Linearized theory and gravitational waves

Question

I've been reading the chapter about gravitational radiation of Schutz's book. In one of the sections, he begins with the linearized Einstein's equations and tries to find an intuitive solution:

$$(-\frac{\partial^2}{{\partial t}^2}+\nabla^2) \bar{h}_{\mu \nu}=-16\pi T_{\mu \nu}$$

With the form of wave equations, assuming the source has an oscillatory time dependence $T_{\mu \nu}=S_{\mu \nu}e^{-i \Omega t}$, we can reach a solution:

$$\bar{h}_{\mu \nu}=B_{\mu \nu}e^{-i\Omega t}$$

Where B satisfies $(\nabla^2+\Omega^2)B_{\mu \nu}=-16\pi S_{\mu \nu}$:

$$B_{\mu \nu}=\frac{A_{\mu \nu}}{r}e^{i\Omega r}$$

It has the form of an outgoing spherical wave.

Now here is the problem: in order to determine the coefficient $A_{\mu \nu}$, we have to make an integration over the source which generates the wave:

$$A_{\mu \nu}=4\int S_{\mu \nu}d^3x$$

My problem is that within the source, like binary inspiral neutron stars, where the field is strong enough that the linearized theory is no longer valid, is this kind of method about determining the coefficient legitimate when we want to determine the gravitational wave far away from the source?

Answer 1

There are a few scales in the problem.

• The size of the black hole or neutron stars in the binary, $r$.
• The instantaneous size of the orbit, or the semimajor axis $a$.
• The wavelength of the gravitational radiation. This follows from Kepler's law, $\Omega \sim GM a^{-3/2}$ and the virial theorem $v^2 \sim GM/a$, leading to $\lambda \sim c/\Omega \sim (GM)^{-1/2} a^{3/2} c \sim (c/v) a$.

There is a natural hierarchy between these scales, when $v/c \ll 1$. In this limit, we have the hierarchy of scales \begin{equation} \lambda \gg a \gg r \end{equation} In this limit, the relevant scale for computing radiation is the wavelength $\lambda$. This hierarchy of scales allows us to perform the post-Newtonian expansion, which is an expansion in powers of $v/c$. The formalism you are using assumes that this parameter is small; near merger, when $v/c\sim 1$, analytic techniques do not do a good job of predicting the gravitational waveform. The internal structure of the source ($r$) is not relevant (it enters only at a high order in the expansion in $v/c$, $O((v/c)^{10})$.

One place you can read more about this point of view are the following Les Houches lectures by Goldberger on the effective field theory of gravitational radiation: https://arxiv.org/abs/hep-ph/0701129

Answer 2

The approximate solution that you are talking about is called the slow-motion assumption of the source of the gravitational wave, and can be used when the oscillatory source moves considerably slower than the speed of light, i.e. the wavelength of the emitted wave is much larger than the characteristic size $L$ of the source: $2\pi c/\Omega\gg L$.

From what we know, this is a sensible assumption, with just the exception of the instants of the merger of a binary black hole system, and possibly some other super-highly energetic processes.

Equation $$A_{\mu\nu}=4\int S_{\mu\nu}d^3x$$ can be actually solved by making use of the conservation laws for the components of $T_{\mu\nu}$ and the Gauss' theorem, and leads you to the sought after approximate solution, namely the quadrupole formula.