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Connecting 2-Forms, Conformal Transformations, Curvature Invariants and Topological Classes in Einstein Spacetimes

Jack C. M. Hughes, Fedor V. Kusmartsev

Abstract

The unique Nature of the Lorentz group in four dimensions is the root cause of the many remarkable properties of the Einstein spacetimes, in particular their operational structure on the 2-forms. We show how this operational structure can be used for two ends. First, it allows for a simple generalization of the Birkhoff theorem to Schwarzschild (A)de-Sitter spacetime. Second, it provides the means to construct an Abelian endomorphism group on the space of 2-forms. It is observed that taking the trace over this group element-wise induces a further Abelian group which may be identified with a tensor representation of conformal transformations, giving Einstein spacetimes access to their own conformal equivalence class. A further trace over the group yields the curvature invariants of the spacetime. The Kretschmann scalar becomes the topological Euler density, which may be linked in a simple way to the Hawking temperature of horizons.

Connecting 2-Forms, Conformal Transformations, Curvature Invariants and Topological Classes in Einstein Spacetimes

Abstract

The unique Nature of the Lorentz group in four dimensions is the root cause of the many remarkable properties of the Einstein spacetimes, in particular their operational structure on the 2-forms. We show how this operational structure can be used for two ends. First, it allows for a simple generalization of the Birkhoff theorem to Schwarzschild (A)de-Sitter spacetime. Second, it provides the means to construct an Abelian endomorphism group on the space of 2-forms. It is observed that taking the trace over this group element-wise induces a further Abelian group which may be identified with a tensor representation of conformal transformations, giving Einstein spacetimes access to their own conformal equivalence class. A further trace over the group yields the curvature invariants of the spacetime. The Kretschmann scalar becomes the topological Euler density, which may be linked in a simple way to the Hawking temperature of horizons.
Paper Structure (12 sections, 1 theorem, 58 equations)

This paper contains 12 sections, 1 theorem, 58 equations.

Table of Contents

  1. Introduction
  2. 2-Forms in Four Dimensions
  3. Birkhoff Theorem for Einstein spacetimes
  4. Constructing the Endomorphism Group on 2-forms
  5. Properties of G and Some Examples
  6. Conformal Transformations as Trace Over G
  7. The Kronecker-delta Property of the Trace
  8. Invariants as Second-Order Contractions over G
  9. Topological Properties of the Hawking Temperature
  10. Conclusions
  11. Open Questions
  12. Proof of Theorem 1

Key Result

Theorem 1

Any spherically symmetric four-dimensional Einstein spacetime is necessarily Schwarzschild-(A)de Sitter.

Theorems & Definitions (1)

  • Theorem 1