Why is Schwarzschild metric taught at all?

Question

It seems like the Schwarzschild metric only has historical relevance. Since other coordinates like the (ingoing) Eddington-Finkelstein coordinates are far superior.

It seems like the condition on the coordinates that all lightcones must point in the same coordinate axis leads to the problems of the metric at the horizon.

Allowing the lightcones to tilt in the coordinate system like in EF coordinates solves this in a realistic way and seems like the best coordinates for a black hole formed from a collapsed star.

Further the vierbein in EF coordinates can be put in a very simple form:

$$e^I_\mu(r,\theta,\phi,t) = \begin{bmatrix} 1-\frac{m}{r} & 0 & 0 & \frac{m}{r} \\ 0 & r & 0 & 0 \\ 0 & 0 & r \sin(\theta) & 0 \\ -\frac{m}{r} & 0 & 0 & 1 + \frac{m}{r} \end{bmatrix}$$

This makes me wonder why Schwarzschild coordinates are ever used? Is it just because they seem simpler as they have no diagonal entries? (Where the fact it has no diagonal entries leads to its problems).

Answer 1

Schwarzschild coordinates work

Most astrophysical calculations involve things that are outside the event horizon. In these cases Schwarzschild coordinates work just fine. Observed phenomena like gravitational lensing, orbital changes relative to Newtonian gravity, and gravitational redshift can all be understood using Schwarzschild coordinates.

Schwarzschild coordinates look like spherical coordinates

From a pedagogical standpoint its helpful that Schwarzschild coordinates look like spherical coordinates. Most students learning general relativity have seen Newtonian gravity and electromagnetism described in spherical coordinates. We can draw on this experience to reinforce new results by comparing them to results students are already familiar with. In particular Schwarzschild coordinates are great for exploring weak-field limits of GR.

Schwarzschild coordinates are not really spherical coordinates, so there is some danger of misinterpretations. I would argue the benefits outweighs the risk.

Schwarzschild coordinates describe what distant observers see

The time coordinate $t$ of Schwarzschild coordinates is defined as the proper time for an observer at $r\rightarrow\infty$. This makes Schwarzschild coordinates the natural coordinate system for us when we look at astrophysical events occurring near distant black holes.

Answer 2

Schwarzschild (Droste) coordinates have a number of advantages over other coordinates systems for the Schwarzschild metric such as Eddington-Finkelstein (EF) or Gullstand-Painlevé (GP) coordinates.

1. Symmetry

One is that in Schwarzschild coordinates the time translation and axial symmetries are explicitly manifest. In these coordinates the vector fields $\tfrac{\partial}{\partial t}$ and $\tfrac{\partial}{\partial \phi}$ are Killing vector fields. (This applies to EF and GP as well.)

In addition it has manifest time reversal symmetry $t \mapsto -t$. This sets Schwarzschild coordinates apart from EF and GP coordinates.

2. Simplicity

The Schwarzschild metric takes on its simplest form in Schwarzschild coordinates. It has the least number of non-zero components. This makes it particularly easy to work with. This also makes it a good set of coordinates to start with in a GR class.

This simplicity is very beneficial in proving things like the Birkhoff's theorem.

3. Sufficiency

Schwarschild coordinates smoothly cover the entire region outside the event horizon. In many (astro)physical scenarios this is the only region that we care about. These coordinates are therefore sufficient for many practical applications.

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Of course, for any calculation we should use coordinates suited to the purpose. For example, if we are interested in understanding something falling into a black hole, ingoing Eddington-Finkelstein coordinates are a good choice. If we are interested in the causal structure of the maximally extended solution, we typically want Kruskal coordinates, etc.