The two-way speed of light

Up until now I have chiefly looked at some conceptual issues relating to special relativity: How do physicists choose to adjust distant clocks? What do they, therefore, mean when they state that the one-way speed of light is constant for every observer? What consequences do their decisions and definitions have for the applicability of concepts such as simultaneous existence, one-way speed and causality in the framework of special relativity?

Interesting and important as I believe these issues are, it is now time to turn to the physical substance of special relativity. At the heart of this lies the empirical finding that the two-way or round-trip speed of light is the same in every spatial inertial coordinate system.

Albert Einstein showed that the constancy of the two-way speed of light implies all of special relativity, including length contraction and time dilation by the relativistic factors for moving bodies in any spatial inertial frame of reference in which clocks are Einstein-adjusted.

Helmut Günther has shown that, conversely, length contraction and time dilation by the relativistic factors for moving bodies in any particular spatial inertial frame of reference in which clocks are Einstein-adjusted implies all of special relativity, including the constancy of the two-way speed of light.

To see more directly that length contraction and time dilation implies the constancy of the two-way speed of light, consider a first spatial inertial frame of reference Σ in which light propagates in symmetrical conditions in opposite directions so that we can use Einstein's clock adjustment procedure to synchronize clocks in that frame.

The two-way speed of light c' in a second frame of reference S moving at v relative to Σ can then be worked out as follows:

In


we know that

where t1 and t2 are the to-and-fro times in Σ which the light needs to travel along the to-and-fro path 2x' in S.
From

and


we find that

It follows that


In this blog, I will follow Günther in regarding length contraction and time dilation as fundamental and the constancy of the two-way speed of light as a consequence. Speed measurements are, after all, based on measurements of length and time, so to explain the constancy of the two-way speed of light it seems useful to explore what we know about the behaviour of measuring rods and clocks in different spatial inertial coordinate systems.

The question of how the constancy of the two-way speed of light can be explained can then be rephrased as follows: how can we explain length contraction and time dilation for moving bodies in any spatial inertial frame of reference in which light propagates in symmetrical conditions and clocks have been synchronized using Einstein's method?

Moving bodies move relative to a frame of reference in which light propagates in symmetrical conditions because at some point they have undergone local acceleration relative to that frame, and that is the only action that has been performed on them that could explain length contraction and time dilation. My question could thus finally be expressed as follows: how does local acceleration in such frames cause bodies to contract in the direction in which they have been accelerated, and how does it cause clocks to go slow?

Expressed in this way, it seems to me that, once conceptual issues have been sorted out, much of the remaining mysteriousness of special relativity vanishes. As discussed in previous posts, the local acceleration of any particle surrounded by, for example, an electromagnetic field distorts that field and may thus well have an effect on the length of any body containing such particles, and on the duration of any periodic processes involving that particle.

What is more, the speed at which field disturbances propagate through the field, in other words the speed of light c, can be expected to influence those length and time effects because it has a bearing on the distortion of the field.

The remaining question is then simply how acceleration to a particular speed v in a frame of reference in which light signals propagate in symmetrical conditions changes length and time measurements precisely in accordance with the relativistic length contraction and time dilation factors. And this is the question to which I now intend to turn.