Physicists believe that some galaxies are moving away from us at faster than the speed of light. A galaxy that is moving away from us at faster than the speed of light would be moving backwards in time relative to our galaxy, the Milky Way. That means we would be moving forward in time relatively. However, from the perspective of someone living in that far away galaxy, we would be moving away from it at faster than the speed of light and we would be going backwards in time therefore, they would be moving forward in time. How can all of this be reconciled?

• You may find this question interesting Commented Jun 12 at 23:28
• I don't see a paradox here, changing reference frame will cause this to happen, it also applies in Newtonian physics, say I am moving and you are not (relative to the ground), from your perspective I am moving but from my perspective you are moving. Commented Jun 13 at 0:02
• Note that the elapsed proper time for faster-than-light motion is not negative, it's imaginary.
– Sten
Commented Jun 13 at 1:21
• Also, recession rates aren't actual velocity, it's the expansion of space. It only comes out to have the same units of velocity, not actually implying physical velocity or momentum in either galaxy's rest frame. Commented Jun 13 at 2:09
• Comment to the title (v2): Consider to make the title more informative. Commented Jun 13 at 5:51

The recession rates you are talking about are not relative velocities, so it's not meaningful to compare them to the speed of light, and it's incorrect to use them to calculate relativistic effects.

These recession rates can be defined as follows. Conceptually, imagine a chain of galaxies that leads to some target galaxy. Each galaxy along the chain has some small velocity relative to the galaxy before it. If you add all of those relative velocities together, that should give you the velocity of the target galaxy, right? However, velocities in relativity add in a special way; see the relativistic velocity addition formula. The cosmological recession rate is computed by instead just adding the relative velocities naively, without properly using relativistic velocity addition. That's why we should not be concerned that it can exceed the speed of light.

It's also worth noting that the relative velocity between cosmologically distant objects doesn't even have a unique meaning. You can just as well say that it is very high or that it is zero. This is part of why we talk about recession rates: there is not even a unique relative velocity to speak of.

[partly copied from one of my answers on Astronomy.SE]

This is because of general relativity. GR says that spacetime is not fixed, but can "curve" and change depending on what's living on it. The universe is expanding and this means that the physical distance between those two galaxies is increasing. How is it even possible to define velocity when the scale of the universe is changing constantly?

The trick is to use small enough patches of spacetime such that the effects of curvature don't matter anymore. For example, between someone on the ground and a satellite in orbit there is time dilation, because the gravitational field is weaker there. If we confine ourselves to a small space, for example just a lab, the effects of GR are not noticable and we can use special relativity.

In this small patch we can confidently say that we can't break the speed of light, which is a result of special relativity.

SR also says is the distance between two oberservers can never increase faster than the speed of light. In GR this does not hold anymore. While in GR each observer can never move faster with respect to theirlocal neighbourhood, the distance between them does not have such a bound.

A nice analogy is that of two ants on the surface of a rubber balloon. The 2D surface of the balloon represents our 3D space. Each ant is limited by how fast it can move around the balloon, but if we inflate the balloon, the distance between two ants can increase arbitrarily fast.