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What Gravitates?

A few weeks ago, Nate mentioned the fact that, when estimating the Newtonian attraction of a black-hole binary, you need to include the kinetic energy of the holes. This opens up a whole range of interesting points about the fundamentals of GR — especially in the general regime, where we don’t have exact solutions, but can still say interesting things. These (in my mind) tie nicely into another thing he talked about: spin kicks. I briefly discuss what these are, why they happen, and how to understand them on an analytic level.

Definitions of mass

Horizon mass

Area theorem

Draw a Penrose diagram of Minkowski spacetime. Illustrate $i^+$, $i^-$, $i^0$, $\mathcal{I}^+$, and $\mathcal{I}^-$, as well as outgoing and ingoing null rays, and spacelike surfaces. Explain that $\mathcal{I}^+$, and $\mathcal{I}^-$ are generated by null rays.

Now, add in a black hole by erasing the center of the universe, and drawing the hole’s generator. Note that the generator does not actually touch future null infinity; rather, it touches $i^+$. Define the event horizon as the boundary of the causal past of future null infinity. Our typical picture of an event horizon is a two-dimensional sphere — this is just the event horizon on a spacelike slice through the spacetime. Given certain assumptions about the nature of spacetime (predictability, positive energy density), you can prove that the generators have non-negative expansion. This says that if you take any spacelike slice of the spacetime, followed by any other spacelike slice, the area of the event horizon on the second slice will not be smaller than the area of the event horizon on the first. (It may be the same or larger, but it won’t be smaller.)

Now, if we have two black holes in our spacetime, the “boundary of the causal past of future null infinity” at a single point in time is given by two disconnected spheres. [Draw the legs of the pair of pants.] In this case, we can just take the areas of the two separate event horizons. The area theorem then holds for the sum of these two areas. [Draw the pair of pants, and explain what the above means for merging black holes.] If the holes now merge, the area theorem says
\begin{displaymath} A_{\text{H},1} + A_{\text{H},2} \leq A_{\text{H},3}\ . \end{displaymath}

Defining horizon mass

For Schwarzschild, we have the relations $r_{\text{H}}=2M$, and $A_{\text{H}}=4\pi r_{\text{H}}^2 = 16\pi M^2$. We can measure the surface area of any black-hole horizon and, by analogy with Schwarzschild, define its mass as
\begin{displaymath}M_{\text{H}} \equiv \sqrt{A_{\text{H}}/16\pi}\ . \end{displaymath}
By the area theorem, for two merging black holes, we have
\begin{displaymath} M_{\text{H},1} + M_{\text{H},2} \leq M_{\text{H},3}\ . \end{displaymath}
Thus, the “horizon mass” is more commonly referred to as the irreducible mass.

Christodoulou mass


\begin{displaymath} M_\chi \equiv \sqrt{M_{\text{H}}^2 + S^2/4M_{\text{H}}^2}\ . \end{displaymath}

GR as nonlinear field theory on a background spacetime

Brief review of linearized gravity and waves

  • The basic quantities
    • We assume that spacetime is flat to a very good approximation, so we can lay down nice coordinates in the usual way
    • But, we assume that spacetime is not perfectly flat, so we define a nontrivial quantity $\bar{h}^{\alpha \beta} \equiv \eta^{\alpha \beta} - \sqrt{-g} g^{\alpha \beta}$.
    • In linear gravity, we keep only the first term in the perturbation to the metric $h^{\alpha \beta} \equiv g^{\alpha \beta} - \eta^{\alpha \beta}$
    • In a particular coordinate system, it’s not hard to show that $\bar{h}^{\alpha \beta} \rightarrow h^{\alpha \beta} - \frac{1}{2} \eta^{\alpha \beta} h^\gamma_{\phantom{\gamma}\gamma}$
    • This is almost a geometric nonequation (rather than a coordinate-dependent equation), so it must be true in every coordinate system
  • Gauge
    • It’s not hard to show that it is always possible (in nonlinear theory) to choose coordinates in such a way that $\bar{h}^{\alpha \beta}_{\phantom{\alpha \beta}, \beta} = 0$
    • This is analogous to Lorenz gauge in E&M; here, it is called Lorentz gauge, and this may or may not be legitimate.
  • Einstein tensor
    • Writing out the Einstein tensor in terms of the metric, in the usual way, and keeping only first-order terms, we have
      \begin{displaymath} G_{\mu\nu} = 8\pi T_{\mu \nu} \Rightarrow \boxdot\bar{h}_{\mu\nu} = (-\partial_t^2 + \nabla^2) \bar{h}_{\mu\nu} = - 16\pi T_{\mu \nu}\ ,\end{displaymath}
      where we define $\boxdot$ to be the nice flat-space wave operator.
    • The fact that this is a wave operator is very nice, and is the reason for gravitational waves given off in linearized theory.
  • Application of the linearized Einstein equation to the linearized Schwarzschild metric
    • To first order in the Newtonian potential, the metric around a spherically symmetric and stationary body is given by $g_{00} = -(1-2\Phi)$; $g_{ij} = (1+2\Phi)\delta_{ij}$
    • thus, the metric perturbation is $h_{00} = 2\Phi$; $h_{ij} = 2\Phi \delta_{ij}$
    • the trace is $h = h_{\mu \nu} \eta^{\mu \nu} = -2\Phi + 2\Phi \delta = 4\Phi$
    • thus, $\bar{h}_{\mu\nu} = 4\Phi \delta_{\mu 0} \delta_{\nu 0}$
    • Einstein’s equation says that $\boxdot \bar{h}_{00} = \nabla^2 4\Phi = -16\pi T_{00} = -16\pi \rho$, or $\nabla^2 \Phi = -4\pi \rho$, which is Poisson’s equation
    • Wave hands about spherical symmetry
    • Now, we can use a Green’s function to write down the solution to this equation:
      \begin{displaymath} G(\mathbf{x}, \mathbf{x}') = - \frac{1} {4\pi} \frac{1}{|\mathbf{x} - \mathbf{x}'|}\ ,\end{displaymath}
      so
      \begin{displaymath} \Phi(\mathbf{x}) = - \int \frac{\rho(\mathbf{x}')} {|\mathbf{x} - \mathbf{x}'|}\, dV'\ .\end{displaymath}

Mass in this theory

Some discussion of why mass doesn’t carry through so easily into GR.

 
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lunchtalks/gravitating.txt · Last modified: 2008/05/29 10:55 by boyle