Schwarzschild Light Anisotropy: Explained

[smoothoperator]

I was just reading an article on wikipedia about the Schwarzschild metric and I've red that these coordinates imply that the coordinate speed of light is anisotropical in some way. Can somebody explain to me when does this occur and how does it exactly occur, in which directions etc.?

Thank you.

 

[Nugatory]

The speed of light is always locally $c$; plunk down an observer anywhere on spacetime and have him run an experiment that measures the speed of light anywhere near enough to him and he'll get $c$. This is true for all metrics always.

However, in any curved spacetime, there is no way of consistently defining the speed of anything that is not sufficiently near the observer.

The classic example from the schwarzschild metric: position observers at various distances from the center. (For them to be hovering at a fixed r, they must be blasting downwards with powerful rocket engines or equivalent). Each one of them will report the speed of a passing flash of light to be c. However, if you try to calculate the speed of light using good old "speed equals distance traveled divided by time to travel" $v=\Delta{r}/\Delta{t}$ between any two of these observers... You are measuring the coordinate speed of light, and it won't come out to be exactly c.

 

[smoothoperator]

The speed of light is always locally $c$; plunk down an observer anywhere on spacetime and have him run an experiment that measures the speed of light anywhere near enough to him and he'll get $c$. This is true for all metrics always.

However, in any curved spacetime, there is no way of consistently defining the speed of anything that is not sufficiently near the observer.

The classic example from the schwarzschild metric: position observers at various distances from the center. (For them to be hovering at a fixed r, they must be blasting downwards with powerful rocket engines or equivalent). Each one of them will report the speed of a passing flash of light to be c. However, if you try to calculate the speed of light using good old "speed equals distance traveled divided by time to travel" $v=\Delta{r}/\Delta{t}$ between any two of these observers... You are measuring the coordinate speed of light, and it won't come out to be exactly c.

I understand this part, but anisotropy means different speeds in different directions. I know that the coordinate speed may be lower than c closer to gravitational field because of gravitational time dilation, but how does this imply any kind of anisotropy?

 

[DrGreg]

I was just reading an article on wikipedia about the Schwarzschild metric and I've red that these coordinates imply that the coordinate speed of light is anisotropical in some way.

I guess you are referring to Deriving the Schwarzschild solution § Alternative form in isotropic coordinates?

Strictly speaking, it's meaningless to talk about isotropy in Schwarzschild coordinates, because 2 of the coordinates are angles, not distances, and so provide the angular velocity of light (e.g. in radians per second), which can't be compared with linear radial velocity. But if you treat the 3 spacelike Schwarzschild coordinates as if they were Euclidean spherical polar coordinates, to calculate linear velocity, (i.e. components $dr / dt$, $r \, d\theta / dt$ and $r \sin \theta \, d\phi / dt$) you will find the radial coordinate velocity of light isn't the same as the the tangential coordinate velocity of light. The Wikipedia article gives a change of coordinates to make the coordinate velocity of light isotropic in this sense.

 

[smoothoperator]

I guess you are referring to Deriving the Schwarzschild solution § Alternative form in isotropic coordinates?

Strictly speaking, it's meaningless to talk about isotropy in Schwarzschild coordinates, because 2 of the coordinates are angles, not distances, and so provide the angular velocity of light (e.g. in radians per second), which can't be compared with linear radial velocity. But if you treat the 3 spacelike Schwarzschild coordinates as if they were Euclidean spherical polar coordinates, to calculate linear velocity, (i.e. components $dr / dt$, $r \, d\theta / dt$ and $r \sin \theta \, d\phi / dt$) you will find the radial coordinate velocity of light isn't the same as the the tangential coordinate velocity of light. The Wikipedia article gives a change of coordinates to make the coordinate velocity of light isotropic in this sense.

But the total velocity product of these velocities (the radial and the tangential velocity) isn't anisotropic? So if I say the coordinate velocity of light is, for instance, c/2 at some point, that product is istropical. I mean the situation is a bit different than choosing different parameters in defining simultaneity relative to an inertial frame where the speed of light can be found to be anysotropical under other conventions than Einstein Synchronisation.

 

[Nugatory]

I understand this part, but anisotropy means different speeds in different directions. I know that the coordinate speed may be lower than c closer to gravitational field because of gravitational time dilation, but how does this imply any kind of anisotropy?

Try it in the outwards direction and the inwards direction.

 

[smoothoperator]

Try it in the outwards direction and the inwards direction.

I don't get it. Could you please explain it to me? Are you referring to an average speed of light when leaving and entering the gravitational field over some finite distance or at some spacetime point that is affected by gravitational time dilation?

My line of reasoning always went like this. If a distant clock is slowed down by gravitational time dilation by half relative to our local, proper time, than the coordinate speed of light is also slowed down by a factor of 2, independently of direction. Now please correct where I'm wrong in this reasoning and how exactly light behaves anisotropically in this regard.

 

[Ben Niehoff]

The "coordinate speed of light" is a meaningless concept that, as you see here, serves only to confuse matters.

To measure the speed of something, you need a standard set of measuring rods and clocks: i.e., an orthonormal frame. The speed of light is always c.

Physical quantities have nothing to do with coordinates.