B One-Way Speed of Light

[shawnhcorey]

TL;DR Summary

The one-way speed of light can be measure with the use of a transparent medium.

To measure the two-way speed of light, a pulse of light is sent thru a vacuum to bounce off a mirror and return. By knowing the distance to the mirror and how long the pulse takes to travel, the two-way speed of light can be calculated. To measure the one-way speed of light, place a transparent medium in one side of the path and measure the time. Then repeat for the other side. The difference in the times can be use to calculate the difference in the speed of light (if any) in both directions.

Example

Set up the mirror so that it is 29.9792458 cm from the emitter and the detector. It will take 2.00 ns for the pulse to travel to the mirror and back. Place a transparent medium in one side that slows light by 10%.

Same Speed

If light travels at the same speed in both directions, it takes 1.00 ns for one direction. By slowing the light by 10% on one side, it will take 1.10 ns. Same for the other side. Total travel time is 2.10 ns.

Different Speed

For the sake of argument, assume light takes 0.50 ns in one direction and 1.5 ns in the other. Total round-trip time is 2.00 ns.

Place the medium in the side of 0.50 ns. This slows the speed to 0.55 ns and has a total round-trip time of 2.05 ns. Placing it in the other side slows light to 1.65 ns for a round-trip time of 2.15 ns.

If the times for the medium in each side is the same, the light travels the same speed in both directions. But what if the amount the medium slows light down is dependent on the direction?

Snell's Law

Snell's law states that the ratio of the sine of angle of incident to the sine of the angle of refraction is equal to the ratio of the corresponding speed of light in the mediums. Set up an apparatus so that the light in the vacuum is in the same direction as one direction of the pulse of light and then in the other direction. If the angles remain the same, then the medium slow light by the same factor.

 

[Ibix]

No. The choice of the one-way speed of light represents your freedom to choose non-orthogonal axes in your reference frame. It has no detectable physical consequences because it's a personal choice.

I suspect you haven't realised that selecting a different one-way speed of light also changes your definition of the speeds of everything else, including the speed of light in a medium. If you account for that speed change (which is also anisotropic, so needs to be handled differently for the different directions) correctly then the flight time differences will disappear.

 

[shawnhcorey]

No. The choice of the one-way speed of light represents your freedom to choose non-orthogonal axes in your reference frame. It has no detectable physical consequences because it's a personal choice.

I suspect you haven't realised that selecting a different one-way speed of light also changes your definition of the speeds of everything else, including the speed of light in a medium. If you account for that speed change (which is also anisotropic, so needs to be handled differently for the different directions) correctly then the flight time differences will disappear.

Physics is not about choice. It's about measurements. Your comment is unclear. Please specify which measurements you think are wrong.

 

[shawnhcorey]

It is easy to accept a one-way speed of light if you are willing to assume that the clocks on both ends are perfectly synchronized. But that is not a legitimate assumption to people in another inertial reference frame. You should consider the relativity of simultaneity.

True, the one-way speed of light cannot be measured with two clocks. The above experiment uses only one clock.

 

[Ibix]

Physics is not about choice.

Correct. But you may choose to use orthogonal or non-orthogonal axes to express the maths, and the one-way speed of light is an artefact of that choice.

Please specify which measurements you think are wrong.

None of them are wrong, but you implicitly chose to use orthogonal axes to interpret your measurements. Thus you will get an isotropic one-way speed of light because that is an assumption hidden in your analysis.

 

[Ibix]

The above experiment uses only one clock.

...but assumes that the one-way speed of light in glass (or perhaps the refractive index) is isotropic, thereby assuming that the one-way speed of light in vacuum is isotropic

 

[FactChecker]

True, the one-way speed of light cannot be measured with two clocks. The above experiment uses only one clock.

Oh. I stand corrected. I didn't understand the experiment.

 

[PeroK]

Physics is not about choice. It's about measurements. Your comment is unclear. Please specify which measurements you think are wrong.

The one-way speed of anything depends on your synchronization convention for spatially separated clocks.

The question is whether the Einstein synchronization convention - which is the default- is the only option. It isn't the only option. Other coordinate systems, where the one-way speed of light is not isotropic, are regularly used in physics.

 

[shawnhcorey]

Correct. But you may choose to use orthogonal or non-orthogonal axes to express the maths, and the one-way speed of light is an artefact of that choice.

None of them are wrong, but you implicitly chose to use orthogonal axes to interpret your measurements. Thus you will get an isotropic one-way speed of light because that is an assumption hidden in your analysis.

Most physics is described using orthogonal axes. You still haven't specified what is wrong.

 

[shawnhcorey]

...but assumes that the one-way speed of light in glass (or perhaps the refractive index) is isotropic, thereby assuming that the one-way speed of light in vacuum is isotropic

By using Snell's law, this possibility is removed.

 

[shawnhcorey]

The one-way speed of anything depends on your synchronization convention for spatially separated clocks.

The question is whether the Einstein synchronization convention - which is the default- is the only option. It isn't the only option. Other coordinate systems, where the one-way speed of light is not isotropic, are regularly used in physics.

There is only one clock in the experiment.

 

[PeroK]

Most physics is described using orthogonal axes. You still haven't specified what is wrong.

The difference between the time taken for light to travel in vacuum and through a medium with refractive index of 1.1, say, will not be a percentage of the speed but a fixed amount. If you use the Einstein convention and find the difference to be 1 ns, say. Then, with a different synchronization convention the difference must still be 1 ns. This time interval cannot depend on the clock synchronization.

 

[Ibix]

None of them are wrong, but you implicitly chose to use orthogonal axes to interpret your measurements.

To be a bit more precise, you are assuming an isotropic refractive index. That means you are assuming orthogonal time- and space-like axes and hence an isotropic speed of light. To get an anisotropic speed of light you need to consider non-orthogonal coordinates, which would (if done correctly) lead to an anisotropic refractive index and probably a modified version of Snell's law.

You are simply assuming that you can have an anisotropic speed of light without carrying through all of the other changes. That might be consistent with something like a mechanical ether theory with an ether wind, but is not consistent with relativity.

You still haven't specified what is wrong.

I have, at least twice. You are assuming you can use orthogonal axes for some of your maths and non-orthogonal axes for other parts without accounting correctly for the different expressions of physics in the two systems. That mis-match is where your analysis is in error.

 

[PeroK]

There is only one clock in the experiment.

Yes, but if your analysis were valid, it would establish the Einstein synchronization convention as the only viable convention. Which it isn't. So, something must be wrong with your analysis.

The mistake is the assumption about refractive index for non-isotropc coordinates.

 

[PeroK]

Here's a thought experiment. Imagine we have a Moon base. A light signal is sent from Earth saying "the time is $t =0$". The signal is received on the Moon and they have to decide what to do in terms of setting their clocks.

One option is to set the Moon clock to $t' =0$ when the signal arrives. The other option is to set it to $t' = 1.2s$, because the Moon is $1.2$ light-seconds from Earth.

The second option is the Einstein synchronization convention. What about the first option? Is that just totally wrong?

It's not immediately obvious whether that option is valid or not. But, it turns out that it is valid. However, it means that one way speeds between the Earth to the Moon are not the same. That includes most clearly the one way speed of light - which is effectively measured to be infinite in one direction.

No experiment with mirrors and refractive media is going to change this.

Now, you might imagine that for most practical purposes the first convention is awkward and makes life difficult. E.g. it changes how to calculate the effect of a refractive medium on the speed of light.

But, in another context (experiments involving light or objects falling into a black hole) having an alternative synchronization convention is invaluable.

In any case, you can read more about this here:
https://en.m.wikipedia.org/wiki/One-way_speed_of_light

 

[shawnhcorey]

The difference between the time taken for light to travel in vacuum and through a medium with refractive index of 1.1, say, will not be a percentage of the speed but a fixed amount. If you use the Einstein convention and find the difference to be 1 ns, say. Then, with a different synchronization convention the difference must still be 1 ns. This time interval cannot depend on the clock synchronization.

Snell's law states that it is a percentage.

 

[shawnhcorey]

To be a bit more precise, you are assuming an isotropic refractive index. That means you are assuming orthogonal time- and space-like axes and hence an isotropic speed of light. To get an anisotropic speed of light you need to consider non-orthogonal coordinates, which would (if done correctly) lead to an anisotropic refractive index and probably a modified version of Snell's law.

You are simply assuming that you can have an anisotropic speed of light without carrying through all of the other changes. That might be consistent with something like a mechanical ether theory with an ether wind, but is not consistent with relativity.
I have, at least twice. You are assuming you can use orthogonal axes for some of your maths and non-orthogonal axes for other parts without accounting correctly for the different expressions of physics in the two systems. That mis-match is where your analysis is in error.

Snell's law states that the refractive index is isotropic.

 

[shawnhcorey]

Yes, but if your analysis were valid, it would establish the Einstein synchronization convention as the only viable convention. Which it isn't. So, something must be wrong with your analysis.

The mistake is the assumption about refractive index for non-isotropc coordinates.

Snell's law measures if it is isotropic.

 

[shawnhcorey]

Here's a thought experiment. Imagine we have a Moon base. A light signal is sent from Earth saying "the time is $t =0$". The signal is received on the Moon and they have to decide what to do in terms of setting their clocks.

One option is to set the Moon clock to $t' =0$ when the signal arrives. The other option is to set it to $t' = 1.2s$, because the Moon is $1.2$ light-seconds from Earth.

The second option is the Einstein synchronization convention. What about the first option? Is that just totally wrong?

It's not immediately obvious whether that option is valid or not. But, it turns out that it is valid. However, it means that one way speeds between the Earth to the Moon are not the same. That includes most clearly the one way speed of light - which is effectively measured to be infinite in one direction.

No experiment with mirrors and refractive media is going to change this.

Now, you might imagine that for most practical purposes the first convention is awkward and makes life difficult. E.g. it changes how to calculate the effect of a refractive medium on the speed of light.

But, in another context (experiments involving light or objects falling into a black hole) having an alternative synchronization convention is invaluable.

In any case, you can read more about this here:
https://en.m.wikipedia.org/wiki/One-way_speed_of_light

Your example has two clocks. The above experiment has only one clock. The Wikipedia page discusses the problem of synchronizing two clocks. Not a problem here since there is only one clock.

 

[Ibix]

Snell's law states that the refractive index is isotropic.

In orthogonal coordinates, yes. But in those coordinates the one-way speed of light is isotropic. You'd need to re-write Snell's Law in non-orthogonal coordinates if you want to consider an anisotropic speed of light.

If you want to consider an anisotropic speed of light you need to consider all of the consequences of your coordinate choice, not just the ones you want to.

 

[Ibix]

Your example has two clocks. The above experiment has only one clock. The Wikipedia page discusses the problem of synchronizing two clocks. Not a problem here since there is only one clock.

But if you could measure the one-way speed of light in an assumption-free way then you could synchronise clocks remotely in an assumption-free way. The two things are two sides of the same coin.

If you accept that you cannot synchronise remote clocks without assumptions then you accept that your experiment cannot measure the one-way speed of light without assumptions. Or you contradict yourself.

 

[Ibix]

Snell's law measures if it is isotropic.

It's worth noting, of course, that Fizeau's discovery of the failure of the refractive index to behave in a Newtonian fashion is now understood to be one of the first pieces of evidence for relativity.

 

[shawnhcorey]

In orthogonal coordinates, yes. But in those coordinates the one-way speed of light is isotropic. You'd need to re-write Snell's Law in non-orthogonal coordinates if you want to consider an anisotropic speed of light.

If you want to consider an anisotropic speed of light you need to consider all of the consequences of your coordinate choice, not just the ones you want to.

If Snell's law wasn't isotropic, then all the optics in the world would need constant adjustment as the world spins around. Since this is not necessary, Snell's law is isotropic.

 

[shawnhcorey]

But if you could measure the one-way speed of light in an assumption-free way then you could synchronise clocks remotely in an assumption-free way. The two things are two sides of the same coin.

If you accept that you cannot synchronise remote clocks without assumptions then you accept that your experiment cannot measure the one-way speed of light without assumptions. Or you contradict yourself.

No, you're assume that the one-way speed of light cannot be measured without two clocks.

 

[shawnhcorey]

It's worth noting, of course, that Fizeau's discovery of the failure of the refractive index to behave in a Newtonian fashion is now understood to be one of the first pieces of evidence for relativity.

Fizeau's experiment is about a moving medium. The medium does not move in the above. Not relevant.

 

[PeroK]

Fizeau's experiment is about a moving medium. The medium does not move in the above. Not relevant.

Well, then, you'd better publish and claim the Nobel Prize!

 

[PeterDonis]

place a transparent medium in one side of the path

If you are using only one clock, as you say, that means the light has to return to the same point that it started from, since it has to come back to the same clock. Which means you can't place a transparent medium in "one side" of its path and not the other. Both legs of the light's path--outbound and return--cover the same path in space. So if the medium is there on one leg, it's there on the other as well.

If, on the other hand, you insist on having the light hit the mirror at an angle so that the two legs of its path are in different parts of space, then you can't measure the light's flight time using just one clock. You need two.

Either way, your claim to have discovered a way to make an invariant (independent of any conventions) measurement of the one-way speed of light is not valid.

 

[Ibix]

If Snell's law wasn't isotropic, then all the optics in the world would need constant adjustment as the world spins around. Since this is not necessary, Snell's law is isotropic.

You fail to understand. In a system where the speed of light is anisotropic, you need an anisotropic Snell's Law to balance the anisotropic speed of light and avoid exactly those adjustments that, as you yourself note, we do not see.

 

[Ibix]

No, you're assume that the one-way speed of light cannot be measured without two clocks.

Again, you aren't understanding. If you can measure the one-way speed of light then you can synchronise remote clocks without assumptions. You seem to understand that this synchronisation is impossible, but then contradict yourself by claiming a one-way speed measure that would allow it.

 

[Nugatory]

If Snell's law wasn't isotropic, then all the optics in the world would need constant adjustment as the world spins around. Since this is not necessary, Snell's law is isotropic.

The form of Snell’s law is isotropic when expressed in isotropic coordinates, not isotropic when expressed in non-isotropic coordinates. That’s how we can analyze the behavior of light using either coordinates and arrive at the same physical result (namely, that these optics adjustments are not required).

This might also be a good time to mention that angles are inherently frame-dependent; the classical example is a ball bouncing up and down on the deck of a ship.

 

[Orodruin]

If Snell's law wasn't isotropic, then all the optics in the world would need constant adjustment as the world spins around. Since this is not necessary, Snell's law is isotropic.

It is isotropic only if you have orthogonal temporal and spatial axes. In essence, you are making the assumption of using coordinates where the speed of light is isotropic because that is what your version of Snell’s law states. Only in such coordinates, where the one-way speed of light is the same as the two-way speed of light, does your conclusion make sense.

 

[shawnhcorey]

Again, you aren't understanding. If you can measure the one-way speed of light then you can synchronise remote clocks without assumptions. You seem to understand that this synchronisation is impossible, but then contradict yourself by claiming a one-way speed measure that would allow it.

I said synchronization is impossible using two clocks. That does not imply it is impossible using one clock.

 

[shawnhcorey]

It is isotropic only if you have orthogonal temporal and spatial axes. In essence, you are making the assumption of using coordinates where the speed of light is isotropic because that is what your version of Snell’s law states. Only in such coordinates, where the one-way speed of light is the same as the two-way speed of light, does your conclusion make sense.

That is also how they measure the two-way speed of light. They ignore any acceleration and therefore assume orthogonal axes. If they do it for the two-way speed of light, it should be done the same for the one-way speed of light.

 

[Orodruin]

That is also how they measure the two-way speed of light. They ignore any acceleration and therefore assume orthogonal axes. If they do it for the two-way speed of light, it should be done the same for the one-way speed of light.

That’s just ignorant. Further discussion is useless until you have actually learned relativity and are ready to accept that you may be wrong.

 

[Ibix]

That does not imply it is impossible using one clock.

What are you synchronising if you only have one clock..?

 

[PeterDonis]

I said synchronization is impossible using two clocks. That does not imply it is impossible using one clock.

As I said in post #27, if your experiment only has one clock, it can't be done the way you describe it--you can't put the transparent medium in only one leg of the light beam's travel. You can only do that if there are two separate clocks.

 

[PeterDonis]

That is also how they measure the two-way speed of light. They ignore any acceleration and therefore assume orthogonal axes. If they do it for the two-way speed of light, it should be done the same for the one-way speed of light.

This is simply wrong. No "axes" have to be assumed at all to measure the round-trip speed of light. All you need is a clock and a mirror at rest relative to it, with a ruler extended between the clock and the mirror to read off the distance between them. You don't even need coordinates; everything in the experiment is a direct observable.

 

[javisot]

If you are using only one clock, as you say, that means the light has to return to the same point that it started from, since it has to come back to the same clock. Which means you can't place a transparent medium in "one side" of its path and not the other. Both legs of the light's path--outbound and return--cover the same path in space. So if the medium is there on one leg, it's there on the other as well.

If, on the other hand, you insist on having the light hit the mirror at an angle so that the two legs of its path are in different parts of space, then you can't measure the light's flight time using just one clock. You need two.

Either way, your claim to have discovered a way to make an invariant (independent of any conventions) measurement of the one-way speed of light is not valid.

If I understand you correctly, Peter, you're saying that "measuring the one-way speed of light" means that we have emitted a photon, overtaken it, and received it ourselves without any intermediate interaction.

Since the speed of light is the maximum speed in the universe, it is impossible to overtake a photon that we have emitted ourselves, and this is equivalent to "not being able to measure the one-way speed of light."

Is this correct?

 

[PeterDonis]

If I understand you correctly, Peter, you're saying that "measuring the one-way speed of light" means that we have emitted a photon, overtaken it, and received it ourselves without any intermediate interaction.

No. I have no idea where you got any of this from.

Also, please do not use the term "photon" in this context. This is the relativity forum, not the quantum physics forum. (Even in a quantum context, the term "photon" doesn't mean what you think it means--but that's a discussion for some other thread, not this one.)

 

[javisot]

If you are using only one clock, as you say, that means the light has to return to the same point that it started from, since it has to come back to the same clock. Which means you can't place a transparent medium in "one side" of its path and not the other. Both legs of the light's path--outbound and return--cover the same path in space. So if the medium is there on one leg, it's there on the other as well.

If, on the other hand, you insist on having the light hit the mirror at an angle so that the two legs of its path are in different parts of space, then you can't measure the light's flight time using just one clock. You need two.

Either way, your claim to have discovered a way to make an invariant (independent of any conventions) measurement of the one-way speed of light is not valid.

I could say light instead of photon (like you) if that's more appropriate, ok. I'm trying to understand this. Could you explain more?

 

[PeroK]

That is also how they measure the two-way speed of light. They ignore any acceleration and therefore assume orthogonal axes. If they do it for the two-way speed of light, it should be done the same for the one-way speed of light.

When you say "they" I guess you mean professional physicists. Given that @Orodruin is a physics professor, you're now actually telling him how he does things!

 

[Nugatory]

That is also how they measure the two-way speed of light. They ignore any acceleration and therefore assume orthogonal axes.

The two way measurement uses the proper time along the clock's worldline between the emission event and the reception event; no coordinates and coordinate axes enter into the computation at any time.

 

[Nugatory]

If I understand you correctly, Peter, you're saying that "measuring the one-way speed of light" means that we have emitted a photon, overtaken it, and received it ourselves without any intermediate interaction.

A one way measurement is a measurement between two different points in space, with a clock at each point. We say the flash of light (not photon! photons are not what you think they are and light is not photons moving from one point to another) left point A when the clock there read zero, arrived at point B when the clock there read $T$, if the two points are separated by distance $D$ then the one-way speed of light is $D/T$.
It's the same logic as if leave my house at noon, arrive at my destination 30 miles away when the clock there reads 1:00, therefore my average speed was 30 miles per hour.

There's no catching up or overtaking involved.

 

[Ibix]

If I understand you correctly, Peter, you're saying that "measuring the one-way speed of light" means that we have emitted a photon, overtaken it, and received it ourselves without any intermediate interaction.

I think Peter's saying that you can only measure a one-way speed with one clock if you can overtake the light. You can do one-way measures using two clocks. The setup under discussion here is really a two-way measure - the OP only thinks it's a one-way measure because he's failing to use the refractive index correctly in his anisotropic system.

 

[javisot]

I think Peter's saying that you can only measure a one-way speed with one clock if you can overtake the light.

That's what I said, but Peter comments that's not correct. I have asked him to explain more about this.

 

[Ibix]

Hold on that's what I said, but Peter comments that's not correct. I have asked him to explain more about this.

Not quite - you omitted the "with one clock" bit. You can measure the one-way speed with two clocks without travelling faster than light (obviously the answer is assumption dependent).

Or at least, that's what I think Peter's point is.

 

[PeterDonis]

I think Peter's saying that you can only measure a one-way speed with one clock if you can overtake the light.

That's not what I was saying, no. @Nugatory in post #43 correctly described what I was saying.

 

[PeterDonis]

That's what I said, but Peter comments that's not correct. I have asked him to explain more about this.

@Nugatory in post #43 correctly explained what I was saying. However, there's also an additional point.

Since a clock can't move faster than light, consider trying to measure the speed of a bullet instead. Does it even make sense to send a clock along the bullet's path faster than the bullet in order to measure the bullet's speed? The clock is moving. That means, whatever you think you're measuring, it isn't the speed of the bullet relative to clocks at rest, which is what a one-way speed measurement is supposed to be measuring.

Add to that the fact that clocks can't move faster than light, or even at the speed of light, to begin with, and what does that leave?

 

[javisot]

Add to that the fact that clocks can't move faster than light, or even at the speed of light, to begin with, and what does that leave?

That we can't measure the one-way speed of light "with a single clock".

We can bounce light off a mirror, we can bend geometry to make it back, we can use two clocks and send a signal back when the light reaches point B, but that's not measuring the one-way speed of light with a single clock (at rest).

 

[Dale]

TL;DR Summary: The one-way speed of light can be measure with the use of a transparent medium.

To measure the two-way speed of light, a pulse of light is sent thru a vacuum to bounce off a mirror and return. By knowing the distance to the mirror and how long the pulse takes to travel, the two-way speed of light can be calculated. To measure the one-way speed of light, place a transparent medium in one side of the path and measure the time. Then repeat for the other side. The difference in the times can be use to calculate the difference in the speed of light (if any) in both directions.

Example

Set up the mirror so that it is 29.9792458 cm from the emitter and the detector. It will take 2.00 ns for the pulse to travel to the mirror and back. Place a transparent medium in one side that slows light by 10%.

Same Speed

If light travels at the same speed in both directions, it takes 1.00 ns for one direction. By slowing the light by 10% on one side, it will take 1.10 ns. Same for the other side. Total travel time is 2.10 ns.

Different Speed

For the sake of argument, assume light takes 0.50 ns in one direction and 1.5 ns in the other. Total round-trip time is 2.00 ns.

Place the medium in the side of 0.50 ns. This slows the speed to 0.55 ns and has a total round-trip time of 2.05 ns. Placing it in the other side slows light to 1.65 ns for a round-trip time of 2.15 ns.

If the times for the medium in each side is the same, the light travels the same speed in both directions. But what if the amount the medium slows light down is dependent on the direction?

Snell's Law

Snell's law states that the ratio of the sine of angle of incident to the sine of the angle of refraction is equal to the ratio of the corresponding speed of light in the mediums. Set up an apparatus so that the light in the vacuum is in the same direction as one direction of the pulse of light and then in the other direction. If the angles remain the same, then the medium slow light by the same factor.

In order to measure the one-way speed of light you cannot assume it. You have to use a framework where the one-way speed of light is a variable, and then show how your measurement depends on that variable.

So, we can use Anderson's approach where the one way speed of light is determined by a variable $\kappa$. The transform between an Einstein synchronized frame and an Anderson frame is: $$T=t-\kappa x/c$$$$X=x$$$$Y=y$$$$Z=z$$ where the capitalized coordinates are Anderson’s and the lower-case coordinates are Einstein’s. From this you can calculate that $$V=\frac{dX}{dT}=v\frac{1}{1-\kappa v/c}$$ where $v=dx/dt$.

For this problem the one way time for each leg is $$\Delta T = \frac{\Delta X}{V}=\frac{\Delta x}{v/(1-\kappa v/c)}$$ For the forward path $\Delta x = \Delta X = 29.9792458 \mathrm{\ cm} = 1 \mathrm{\ ns}\ c$ and for the reverse path $\Delta x = \Delta X = -29.9792458 \mathrm{\ cm}= 1 \mathrm{\ ns}\ c$.

When the forward path is empty the velocity is $v=c$ which is $V=c/(1-\kappa)$.

When the forward path has the medium the velocity is $v=0.9 c$ which is $V=0.9 c/(1-0.9 \kappa)$.

When the reverse path has the medium the velocity is $v=-0.9 c$ which is $V=-0.9c/(1+0.9\kappa)$.

When the reverse path is empty the velocity is $v=-c$ which is $V=-c/(1+\kappa)$.

Each individual leg depends on the one-way speed of light parameter $\kappa$. Now, let's sum the times to find the result of the experiment. For the medium in the forward path we get $$\Delta T = \frac{1 \mathrm{\ ns}\ c}{0.9 c/(1-0.9 \kappa)}+\frac{-1 \mathrm{\ ns}\ c}{-c/(1+\kappa)} = 2.11 \mathrm{\ ns}$$ And for the medium in the reverse path we get $$\Delta T = \frac{1 \mathrm{\ ns}\ c}{ c/(1- \kappa)}+\frac{-1 \mathrm{\ ns}\ c}{-0.9 c/(1+0.9 \kappa)} = 2.11 \mathrm{\ ns}$$

So although each leg does depend on $\kappa$, the overall experiment does not. It turns out that there is no clever way around this. It is not possible for any experimental measurement to depend on $\kappa$ without assuming it first, typically through clock synchronization or some assumption of isotropy.

Explicitly, as was mentioned before, assuming that Snell's law is isotropic is not consistent with assuming that the one way speed of light is anisotropic. The correct formula for determining one way speeds is given above. It is not just the one way speed of light that is affected, as the math clearly shows.