Lovelock's theorem

General relativity
${\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}$
Introduction History Mathematical formulation Tests
Fundamental concepts Equivalence principle Special relativity World line Pseudo-Riemannian manifold
Phenomena Kepler problem Gravitational lensing Gravitational waves Frame-dragging Geodetic effect Event horizon Singularity Black hole Spacetime Spacetime diagrams Minkowski spacetime Einstein–Rosen bridge
Equations Formalisms Equations Linearized gravity Einstein field equations Friedmann Geodesics Mathisson–Papapetrou–Dixon Hamilton–Jacobi–Einstein Formalisms ADM BSSN Post-Newtonian Advanced theory Kaluza–Klein theory Quantum gravity
Solutions Schwarzschild (interior) Reissner–Nordström Gödel Kerr Kerr–Newman Kasner Lemaître–Tolman Taub-NUT Milne Robertson–Walker pp-wave van Stockum dust Weyl−Lewis−Papapetrou
Scientists Einstein Lorentz Hilbert Poincaré Schwarzschild de Sitter Reissner Nordström Weyl Eddington Friedman Milne Zwicky Lemaître Gödel Wheeler Robertson Bardeen Walker Kerr Chandrasekhar Ehlers Penrose Hawking Raychaudhuri Taylor Hulse van Stockum Taub Newman Yau Thorne others
Physics portal  Category
v t

Lovelock's theorem of general relativity says that from a local gravitational action which contains only second derivatives of the four-dimensional spacetime metric, then the only possible equations of motion are the Einstein field equations.^[1][2][3] The theorem was described by British physicist David Lovelock in 1971.

Statement[]

In four dimensional space, any tensor ${\displaystyle A^{\mu \nu }}$ whose components are function of metric tensor ${\displaystyle g^{\mu \nu }}$ and its first and second derivatives (but linear in the second derivatives of ${\displaystyle g^{\mu \nu }}$), and also symmetric and divergenceless, then field equation in vacuum ${\displaystyle A^{\mu \nu }=0}$, then only possible form of ${\displaystyle A^{\mu \nu }}$ is

${\displaystyle A^{\mu \nu }=aG^{\mu \nu }+bg^{\mu \nu }}$

where ${\displaystyle a}$ and ${\displaystyle b}$ are just simple constant numbers and ${\displaystyle G^{\mu \nu }}$ is the Einstein tensor.^[3]

The only possible second-order Euler–Lagrange expression obtainable in a four-dimensional space from a scalar density of the form ${\displaystyle {\mathcal {L}}={\mathcal {L}}(g_{\mu \nu })}$ is^[1] ${\displaystyle E^{\mu \nu }=\alpha {\sqrt {-g}}\left[R^{\mu \nu }-{\frac {1}{2}}g^{\mu \nu }R\right]+\lambda {\sqrt {-g}}g^{\mu \nu }}$

Consequences[]

Lovelock's theorem means that if we want to modify the Einstein field equations, then we have five options.^[1]

• Add other fields rather than the metric tensor;
• Use more or fewer than four spacetime dimensions;
• Add more than second order derivatives of the metric;
• Non-locality, e.g. for example the inverse d'Alembertian;
• Emergence – the idea that the field equations don't come from the action.

See also[]

• Lovelock theory of gravity
• Vermeil's theorem

References[]

This relativity-related article is a stub. You can help Wikipedia by .

• v
• t