General relativity
${\displaystyle G_{\mu \nu }+\mathrm{\Lambda }{g}_{\mu \nu }=\kappa T_{\mu \nu }}$
Introduction History Timeline Tests Mathematical formulation
Fundamental concepts Equivalence principle Special relativity World line Pseudo-Riemannian manifold
Phenomena Kepler problem Gravitational lensing Gravitational redshift Gravitational time dilation 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 Raychaudhuri Formalisms ADM NP BSSN Post-Newtonian Advanced theory Kaluza–Klein theory Quantum gravity
Solutions Schwarzschild (interior) Reissner–Nordström Einstein–Rosen waves Wormhole Gödel Kerr Kerr–Newman Kerr–Newman–de Sitter Kasner Kantowski-Sachs Lemaître–Tolman Wahlquist Taub–NUT Milne Robertson–Walker Oppenheimer–Snyder pp-wave van Stockum dust Hartle–Thorne Vaidya Peres De Sitter-Schwarzschild McVittie Weyl Kerr-Newman-de-Sitter
Scientists Einstein Lorentz Hilbert Poincaré Schwarzschild de Sitter Reissner Nordström Weyl Eddington Friedmann Milne Zwicky Lemaître Oppenheimer 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 e

Ingeneral relativity, amanifestly covariant equation is one in which all expressions aretensors. The operations of addition,tensor multiplication,tensor contraction,raising and lowering indices, andcovariant differentiation may appear in the equation. Forbidden terms include but are not restricted topartial derivatives.Tensor densities, especially integrands and variables of integration, may be allowed in manifestly covariant equations if they are clearly weighted by the appropriate power of thedeterminant of the metric.

Writing an equation in manifestly covariant form is useful because it guaranteesgeneral covariance upon quick inspection. If an equation is manifestly covariant, and if it reduces to a correct, corresponding equation inspecial relativity when evaluated instantaneously in alocal inertial frame, then it is usually the correct generalization of the special relativistic equation in general relativity.

An equation may beLorentz covariant even if it is not manifestly covariant. Consider theelectromagnetic field tensor

${\displaystyle F_{ab}={\mathrm{\partial }}_a{A}_b-{\mathrm{\partial }}_b{A}_a}$

where${\displaystyle A_a}$ is theelectromagnetic four-potential in theLorenz gauge. The equation above contains partial derivatives and is therefore not manifestly covariant. Note that the partial derivatives may be written in terms of covariant derivatives andChristoffel symbols as

${\displaystyle {\mathrm{\partial }}_a{A}_b={\mathrm{\nabla }}_a{A}_b+{\mathrm{\Gamma }}_{ab}^c{A}_c}$

${\displaystyle {\mathrm{\partial }}_b{A}_a={\mathrm{\nabla }}_b{A}_a+{\mathrm{\Gamma }}_{ba}^c{A}_c}$

For atorsion-free metric assumed in general relativity, we may appeal to the symmetry of the Christoffel symbols

${\displaystyle {\mathrm{\Gamma }}_{ab}^c-{\mathrm{\Gamma }}_{ba}^c=0,}$

which allows the field tensor to be written in manifestly covariant form

${\displaystyle F_{ab}={\mathrm{\nabla }}_a{A}_b-{\mathrm{\nabla }}_b{A}_a.}$

• Lorentz covariance
• Introduction to the mathematics of general relativity

• C. B. Parker (1994).McGraw Hill Encyclopaedia of Physics (2nd ed.). McGraw Hill.ISBN 0-07-051400-3.
• John Archibald Wheeler;C. Misner;K. S. Thorne (1973).Gravitation. W.H. Freeman & Co.ISBN 0-7167-0344-0.

• General relativity
• Tensors

• Articles with short description
• Short description matches Wikidata
• Pages using sidebar with the child parameter