Let us first recall the evolution equation of in this note

which can be rewritten as the following

for all since in our setting for all . The quantities and come from the Codazzi and Ricci equations. To be precise, we have, by the Codazzi equation,

and, by the Ricci equation,

It is clear that the infinitesimal variation of the Ricci curvature corresponding to an infinitesimal variation of the space metric is

where the notation is nothing but . This formula can be applied to and to get

Thanks to the evolution equation for the metric , i.e. , we get

We now exchange the order of covariant derivatives of the first term on the right hand side of the preceding equation as follows

This can be proved as follows: First we have

we obtain

Therefore,

Consequently,

Exchanging and and adding together we get the estimate.

This helps us to write

We shall eliminate at the same time the following two terms and . Indeed, since

and

we find that

Therefore, the third order terms in and the second order terms in in the equation for can be written in terms of

Consequently, the condition should hold which yields the following condition

Following the previous note, we can easily check that the lapse function has to verify the condition

with satisfying

Once we can remove all high order derivatives, we see that the Einstein equations imply the following wave equation

where we set

and

and

In the final step, we show that any solution of the above wave equation verifies the Einstein equation. Indeed, we first write the stress-energy tensor as the following

Then we transform the Einstein equation into the following from with

Clearly, solving the equation is equivalent to solving

where is linear and homogeneous in . By using the Bianchi identities, we finally obtain a quasi third order system for with principal operator the hyperbolic operator . The vanishing for of results from the constraint equations. Moreover, one can see that all derivatives of order of also vanish. It follows from the uniqueness theorem for hyperbolic systems that all the time.

See also:

- Why do the Einsteins equations describe the propagation of wavelike phenomena?
- Derivation of the Einstein constraint equations: The Hamitonian constraint
- Derivation of the Einstein constraint equations: The momentum constraint
- Construction of spacetimes via solutions of the vacuum Einstein constraint equations and the propagation of the gauge condition
- The evolution equations for the constraint equations
- decomposition of the spacetime metric
- The harmonic slicing
- The stress-energy tensor and Einstein constraint equations in the presence of scalar fields

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