Very little that I know about general relativity is that there are solutions of its equations with singularities, and these are interpreted as black holes.

Mathematically, the most widespread kind of singularities are poles - e. g., for a single complex variable, if $z$ is a coordinate in the vicinity of the singularity, something like $1/z^n$ for $n=1,2,3,...$

Are the singularities of general relativity like that? There are singular points of different kind like $\sqrt z$ which make branching inevitable. Physically this would mean that orbiting around such point you could get to different sheets of the spacetime. Or, if it is $\log z$ or something even more nasty, the branching might be infinite, and interconnections between the sheets might be quite complicated.

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Does such kind of spacetime branching occur? What is the physical meaning of its presence or absence?


2 Answers 2


As far as I understand, such branching can only occur for functions of complex arguments. However, the metric tensor is a function of four real coordinates so no such branching will occur even if a root or log appeared in one of the metric tensor's components.

It is still interesting to think about whether "nasty" functions like a logarithm can appear in a metric tensor. The thing is, however, that it is easy to introduce any kind of functions by a change of coordinates. But why making life more complicated than it already is?

Here is a list of examples for GR metrics: https://en.wikipedia.org/wiki/Metric_tensor_(general_relativity)#Examples As you can see, if there are poles, they are of the form $\frac{1}{r^n}$ (like in Schwarzschild metric) or, in worst case, $\frac{1}{\sqrt{1-r^2}}$ (in FLRW metric) where $r>0$.

  • $\begingroup$ Well, FLRW looks like it should have branching, no? $\endgroup$ May 20, 2021 at 7:12
  • $\begingroup$ If you promote r to a complex variable, it will. Maybe my differentiation between real and complex variables is an even more naive answer to your naive question. :) $\endgroup$
    – Photon
    May 20, 2021 at 7:16

The simplest general relativity solution (other than the trivial Minkowski metric which is the solution when there is no gravity present) is the Schwarzschild metric. It is the metric when there is an isolated mass point $M$ present at the origin. It is given by

${\displaystyle ds^2=-c^{2}\,{d\tau }^{2}=-\left(1-{\frac {r_{\mathrm {s} }}{r}}\right)c^{2}\,dt^{2}+\left(1-{\frac {r_{\mathrm {s} }}{r}}\right)^{-1}\,dr^{2}+r^{2}{\displaystyle \left(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}\right)}}$

Where $r_s=\frac{2GM}{c^2}$.

Here you can see that at $r=r_s$ this metric is not defined. But this is just due to the coordinates we chose. It is called a coordinate singularity and we can remove this by using different coordinates. An example for this is polar coordinates. Here the origin is a singularity but we can remove that by using Cartesian coordinates. Eddington–Finkelstein coordinates is an example for the coordinate system in which $r=r_s$ is not a singularity.

But the origin is different, it is a singularity in all possible coordinate systems. One simple way to see this is by calculating the Kretschmann scalar. For Schwarzschild metric the Kretschmann scalar after calculation will give $\frac{48G^2M^2}{r^6}$, which is not finite at the origin. Unlike the metric $g_{\mu\nu}$, scalars are not dependent on coordinate system. Infinite scalars are not physically possible.

Also the branching in complex analysis is due to multivalued nature of some of the complex functions. In General relativity we only work with real values. So such thing wont occur. Space-time singularities are similar to having poles in real analysis.

  • $\begingroup$ Well, you can view complex variables as pairs of reals satisfying Cauchy-Riemann, or equivalently just one real function of two variables satisfying Laplace equation. So, solutions of Einstein equations are quite like complex functions in this respect, and thus must have their own classifiable roster of singularities $\endgroup$ May 20, 2021 at 7:43
  • 2
    $\begingroup$ @მამუკაჯიბლაძე If we assume that there are logarithmic like physical functions of r and if we assume r can take complex values this sheets of space-times will arise. But as far as I know they do not have any physical application. $\endgroup$ May 20, 2021 at 7:54

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