Physical meanings of universal coordinates in schwarzschild metric

[dd331]

The metric due to the gravitational field of a spherical mass is described by the schwarzschild metric

ds² = c² (1 - R/r) dt² - (1 - R/r)^-1 dr² - r² d$$\Omega$$ ²

Where $$\Omega$$ is the solid angle, and R is the schwarzschild radius.

What are the physical meanings of the coordinates t and r? My understanding is that r is simply the radial distance and t is the proper time of a STATIONARY observer at infinity.

If two events are separated by dt, dr, d$$\Omega$$, how would they seem to a stationary observer at some arbitrary distance r₀? What is dr' and dt' (measured in the stationary frame of the observer) in terms of dr and dt? How about for a non-stationary observer?

 

[atyy]

Coordinates are "meaningless" in GR. Experimental results are coordinate independent quantites, so one has to ask a question about the result of a specific experiment, and calculate the result which should be a coordinate independent quantity such as the proper time of a particular worldline.

So for example, you could ask if a particular worldline A emits 2 light pulses separated by A's proper time t, and if the two light pulses intersect another wordline B, how much proper time will elapse for B between its reception of the 2 light pulses?

 

[yuiop]

The metric due to the gravitational field of a spherical mass is described by the schwarzschild metric

ds² = c² (1 - R/r) dt² - (1 - R/r)^-1 dr² - r² d$$\Omega$$ ²

Where $$\Omega$$ is the solid angle, and R is the schwarzschild radius.
...
If two events are separated by dt, dr, d$$\Omega$$, how would they seem to a stationary observer at some arbitrary distance r₀? What is dr' and dt' (measured in the stationary frame of the observer) in terms of dr and dt?

For a stationary observer located at r₀ you can use this modified metric:

$$ds^2 = c^2\frac{(1-R/r)}{(1-R/r_o)} dt '^2 - \frac{(1-R/r_o)}{(1-R/r)}dr '^2 -r^2 d\Omega ^2$$

where dt' and dr' are now the time and distance intervals measured by the observer at $r_o$ rather than as measured by the observer at infinity in the classic metric.

(Not totally sure whether the omega factor has to be adjusted, but offhand I think it is self compensating.)

Note that if you set r₀ to infinity, the above equation reduces to the regular Schwarzschild metric.

Since $ds^2$ for the event is invariant for any observer we can say:

$$c^2(1-R/r) dt^2 - (1-R/r)^{-1}dr^2 -r^2 d\Omega ^2 = c^2\frac{(1-R/r)}{(1-R/r_o)} dt '^2 - \frac{(1-R/r_o)}{(1-R/r)}dr '^2 -r^2 d\Omega ^2$$

and so if you know the measurements dt and dr according to a stationary observer at infinity, you can work out the measurements dt' and dr' according to a stationary observer located at r₀.

... What are the physical meanings of the coordinates t and r? My understanding is that r is simply the radial distance and t is the proper time of a STATIONARY observer at infinity.

Basically correct, but some clarification is needed on the physical meaning of r as there are different ways of measuring distance that give different results. If a physical ring was centred on the gravitational mass, then r is the distance computed by measuring the circumference of the ring using a ruler and dividing by 2pi. A stationary observer on the ring could also multiply the velocity of an object following the path of the ring by the time it takes the object to complete a full orbit and then divide by 2pi to get r. All these measurements are made using the clocks and rulers of the observer local to the ring. These two methods would be consistent. If he measured the radial distance between two concentric rings with a ruler, he would get a different result and if he measured the distance between two concentric rings by timing reflected light signals (radar), he would obtain yet another result.

Hope that helps.