# What is wrong with these ways of measuring one-way speed of light? [duplicate]

I thought of 2 ways of measuring one-way speed of light. Can somebody explain why these won't work please?

1. Create a machine that emits a pulse of light and a sound from far away. Have a clock ready and start the clock when the pulse of light reaches it and stop it when the sound reaches it. (All this done a in controlled air tight environment) Then we use the known facts (distance, speed of the sound pulse & the difference in time of the two (light and the sound)) to figure out the speed of light.

2. Create a 2 clock system with a synchroniser in the middle that emits sound waves to both clocks.

clock----------synchroniser----------clock

When the sound wave reaches the clocks they start timing and the first clock emits a beam of light and when the beam of light hits the second clock it stops. Therefore measuring the time taken. Then use the distance and the time to calculate the speed.

• Does this answer your question? Can One-Way Speed of Light be Instantaneous?
– Dale
Commented Jun 8, 2022 at 10:39
• The fundamental fact to remember here, and Einstein's starting point in his 1905 paper, is that any information ascertained about a location far away is dependent upon receiving a signal from that location. And to interpret that signal you must make some assumption about the one way speed of light. Commented Jun 8, 2022 at 15:06
• How do you know the one way speed of sound? Commented Jun 8, 2022 at 15:13

There is no way to measure the one way speed of light (OWSOL), so this experiment is no exception.

To measure the OWSOL it must be treated as a variable and then the equations of the experiment need to be solved to find that variable. Here, I will use Anderson's convention and use units such that the two-way speed of light is $$c=1$$. Anderson's $$\kappa$$ is related to the more famous Reichenbach $$\epsilon$$ by $$\kappa = 2 \epsilon -1$$, but I find Anderson's approach more convenient to work with.

Anderson's $$\kappa$$ is a vector field where the one-way speed of light in the direction $$\hat n$$ is given by: $$c(\hat n) = \frac{c}{1-\kappa \cdot \hat n}$$

Therefore, to measure the OWSOL we would need to construct an experiment whose outcome depends on $$\kappa$$.

Such an experiment does not exist and cannot exist. The issue is that $$\kappa$$ is set by a coordinate transform. Specifically, the coordinate transform from a standard inertial coordinate system $$(t,x,y,z)$$ to an Anderson coordinate system $$(T,X,Y,Z)$$ is given by $$T=t-\kappa \cdot (x,y,z)$$ $$X=x$$ $$Y=y$$ $$Z=z$$

This is just an ordinary coordinate transform, and $$\kappa$$ is merely a parameter used in the coordinate transform. However, no experiment ever depends on the choice of coordinates. Therefore no experiment can depend on $$\kappa$$.

To emphasize the fact that no experiment ever depends on the choice of coordinates, recall that all of the known laws of physics can be encapsulated in this single equation: This equation, which represents all known physics, is written in terms of tensors and is therefore manifestly invariant meaning that it is manifestly independent of the coordinate system. Since it is manifestly invariant then it does not depend on $$\kappa$$.

There is therefore no law of physics which can be used to measure $$\kappa$$ and therefore no law of physics which can measure the OWSOL. It doesn't matter if you use a clever arrangement of light, or a clever arrangement of clocks, or a clever arrangement of sound, or a clever arrangement of matter or fields of any kind. No possible measurement you can ever do will depend on $$\kappa$$ so no experiment can determine the OWSOL.

• Wow, that looks pretty deep, clearly well above my pay grade ;) Does it cover the laws of thermodynamics? Commented Mar 25, 2023 at 12:57
• The laws of thermodynamics can be derived from the laws above. Indeed, many of the results of statistical mechanics do not actually depend on the form of the underlying laws
– Dale
Commented Mar 25, 2023 at 14:00

## Accuracy

The obvious issue is that since speed of light is many orders of magnitude larger than the speed of sound, any inaccuracies in measuring the sound propagation time will have a disproportional effect on the calculated estimate of the speed of light, so a small inaccuracy in your estimate of speed of sound or any tiny delay in the process of determining when exactly "sound wave reaches the clocks" will make the whole endeavor useless.

Speed of sound depends on all kinds of factors such as temperature, humidity and pressure, so you would need very accurate knowledge of the atmospheric conditions (and very accurate information about their exact effect on speed of sound) throughout your experimental area - since speed of sound is approx million times smaller than the speed of light, a 1-in-million error in the estimate of speed of sound means +/- 100% error in your calculation of speed of light. The same also applies for your second scenario unless you can somehow control conditions ensure that the speed of sound is equal in both directions from the synchronizer, which won't be true by default.

Propagation of sound also is diffuse enough to limit you to relatively small distances (it's tricky to accurately transmit a pulse of sound over 10+ miles) so the delay of sound propagation across a part of the receiving apparatus can be larger than the elapsed time of light propagation you're trying to measure.

So the conclusion is that you technically could try to do it this way, but it's simply not the most practical way to construct an accurate apparatus to make this measurement useful. Like, as you say, having a "controlled air tight environment" removes some sources of inaccuracy, but it's not trivial to have controlled air tight environment that's, say, a mile long, and it's difficult to get an accurate measurement of the speed of light if you want to limit the experiment to something room-sized since the elapsed time would be so tiny.

And of course, if you would postulate that there's some difference in light speed one way over the other, then this would likely also affect the directionality of the speed of sound, in that case you can't simply assume that the speed of sound is equal in all directions.