Beta Phase: Square45 is currently in beta testing. Expect some features or content to be incomplete or missing.
45

Einstein Field Equations

Field: General Relativity

A set of ten equations in General Relativity that describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy.

Sequence of Expressions

Definition

Metric Tensor

Define the metric tensor gμνg_{\mu\nu} as the fundamental object determining the spacetime interval ds2ds^2 via the quadratic form:\nds2=gμνdxμdxνds^2 = g_{\mu\nu} dx^{\mu} dx^{\nu} \nIn a local coordinate system, gμνg_{\mu\nu} defines the inner product on the tangent space, allowing the calculation of distances and time intervals.
The stress-energy tensor TμνT_{\mu\nu} is a rank-2 symmetric tensor defined by the variation of the Lagrangian density L\mathcal{L} with respect to the metric gμνg_{\mu\nu}, representing the density and flux of energy and momentum:\nTμν=2−gδLδgμνT_{\mu\nu} = \frac{2}{\sqrt{-g}} \frac{\delta \mathcal{L}}{\delta g^{\mu\nu}} \nIt satisfies the conservation law:\nabla^{\mu} T_{\mu\nu} = 0$.
Gμν=8πGc4TμνG_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} \nWhere Gμν=Rμν−12RgμνG_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} is the Einstein tensor, derived from the Ricci curvature tensor RμνR_{\mu\nu} and the scalar curvature R=gμνRμνR = g^{\mu\nu} R_{\mu\nu}, and TμνT_{\mu\nu} is the stress-energy tensor describing the source of gravity.
The path xμ(λ)x^\mu(\lambda) of a particle moving freely in curved spacetime (a geodesic) is parameterized by λ\lambda and satisfies the equation:\nd2xμdλ2+Γνρμdxνdλdxρdλ=0\frac{d^2x^{\mu}}{d\lambda^2} + \Gamma^{\mu}_{\nu\rho} \frac{dx^{\nu}}{d\lambda} \frac{dx^{\rho}}{d\lambda} = 0 \nWhere Γνρμ\Gamma^{\mu}_{\nu\rho} are the Christoffel symbols of the second kind, defined by the metric tensor gμνg_{\mu\nu} as:\Γνρμ=12gμσ(∂νgσρ+∂ρgσν−∂σgνρ)\Gamma^{\mu}_{\nu\rho} = \frac{1}{2} g^{\mu\sigma} (\partial_{\nu} g_{\sigma\rho} + \partial_{\rho} g_{\sigma\nu} - \partial_{\sigma} g_{\nu\rho})\n
Let gμνg_{\mu\nu} be the metric tensor defining the spacetime manifold (M,g)(M, g). The Riemann curvature tensor RρσμνR^{\rho}{}_{\sigma\mu\nu} is defined by the commutator of covariant derivatives acting on a vector field VV: \n\nRρσμνVσ=[∇μ,∇ν]Vρ−RρσμνVσR^{\rho}{}_{\sigma\mu\nu} V^{\sigma} = \left[ \nabla_{\mu}, \nabla_{\nu} \right] V^{\rho} - R^{\rho}{}_{\sigma\mu\nu} V^{\sigma} \n\nAlternatively, using the Christoffel symbols Γμρ\Gamma^{\rho}_{\mu} derived from gμνg_{\mu\nu}, the components are given by:\n\nRρσμν=∂μΓνρσ−∂νΓμρσ+ΓνλσΓμρ−ΓμλσΓνρR^{\rho}{}_{\sigma\mu\nu} = \partial_{\mu}\Gamma^{\rho}_{\nu}\sigma - \partial_{\nu}\Gamma^{\rho}_{\mu}\sigma + \Gamma^{\lambda}_{\nu}\sigma\Gamma^{\rho}_{\mu} - \Gamma^{\lambda}_{\mu}\sigma\Gamma^{\rho}_{\nu}
Define the Ricci tensor RμνR_{\mu\nu} as the contraction of the Riemann tensor: \n\nRμν=RλμλνR_{\mu\nu} = R^{\lambda}{}_{\mu\lambda\nu} \n\nLet RR be the Ricci scalar, defined as the trace of the Ricci tensor: \n\nR=gμνRμνR = g^{\mu\nu} R_{\mu\nu} \n\nThe Einstein tensor GμνG_{\mu\nu} is then defined by the combination:\n\nGμν=Rμν−12gμνRG_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R
The Schwarzschild metric gμνg_{\mu\nu} describes the spacetime around a non-rotating, spherically symmetric mass MM. In standard coordinates (t,r,θ,ϕ)(t, r, \theta, \phi), the line element ds2ds^2 is given by:\n\nds2=−(1−2GMc2r)c2dt2+(1−2GMc2r)−1dr2+r2(dθ2+sin⁡2θdϕ2)ds^2 = -\left(1 - \frac{2GM}{c^2 r}\right) c^2 dt^2 + \left(1 - \frac{2GM}{c^2 r}\right)^{-1} dr^2 + r^2 (d\theta^2 + \sin^2\theta d\phi^2) \n\nHere, GG is the gravitational constant, cc is the speed of light, and rr is the radial coordinate.
The Principle of Equivalence asserts that in any sufficiently small, freely falling (local inertial) frame, the gravitational field effects vanish, allowing the spacetime to be approximated by flat Minkowski spacetime (ημν\eta_{\mu\nu}). Mathematically, this implies the existence of a coordinate transformation x′μ=x′μ(x)x'^\mu = x'^\mu(x) such that the connection coefficients vanish locally:\Γνρ′μ=0\Gamma'^{\mu}_{\nu\rho} = 0 \nThis condition ensures that the local physics is indistinguishable from that in the absence of gravity.
Let gμνg_{\mu\nu} be the metric tensor defining the geometry of a four-dimensional spacetime manifold (M,g)(\mathcal{M}, g). The curvature of this spacetime is quantified by the Riemann curvature tensor RμνρσR_{\mu\nu\rho\sigma}. The Ricci tensor, RμνR_{\mu\nu}, is obtained by contracting the Riemann tensor: Rμν=RμρνρR_{\mu\nu} = R^{\rho}_{\mu\rho\nu}. The scalar curvature RR is the trace of the Ricci tensor: R=gμνRμνR = g^{\mu\nu} R_{\mu\nu}. The Einstein tensor, GμνG_{\mu\nu}, which represents the geometric side of the field equations, is defined as: Gμν=Rμν−12gμνRG_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R \n\nThis geometric curvature is directly related to the distribution of mass and energy, represented by the Stress-Energy-Momentum tensor TμνT_{\mu\nu}. The Einstein Field Equations (EFE) establish this fundamental relationship:\n\nGμν=8πGc4TμνG_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} \n\nwhere GG is Newton's gravitational constant and cc is the speed of light. The equation dictates that the curvature of spacetime (GμνG_{\mu\nu}) is proportional to the density and flux of energy and momentum (TμνT_{\mu\nu}).
Let TμνT_{\mu\nu} be the stress-energy tensor describing the distribution of energy and momentum. The conservation of energy and momentum in curved spacetime is expressed by the vanishing of the covariant divergence of TμνT_{\mu\nu}:\n\n∇μTμν=0\nabla_{\mu} T^{\mu\nu} = 0 \n\nWhere ∇μ\nabla_{\mu} is the covariant derivative associated with the metric gμνg_{\mu\nu}, ensuring that the energy-momentum flux is locally conserved.