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
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.
- 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.