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Properties of general stationary axisymmetric spacetimes: circularity and beyond

Eugeny Babichev, Jacopo Mazza

TL;DR

This work addresses how to describe general stationary, axisymmetric spacetimes without assuming circularity, a feature often present in Kerr but not guaranteed in modified gravity. It establishes a local Kerr-like (and orthogonal) gauge, proves local existence, and derives two algebraic circularity conditions among inverse-metric components, enabling analytic non-circular deformations of Kerr. By constructing two explicit Kerr-like non-circular examples, it shows how horizons, rotosurfaces, and surface gravity can differ from Kerr while preserving tractable analytic control. The results provide a framework for systematic exploration of non-circular geometries and their phenomenology, with potential implications for horizon thermodynamics and astrophysical signatures in beyond-GR theories.

Abstract

We analyse properties of general stationary and axisymmetric spacetimes, with a particular focus on circularity -- an accidental symmetry enjoyed by the Kerr metric, and therefore widely assumed when searching for rotating black hole solutions in alternative theories of gravity as well as when constructing models of Kerr mimickers. Within a gauge specified by seven (or six) free functions, the local existence of which we prove, we solve the differential circularity conditions and translate them into algebraic relations among the metric components. This result opens the way to investigating the consequences of circularity breaking in a controlled manner. In particular, we construct two simple analytical examples of non-circular deformations of the Kerr spacetime. The first one is "minimal", since the horizon and the ergosphere are identical to their Kerr counterparts, except for the fact that the horizon is not Killing and its surface gravity is therefore not constant. The second is "not so minimal", as the horizon's profile can be chosen arbitrarily and the difference between the horizon and the so-called rotosurface can be appreciated. Our findings thus pave the way for further research into the phenomenology of non-circular stationary and axisymmetric spacetimes.

Properties of general stationary axisymmetric spacetimes: circularity and beyond

TL;DR

This work addresses how to describe general stationary, axisymmetric spacetimes without assuming circularity, a feature often present in Kerr but not guaranteed in modified gravity. It establishes a local Kerr-like (and orthogonal) gauge, proves local existence, and derives two algebraic circularity conditions among inverse-metric components, enabling analytic non-circular deformations of Kerr. By constructing two explicit Kerr-like non-circular examples, it shows how horizons, rotosurfaces, and surface gravity can differ from Kerr while preserving tractable analytic control. The results provide a framework for systematic exploration of non-circular geometries and their phenomenology, with potential implications for horizon thermodynamics and astrophysical signatures in beyond-GR theories.

Abstract

We analyse properties of general stationary and axisymmetric spacetimes, with a particular focus on circularity -- an accidental symmetry enjoyed by the Kerr metric, and therefore widely assumed when searching for rotating black hole solutions in alternative theories of gravity as well as when constructing models of Kerr mimickers. Within a gauge specified by seven (or six) free functions, the local existence of which we prove, we solve the differential circularity conditions and translate them into algebraic relations among the metric components. This result opens the way to investigating the consequences of circularity breaking in a controlled manner. In particular, we construct two simple analytical examples of non-circular deformations of the Kerr spacetime. The first one is "minimal", since the horizon and the ergosphere are identical to their Kerr counterparts, except for the fact that the horizon is not Killing and its surface gravity is therefore not constant. The second is "not so minimal", as the horizon's profile can be chosen arbitrarily and the difference between the horizon and the so-called rotosurface can be appreciated. Our findings thus pave the way for further research into the phenomenology of non-circular stationary and axisymmetric spacetimes.
Paper Structure (17 sections, 22 equations, 1 figure)

This paper contains 17 sections, 22 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Circularity vs. non-circularity
  3. Why circularity?
  4. Why not circularity?
  5. General axisymmetric stationary metric and gauge choice
  6. Gauge choices for the metric
  7. Proof of local existence of the gauge choice
  8. Residual gauge freedom
  9. Properties of the metric in Kerr-like coordinates
  10. Notable surfaces: static limit, rotosurface, event horizon
  11. Null tetrad
  12. Geodesics
  13. Solving the circularity conditions
  14. Examples
  15. "Minimal" breaking of circularity
  16. ...and 2 more sections

Figures (1)

  • Figure 1: Horizon, rotosurface, and static limit for the specific "non-so-minimal" deformation determined by \ref{['eq:HEg', 'eq:muChoice']}. Here, $a=0.9M$ and $\epsilon = 0.2$; for comparison, we show the equivalent profiles for a Kerr spacetime with the same spin.