# How to measure the speed of light, and is it a constant?

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TL;DR Summary
Simple, natural and understandable imaginary experiments on measuring the speed of light and calculating the prediction of the theory of GR (Schwarzschild metric)
The question constantly arises how the speed of light is measured and what does it mean that the speed is constant, including at remote points for the observer, including at points beyond the local frame of reference, as you understand it in general relativity (GR).

First of all, it should be noted that the speed cannot be directly measured. It is always calculated! Actually, this is indicated by its dimension - length / time.
Changing the speed of light in Euclidean space

To begin with, we need to formulate ways to measure the speed of light at remote points in Euclidean space.
1. Transverse method

A reference sample is taken, of a given length, for example, a meter.
The center of the standard is placed along the geodesic from the observer to any point in space.
It is located across the geodesic. You can do this in advance, before moving along the geodesic.

It turns out that the picture is symmetrical relative to the observer.

Further:

Photons are emitted from one end simultaneously along geodesics to the observer (f2) and along the reference (f2).
When the photon f1 reaches the other end of the standard, a third photon (f3) is emitted from this 2nd end towards the observer along its geodesic.
The observer registers the difference in the time of photon reception (f2, f3) from different ends of the standard and calculates the speed of light from it: c = L / Δt, where L is the length of the standard.

Note 1: This method is applicable only for points with axial symmetry, for example, for points lying on a straight line connecting the center of the metric with the observer!
2. Longitudinal method

In this variant, the standard, after moving away from the observer, is located along the geodesic. Again, orientation can be set before moving along the geodesic.

The end closest to the observer emits photons, in opposite directions along the standard, to the observer (f2) and from the observer (f1).
When the photon f1 reaches the 2nd end, a photon (f3) is emitted from it towards the observer.
The observer registers the difference in the time of photon reception (f2, f3) and calculates the speed of light from it: c = 2L / Δt.

3. Direct method

In this method, not a standard is used, but a calculated three-dimensional distance between 2 points, based on:

Visible to a dedicated observer of the angular difference between them
The known removal of each point.
The distance is measured by the time the light signal travels back and forth to each point along geodesics from/to the observer.

It is obvious that the speed of light calculated in such ways in the remote region will coincide and be a constant for all regions, for Euclidean space.

Physically, such measurements can be made, technical capability, as they say, is not included in the task.

Note:
Methods 1 and 2 are actually "halves" of the Michelson–Morley experiment, but not for observing the effect of the dependence of the speed of light on the rotations of the reference frame and the speed of its movement,
and the dependence of the speed of light on the distance from the observer (parallel transfer) and (in general) the presence of a gravitational field / curvature of space. The latter is discussed below.
Change in the speed of light in curved space (GRT)

And now imagine that the space was not flat, but curved. Let's say it corresponds to the Schwarzschild metric. The coordinates of the point are set to 0 < r < ∞, 0 < θ < π, 0 < φ < 2π. Rg is the radius of the horizon.

What results will the observer (experimenter) get in each measurement method? Including for points of space near the event horizon?

In order to somehow narrow down and concretize the task, we will set the specific coordinates of the observer (R0. π/2, 0).

And the centers of the standards/observed segment lie on the lines (r1, π/2, 0) and (r2, π/2, π/2), where Rg < r1, r2 < R0.

That is, these are the points lying on the lines from the center of the metric to:

the observer
perpendicular to the connecting center of the metric and the observer.

The area of determination of r2 can be further limited from below, due to the fact that some of the points may not be "visible" to the observer (the effect of the Earth horizon) and taking into account Remark 1.
But it doesn't matter.

Also, it is probably worth adding the condition that Rg > L , where L, let me remind you, is the length of the standard / segment.

Physically, such measurements can be made, the technical possibility, as they say, is not included in the task, but in reality there is no such possibility yet.

BUT WHAT DO THE CALCULATIONS BASED ON THE SCHWARZSCHILD METRIC SAY?

Fundamentally there are two situations.
The speed of light calculated in this way:

Coincides with the speed of light as a constant.
Different from her.

In the second case, you can do not with an exact calculation, but with an approximation, showing that the answer will not be equal to the speed of light.

You can also limit yourself to separate "convenient" points for calculation from the domain of definition of r1 and r2, assuming that the effect monotonically depends on the magnitude of r1 and r2.

I hope that the problem statement is both physically (how hypothetical observations are carried out) and geometrically formulated correctly and fully and will not generate "tricky" questions like - how do you determine distances or in what frame of reference do you observe it.

Motore

There are a few facts here that cut through your rather long experimental descriptions.

First, the one way speed of light can take any value you like, as long as the speed of light in the reverse direction is defined so that the two way speed of light is ##c##. This is a choice that is free of physical consequences, so cannot be measured even in principle.

Second, the two way speed of light is a defined constant in the SI system. You can no longer measure it without some circularity. For example, you might ask how long it takes light to travel 1m - but 1m is defined to be the distance it takes light ##1/c## seconds to travel so the answer inevitably gives you the defined constant, limited only by your experimental precision.

Third, you cannot unambiguously measure speeds in curved spacetime except if you do so locally. So your procedures are either local processes monitored remotely (so the answer will be ##c##) or depend greatly on the assumptions you make (so the answer will depend on your assumptions).

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glappkaeft, Orodruin, cianfa72 and 6 others
Summary:: Simple, natural and understandable imaginary experiments on measuring the speed of light and calculating the prediction of the theory of GR (Schwarzschild metric)

A reference sample is taken, of a given length, for example, a meter.
It doesn’t really make sense to measure the speed of light in meters. If you use meters as your length standard then the speed of light is an exact defined constant, not subject to measurement.

Edit: I see @Ibix made this point as well as other more important points.

I wouldn’t worry too much about measuring the speed of anything in curved spacetime. What is more important is the invariant fact that at each event you can form a light cone of all of the lightlike tangent lines, and this light cone defines the causal structure of the spacetime

Ibix
First, the one way speed of light can take any value you like, as long as the speed of light in the reverse direction is defined so that the two way speed of light is ##c##. This is a choice that is free of physical consequences, so cannot be measured even in principle.
The problem is different.
As far as I understand, when creating a GR, it is assumed that with a constant.
If you solve an equation with parameter a, assuming that it is equal to c or, say, less than zero, but the solution turns out to be such that it is not equal to c or greater than zero, then such solutions are discarded.
It seems to me that this is very correct and understandable.
If it turns out that the original conditions of the constancy of the speed of light are violated somewhere, there is no need to hide for the fact that we cannot measure it in reality, such solutions in such areas or in general the whole theory should be discarded!

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Second, the two way speed of light is a defined constant in the SI system. You can no longer measure it without some circularity. For example, you might ask how long it takes light to travel 1m - but 1m is defined to be the distance it takes light ##1/c## seconds to travel so the answer inevitably gives you the defined constant, limited only by your experimental precision.
I am not against such a definition. only I would VERY much like the ebb in the metal in its reference frame to have a length not equal to how much light passes in t seconds and transferring the length to any available point in space, it remains that way both in the local system remote from the original, and when measuring this length remotely from the original reference frame.
Why it's important.
because we sit in our reference frame on Earth and calculate distances in distant galaxies based on a constant.
At the same time, if there are different gravitational fields and there is no constant REMOTELY, AND WE CAN'T KNOW ANYTHING AT ALL OUTSIDE OF OUR EARTH, the maximum of the solar system!

It doesn’t really make sense to measure the speed of light in meters. If you use meters as your length standard then the speed of light is an exact defined constant, not subject to measurement.

Edit: I see @Ibix made this point as well as other more important points.

I wouldn’t worry too much about measuring the speed of anything in curved spacetime. What is more important is the invariant fact that at each event you can form a light cone of all of the lightlike tangent lines, and this light cone defines the causal structure of the spacetime
I have already written about our study of other galaxies, and my own ...
maybe there is no dark matter, just the speed of light is different outside the solar system?))
If I understood your words correctly, such solutions appear in theory))
! How is a light cone used in real astronomical observations?

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maybe there is no dark matter, just the speed of light is different outside the solar system?
Do you have a peer reviewed reference that states this as a possibility? Personally, I cannot see how that could possibly account for the observations that support dark matter.

Also, as I said above it makes no sense to discuss changes to the speed of light in terms of SI units. So what you are actually interested in is variations in the fine structure constant. There is already good evidence supporting that the fine structure constant is indeed constant, and phrased in terms of the fine structure constant it seems even less likely that variations in it could account for dark matter observations.

! How is a light cone used in real astronomical observations?
The light cone is what connects every astronomical event to the corresponding astronomical observation.

As far as I understand, when creating a GR, it is assumed that with a constant.
No. GR can be done in units in which ##c## does not even appear at all in the equations.

The correct underlying assumption in GR is that spacetime is a geometric object. What you are thinking of as "the speed of light" is actually just part of spacetime being a geometric object. But that underlying assumption affects everything in GR, not just "the speed of light". You couldn't just re-do the GR equations with "the speed of light as a function".

(There are alternative theories of gravity, not GR, in which what you are calling "the speed of light" can depend on things other than the geometry of spacetime. None of those theories have stood up when tested against observations.)

we sit in our reference frame on Earth and calculate distances in distant galaxies based on a constant.
No, we don't. We calculate distances to distant galaxies using a combination of several things:

(1) Their apparent brightness, or more precisely the apparent brightness of particular regions or objects within them), as compared with some known absolute brightness for particular types of objects, such as Cepheid variable stars or Type II supernovas.

(2) Their apparent angular size, as compared with some known absolute size.

(3) Their redshift, using some predicted relationship between redshift and distance from a particular cosmological model.

Distance estimates for distant galaxies are typically made by combining two or all three of the above measurements, taking into account various sources of error. Note that "the speed of light" did not appear in any of the above. (To calculate the distance to something using the speed of light, you have to know the time the light took to travel; but we don't know that because we don't know when light from a distant galaxy that we are observing now was emitted.)

PeroK
No. GR can be done in units in which ##c## does not even appear at all in the equations.

No, we don't. We calculate distances to distant galaxies using a combination of several things:

(1) Their apparent brightness, or more precisely the apparent brightness of particular regions or objects within them), as compared with some known absolute brightness for particular types of objects, such as Cepheid variable stars or Type II supernovas.

(2) Their apparent angular size, as compared with some known absolute size.

(3) Their redshift, using some predicted relationship between redshift and distance from a particular cosmological model.
Everything is correct and what is the distance to some galaxy?...in light years))

Do you have a peer reviewed reference that states this as a possibility? Personally, I cannot see how that could possibly account for the observations that support dark matter.

Also, as I said above it makes no sense to discuss changes to the speed of light in terms of SI units. So what you are actually interested in is variations in the fine structure constant. There is already good evidence supporting that the fine structure constant is indeed constant, and phrased in terms of the fine structure constant it seems even less likely that variations in it could account for dark matter observations.

The light cone is what connects every astronomical event to the corresponding astronomical observation.
That is, if there are some solutions with superluminal speed, we should not dismiss them as wrong?))

The light cone is what connects every astronomical event to the corresponding astronomical observation.
Show me at least one light cone not in your head, but in nature)) ... in the sky))

Do you have a peer reviewed reference that states this as a possibility? Personally, I cannot see how that could possibly account for the observations that support dark matter.
Do you have a link confirming that this is not the case?
But in reality, what is REALLY being observed
1. linear velocity of stars

2. estimation of the total mass of stars

Everything else is SPECULATION

Thank you all, but I would like to receive in response not reasoning, but CALCULATION...I think I can understand him

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Do you have a link confirming that this is not the case?
Since you are engaging in personal speculation, this thread is closed. Please review the forum rules that you agreed to when you signed up for the site.

As far as my evidence for the point I was making, see here: https://arxiv.org/abs/2008.10619
Be aware that this is an active area of research and some other authors have published contradictory papers showing a small variation. As far as I know, nobody (including the authors that do find such variations) has proposed such variations as obviating the need for dark matter.

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