Speed of light when accelerating

Question

I'm studying special relativity and saw the 2 postulates of Einstein. The most remarkable one for me is the universal speed of light. Einstein postulated that the speed of light in vacuum is the same for all inertial observers, regardless of the motion of the source. But what if we are accelerating and we are changing from one inertial frame to another, how will we observe the speed of light? Is it still the same? I suggest it's not? Because I alread saw a small introduction to general relativity where the light will bend in a gravitational field and therefore also when you accelerate. So does the speed of light change because it bends?

Answer 1

To understand this you need to understand what we mean by speed.

If I want to measure positions and times I need to set up a coordinate system. For example I can take my stopwatch and my metre rule and construct some Cartesian axes $t$, $x$, $y$ and $z$, then I can describe every point in spacetime by its position in my coordinates $(t,x,y,z)$. Once I have done this I can calculate the velocity of some object by watching how its position measured using my coordinates changes with the time measured using my coordinates. So for example if something is moving along my $x$ axis the speed is just:

$$ v = \frac{dx}{dt} $$

All very well, but why did I repeatedly use the phrase measured using my coordinates in the paragraph above? Well, it's because my coordinate system is just one way of measuring out spacetime and it doesn't necessarily have any fundamental physical significance. That means the speeds determined by my coordinates don't necessarily have any fundamental physical significance either.

To look into this a bit farther let's stick to flat spacetime so we don't have the complications introduced by general relativity. The obvious coordinates to choose are those of an inertial frame, and you'll meet the phrase inertial frame over and over again in studying relativity. In these coordinates the speed of light is always $c$.

But now suppose I'm accelerating with some acceleration $a$. There's nothing to stop me choosing coordinates where I remain at the origin i.e. I measure all distances and times relative to me. This would be a non-inertial frame, and as you'd expect if I use a non-inertial frame all sorts of weird things can happen e.g. Newton's laws no longer apply.

To make this concrete suppose I am accelerating along the $x$ axis and I measure the velocity of a light beam travelling along the $x$ axis using my non-inertial coordinates. I get the result:

$$ v_\text{light} = c\,\left(1 + \frac{a}{c^2}x \right) \tag{1} $$

I won't go thorough the derivation, but the accelerating coordinates are known as Rindler coordinates and the speed of light is calculated using an equation called the Rindler metric.

Anyhow, what I find is that the velocity of light now changes with the distance along the $x$ axis away from me. When the product $ax$ is positive (i.e. ahead of me) the velocity of light is greater than $c$ and when $ax$ is negative (behind me) the velocity of light is than than $c$. In fact when $ax = -c^2$ the velocity of the light slows to zero and there is an event horizon there (called the Rindler horizon).

But it's the same light in the same spacetime, just described using two different sets of coordinates. So what then is the real speed of the light. And here we reach the key point: there is no real speed. Relativity tells us that any set of coordinates is as valid as any other set of coordinates - you cannot say the inertial coordinates are right and the accelerating coordinates are wrong because they are equally valid ways of describing the same spacetime.

There is one last point to make. Suppose we take my equation (1) for the speed in the accelerating coordinates and calculate the speed at my position i.e. at $x=0$. The value is:

$$ v_\text{light} = c\,\left(1 + \frac{a}{c^2}\,0 \right) = c $$

So even though the speed of light is variable, at my position the speed is equal to $c$. And this is another absolutely key point: although the speed of light may vary in some coordinate systems, if you measure the speed of light at your position you will always get the value $c$. So the speed of light is always locally constant, it's just not globally constant.