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| In [[general relativity]], '''Gauss–Bonnet gravity''', also referred to as '''Einstein–Gauss–Bonnet gravity''',<ref>{{Citation | last = Lovelock | first = David | title = The Einstein tensor and its generalizations | journal = J. Math. Phys. | volume = 12 | issue = 3 | pages = 498 | year = 1971 | url = http://link.aip.org/link/JMAPAQ/v12/i3/p498/s1}}</ref> is a modification of the [[Einstein–Hilbert action]] to include the [[generalized Gauss–Bonnet theorem|Gauss–Bonnet term]] (named after [[Carl Friedrich Gauss]] and [[Pierre Ossian Bonnet]]) <math>G= R^2 - 4R^{\mu\nu}R_{\mu\nu} + R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma} </math>
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| :<math>\int d^Dx \sqrt{-g}\, G</math>
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| 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 [[divergence theorem|surface term]]. This follows from the Gauss-Bonnet theorem on a 4D manifold
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| :<math>\frac{1}{8\pi^2}\int d^4x \sqrt{-g}\, G = \chi(M)</math>.
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| Despite being quadratic in the [[Riemann tensor]] (and [[Ricci tensor]]), terms containing more than 2 partial derivatives of the [[metric tensor|metric]] cancel out, making the [[Euler–Lagrange equations]] [[partial differential equation#Equations of second order|second order]] [[quasilinear]] [[partial differential equations]] in the metric. Consequently, there are no additional dynamical degrees of freedom, as in say [[f(R) gravity]].
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| More generally, we may consider
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| :<math>\int d^Dx \sqrt{-g}\, f\left( G \right)</math>
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| term for some function ''f''. Nonlinearities in ''f'' render this coupling nontrivial even in 3+1D. However, fourth order terms reappear with the nonlinearities.
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| == See also ==
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| * [[Einstein–Hilbert action]]
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| * [[f(R) gravity]]
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| * [[Lovelock gravity]]
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| == References ==
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| {{reflist}}
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| {{Theories of gravitation}}
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| {{DEFAULTSORT:Gauss-Bonnet gravity}}
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| [[Category:Theories of gravitation]] | |
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| {{relativity-stub}}
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