# Gravitational waves: Computing full metric

Write down the full metric (flat metric plus the perturbation) which represents a sinusoidal wave of frequency 200Hz traveling in the z direction with the cross polarization. The maximum strain of 10^-21 occurs when t=0.

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#### Solution Preview

The full metric in the most generic case of gravitational perturbation is:

(see attached file)

Where the traceless tensor (see attached file) is known as the strain tensor. (see attached file) is the lat background metric and (see attached file) is the perturbation. In the transverse gauge:

(see attached file)

and in weak ﬁeid approximation where we keep time derivatives in Einstein's equations but

put (See attached file) and working in transverse traceless gauge in which

(see attached file)

Thus the full metric (1) reduces to:

(see attached file)

We can write the strain tensor as:

(see attached ...

#### Solution Summary

A computation of the full metric and perturbation, resulting in a specific type of gravitational wave.