Moving through space vs sitting in an expanding space

[Happiness]

Suppose an object is moving away from you. Is there a way to determine, experimentally or otherwise, whether it is moving through space or whether it is sitting still but appears moving because the space between you and the object is expanding?

Galaxies that are sufficiently far away from us move away faster than the speed of light. If there is no way of distinguishing between the two cases in the above question, then we can equivalently say these galaxies move through space faster than the speed of light, violating special relativity. So there must be a way to distinguish these cases. Is this correct?

 

[andrewkirk]

It's a good question, and one I used to think about a lot. Here's the resolution I came to, which may or may not help.

There is no such thing as moving 'through space' as space is not a substance. All motion is relative, so there is no fundamental difference between things moving relative to space and space expanding.

However, we can define motion relative to a coordinate system, rather than by reference to a body. The coordinate system mostly used for cosmology is the FLRW system. When we say space is expanding, we mean that the distance between two points with fixed coordinates in that system is increasing over time. Motion relative to the worldline of a fixed set of spatial coordinates in that system is called 'peculiar motion', and corresponds loosely to what people think of as 'moving through space'.

When we say that a far galaxy is receding at a speed faster than light, what we actually mean is that the proper distance between Earth and that galaxy is increasing at a rate greater than $3\times 10^8ms^{-1}$. In the FLRW system, that is not 'peculiar motion' but just the expansion of space.

 

[Happiness]

It's a good question, and one I used to think about a lot. Here's the resolution I came to, which may or may not help.

There is no such thing as moving 'through space' as space is not a substance. All motion is relative, so there is no fundamental difference between things moving relative to space and space expanding.

However, we can define motion relative to a coordinate system, rather than by reference to a body. The coordinate system mostly used for cosmology is the FLRW system. When we say space is expanding, we mean that the distance between two points with fixed coordinates in that system is increasing over time. Motion relative to the worldline of a fixed set of spatial coordinates in that system is called 'peculiar motion', and corresponds loosely to what people think of as 'moving through space'.

When we say that a far galaxy is receding at a speed faster than light, what we actually mean is that the proper distance between Earth and that galaxy is increasing at a rate greater than $3\times 10^8ms^{-1}$. In the FLRW system, that is not 'peculiar motion' but just the expansion of space.

Since there is no fundamental difference between things moving relative to space and space expanding, does that mean that we can violate special relativity as long as we make a sufficient number of things in the universe move faster than the speed of light such that their motion becomes "non-peculiar"?

 

[andrewkirk]

Since there is no fundamental difference between things moving relative to space and space expanding, does that mean that we can violate special relativity as long as we make a sufficient number of things in the universe move faster than the speed of light such that their motion becomes "non-peculiar"?

No we can't. Always keep in mind the notion that motion expressed as a number ('speed') is relative. When you say 'move faster than the speed of light', relative to what is that speed measured? Perhaps you mean relative to the FLRW spatial coordinate frame. If so, that idea of most particles moving at that speed, all in the same direction, is impossible because the FLRW is applied in a context where the Cosmological Principle is assumed, that space is at the large scale homogeneous and isotropic. That means that the average peculiar velocity of everything in the universe must be zero. Indeed, the 'stationary' (or, more accurately, 'co-moving') worldlines in the FLRW system are defined as those whose average peculiar velocity relative to everything else in the universe is zero.

I find the best way to state the relativistic speed limit is that no particle can have a spacelike velocity vector, and massive particles must have timelike velocity vectors (leaving aside the theoretical possibility of tachyons, for the sake of brevity). That way, the limit does not have to be expressed either relative to other particles, or to a coordinate system.

 

[mfb]

Special relativity only makes a statement about the speed limit for things at the same place, because that is the only setting where you can define speed in an unambiguous way. In the absence of gravity or expanding/contracting space you can extend this statement to the whole universe, but that's not the universe we live in.

 

[Happiness]

Special relativity only makes a statement about the speed limit for things at the same place, because that is the only setting where you can define speed in an unambiguous way. In the absence of gravity or expanding/contracting space you can extend this statement to the whole universe, but that's not the universe we live in.

Suppose galaxy X recedes from us at 0.8c (due to the expansion of space) and planet Y in galaxy X travels at a speed of 0.7c relative to galaxy X (due to its movement through space) in the direction away from us, then is Y's speed 1.5c or 1.5/1.56 c (obtained by using special relativity's velocity-addition formula)?

 

[mfb]

It depends on your choice of coordinates. If you imagine space to be filled with rulers, the answer is 1.5.