Gauss–Bonnet gravity

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In general relativity, Gauss–Bonnet gravity, also referred to as Einstein–Gauss–Bonnet gravity,^[1] is a modification of the Einstein–Hilbert action to include the Gauss–Bonnet term (named after Carl Friedrich Gauss and Pierre Ossian Bonnet) $G=R^{2}-4R^{{\mu \nu }}R_{{\mu \nu }}+R^{{\mu \nu \rho \sigma }}R_{{\mu \nu \rho \sigma }}$

$\int d^{D}x{\sqrt {-g}}\,G$

This term is only nontrivial in 4+1D or greater, and as such, only applies to extra dimensional models. In 3+1D and lower, it reduces to a topological surface term. This follows from the Gauss-Bonnet theorem on a 4D manifold

${\frac {1}{8\pi ^{2}}}\int d^{4}x{\sqrt {-g}}\,G=\chi (M)$ .

Despite being quadratic in the Riemann tensor (and Ricci tensor), terms containing more than 2 partial derivatives of the metric cancel out, making the Euler–Lagrange equations second order quasilinear partial differential equations in the metric. Consequently, there are no additional dynamical degrees of freedom, as in say f(R) gravity.

More generally, we may consider

$\int d^{D}x{\sqrt {-g}}\,f\left(G\right)$

term for some function f. Nonlinearities in f render this coupling nontrivial even in 3+1D. However, fourth order terms reappear with the nonlinearities.

References

↑ Lovelock, David (1971), "The Einstein tensor and its generalizations", J. Math. Phys. 12 (3): 498

Theories of gravitation

Standard

Newtonian gravity (NG)	Newton's law of universal gravitation History of gravitational theory

General relativity (GR)	Introduction History Mathematics Resources Tests Post-Newtonian formalism Linearized gravity ADM formalism

Alternatives to
general relativity

Paradigms	Classical theories of gravitation Quantum gravity Theory of everything

Early modifications	Einstein–Cartan Bimetric theories Gauge theory gravity Teleparallelism Composite gravity f(R) gravity Massive gravity Modified Newtonian dynamics (MOND) Nonsymmetric gravitation Scalar–tensor theories Brans–Dicke Scalar–tensor–vector Conformal gravity Scalar theories Nordström Whitehead Geometrodynamics Induced gravity Tensor–vector–scalar

Quantisation attempts	Euclidean quantum gravity Canonical Quantum Gravity Wheeler–DeWitt equation Loop quantum gravity Causal dynamical triangulation Causal sets DGP model

Unification attempts	Kaluza–Klein theory Dilaton Supergravity

Unification and quantisation attempts	Noncommutative geometry Self-creation cosmology Semiclassical gravity Superfluid vacuum theory Logarithmic BEC vacuum String theory M-theory F-theory Heterotic string Type I string theory Type 0 string theory Bosonic string theory Type II string theory Little string theory Twistor theory Twistor String theory

Generalisations/Extensions of GR	Liouville gravity Lovelock theory 2+1D topological gravity Gauss–Bonnet gravity Jackiw–Teitelboim gravity

Pre-Newtonian theories
and Toy models

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Gauss–Bonnet gravity

See also

References