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# Einstein Field Equation

## What is Einstein Field Equation?

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

Following is the Einstein Field Equation:

 $$\begin{array}{l}G_{\mu \upsilon } + g_{\mu \upsilon }\Lambda = \frac{8 \pi G}{c^{4}}T_{\mu \upsilon }\end{array}$$

Where,

• Gμ𝜐 is the Einstein tensor which is given as Rμ𝜐-½ Rgμ𝜐
• Rμ𝜐 is the Ricci curvature tensor
• R is the scalar curvature
• gμ𝜐 is the metric tensor
• 𝚲 is a cosmological constant
• G is Newton’s gravitational constant
• c is the speed of light
• Tμ𝜐 is the stress-energy tensor

## Einstein Field Equations Derivation

Following is the derivation of Einstein Field Equations. Einstein wanted to explain that measure of curvature = source of gravity.

The source of gravity is the stress-energy tensor. The stress-energy tensor is given as:

$$\begin{array}{l}T^{\alpha \beta }=\begin{bmatrix} \rho & 0 &0 & 0\\ 0&P &0 &0 \\ 0 &0 &P &0 \\ 0&0 &0 &P \end{bmatrix}\rightarrow \begin{bmatrix} \rho &0 &0 &0 \\ 0& 0 &0 &0 \\ 0 & 0& 0& 0\\ 0& 0 &0 &0 \end{bmatrix}\end{array}$$

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

The equation of motion is given as:

$$\begin{array}{l}\frac{du^{i}}{d\tau }+\Gamma _{v\alpha }^{i}u^{v}u^{\alpha }=0\end{array}$$

$$\begin{array}{l}\frac{du^{i}}{d\tau }+\Gamma _{i}^{00}=0\end{array}$$

$$\begin{array}{l}\frac{du^{i}}{d\tau }+\frac{1}{2}\frac{\partial g_{00}}{\partial x^{i}}=0\end{array}$$

$$\begin{array}{l}\frac{du^{i}}{d\tau }+\frac{\partial \phi }{\partial x^{i}}=0\end{array}$$

$$\begin{array}{l}g_{00}=-(1+2\phi )\end{array}$$

But we know that

$$\begin{array}{l}\bigtriangledown ^{2}\phi =4\pi G\rho\end{array}$$

Therefore,

$$\begin{array}{l}R^{\mu v}=-8\pi GT^{\mu v}\end{array}$$

Where

$$\begin{array}{l}-8\pi GT^{\mu v}\end{array}$$
is the constant.

### What is Einstein Tensor?

Einstein tensor is also known as trace-reversed Ricci tensor. In Einstein Field Equation, it is used for describing spacetime curvature such that it is in alignment with the conservation of energy and momentum.  It is defined as:

 G = R-½ gR

Where,

• R is the Ricci tensor
• g is the metric tensor
• R is the scalar curvature

### What is stress-energy tensor?

Stress-energy tensor is defined as the tensor Tαβ is a symmetrical tensor which is used for describing the energy and momentum density of the gravitational field. It is given as:

 Tαβ = Tβα

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#### 1 Comment

1. Aanshumaan Shrijai

This was very helpful, thank you so much.