Equivalence principle and the meaning of the coordinate speed of light

Question

Short version: Does the equivalence principle give us a means to tell if variations in the coordinate speed of light have absolute or only relative significance?

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Background

In general relativity the local speed of light is a constant and has the usual value $c$, but the speed of light that we measure from here for a part of space over there (called the coordinate speed) may differ from the accepted value.

This is one way to structure arguments about gravitational red/blue shift and the curvature of light paths relative coordinate systems fixed to a particular observer. It is a common way of explaining the Shapiro delay.

Indeed in that kind of context this point of view is successful enough that it is tempting to take it as definitive. To say

"the speed of light really does vary from place to place and the constancy of the local speed is an artifact of using the motion of light to define our measure of time."

How does the equivalence principle come into play?

Not being a relativist in any serious way myself (my sole course on general relativity is more than twenty years in the past!) I've been wondering about how that notion gets along with the equivalence principle.

I try to isolate the question with the following thought experiment.

Two (identical) space craft are commanded from their geometric centers and also feature a pair of transverse light clocks a height $L/2$ "above" and "below" the cockpit. While accelerating the "high" clock should accumulate more time than the "low" clock just as they would if the ship was grounded upright on Earth.

Now we imaging these two craft hurtling toward each other in deep space while they both employ a stead thrust to reduce their closing velocity in such a way that they arrive at relative rest just as their cockpits come alongside one other. At this instant, the two occupants of the two craft momentarily share a single co-moving frame of reference.

Naive "paradox"

However, occupants of each craft will report a different expectation for the relative timing of the clocks. In particular occupants of craft $A$ see clocks $A_\text{high}$ and $B_\text{low}$ as above them and therefore running fast while clocks $A_\text{low}$ and $B_\text{high}$ are below them and therefore running slow. Occupants of craft $B$ of course have the opposite expectations.

Inertial view

Of course the occupants of both crafts are in non-inertial frames, an observer floating freely nearby will report that both ships exist in a flat space-time for which the coordinate speed of light is everywhere equal to the local speed of light.

What's the point

In my naive view this scenario demolishes claims that the coordinate speed has some absolute significance because

• It arranges a paradox if we believe in absolute significance of the coordinate speed.

• Viewing differences in the coordinate speed of light as having only relative consequence would seem to have no problem with the described scenario.

Is this a sustainable conclusion or is there something that I am missing (that is: is there a correction I'm failing to make that prevent the "paradox" from coming up in the first place thereby leaving the ontological question unresolved)?

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Neither of

• Gravity, Acceleration, Time Dilation and the Equivalence Principle
• Time dilation in a gravitational field and the equivalence principle

address the question of what significance should be understood for variations in the coordinate speed of light.

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¹ If the clocks are on gantries that stick out from the sides of the ships they could even be built so that the make their measurement through the same region of space as their partner.

Answer 1

Yes, and I think you have put your finger on a nice simple way to show that coordinate speed has no absolute significance. In fact now I think about this would make a nice answer to Does light really travel more slowly near a massive body? However I would be inclined to simplify the scenario somewhat.

If we consider flat spacetime then we have an unambiguous interpretation of the speed of light. There is no spacetime curvature to complicate the issue and Einstein synchronisation means we can maintain our clocks in synch everywhere in an inertial frame so we can all agree on timings. The speed of light is simply $c$ everywhere and all observers will agree on this.

Technically the metric that describes the flat spacetime is the Minkowski metric:

$$ ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2 \tag{1} $$

Light travels on null geodesics that have $ds^2=0$. If we substitute this into equation (1), and for convenience we take the light to be travelling in the $x$ direction then we get:

$$ 0 = -c^2dt^2 + dx^2 $$

or:

$$ \frac{dx}{dt} = \pm c $$

This is the situation for your inertial observer.

But the occupants of your two spaceships are in a frame with a constant proper acceleration, and for such observers their spacetime geometry is described by the Rindler metric. If we choose our axes so that the proper acceleration $a$ is along the $x$ axis the metric is:

$$ ds^2 = - \left(1 + \frac{a}{c^2}x \right)^2 c^2 dt^2 + dx^2 + dy^2 + dz^2 \tag{2} $$

where the observer is positioned at the origin $x=y=z=0$. As before we consider the trajectory of a light ray moving along the $x$ axis we get the equation for the speed of light to be:

$$ \frac{dx}{dt} = \pm c \left(1 + \frac{a}{c^2}x \right) $$

And we have an immediate contradiction. The occupants of the decelerating spaceships will observe the speed of light to be greater than $c$ in one direction and less than $c$ in the other direction. What's more the two spaceships will have exactly opposite views about the directions in which the speed increases and decreases. And of course both disagree with the inertial observer.

But, but, hang on. This is flat spacetime - there is no curvature here. How can observers be disagreeing about the speed of light? The simple answer is that what we are calling the speed of light is a coordinate dependent quantity so it naturally differs between different observers. As it happens all inertial observers will agree that the speed of light is $c$ everywhere, but all non-inertial observers will disagree.

And this is where the equivalence principle comes in. At any point in a curved spacetime we can expand the metric to first order and we find that to first order it is the same as the Rindler metric for some value of $a$. This means the curved spacetime is locally the same as a Rindler spacetime, which means that just as we found in your example the coordinate speed of light will vary. Furthermore it will vary by different amounts for different users depending on their proper acceleration. Just as you found in your thought experiment, in curved spacetime the coordinate speed of light is an observer dependent quantity not an absolute one.

Footnotes:

David referred to the clocks either side of the spaceships running at different rates, and we can immediately see why this is so from the metric. To do this we rewrite the metric in terms of the proper time:

$$ c^2 d\tau^2 = \left(1 + \frac{a}{c^2}x \right)^2 c^2 dt^2 - dx^2 - dy^2 - dz^2 $$

If we consider a stationary clock at a height $h$ from the spaceship then $dx=dy=dz=0$ because the clock is stationary and the metric gives us:

$$ d\tau = \left(1 + \frac{a}{c^2}h \right) dt $$

where $d\tau$ is the time interval recorded by the clock and $dt$ is the time interval recorded by the observer on the spaceship. This gives us the time dilation:

$$ \frac{d\tau}{dt} = 1 + \frac{a}{c^2}h $$

Above the spaceship, i.e. in the region where $a$ and $h$ have the same sign $d\tau/dt \gt 1$ and the clock runs faster. Below the spaceship $d\tau/dt \lt 1$ and the clock runs slower. And this is exactly what we measure for the gravitational time dilation on Earth.

Answer 2

This might not be a completely satisfying answer but I'll give it a shot. In my point of view, we can certainly use a variable speed of light for calculations, but no one actually believes (or should believe) that the variation of the speed of light has an absolute significance (though I can't speak for Shapiro). The reason is that we can change coordinates to get a different speed of light; the answer to any problem will be the same, but the details will be different.

(Edit: I should add that we can also change coordinates in special relativity, but there we have global inertial frames, in which the speed of light is constant. When spacetime is curved no set of coordinates is priviliged.)

Shapiro calculated his time delay using the variation of $dr/dt$ along the light's path. But I know if that I please I can use another coordinate $R(r)$; the speed of light is now $dR/dt = (dR/dr) (dr/dt)$, which is clearly not the same as before at a given spacetime point. And yet the time taken for light to go somewhere and return is the same in either case. Why should the Schwarzschild $r$ coordinate be privileged? What do you do if you don't have symmetries that let you use an obvious set of coordinates?

In this sense I think your proposed "paradox" does indeed achieve its goal, though at least to me it does so in a roundabout way: the key fact is that you can use different sets of coordinates, and while the coordinate speed of light can change, the physics doesn't. Your example then constructs two coordinate systems in flat spacetime by using two ships with opposite accelerations.