14. Field Equation

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,                                                                

                                                                                                           $(1)$

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.