# Schwarzschild geometry, what is physical meaning of coordinates?

A past exam has a question:

For the Schwarzschild metric external to a non-spinning spherical mass, what is the physical significance to the coordinates $t,r,\theta,\phi$?

Not sure how to answer this question, I am thinking there is some obvious canonical answer, but it feels very non-specific.

Is the answer something like, an observer at $r\to \infty$ has that $t$ and $\tau$ are the same, and $r,\theta,\phi$ are all to some extent arbitary?

• If you like this question you may also enjoy reading this Phys.SE post. – Qmechanic Mar 14 '18 at 11:16

Well, since the metric is assymptotically flat, in $\infty$, $t$ and $\tau$ do indeed coincide, so you can view the time coordinate $t$ as the time measured by an inertial observer at infinity.

The radial coordinate $r$ is actually more of an "areal" coordinate. Consider 2-surfaces of constanat $t=T_0$ and $r=R_0$. Then the induced metric on the 2-surfaces are just the spherical metrics $$ds^2=R_0^2(d\vartheta^2+\sin^2\vartheta d\varphi^2),$$ which implies the area of the 2-surfaces are $$\text{Area}(t=T_0,r=R_0)=\iint R_0^2\sin\vartheta\ d\vartheta d\varphi=2\pi R_0^2\cdot[-\cos\vartheta]_{0}^{\pi}=4\pi R_0^2,$$ which is of course, the surface area of 2-spheres in euclidean space.

Therefore, we can say that the $r$ coordinate denotes the points occupied by origin-centric spheres whose surface areas are $4\pi r$.

Because the part of the metric that contains the angular coordinates $\vartheta$ and $\varphi$ is the same as the spherical metric in euclidean space, the angular coordinates have the usual meaning.

$t$ and $\phi$ are adapted coordinates to the generators of unidimensional translation symmetry in time and rotation symmetry in $\phi$. Noether's theorem, in this particular setting the Killing equation, dictates that conserved currents are related to those symmetries. The related charges are the Komar mass and Komar angular momentum. The first one is the canonical mass of the Schwarzschild metric and the latter is zero. So one "physical" significance of $t$ and $\phi$ are those two conserved quantities.

The typical coordinate choice of $r$ and $\theta$ has the consequence that $g_{\phi\phi}=\sin^2(\theta)g_{\theta\theta}$ and $g_{\theta\theta}=r^2$ which makes the angular part of the metric the canonical metric of a sphere and $r$ is the areal radius. This has been pointed out by @Uldreth.

So in a way the typical coordinate choice is neatly tied to the spacetime symmetries and the metric is relatively simple. That being said depending of the application other coordinate choices (isotropic, Gullstrand–Painlevé coordinates) are better suited.

A decisive answer is difficult because I personaly find that question rather vague because "physical significance" is in my opinion a rather board term and in the end all physics has to independent of coordinate choice.

The physical distance between the two radial coordinates $r_1$ and $r_2$ is not $r_2-r_1$, but

$$\Delta R=\int_{\text{r1}}^{\text{r2}} \frac{{\rm d}r}{\sqrt{1-\frac{r_s}{r}}}$$

(whis is larger than $r_2-r_1$). Nevertheless, the physical circumference of a shell around the black hole is still $2 \pi r$, so the angles still span $2\pi$ radians for a whole circle in a sperically symmetric situation, so

1) The radial distance $r$ is lenght contracted,

2) The angles $\phi$ and $\theta$ have the same meaning as in standard spherical coordinates,

3) The time coordinate $t$ is that of a stationary observer far away from the black hole whose clocks run fastest.

## protected by Qmechanic♦Jun 9 '17 at 11:03

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