general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...

, Gauss–Bonnet gravity, also referred to as Einstein–Gauss–Bonnet gravity, is a modification of the

Einstein–Hilbert action The Einstein–Hilbert action (also referred to as Hilbert action) in general relativity is the action that yields the Einstein field equations through the stationary-action principle. With the metric signature, the gravitational part of the ac ...

to include the Gauss–Bonnet term (named after

Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...

and

Pierre Ossian Bonnet Pierre Ossian Bonnet (; 22 December 1819, Montpellier – 22 June 1892, Paris) was a French mathematician. He made some important contributions to the differential geometry of surfaces, including the Gauss–Bonnet theorem. Biography Early ye ...

) :

\int d^Dx \sqrt\, G

, where :

G= R^2 - 4R^R_ + R^R_

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

generalized Gauss–Bonnet theorem A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characte ...

on a 4D manifold :

\frac\int d^4x \sqrt\, G = \chi(M)

. In lower dimensions, it identically vanishes. Despite being quadratic in the

Riemann tensor In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. ...

(and

Ricci tensor In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...

), terms containing more than 2 partial derivatives of the

metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem ...

cancel out, making the Euler–Lagrange equations second order

quasilinear Quasilinear may refer to: * Quasilinear function, a function that is both quasiconvex and quasiconcave * Quasilinear utility, an economic utility function linear in one argument * In complexity theory and mathematics, O(''n'' log ''n'') or some ...

partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...

in the metric. Consequently, there are no additional dynamical degrees of freedom, as in say

f(R) gravity () is a type of modified gravity theory which generalizes Einstein's general relativity. () gravity is actually a family of theories, each one defined by a different function, , of the Ricci scalar, . The simplest case is just the function bei ...

. Gauss–Bonnet gravity has also been shown to be connected to

classical electrodynamics Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model; It is, therefore, a classical fi ...

by means of complete gauge invariance with respect to

Noether's theorem Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...

. More generally, we may consider a :

\int d^Dx \sqrt\, f\left( G \right)

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

References

Theories of gravity {{relativity-stub

See also

References