# I Apparent Superluminal Velocity of Fixed Stars?

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1. Jun 7, 2016

### KingSnackMan

If I spin around in an open field at night and look up to the stars they appear to be moving relative to me. Additionally, they are very far away and trace out a giant arc length in a very short time (S=rθ). With respect to me, these stars are moving faster than light. Is this a problem? Has the cosmic speed limit been broken? Do I have to abandon relative motion?

2. Jun 7, 2016

3. Jun 7, 2016

### Orodruin

Staff Emeritus
To summarise, your rotating frame is not an inertial frame. The coordinate speed of light is only limited and fixed in inertial frames.

The physical relative velocity between two objects is a different matter. It is bounded by c regardless of the coordinates you use. It also gets slightly more complicated in GR.

4. Jun 7, 2016

### Ibix

The speed of light is invariant and cannot be exceeded locally. If you use a simple Cartesian coordinate system then this concept is simple to express everywhere just by dropping the word "locally" (at least for SR).

If you use a more complex coordinate system, such as rotating polar coordinates, then this concept is not simple to express in general. You find that the speed of light expressed in these coordinates is different at different points and in different directions. However, all the additional complexity is down to a silly choice of coordinates. You are still expressing the same notion of the invariance of light speed, just hiding the invariance behind a layer of complicated maths.

There can be good reasons to do this (see GR for one). But don't if you don't have to, would be my advice.

5. Jun 7, 2016

### quickAndLucky

This is very interesting, specifically that it is only local speeds that need to be c. Is this in any well known books? Carroll? Wald?

6. Jun 7, 2016

### Ibix

Any text covering general relativity will mention it, I should imagine. In curved spacetime it isn't necessarily possible to make an unambiguous comparison of velocities (or any other vector) at one point with velocity (or whatever) at another point. So you can only insist that local relative speeds cannot exceed c bcause you can't really define relative speed between spatially separated objects.

You can define relative speed for spatially separated objects in flat spacetime (i.e. in special relativity), and hence insist that it must be less than c. If you choose a complicated set of coordinates, you can hide that simple statement quite thoroughly.

7. Jun 7, 2016

### Orodruin

Staff Emeritus
Let me add that this is very similar to using curvilinear coordinates. Unless you have a special reason not to, using Cartesian coordinates is almost always the easiest. In the same fashion, using Minkowski coordinates in SR is most often the easiest.