# What is the source of the perturbation $h_{\mu\nu}$ in linearized Einstein field equations in vacuo?

In linearized field equations, the metric tensor is writen as $$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$$, where $$|h_{\mu\nu}|\ll 1$$ is a small perturbation of the flat Minkowski metric $$\eta_{\mu\nu}$$ such that, the linearlized Einstein equations then take the form.

\begin{align} \label{eq:linearized_EFEs2} G_{\mu\nu} = \tfrac{1}{2}\left(-\partial_\mu\partial_\nu h - \square h_{\mu\nu} + \partial_\nu\partial_\rho h^\rho_\mu + \partial_\mu\partial_\rho h^\rho_\nu + \eta_{\mu\nu}\left(\square h - \partial_\rho\partial_\sigma h^{\rho\sigma}\right)\right). \end{align}

Taking a gauge transformation as below,

\begin{align} \partial_\mu h^\mu{}_\lambda - \frac{1}{2}\partial_\lambda h = 0, \qquad h \equiv \eta^{\mu\nu}h_{\mu\nu}. \end{align}

we are left with

\begin{align} G_{\mu\nu} = -\frac{1}{2}\left(\square h_{\mu\nu} -\frac{1}{2}\eta_{\mu\nu}\square h\right). \end{align}

The Einstein field equation $$G_{\mu\nu} = \kappa T_{\mu\nu}$$ becomes

\begin{align} \square h_{\mu\nu} -\frac{1}{2}\eta_{\mu\nu}\square h = -2\kappa T_{\mu\nu}. \end{align}

In vacuum (i.e. $$T_{\mu\nu}=0)$$ the trace of this equation shows that $$\square h =0$$, and hence the vacuum field equation becomes

\begin{align} \boxed{ \square h_{\mu\nu} = 0,} \end{align}

which is a homogeneous wave equation for each component of the metric.

My question is: What exactly is the $$T_{\mu\nu}$$ here; is it the energy-momentum tensor of the source (say a neutron star which is emitting gravitational waves)? Now if $$T_{\mu\nu}=0$$,then does that mean we are speaking of the spacetime metric in absence of the neutron star? If so, then what is the source of the perturbation $$h_{\mu\nu}$$ in absence of $$T_{\mu\nu}$$?
P.S.: I am self studying the basics of gravitational waves and had this doubt while reading Section 17.5 of "General Relativity: An Introduction for Physicists" by M.P. Hobson).

• It's the same as Maxwell's equations, in a vacuum where there is no charge and current, we find the wave equation or a vibrating string without damping ...., In the harmonic gauge, the linear Einstein equation is analogous to the propagation equation of the 'electromagnetic' 4-potential. Jun 22 at 9:44

1 - What exactly is the $$T_{\mu \nu}$$ here; is it the energy-momentum tensor of the source (say a neutron star which is emitting gravitational waves)?

Yes.

2 - Now if $$T_{\mu \nu}=0$$, then does that mean we are speaking of the spacetime metric in the absence of the neutron star?

Do not forget space dependence: your source (e.g. the neutron star) is localized. However, you may want to study the propagation of $$h_{\mu \nu}$$ far from it, in a region where $$T_{\mu \nu}=0$$.

3 - If so, then what is the source of the perturbation $$h_{\mu \nu}$$ in absence of $$T_{\mu \nu}$$?

The source is matter (or, e.g., colliding black holes) in a region of spacetime where $$T_{\mu \nu}\neq 0$$. After emission, the perturbation propagates far from this region. This is what caused the perturbation in the first place, now you are studying the perturbation in empty space, after its emission and propagation.

A more extended comment: Imagine you have an asteroid moving in vacuum: you assign an initial condition and solve the dynamics, even if you do not know where the asteroid come from and what generated it. This is the so-called initial value problem (or partial differential equations, like the wave equation, it typically goes under the name of Cauchy problem).

Same here for $$h_{\mu \nu}$$. Another example would be the electromagnetic field propagating in empty space, i.e. no 4-current in Maxwell's equations. The wave equation (or any other differential equation with source terms) does not necessarily need to contain the source (see this question): you can assign an initial condition (according to some physical criterion, you choose) and integrate it to find out how the perturbation will evolve after it leaves the source region.

But... if you do not model the source, then how can you impose the initial condition? A solution would be to model the full process including the source but this is often unpractical or very costly and also subject to large uncertain pieces of physics (e.g. supernovae, neutron star mergers, the big bang...). Moreover, most of the time you do not need to know anything about the source, you just want to find out the general properties of propagation in an unperturbed medium (or vacuum), like dispersion relations, the velocity of propagation... This general reasoning is also valid for matter waves (acoustics, hydrodynamics, vibrations/sound in solids), where you study propagation through a previously unperturbed medium with no reference to the source (e.g. sound in air far from the speaker).

A final example: consider the study of wave packets, like in this question. A wave packet is a "bump" in the wave function that evolves under the free particle Schrödinger equation. But what created the "bump" in the first place? It is not important (or, at least, it's not part of the problem one wants to study!), it's just, again, just a Cauchy problem.

Edit: I found a closely related question that is worth checking, Gravitational waves energy source in linearized theory.

• This answers my question very clearly. Thanks! Jun 24 at 2:47