section_path: Modern research/Einstein field equations
section_name: Einstein field equations
paragraph_text: Einstein field equations
  
  The Einstein field equations are a system of 10 partial differential equations which describe how matter affects the curvature of spacetime. The system is often expressed in the form G μ ν + Λ g μ ν = κ T μ ν, {\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu },} where G is the Einstein tensor, g is the metric tensor, T is the stress–energy tensor, Λ is the cosmological constant, G {\displaystyle G} is the Newtonian constant of gravitation and c {\displaystyle c} is the speed of light. The constant κ = 8 π G c 4 {\displaystyle \kappa ={\frac {8\pi G}{c^{4}}}} is referred to as the Einstein gravitational constant.
  
  A major area of research is the discovery of exact solutions to the Einstein field equations. Solving these equations amounts to calculating a precise value for the metric tensor (which defines the curvature and geometry of spacetime) under certain physical conditions. There is no formal definition for what constitutes such solutions, but most scientists agree that they should be expressable using elementary functions or linear differential equations. Some of the most notable solutions of the equations include:
  
  Today, there remain many important situations in which the Einstein field equations have not been solved. Chief among these is the two-body problem, which concerns the geometry of spacetime around two mutually interacting massive objects, such as the Sun and the Earth, or the two stars in a binary star system. The situation gets even more complicated when considering the interactions of three or more massive bodies (the " n -body problem"), and some scientists suspect that the Einstein field equations will never be solved in this context. However, it is still possible to construct an approximate solution to the field equations in the n -body problem by using the technique of post-Newtonian expansion. In general, the extreme nonlinearity of the Einstein field equations makes it difficult to solve them in all but the most specific cases.