We can now obtain the Einstein’s
Field Equation. The non-zero curvature of the curvilinear system in general is
given by the Riemann curvature tensor. The source of the curvature is given by
the energy-momentum tensor. But one, can immediately notice that we can equate
these two tensor object followed by some proportionality constant. One
difficulty we come across is the mismatch in the rank of the tensor. To be
precise, the Riemann Tensor is a $4^{th}$ rank tensor whereas the
energy-momentum tensor is the $2^{nd}$ rank tensor. Thus we cannot obtain the
correct equation from the Riemann curvature tensor itself. However, what we can
do and have earlier shown that the reduced form of the Riemann Tensor takes the
form of the $2^{nd}$ rank tensor (Ricci Tensor). Now we can guess the equation
to be the form as follows,

This is actually the wrong equation.
Einstein himself first used this equation but soon found that this equation
violated the very fundamental law (conservation law).

For instance if we take a covariant
derivative of the equation $(1)$ then it follows that,

But in this equation the LHS is not
equal to zero, while the conservation law of energy-momentum says that this has
to be zero. Thus equation $(1)$ is not the correct equation.

We fix this by a suitable candidate
which satisfies the conservation law, the one which does is the (Bianchi
Identities) which is as follows,

We can now write the correct
equation, which is;

Or, in more compact form,

These are then the Einstein’s Field
Equation. Here, $G_{\mu \nu }$ is called the Einstein’s curvature Tensor.

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