What is it about?

In this paper we reconsider the Schwarzschild black hole solution of General Relativity in the Palatini formulation, where the connection is a priori seen as independent of the metric. A direct result of this independence is that the theory turns out to be invariant under the so-called projective transformations, which only modify the connection, leaving the metric unchanged. Though these transformations do not change the Einstein tensor, they do have an impact on the Riemann curvature tensor, which picks up a new term under projective shifts of the connection.

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Why is it important?

It turns out that the so-called Kretschmann scalar is not invariant under projective transformations. This scalar is a quadratic object constructed with the Riemann tensor and used in many standard textbooks to illustrate that the Schwarzschild solution has an intrinsic curvature singularity at the center of the black hole. However, the lack of projective invariance can be used to gauge away this divergence. In fact, we find a projective transformation that makes the Kretschmann vanish everywhere, leaving the metric unaffected. Thus, if the metric singularity at the horizon can be removed by a change of coordinates, the singularity in the Kretschmann can be removed by a projective transformation.


This paper illustrates that the traditional interpretation of gravitation as an effect of curvature may be misleading in some cases. This should not be surprising given that alternative formulations of general relativity based on torsion, for instance, are possible. Though there are other quadratic scalars which are still divergent in the Schwarzschild case, it is important to realize that some geometric infinities can be eliminated by mere gauge transformations. The study of alternative formulations of GR with new variables and symmetries could help progress further in this direction.

Gonzalo J Olmo
Universitat de Valencia

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This page is a summary of: Geometric inequivalence of metric and Palatini formulations of General Relativity, Physics Letters B, March 2020, Elsevier, DOI: 10.1016/j.physletb.2020.135275.
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