In this post, we will derive the Einstein’s field equation from the least action principle. Let’s just construct an action with the Ricci scalar, Lagrangian and an invariant space-time volume element( ; g- metric) : All these quantities have the same value in all frames of reference.

Let’s call and also subsitute in the above equation.

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To simplify further we employ Jacobi’s formula ** and also note that the boundary terms that arise out of vanish according to Stoke’s theorem.

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This tells us that under a variation of the metric if we would like a stationary action then it has to be true that :

where – Riemann Tensor, – Ricci scalar, – Energy-momentum Tensor and – Metric Tensor

This is the Einstein’s field equation. Inserting the constants at the right places yields the more familiar form:

#### What would happen ?

As a fun little exercise you can start off with the invariant volume given by the inverse metric instead. i.e and you will obtain the same equation. BUT, you should be careful when employing the Jacobi’s formula else you might ending up with a extra minus sign with the Ricci scalar term.

**Some more fun questions to ponder: **

- Why is called the invariant space-time volume element ?
- Why is the action defined as ? Why not any other way?
- How do you prove the Jacobi’s formula ?