Coordinate systems in General Relativity

Question

In General Relativity spacetime is a four dimensional Lorentzian manifold $M$ with metric tensor $g$. Being a smooth manifold, we can use coordinate systems on spacetime.

And in reality, most traditional GR exposures rely heavily on coordinate systems, so that everything is written in a specific coordinate system $(x^0,x^1,x^2,x^3)$ always. This is so common that one usually referes to the metric tensor just as $g_{\mu\nu}$, that is, by its components in a particular chart.

The trouble is that we don't know what $M$ is! I mean, we don't know at first what manifold spacetime is. This brings a problem: how does one define coordinate systems in the first place if one doesn't know spacetime fully?

A coordinate system is a chart $(U,x)$ on the open set $U\subset M$, being $x : U\to \mathbb{R}^4$ a homeomorphism. How can one build such a coordinate system if one doesn't know what actually is $M$?

I'll give just a simple example. The Schwarzschild metric is usually written as:

$$ds^2=-\left(1-\dfrac{2GM}{r}\right)dt^2+\left(1-\dfrac{2GM}{r}\right)^{-1}dr^2+r^2d\theta^2+r^2\sin^2\theta d\phi^2.$$

Since we don't know what manifold is spacetime, how can we make sense of the coordinate functions $t,r,\theta,\phi : M\to \mathbb{R}$?

This resembles spherical coordinates, but this isn't $\mathbb{R}^3\times \mathbb{R}$ after all!

So on the one hand it seems hard to define coordinate systems because we don't know what manifold is spacetime.

On the other hand, given one observer $\gamma : I\subset \mathbb{R}\to M$, we always have one coordinate system adapted to $\gamma$. In the same way, given a reference frame $e_\mu : M\to TM$ we can also construct a coordinate system adapted to it if I'm not wrong.

So since we don't know what manifold spacetime is, all the relevant coordinate systems for Physics in GR always comes from observers and reference frames?

If not, how to make sense of coordinate systems in GR, considering all these points?

Answer 1

Given a $n+1$ dimensional manifold $M$ one by definition has charts or coordinates that are homeomorphic to $\mathbb{R}^{n+1}$. This is independent of any Lorentzian or Riemaniann metric on $\cal{M}$.

Now if the manifold $M$ admits a Lorentzian metric $g$ then the coordinates use to define the manifold also define the components of the metric.

A common way to define a spacetime even if it is not know the whole manifold $M$ is to work locally. Even if the topology of $M$ is not $\mathbb{R}^{4}$, locally it is. Then using the local (flat) coordinates one define locally a metric and imposing some symmetries and physical conditions one can arrive to relevant metrics such as the Schwarzschild metric.

However, another way to find solutions to Einstein's equations is to see them as an initial value problem. That is given on a hypersurface $\Sigma$ the first and second fundamental form one can determine a Lorentzian manifold $M$ using Einstein's equations.

Regarding your comment of how to make the sense of coordinates. In this case we have chosen an initial n-dimensional $\Sigma$ where we know the coordinates by definition. Then, also one define a lapse function $N$ and shift vector $\beta$ on $\Sigma$. These choices define locally a chart with topology $M={\mathbb{R}} \times {\Sigma}$. The choice of the lapse and shift sometimes are related with physical observers, but others are used for mathematical convenience such as the harmonic slicing. See section 9 This define the $n+1$- dimensional coordinates on $M$. In fact this is the topology of all globally hyperbolic spacetimes.

However, this is only a local characterization. If one wants the total manifold then one is interested in maximal extensions. For example, the chart you used has a well know coordinate singularity at $r=2M$ and therefore one needs to change the chart and coordinates to cover the full spacetime. Notice that the existence of the singularity is responsible for the change in the global topology.

The concept of maximal extension is related to geodesic completeness or the well-posedness of Einstein equations (no curvature singularities). If this two criteria are equivalent is part of the Strong Cosmic Censorship. The global case which correspond to the question of the maximal extension of the space-time as as a well-posed problem of Einstein's equations is an active area of research. The Strong Cosmic Censorship deals with the uniqueness of global solutions.