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Analysis of the circular track experiment measuring the one-way speed of light

All experiments attempting to verify the invariance of speed of light directly are based on two-way speed measurement. The challenge in one-way speed measurement, the requirement of spatially separated synchronised clocks, can be possibly circumvented by measuring the speed of light travelling in a closed path. An apparent violation of the invariance principle has been recently reported in the first experiment attempting to measure the one-way speed of light utilising this concept. This experiment is reanalysed here. It is found that the results of the experiment can be explained within the framework of relativity, without requiring any violation of the invariance principle.


I. INTRODUCTION
The invariance of the speed of light for all inertial observers is a basic postulate of the special theory of relativity. 1 However, this postulate has not been experimentally established directly. It has only been experimentally established that the two-way speed of light is independent of the velocity of the inertial frame in which it is measured. 2 One of the major challenges in performing direct measurement of the speed of light during one-way transmission is the requirement of spatially separated, synchronised clocks.
A possible way of circumventing this challenge is measuring the speed of light travelling in a closed path. This will enable the use of the same clock for the initial and final measurement, thereby eliminating the need for synchronisation.
The speed of light in an experiment based on this idea, conceived and performed by C. S. Unnikrishnan, was found to have an apparent dependence on the velocity of the observer. 3 We present a reanalysis of this experiment and show that the experimental results can be explained within the framework of special relativity, without requiring any violation of the principle of invariance of speed of light.

II. THE EXPERIMENT
A brief description of Unnikrishnan's experiment is given below to facilitate the understanding of this paper. According to the invariance principle, the measured speed of light must be independent of the constant velocity the two clocks may be moving with. It was argued that, even if the path followed by the light pulse is not straight as in Figure 1  The path may be bent to make the initial and final clock the same, like in Figure 2. This maintains the total path length and eliminates the concern about synchronisation.
The invariance principle was to be validated by comparing the speeds of two light pulses, travelling in opposite directions, in a closed path, as schematically shown in Figure 2

III. EXPLANATION OF THE RESULTS
We will show below that this apparent violation of the postulate of invariance of speed of light arises because, according to the special theory of relativity, from the frame of reference of the inertial observer, the path travelled by the two light pulses are of unequal length.
This can be seen easily by examining the Minkowski space-time diagram in Figure 4.
If we consider the frame of reference of the observer, the mirrors keep moving while the light pulses travel from one mirror to another. From the start of the pulse to the detection of the pulse, this results in a change in the effective distance between the mirrors. Contrary to naive expectation, according to the special theory of relativity, the total distance travelled by the two light pulses is seen to be unequal (see Figure 4). It turns out that it was erroneous to expect A and B of Figure 1 to yield the same result. We are expecting an effect which is first order in velocity. Since the velocity of the inertial observer is much less than the speed of light, all contributions arising from higher orders of velocity can be ignored.