Gravitational Wave - Mathematics

Mathematics

Einstein's equations form the fundamental law of general relativity. The curvature of spacetime can be expressed mathematically using the metric tensor — denoted . The metric holds information regarding how distances are measured in the space under consideration. Because the propagation of gravitational waves through space and time change distances, we will need to use this to find the solution to the wave equation.

Spacetime curvature is also expressed with respect to a covariant derivative, in the form of the Einstein tensor — . This curvature is related to the stress-energy tensor — — by the key equation

,

where is Newton's gravitational constant, and is the speed of light. We assume geometrized units, so .

With some simple assumptions, Einstein's equations can be rewritten to show explicitly that they are wave equations. To begin with, we adopt some coordinate system, like . We define the "flat-space metric" to be the quantity which — in this coordinate system — has the components we would expect for the flat space metric. For example, in these spherical coordinates, we have

.

This flat-space metric has no physical significance; it is a purely mathematical device necessary for the analysis. Tensor indices are raised and lowered using this "flat-space metric".

Now, we can also think of the physical metric as a matrix, and find its determinant, . Finally, we define a quantity

.

This is the crucial field, which will represent the radiation. It is possible (at least in an asymptotically flat spacetime) to choose the coordinates in such a way that this quantity satisfies the "de Donder" gauge conditions (conditions on the coordinates):

,

where represents the flat-space derivative operator. These equations say that the divergence of the field is zero. The linear Einstein equations can now be written as

,

where represents the flat-space d'Alembertian operator, and represents the stress-energy tensor plus quadratic terms involving . This is just a wave equation for the field with a source, despite the fact that the source involves terms quadratic in the field itself. That is, it can be shown that solutions to this equation are waves traveling with velocity 1 in these coordinates.