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Birkhoff implies Quasi-topological

Pablo Bueno, Robie A. Hennigar, Ángel J. Murcia

TL;DR

The work unifies the landscape of quasi-topological gravities in $D\geq 5$ by showing all past constructions collapse into three inequivalent types: I, II, and III. It proves a rigorous nesting: type II QTGs are contained in type I, and type III QTGs are contained in type II modulo pure Weyl invariants, with type II theories preserving second-order dynamics on general SSS backgrounds and satisfying a Birkhoff theorem (up to a zero-measure exception). Consequently, theories obeying Birkhoff have static, Schwarzschild-like spherically symmetric solutions described by a single function $f(r)$ solving an algebraic mass equation, tying the global solution structure to a simple master equation. The results clarify when higher-curvature theories admit Birkhoff behavior, relate traced-field-order constraints to Weyl invariants, and suggest broad directions for extending QTGs to EM settings, generalized and non-polynomial Lagrangians, with potential implications for holography and beyond-GR phenomenology.

Abstract

Quasi-topological gravities (QTGs) are higher-curvature extensions of Einstein gravity in $D\geq 5$ spacetime dimensions. Throughout the years, different notions of QTGs constructed from analytic functions of polynomial curvature invariants have been introduced in the literature. In this paper, we show that all such definitions may be reduced to three distinct inequivalent notions: type I QTGs, for which the field equations evaluated on a single-function static and spherically symmetric ansatz are second order; type II QTGs, whose field equations on general static and spherically symmetric backgrounds are second order; and type III QTGs, for which the trace of the field equations on a general background is second order. We show that type II QTGs are a subset of type I QTGs and that type III QTGs are a subset of type II QTGs modulo pure Weyl invariants. Moreover, we prove that type II QTGs possess second-order equations on general spherical backgrounds. This allows us to prove that any theory satisfying a Birkhoff theorem is a type II QTG, and that the reverse implication also holds up to a zero-measure set of theories. For every theory satisfying Birkhoff's theorem, the most general spherically symmetric solution is a generalization of the Schwarzschild spacetime characterized by a single function which satisfies an algebraic equation.

Birkhoff implies Quasi-topological

TL;DR

The work unifies the landscape of quasi-topological gravities in by showing all past constructions collapse into three inequivalent types: I, II, and III. It proves a rigorous nesting: type II QTGs are contained in type I, and type III QTGs are contained in type II modulo pure Weyl invariants, with type II theories preserving second-order dynamics on general SSS backgrounds and satisfying a Birkhoff theorem (up to a zero-measure exception). Consequently, theories obeying Birkhoff have static, Schwarzschild-like spherically symmetric solutions described by a single function solving an algebraic mass equation, tying the global solution structure to a simple master equation. The results clarify when higher-curvature theories admit Birkhoff behavior, relate traced-field-order constraints to Weyl invariants, and suggest broad directions for extending QTGs to EM settings, generalized and non-polynomial Lagrangians, with potential implications for holography and beyond-GR phenomenology.

Abstract

Quasi-topological gravities (QTGs) are higher-curvature extensions of Einstein gravity in spacetime dimensions. Throughout the years, different notions of QTGs constructed from analytic functions of polynomial curvature invariants have been introduced in the literature. In this paper, we show that all such definitions may be reduced to three distinct inequivalent notions: type I QTGs, for which the field equations evaluated on a single-function static and spherically symmetric ansatz are second order; type II QTGs, whose field equations on general static and spherically symmetric backgrounds are second order; and type III QTGs, for which the trace of the field equations on a general background is second order. We show that type II QTGs are a subset of type I QTGs and that type III QTGs are a subset of type II QTGs modulo pure Weyl invariants. Moreover, we prove that type II QTGs possess second-order equations on general spherical backgrounds. This allows us to prove that any theory satisfying a Birkhoff theorem is a type II QTG, and that the reverse implication also holds up to a zero-measure set of theories. For every theory satisfying Birkhoff's theorem, the most general spherically symmetric solution is a generalization of the Schwarzschild spacetime characterized by a single function which satisfies an algebraic equation.

Paper Structure

This paper contains 14 sections, 17 theorems, 136 equations, 1 figure.

Table of Contents

  1. Introduction and summary of results
  2. The various notions of QTG
  3. Main results
  4. Theories with $2^{\rm nd}$-order equations on single-function SSS backgrounds
  5. Theories with $2^{\rm nd}$-order equations on general SSS backgrounds
  6. Theories with $2^{\rm nd}$-order equations on general SS backgrounds
  7. Birkhoff implies QTG (of type II)
  8. QTGs and theories with $2^{\rm nd}$-order traced field equations
  9. Future directions
  10. Alternate notions of Birkhoff theorem.
  11. Theories with non-minimally coupled matter.
  12. Generalized quasi-topological theories.
  13. Extension to non-polynomial Lagrangians.
  14. An alternative characterization for type II QTGs

Key Result

Theorem 1

Notions def:1 and def:2 are equivalent.

Figures (1)

  • Figure 1: Schematic representation of the different types of QTGs as classified in this paper. The broadest class, type I QTGs (interior of largest black rectangle), corresponds to theories for which the equations of motion evaluated on the SSS single-function ansatz are second order. It contains the set of type II QTGs (interior of second largest black rectangle), which are theories for which the equations of motion evaluated on a general SSS ansatz are second order. The type II set turns out to be almost identical to the set of theories which satisfy a Birkhoff theorem (region shaded in blue). The difference between both is a zero-measure set of type II QTGs which do not satisfy a Birkhoff theorem (represented by a horizontal thin rectangle bounded by black and blue lines). The set of type III QTGs (interior of the purple rectangle) involves theories with second-order traced equations. Observe that it is not completely contained within the type II or type I classes, as the addition of pure Weyl invariants, which possess second-order traced field equations, may take a theory out of the type II and type I sets. The type III and type II sets also contain Lovelock theories, for which the equations of motion are second order in general backgrounds.

Theorems & Definitions (37)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • ...and 27 more