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within the source, like binary inspiral neutron stars, where the field is strong enough that the linearized theory is no longer valid, is this kind of method about determining the coefficient legitimate when we want to determine the gravitational wave far away from the source?

seVenVo1d
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Jiaxiang Zhu
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  • Related: https://physics.stackexchange.com/q/427110/226902, https://physics.stackexchange.com/q/769218/226902. – Quillo Jul 20 '23 at 15:32

2 Answers2

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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

Andrew
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    Thanks! So treating the binary neutron stars as two mass point is indeed a good approximation for the early stage of merge. The point of view that the internal structure is a higher order is really new and interesting to me. – Jiaxiang Zhu May 17 '22 at 14:26
  • @JiaxiangZhu Yes, indeed. It is actually a very difficult thing to measure (but also very interesting if you can measure it!). There's some information in the following paper: https://arxiv.org/abs/1805.11581 – Andrew May 17 '22 at 14:45
  • A side note about internal structure being higher order. While it is true that these terms come in only at 5PN order, their coefficients tend to scale with $R^5/(GM)^5$ where R is the size of the object. So these terms may become relevant well before on would naively expect based on the PN order alone! – TimRias May 18 '22 at 08:09
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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.

AstoundingJB
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  • I see. So the quadrupole formula is actually a good approximation for binary neutron stars. I've read some references and some of them mention Post-Newton corrections for the early stage of merge. Is it connected with my problem? – Jiaxiang Zhu May 17 '22 at 14:19
  • Yes, definitely. This is the quasi-Newtonian approximation, where you are using a relativistic theory but basically neglecting everything $\mathcal{O}(v/c)$, which implies that the source is moving according to Newton's laws. In the post-Newtonian expansion, you start considering the first order terms in $v/c$(squared), which is appropriate for describing also the motion of the source, especially in a binary system. An excellent reference is the Living review by Luc Blanchet. – AstoundingJB May 17 '22 at 15:27