Einstein Field Equation

The Einstein Field Equation (EFE) are also known as Einstein’s equation. There are the set of ten equations extracted from Albert Einstein’s General Theory of Relativity. The EFE describes basic interaction of gravitation. The equations were first published in 1915 by Albert Einstein as a tensor equation.

Einstein Field Equation

\(G_{\mu \upsilon } + g_{\mu \upsilon }\Lambda = \frac{8 \pi G}{c^{4}}T_{\mu \upsilon }\)


\(G_{\mu \upsilon } = R_{\mu \upsilon } – \frac{1}{2}Rg_{\mu \upsilon }\) = Einstein Tensor

\(R_{\mu \upsilon }\) = Ricci Curvature Tensor

\(R \) = Scalar Curvature

\(g_{\mu \upsilon }\) = Metric Tensor

\(\Lambda\) = Cosmological Constant

\(G\) = Newton’s Gravitational Constant

\(c\) = Speed of Light

\(T_{\mu \upsilon }\) =  Stress-Energy Tensor

The Einstein field equation are determined by the spacetime geometry with the presence of energy-mass and linear momentum. This is similar to the Maxwell’s equation where, the electromagnetic fields are identified by currents and charges.

The Einstein field equation was first formulated in a four-dimensional theory. Further, some theorists explored it in the n dimensions. However, the equation apart from general relativity are also referred as Einstein field equation.

If the equations are fully written out, they are a system of 10 nonlinear, coupled, hyperbolic-elliptic differential equations. The equation defining the Einstein tensor.

\(G_{\mu \upsilon} = R_{\mu \upsilon} – \frac{1}{2}Rg_{\mu \upsilon}\)

\(G_{\mu \upsilon} + \Lambda g_{\mu \upsilon} = \frac{8 \pi G}{c^{4}}T_{\mu \upsilon}\)

\(G_{\mu \upsilon} + \Lambda g_{\mu \upsilon} = 8 \pi T_{\mu \upsilon}\)

These equations with geodesic equation shows how a matter moves through space-time.

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Practise This Question

Three particles A, B and C are situated at the vertices of an equilateral triangle ABC of side 'd' at t = 0. Each of the particles moves with constant speed v. A always has its velocity along AB, B along BC and C along CA. At what time will the particles meet each other?