# Gravitational Waves¶

In the weak field regime of sourceless Einstein’s equation (\(T^{\mu\nu}=0\)), the equation for metric with perturbations is reduced to a wave equation,

where \(\bar h^{\alpha\beta}\) is the trace-reversed perturbation of the metric on top of Minkowski metric background, i.e.,

where \(h^{\alpha\beta} = g^{\alpha\beta} - \eta^{\alpha\beta}\) and \(h\) is the trace of metric perturbation \(h^{\alpha\beta}\).

Trace Reverse

The tensor \(\bar h^{\alpha\beta}\) is called trace reverse of \(h^{\alpha\beta}\) for its trace is \(-h\).

## Gauge¶

To solve the equation we introduce a solution of the form \(\hat h^{\alpha\beta} = A^{\alpha\beta}e^{i k_\mu x^\mu }\), which simiplifies the equation

To solve the amplitude \(A^{\alpha}\) we need constraints on it. We can derive that gravitational waves are always null, that is \(k^\mu k_\mu=0\).

Some of the conditions requires a gauge transformation. In any case, we have the second gauge condition as

which specifies that \(A_{\alpha\beta}\) is orthogonal to the vector we chose \(U^{\beta}\). A practical choice of \(U^\beta\) is a four velocity. This removes another **four degrees of freedom**. For illustration purpose, we choose \(U^{\beta} \to ( 1, 0, 0, 0 )\) since it’s a null vector. The degrees of freedom removed can be visualized as the first rwo and column.

The second one we can think of is a transverse condition,

which removes **another three degrees of freedom**. This specifies that the wave is transverse, i.e., \(A_{\alpha\beta}\) can not have elements that is in the direction of four wavevector. We specify a wavevector \(k^\beta \to (\omega, 0, 0, \omega )\), which leads to the removal of the remaining elements of the fourth row and column.

The matrix we have now becomes

The last gauge condition is traceless condition \(A^\alpha_\alpha = 0\) which also requires the gauge transformation. This condition fixes the phase relations between different spatial directions, that is \(A_{xx} = e^{i\pi} A_{yy} = - A_{yy}\). This conditions insists that the two directions of distance oscillations should be quadrupole-like, i.e., contracts in one direction (say x) while extend in the other direction (say y).

Slicing

The first two conditions are basically specifying slicings of spacetime.

Physical Significance of Transverse-traceless Gauge

Transverse-traceless gauge is the very gauge that determines a coordinate system that a test particle is stationary in terms of coordinates.

To show this we assume that we have a test particle being stationary initially, i.e., \(U^\alpha\vert_{\tau=0} \to (1,0,0,0)^{\mathrm T}\).

The particle should travel on geodesics,

which leads to

The four acceleration is 0 for the test particle. No motion would be detected within the coordinate system.

The same is true for a particle moving in \(z\) direction. However, the conclusion doesn’t hold for other motions. p