This entry is a continuation of the previous entry where we showed in detail the derivation of the Hamiltonian constrain in general relativity. Today, we derive the so-called momentum constraint equation.

First, let us recall the Codazzi equation which is given by the following identity

Sometimes, we call it the Codazzi–Mainardi equation or the Ricci identity, which expresses the curvature of the normal bundle in terms of the second fundamental form. Using this, we obtain

for any tangent vector . Since , for all , belongs to the tangent space, there hold

and thus we can write

Moreover, by the antisymmetric property of the Riemann curvature tensor, i.e., , we know that

which immediately implies

Hence, we get that

Therefore,

We now evaluate the right hand side. In order to achieve that goal, we make use of the formula involving the covariant derivative of the second fundamental form. Indeed, we first write

Using the Leibniz rule and thanks to , we can write

Thus, we get that

By summing, we obtain

For the second term, we do the same way as follows

Without loss of generality, we can assume that we are in normal coordinates at some point, say . This immediately implies that for all . Therefore,

Thus, by summing, we obtain

Thus, we have proved that

By using the Einstein equation and thanks to , we get that

Thus, we arrive at

known as the momentum constraint equation in general relativity.

Remark: Throughout the proof, we have used the fact that the vector belongs to the tangent space, thus giving us . For a proof of this fact, see this entry.

See also:

- Why do the Einsteins equations describe the propagation of wavelike phenomena?
- Derivation of the Einstein constraint equations: The Hamitonian constraint

Reference: Justin Corvino, *Introduction to General Relativity and the Einstein Constraint Equations*, Lecture notes.

## Leave a Reply