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In a Veritasium video, it is claimed that it is impossible to experimentally measure the 'one-way speed' $^1$ of light. The host further stipulates the stronger claim that the laws of physics are unaffected by the division of speed of light between its sent and return journey as long as the average speed remains $c$.

If the speed of light is assumed $kc\,(k\in[\frac{1}{2},1])$ in one direction, what should be the speed on the return journey so that the average speed remains $c$?

eg. the video claims that if the speed of light is assumed to be $\frac{1}{2}c$ one way, $\infty$ the other way, then the average speed still remains $c$.

What evidence is there that the speed of light has the same value in different directions? Is it possible that it can have different speeds in different directions?

$^1$Consider a light beam emanating from a source $S$ heading towards a target destination $D$ then reflecting back towards $S$. The 'one-way speed' of light is the speed of light on the path $SD$. The two-way speed of light, or just speed of light, is the average speed on the path $SDS$

benrg
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Rounak Sarkar
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If the speed of light is kc in a direction where $k\in[\frac{1}{2},1]$, then what would the speed of light be in the opposite direction?

This can be found out by just figuring out what the speed of light must be in the opposite direction to ensure that the average speed of light across the total journey is $c$.

So, if we have the speed during 1st half as $k$c and return speed as, say $v$ and let the distance each way be $d$.

Then, $$\text{average speed across whole journey} = c$$

or, $$\frac{\text{total distance}}{\text{time taken for the first half} + \text{time taken for 2nd half}} = c$$

or, $$\frac{2d}{d/kc + d/v} = c$$

Solving, the above equation, we get, $$v = \frac{kc}{2k-1}$$

So, if the speed of light is kc in a direction where $k\in[\frac{1}{2},1]$, then the speed of light in the opposite direction would be $\frac{kc}{2k-1}$

About your larger question, I think your main issue is summed up by this statement

The video blew my mind that it is impossible to measure the one-way speed of light, especially the fact that no scientists talk about it.

Let me ask you a different question. Have you ever seen a photograph of an electron? Do you think that any scientist has ever taken a photograph of an electron? The answer is no. It is not possible to do this. But still scientists all agree about it, and study its effects, etc., right?

Generally in science, if scientists cannot make a direct observation about something, they work with the simplest "assumption" they can make about it which satisfies all the observations regarding it and all other observations which are based on it. In other terminology, it may be referred to as a postulate, because they are postulating this without direct measurement of it. The more and more observations they are able to explain using this postulate, the less and less reason they have to question this postulate.

It is a similar case, with the speed of light. It is not possible to directly measure the one way speed of light. So, they work with the simplest "assumption"/postulate that it is constant, and since working with this postulate, they have been able to explain, describe and predict all observations in this domain, they have gained confidence in this postulate and hence do not need to question it until the day, they come across some phenomenon where this postulate breaks down. They have not found any such phenomenon yet.

This is the reason behind " the fact that no scientists talk about it." It is not because they are trying to hide anything or pull a fast one.

Peter Mortensen
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silverrahul
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Of course you can measure the one way speed of light, the only "gotcha" is that you need to use two different clocks in different places to do so. So you need to define how to synchronize them. For example, you could synchronize the clocks in the same place, then carry them slowly and carefully equal distances to the final test locations, do the measurement, then carry them back to verify that they are still in sync. If you do that, you'll measure the one way speed of light to be c. Pretty much any other "natural" synchronization method will also yield the same one-way and two-way measurements for the speed of light.

Now, of course, you could arbitrarily decide that eastbound clocks must be moved ahead some small amount based on distance, and westbound clocks moved back the same amount. In that case your one way speed of light in the east-west direction will not be c. But why would you do that? What's the motivation? It's adding an unnecessary complication to your theories.

EDIT: based on comments by Eleite and muhmed20 it's clear I didn't phrase my answer well. What I'm trying to say is that given a synchronization convention, you can measure the one way speed of light relative to that convention. I grant that the choice of convention involves assumptions. In effect, the value you get for the one way speed of light depends on the assumptions you make, i.e the precise synchronization convention you choose. My argument is that the synchronization methods we naturally use in the real world will yield c as the measured one way speed of light, and we might as well assume that what we actually measure with our instruments is what is "real".

Eric Smith
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It's true that the one-way speed of light cannot be measured, but it is always the case that the speed of light is calculated to be $c$. I.e. the two-way speed of light will always average to be $c$ over the total journey. Therefore if say, in a certain direction, the speed is $v_{direction} = \frac{1}{2} c$ then in the opposite direction the speed of light must be infinite! This is the only way to have the total journey average out with $v_{avg} = c$.

To calculate the speed of light in the opposite direction, it's very simple: $$ c_{\pm} = \frac{c}{1 \pm \kappa} $$ where $\kappa \in [0,1]$. The limiting case where $c_{+} = \frac{1}{2} c$ means that $\kappa = 1$. That also means we must have that $c_{-}$ diverges to infinity. If you want to use your notation with $kc$ and $k \in [\frac{1}{2},1]$, which isn't standard, then just use that $k= \frac{1}{1+\kappa}$ and for the opposite direction speed of light you easily find $k_{-}= \frac{1}{1-\kappa}$.

If I'm correct, these questions have been asked many times on here, but you may also find these Wikipedia pages useful:
https://en.wikipedia.org/wiki/Einstein_synchronisation
https://en.wikipedia.org/wiki/One-way_speed_of_light
(Here also includes the formula you ask about)