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To remove the singularity at the horizon we move from Schwarzchild to Eddington Finkelstein coordinate system. Our ingoing null geodesics then become straight lines. Then we move to Kruskal solution where both the ingoing and outgoing null geodesics are straight lines. Is there a problem in E-F coordinate system? Because while doing a transformation we define advanced and retarded time parameter which explains our spacetime geometry very well. Why do we move to Kruskal coordinates?

With respect to which observer are Kruskal coordinates defined? Is it the observer radially infalling into the black hole or is it an observer who is at a far distance from the black hole?

Given a metric described in a particular coordinate system, with respect to which observer are the coordinates defined?

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The Schwarzschild metric is a solution of Einstein's equations, given by $$ ds^2 = -f(r)dt^2+ \frac{dr^2}{f(r)} + r^2d\Omega^2 $$ where $f(r) = (1 -\frac{2M}{r}) $. Here M is a parameter, which could be the mass of a star or a black hole. The above metric describes the empty region outside a spherical source and is not applicable in a region occupied by a source (like a star or a black hole).

If the radius of the source is smaller than $2M$ then we have a problem. This metric should be valid near the region $r = 2M$ but we hit a singularity as $f(2M) = 0$.

Thankfully, this singularity is a coordinate singularity only, in the sense that it arises due to our bad choice of coordinates. (Unlike the singularity at $r = 0$). So, to remove this fictitious singularity we go to the Eddington Finkelstein coordinates that you mentioned.

There is no particular problem with these coordinates, and they serve the purpose they were invented for. But these coordinates are incomplete in a certain sense and can be extended to the Kruskal–Szekeres coordinates. This process is called an extension of the metric, and in this case the new metric is a maximal extension, i.e. one cannot further extend this metric.

This is very similar to the case of extending the Rinder coordinates to the usual inertial Minkowski coordinates. The rindler coordinates describe an accelerating observer, and the metric has a fictitious singularity. When we transform from Rindler to inertial coordinates, a key observation to make is that the spacetime has 'doubled up'. The Rindler coordinates covered only the right wedge and the future wedge of the Minkowski spacetime. Thus the Rindler coordinates are incomplete to describe the complete structure of the flat spacetime, and one has to extend the coordinates.

This explains your question of why we use the Kruskal coordinates: they are the maximal extension of the Schwarzschild coordinates.

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