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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.

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.