7.3 Emission of Gravitational Waves

In Chap. 5 we studied the linearized form of Einstein equations and, after fixing the de Donder gauge, we arrived at the following form:

γμν = −16π GTμν

(7.3.1)

∂μγμν = 0

where γμν is a linear redefinition of the metric deformation around its flat Minkowskian average:

gμν (x) = ημν + hμν (x) + O h2

(7.3.2)

γμν (x) = hμν − 1 ημν hρσ (x)ηρσ

2

Using the retarded Green function (7.2.18) to solve these linearized equations (7.3.1), we obtain the following expression for the field γμν (x) generated by a matter system, whose description is encoded in the stress-energy tensor Tμν (x):

As already emphasized the integral is extended over the past light-cone of the point, whose coordinates are xμ. The above solution represents the disturbance in the gravitational field produced at point xμ by the presence and motion of some matter in another distant region of space-time. Obviously the disturbance is felt only after the time needed for the signal to travel (at the speed of light) the distance separating xμ from the region where Tμν is present.

Relying on the above formula, our main concern is that of evaluating the energy transported by a linearized gravitational wave and relate this latter to the relevant deformations of the source-system. This procedure will produce an evaluation of the emission power of a gravitational wave source. At the end of our elaborations we shall discover that such a power, namely the amount of energy emitted per unit time, is proportional to the modulus squared of the third derivative of the source quadruple moment. This is typically a very small quantity and that is the main reason why gravitational waves have so far escaped direct measurement.

7.3.1 The Stress Energy 3-Form of the Gravitational Field

In order to calculate the energy transported by the gravitational wave we have to define the stress-energy tensor not of matter, but of the gravitational field itself. In the case of general metrics, the definition of mass, energy and momentum of the gravitational field corresponds to an ambiguous problem, since to introduce a stressenergy tensor we need a background reference metric, which is not uniquely given.

Yet, in the case of linearized gravity, the reference metric is uniquely identified by the undeformed Minkowski metric. In this case a simple manipulation of Einstein equations allows to derive the correct expression of the stress-energy tensor.

We start from the field equations in differential form language:

dEa + ωab Ecηbc = 0

(7.3.4)

Rab Ecεabcd = κ Td

where Ea is the vierbein, ωab the spin connection and κ = 16π G/3. We insert the definition of the curvature

can be declared to encode the stress-energy tensor of the gravitational field. Indeed, as a consequence of (7.3.9) the following 3-form

Note that this definition is not invariant with respect to local Lorentz transformations since it depends on the bare field ωab . However it is invariant against global Lorentz transformations and therefore it can be used in the asymptotic region of a space-time which is nearly flat.