Physical meanings of universal coordinates in schwarzschild metric

• dd331
In summary, the conversation discusses the schwarzschild metric and its application in describing the gravitational field of a spherical mass. It also delves into the physical meanings of the coordinates t and r, with r being the radial distance and t being the proper time of a stationary observer at infinity. The conversation also explores the effects of distance and time measurements for a stationary observer at an arbitrary distance r0, as well as the different methods of measuring distance in the context of general relativity.
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?

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?

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dd331 said:
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.
dd331 said:
... 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.

1. What are universal coordinates in the Schwarzschild metric?

Universal coordinates in the Schwarzschild metric refer to a coordinate system used to describe the spacetime around a non-rotating spherically symmetric mass, such as a black hole. These coordinates are useful for studying the properties of spacetime and the behavior of particles near the mass.

2. What is the physical meaning of the universal time coordinate in the Schwarzschild metric?

The universal time coordinate, also known as the Schwarzschild time, represents the time as measured by an observer at infinity. It is important because it is a measure of proper time, which is the time experienced by an object or observer moving along a specific path in spacetime.

3. How are the spatial coordinates defined in the Schwarzschild metric?

In the Schwarzschild metric, the spatial coordinates are defined using angles and distances from the center of the mass. The angle coordinates are known as the polar coordinates, and the distance coordinate is known as the radial coordinate. Together, these coordinates describe the position of an object in spacetime.

4. What does the universal time coordinate approach as the distance from the mass increases?

As the distance from the mass increases, the universal time coordinate approaches the time coordinate of a flat spacetime. This means that at large distances, the effects of the gravitational field become negligible, and time behaves similarly to how it would in a non-gravitational environment.

5. How do the universal coordinates relate to other coordinate systems in the Schwarzschild metric?

The universal coordinates are related to other coordinate systems, such as the Schwarzschild coordinates and the isotropic coordinates, through mathematical transformations. These transformations allow for the conversion between different coordinate systems and are essential for understanding the physical properties of the spacetime around a mass.

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