Einstein Field Equation

As per the theory of relativity, the Einstein field equations relates to the geometry of spacetime. It is up to the distribution of matter within it. The equation was given by Einstein in 1915. It was in the form of a tensor equation that related the local spacetime curvature with the local energy, momentum and stress within spacetime. Learn einstein field equation here.

Einstein field equation

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Introduction to Einstein Field Equation

The way electromagnetic fields are relating to the distribution of charges and currents through Maxwell’s equations, the Einstein field equation relates to the spacetime geometry to the distribution of mass-energy, momentum and stress. That’s the reason why they determine the metric tensor of spacetime for a given arrangement of stress–energy-momentum in spacetime.

The relationship between the metric tensor and the Einstein tensor allows the Einstein field equation to be written as a set of non-linear partial differential equations. The solutions of the Einstein field equation are the components of the metric tensor. Using The geodesic equation, inertial trajectories of particles and radiation in the resulting geometry are calculated.

Implying local energy-momentum conservation, the Einstein field equation reduces Newton’s law of gravitation in the limit of a weak gravitational field and velocities that are much less than the speed of light. The exact solutions for the Einstein field equation can be found under simplifying assumptions such as symmetry.

The mathematical form of the Einstein Field Equation

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

  • Here, \(G_{\mu v}\) is the Einstein tensor which is given as, \(G_{\mu \nu}=R_{\mu \nu}-\frac{1}{2} R g_{\mu \nu}\) where \(R_{\mu \nu}\) is the Ricci curvature tensor and R is the scalar curvature
  • \(g_{\mu v}\) is the metric tensor
  • \(\Lambda\) is a cosmological constant
  • G is Newton’s gravitational constant
  • c is the speed of light
  • \(T_{\mu v}\) is the stress-energy tensor

Einstein Field Equations Derivation

In the derivation of Einstein Field Equations, Einstein tries to explain that measure of curvature = source of gravity. The source of gravity is the stress-energy tensor and it is as follows:

\(T^{\alpha \beta}\) =

\(= \begin{bmatrix} rho & 0 & 0 & 0 \\  0 & P & 0 & 0\\  0 & 0 & P & 0 \\  0 & 0 & 0 & P \end{bmatrix}\)

\(= \begin{bmatrix} rho & 0 & 0 & 0 \\  0 & o & 0 & 0\\  0 & 0 & 0 & 0 \\  0 & 0 & 0 & 0 \end{bmatrix}\)

In the above matrix, P is tending to zero because, for Newton’s gravity, the mass density is the source of gravity.

Now, the equation of motion is

\(\frac{d u^{i}}{d \tau}+\Gamma_{v \alpha}^{i} u^{v} u^{\alpha}=0\)

\(\frac{d u^{i}}{d \tau}+\Gamma_{i}^{00}=0\)

\(\frac{d u^{i}}{d \tau}+\frac{1}{2} \frac{\partial g_{\infty}}{\partial x^{i}}=0\)

\(\frac{d u^{i}}{d \tau}+\frac{\partial \phi}{\partial x^{i}}=0\)

\(g_{00}=-(1+2 \phi)\)

We know that \(\nabla^{2} \phi=4 \pi G \rho\)


\(R^{\mu v}=-8 \pi G T^{\mu 0}\)

Where \(-8 \pi G T^{\mu 0}\) is constant.

FAQ on Einstein Field Equation

Question: What is Einstein Tensor?

Answer: Einstein tensor is a trace-reversed Ricci tensor. In Einstein Field Equation, it is in use for describing spacetime curvature in such a way that it is in alignment with the conservation of energy and momentum.  It is as follows:

G = R-1/2 gR

Here, R is the Ricci tensor, g is the metric tensor and R is the scalar curvature.

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