DR

Why is 'c' not well defined over cosmological distances?

The context is this post by @PeroK; Post #2 in this thread -

https://www.physicsforums.com/threa...fect-the-speed-of-light.1059351/#post-7088760

I did some Googling and couldn't find discussion on this topic.

What are the problems with c at cosmological scales? Does the issue arise at scales larger than the observable universe, or even before that?

 

[PeroK]

TL;DR Summary: Why is 'c' not well defined over cosmological distances?

The context is this post by @PeroK; Post #2 in this thread -

https://www.physicsforums.com/threa...fect-the-speed-of-light.1059351/#post-7088760

I did some Googling and couldn't find discussion on this topic.

What are the problems with c at cosmological scales? Does the issue arise at scales larger than the observable universe, or even before that?

In general, there is no unique global simultaneity convention. Speed generally is not well-defined in a curved spacetime.

In fact, you can setup a simple experiment where a light pulse bounces between two points at different gravitational potentials. The proper distance between the two points is well-defined, but the round-trip time for the light pulse is less as measured by a clock at the point deeper in the well. Think of gravitational time dilation. That gives two different coordinate speeds for the light. Measuring the speed of light locally avoids the problem of spacetime curvature.

 

[Ibix]

Just to emphasise, this is a general problem in curved spacetime. How fast two objects pass each other is well defined, but not how fast they are going relative to each other when they are not in the same place.

A textbook example you can look up is Shapiro Delay. One way of conceptualising it is that light travels at different speeds depending how near the Sun it is.

 

[Vanadium 50]

If you send a pulse from A to B and they are moving apart, what is the distance between them? The distance when the light pulse between them starts? When it is received? When it is halfway?

 

[Grinkle]

If you send a pulse from A to B and they are moving apart, what is the distance between them? The distance when the light pulse between them starts? When it is received? When it is halfway?

So, not thinking about cosmology, just offering the answer I think to be correct - the distance between A and B is always frame dependent.

 

[PeroK]

So, not thinking about cosmology, just offering the answer I think to be correct - the distance between A and B is always frame dependent.

Velocity is a vector. In Newtonian space and time and also flat Minkowski spacetime, you have the concept of moving vectors around from one point to another. It makes sense to say, for example, that objects A and B are moving in the same direction at the same speed, hence have the same velocity.

One major difference when dealing with the curved spacetime of GR is that vectors become local quantities. Mathemetically, a vector is defined in a tangent space that is defined at a particular point in spacetime. It's no longer possible to move a vector from one point in spacetime to another in a simple, unambiguous way.

In GR, therefore, we tend to have coordinate systems that cover a large region of spacetime and things like coordinate velocity and coordinate speed can be defined. But, the vectors we are more familiar with: velocity, momentum, force are defined locally. And these physical quantities become generally ambiguous if we try to defined them non-locally.

This mathematical transition from global vectors to local vectors is one of the conceptual difficulties you must overcome if you want to learn GR. You have to give up the security blanket of position vectors and displacement vectors and relearn physics to some extent in terms of this division into global coordinate systems and local tangent spaces.

 

[Grinkle]

a vector is defined in a tangent space that is defined at a particular point in spacetime.

I hope I have the gist of what you are saying, using only as many dimensions as one can picture in my description.

I have an image of a plane that is tangent to a curved 2-d surface that is the spacetime where observers reside, both plane and surface embedded in a 3-d embedding space, not sure if this embedding space would also be a spacetime or just something invoked by attempts to picture the situation that has no name related to the physics. A and B are points somewhere on this curved surface. Pick A to be where the plane is made tangent.

If spacetime is flat, the tangent plane is the same as the spacetime it is tangent to (or there is no tangent, not sure which is the more correct way to say it), and both A and B are located on this tangent plane, since its the same as the surface itself.

In curved spacetime, if one moves the point of tangency (I hope that is an actual word) from A to B, and maps the initial plane to the translated plane, one sees that the velocity vector is not the same in the embedding 3-d space. At least so far as the simple way I am describing things, the magnitude will be the same but the dot products to each of the 3 dimensions that define the embedding space will all change.

 

[PeroK]

The surface of a sphere is generally the simplest example of curvature. You don't need an embedding space. You can treat the sphere as a manifold in its own right, without embedding it in a higher dimension. The tangent space would be a plane at each point. And, there's no unique way to say when two objects at different points on the sphere are moving with the same velocity. And, of course, if you have a 4D spacetime, then speed also becomes a local quantity.

 

[jbriggs444]

So, not thinking about cosmology, just offering the answer I think to be correct - the distance between A and B is always frame dependent.

So not invariant.
And also not necessarily constant.

So speaking about "the distance" is already erroneous. The definite article is not warranted. We are only talking about "a distance".

 

[Grinkle]

@Ibix The pointer to Shapiro delay was very helpful - thanks!