Today, we study a very interesting property of stress energy tensor of a scalar field. Recall from this topic that Einstein tensor is divergence free, that is

.

We still know from the Einstein equation that

where the stress energy tensor. As a consequence, is also divergence free, that is

where is covariant derivative. We all know from the topic where we define Einstein tensor by contracting the Bianchi identity.

We know consider a special case of the stress energy tensor, that is, stress energy tensor of a scalar field with potential , a function of . In general relativity, in a local frame is given as follows

.

We firstly need to raise indexes . Obviously, one gets

Thus

.

Now we compute . Indeed, we have

.

A simple calculation with the fact that shows that

.

Thus

.

This is why we always assume that the field is supposed to satisfy some semilinear wave equation.

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