Let us introduce the components of the -metric of the hypersurface with respect to the coordinates
From the definition of the shift vector , we can write
For the spacetime metric of , with respect to the coordinates we can also write
Keep in mind that . Thanks to , we obtain
Similarly, one can also obtain
Finally, it is easy to see that
for all .
By collecting the above calculation, we arrive at
Equivalently, we can write
The components of the inverse matrix can be calculated as given below
Using the Cramer rule for expressing the inverse of the matric , we have
where is the element of the cofactor matrix associated with . Clearly, where is the minor of the matrix . Hence . In other words, there holds
Thus, we have shown that
Since , our calculation above also shows that .
- 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