Curvature invariants of static spherically symmetric geometries

TL;DR

This paper systematically constructs and analyzes scalar curvature invariants specific to static spherically symmetric geometries, revealing a significant reduction in complexity compared to general spaces, with implications for gravitational actions.

Contribution

It provides a complete classification of local scalar curvature invariants for static spherically symmetric spacetimes, highlighting their simplified structure and relation to Lovelock invariants.

Findings

Number of invariants is greatly reduced in static spherically symmetric geometries.

All invariants at each order collapse to a single Lovelock invariant.

Implications for gravitational actions and solution spaces are discussed.

Abstract

We construct all independent local scalar monomials in the Riemann tensor at arbitrary dimension, for the special regime of static, spherically symmetric geometries. Compared to general spaces, their number is significantly reduced: the extreme example is the collapse of all invariants ~ Weyl^k, to a single term at each k. The latter is equivalent to the Lovelock invariant L_k. Depopulation is less extreme for invariants involving rising numbers of Ricci tensors, and also depends on the dimension. The corresponding local gravitational actions and their solution spaces are discussed.