Light Speed Paradox: Xavier's Torch on X

[Happiness]

Suppose planet X is so far away from Earth that due to the expansion of the universe, it is moving at a velocity $v > c$ away from us. Xavier stands on planet X and points a torch away from us. To Xavier, the light emitted from the torch is in front of him. But to us, the light emitted from the torch is behind him (since the velocity of light in our frame is also $c < v$). How can this be?

 

[Drakkith]

To Xavier, the light emitted from the torch is in front of him. But to us, the light emitted from the torch is behind him (since the velocity of light in our frame is also c<vc < v). How can this be?

It's not behind him in any frame of reference. The light is not observable to us (and never will be), but would be receding faster than c away from us since the expansion of space would be carrying both it and Xavier away from us. Light does indeed always travel at c in a vacuum, but this refers to local measurements, not to measurements on a cosmological scale where the expansion of space has to be taken into account.

 

[Happiness]

Light does indeed always travel at c in a vacuum, but this refers to local measurements, not to measurements on a cosmological scale where the expansion of space has to be taken into account.

If we take into account the expansion of space, can I say that the velocity of light is $v + c$ (which is more than $2c$) relative to us?

 

[Drakkith]

If we take into account the expansion of space, can I say that the velocity of light is $v + c$ (which is more than $2c$) relative to us?

Certainly, just be clear when saying this that the extra velocity is due to the expansion of space or you're likely to start a riot around here.
Also, remember that we can never detect this light. You will always MEASURE light as traveling at c, because measurements have to be done locally.

 

[PAllen]

There is a more basic point. In relativity, you may speak of two different things:

1) relative velocity
2) recession velocity

Both of these exist for both SR and GR (including cosmological solutions). For both SR and GR, it is only recession velocity not relative velocity that can exceed c. In SR in standard coordinates, recession velocity can reach 2c. In Milne coordinates (which model cosmology in flat spacetime) in SR, it can exceed 2c. Meanwhile, the statement that no relative velocity exceeds c refers - you notice - to (1) not (2). This statement is true without exception, in both SR and GR [ in GR, relative velocity is non-unique; it depends on the path over which vectors are brought together to compare. However, it is always less than c, no matter what the path, even for object's whose recession velocity may be 10c.]

A further key point is that the relative velocity between light and any material body is unambiguously c in GR, including all cosmologies. The reason is that parallel transport of a null vector preserves its character as a null vector, so no matter what material body's velocity vector you compare it to, the relative velocity is always c, no exceptions.

Now, answering your question with correct terminology:

The distant planet's light emitted away from us has relative velocity c, while the planet itself has ambiguous but < c relative velocity. The light will go away from us faster than the distant planet will. The recession velocity of the distant planet will be > c and the recession velocity of the light emitted away from us will also be >c and larger.

To give the very simple SR analog of all this:

In frame S, A is moving .9c to the left, an B is moving .9c to the right. Their recession velocity is 1.8c. Their relative velocity is .9945c (appx). If A emits light to its left, the recession velocity between this light pulse and B is 1.9 c, while its relative velocity to B is c as always.

If I had a penny for all the confusion caused by author's over-using expanding space and totally mixing up relative and recession velocity, I could buy out Warren Buffet.

 

[Happiness]

There is a more basic point. In relativity, you may speak of two different things:

1) relative velocity
2) recession velocity

Why are there two types of velocity? I picture relative velocity to be the velocity of an object moving through space, and recession velocity to be the velocity of an object due to the expansion of space. But Alan Guth mentioned that both pictures about velocity are correct and equivalent.

He said that the redshift due to special relativity agrees with the cosmological redshift due to the expansion of space, for a universe whose $\,a(t)$ is linear, i.e., a universe that expands or contracts at a constant rate. (See the youtube link below @29:30.) When we calculate the redshift due to special relativity, we are taking the position that galaxies are moving at some relative velocity. And when we calculate the cosmological redshift due to the expansion of space, we are taking the position that galaxies are moving at some recession velocity. So these two velocities, in this sense, are equivalent.

But if we move further out into space (so that the velocities of galaxies exceed c), it seems like the picture of relative velocity breaks down and we can't calculate the redshift due to special relativity. Only the picture of recession velocity survives. It's weird that these two pictures started out equivalent but can't give the same result when we move further out.

 

[PAllen]

There are two types of velocity in relativity precisely because of the invariant limiting velocity in relativity. In Newtonian mechanics, if I see Alice going left at .9c, and Bob going right at .9c, then:

a) I see them separating at 1.8c
b) I know (and each can verify for me) Alice and Bob each see the other moving at 1.8c

However, in relativity (and our universe, which it describes), (a) remains true but (b) is false. Alice and Bob each see the other moving at .9945 instead of 1.8c.

Your quote of Alan Guth is simply wrong. I do not have time to look at video and see if his full contextual statement is accurate, but I have no doubt that in a one on one discussion he would fully understand the difference.

Your last paragraph is factually wrong. There is direct extension of SR Doppler calculation to any cosmological solution in GR, for any distance, and it give the same answer as the expansion factor approach. Also note that any degree of red shift at all corresponds to some local relative velocity < c. The interpretation of this as caused by expansion of space is just that - a useful interpretation that depends in part on choice of coordinates. Second, the velocity that exceeds c in cosmology corresponds to (a) above - it is a growth of distance measured in a single coordinate system for two different objects that each perceive to the other as moving. Meanwhile, for each such galaxy, the relative velocity of the other is ambiguous in GR, meaning you can make it come out different depending how you move velocity vectors together to compare them. However, no matter what path you pick to move them together, you always get relative velocity < c. One of the ways of moving them together to compare will be consistent with the local, relative velocity consistent with the observed redshift (always < c).

I find it unfortunate that many authors try to make this much more 'gee whiz' emphasizing 'faster than light' when there is nothing at all unusual about faster than light recession velocity, and such faster than light recession velocity can occur locally, as I have explained.