Energy, Momenta, Mach's Principle, and Compact Binaries

Energy, Momenta, Mach's Principle, and Compact Binaries

I think this talk is an interesting and coherent one. Its objective is an understanding of the radiation given off by a black-hole binary inspiral. I examine fundamental difficulties in the notion of a binary: its evolution, as resolved by the Hamiltonian formulation of GR; how we define a spinning system; how we characterize the energy given off...

Introduction

LIGO, GEO600, and VIRGO are trying to measure gravitational waves.
The singal that comes out of them can be reproduced very nicely as sound: Detector Sounds
The signals we are listening for are totally buried in the noise, and sound like
- A $2\times 10M_{\bigodot}$ inspiral followed by a ringdown
- Ground-based detectors won’t hear them, but we also expect to find
In order to hear these, we have to know just what we’re looking for
In order to know just what we’re looking for, we need to be able to model these things
The basic problem is how to model an inspiral, merger, and ringdown
The basic solution is initial data, along with some evolution method
For initial data, we need two black holes spinning around each other. This is complicated by the notion of what “spinning” really means. What we’re doing is taking an isolated system in an otherwise empty Universe, and saying that it’s spinning. Newton would have been perfectly happy with this idea. Mach, however, didn’t like it at all. His principle states that “matter there influences inertia here“. The appearance of Mach’s principle in modern GR actually has a lot to do with the next point.
For an evolution method, we can fall back on the classic (and quite general) notion of the Hamiltonian
That’s where I’ll start off.

Hamiltonians

We take some system, and identify a degree of freedom—called the position, for simplicity—which has coordinate $q$ (where $q$ represents $n$ coordinates). We then invent some other space (also with $n$ dimensions), and label it with coordinates $p$ . We call this the momentum associated with the position. Though momentum has no interpretation yet, we will eventually see it as something like the time derivative of the position. For now, though, its just some other variable. Thus, at any point in time, our system is represented by the pair $(p,q)$ . We might say that this pair lives in a manifold $M^{2n}$ . We define some function, which we’ll call a Hamiltonian, $H:M^{2n} \rightarrow \mathbb{R}$ , that is $H: (p,q) \mapsto \mathbb{R}$ . The usual interpretation of this is taking a point in phase space, and giving its energy as a real number.

Now, we can define a derivative operator $d$ on our $M^{2n}$ , so that we can take the gradient of the function $H$ and end up with a 1-form. From this 1-form, we can find a vector with a little trickery. Basically, we replace a $dq$ with a $\vec{p}$ , and a $dp$ with a $\vec{q}$ . In some sense, we take a vector which is orthogonal to the 1-form. We take this derivative of $H$ at some point $(p,q)$ , and get a vector. This vector points in the direction of flow from that point. That is, the vector gives the trajectory of the system when the system is at the point $(p,q)$ . We can follow this trajectory along, to find the time evolution of the system.

A concrete example will help beat this horse to death:

Take a single particle in a single dimension, with some potential $V$ . The Hamiltonian will be
$\begin{displaymath} H = \frac{p^2}{2m} + V(q)\ , \end{displaymath}$
where $m$ is the mass, or some parameter. We can take its derivative to find
$\begin{displaymath} dH = \frac{p}{m}\,dp + V'(q)\,dq\ . \end{displaymath}$
Now we draw a stable potential, and the phase space for this particle. We look at the vector when $p=0$ and when $V'(q)=0$ , and see that if we follow the vector around, we get the usual trajectory through phase space.

Now is a good time to note that when we take the derivative of the Hamiltonian function, we get a 1-form which sort of gives the direction of increase

Strictly speaking, this is the extent of the formalism that we need. We just need to create some Hamiltonian function which gives the behavior that is either observed, or predicted by Lagrangian methods, etc. It can be shown that this Hamiltonian flow then implies the usual Hamilton’s equations:
$\begin{eqnarray*} \dot{q} &=& \frac{\partial H}{\partial p}\ , \\ \dot{p} &=& -\frac{\partial H}{\partial q}\ . \end{eqnarray*}$
It can also be shown that the Hamiltonian function can be derived from the Lagrangian, with
$\begin{eqnarray*} p &=& \frac{\partial L}{\partial \dot{q}}\ , \\ H &\equiv& p \dot{q} - L\ . \end{eqnarray*}$
Typically, the Lagrangian is found as some function which, when varied, gives equations found in some other way.

Hamiltonian Formulation of E&M

We don’t bother with the fancy covariant form of E&M, because we need to make a choice of time coordinate anyways, so we just use the familiar $\vec{A}$ and $\varphi$ . (Recall that $\vec{E} = -\dot{\vec{A}} - \vec{\nabla}\varphi$ , and $\vec{B} = \vec{\nabla} \times \vec{A}$ .) At this point, we expect to take $\vec{A}$ and $\varphi$ as our “position” variables, and then invent some corresponding momenta $\vec{p}_A$ and $p_\varphi$ .

We know Maxwell’s (source-free) equations. It’s not hard to show that the Lagrangian which results in them is
$\begin{displaymath} L = \frac{1}{2} \left( \dot{\vec{A}} + \vec{\nabla}\varphi \right)^2 - \frac{1}{2} \left( \vec{\nabla} \times \vec{A} \right)\ . \end{displaymath}$
We can find the momentum conjugate to $\vec{A}$ :
$\begin{displaymath} \vec{p}_A = \frac{\partial L} {\partial \dot{\vec{A}}} = \dot{\vec{A}} + \vec{\nabla} \varphi = - \vec{E}\ . \end{displaymath}$
We next want the momentum conjugate to $\varphi$ :
$\begin{displaymath} p_\varphi = \frac{\partial L} {\partial \dot{\varphi}} = 0\ . \end{displaymath}$
This is identically zero, which is trouble. Rather than treating $\varphi$ as a dynamical variable, we could treat it as another quantity appearing in the Hamiltonian, which acts as a Lagrange multiplier. This will give us a “constrained” Hamiltonian formulation, which is expected to arise from any system with gauge arbitrariness. (We shouldn’t have expected to have a totally determined evolution of $\vec{A}$ and $\varphi$ because of the gauge freedom we had in Maxwell’s equations to add any gradient to $\vec{A}$ .)

Now, we are treating $\vec{A}$ as the “position” variable, and $\varphi$ as some Lagrange multiplier. The Hamiltonian function will be given by
$\begin{displaymath} H = \vec{p}_A \cdot \dot{\vec{A}} - L = \frac{1}{2} \vec{p}_A \cdot \vec{p}_A + \frac{1}{2} \vec{B} \cdot \vec{B} + \varphi \vec{\nabla} \cdot \vec{p}_A\ . \end{displaymath}$
Hamilton’s equations say
$\begin{eqnarray*} \dot{\vec{A}} &=& \frac{\partial H}{\partial \vec{p}_A} = \vec{p}_A - \vec{\nabla} \varphi\ ,\\ \dot{\vec{p}}_A &=& -\frac{\partial H}{\partial \vec{A}} = - \vec{\nabla} \times (\vec{\nabla} \times \vec{A})\ . \end{eqnarray*}$
Finally, the Lagrange multiplier comes in by setting
$\begin{displaymath} \frac{\partial H}{\partial \varphi} = 0 = \vec{\nabla} \cdot \vec{p}_A\ . \end{displaymath}$
By inspection, we can see that these equations (along with the assumption that we can write $\vec{B}$ as the curl of something) are equivalent to Maxwell’s (source-free) equation.

Now, this was a lot of work for just some equation the we already knew. The power here is the general technique for finding a set of evolution equations which describe how our field changes in time (as well as a constraint equation when gauge freedom needs to be dealt with).

Hamiltonian Formulation of GR

3+1 Split

Just as E&M can be written in the full covariant form using $F_{\mu\nu}$ , etc., or the simple form with $\vec{A}$ and $\varphi$ , so too can GR be written in its elegant form with $G=8\pi T$ , or in a split form. We start with a full four-dimensional spacetime $S$ . We define some function into the real numbers $t:S \rightarrow \mathbb{R}$ (time) for which the level sets are spacelike. (That is, if we take some vector tangent to the surface, it will have positive norm.) We can think of these three-dimensional slices as surfaces of simultaneity.

The crucial variable in full GR is, of course, the metric $g_{\mu\nu}$ . We can decompose this metric as follows:

The lapse function $N$ describes how our time function relates to the lapse of proper time for an observer moving directly between hypersurfaces.
The shift vector $N^i$ is a 3-d vector in a hypersurface which describes the velocity of a coordinate observer.
The 3-d metric $h_{ij}$ , which is just the restriction of the 4-d metric to a 3-d surface.

The full 4-d metric can be written in terms of these variables, so we expect that this set also describes the “position” of GR’s field, and that we would have conjugate momenta for each.

Don’t bore them with the details. Draw an analogy between the E&M evolution equations and evolution equations for $h_{ij}$ and its momentum $\pi_{ij}$ ; also between the constraint equation for E&M and the constraints for GR.

Lapse and Shift as Lagrange Multipliers Give Constraint Equations as Elliptic Equations

Go through some formulas from the Caudill, et al., paper.

The Initial Data Problem and Mach's Principle

Boundary conditions are necessary for an elliptic system of differential equations
When specifying initial data, we need to give boundary conditions
Geoff has been working on initial data, trying to

Gravitational Radiation

Wave Equation Form of Einstein's Equations

If we choose some coordinate system, and write the metric as
$\begin{displaymath} \bar{h}_{\alpha \beta} \equiv \sqrt{-g} g_{\alpha\beta} + \eta_{\alpha\beta}\ , \end{displaymath}$
Einstein’s equations can be written in the form
$\begin{displaymath} \Box \bar{h}_{ab} = -16\pi \tau_{ab}\ , \end{displaymath}$
where $\tau_{ab}$ is the stress-energy tensor plus some terms involving derivatives of $\bar{h}$ , and $\Box$ is the flat-space wave operator. This is just a wave equation with some source. For radiation, the source can be written in terms of the mass- and current-multipoles denoted $I_{lm}(t-r)$ and $S_{lm}(t-r)$ . These are analogous to charge and current multipoles in E&M.

I’m working on analyzing simulated inspirals.
Using the phase information from the simulation, and plugging that in to the formulas given above, we can deduce the gravitational radiation we expect to be emitted:

These graphs show the value of $\Psi_4$ (which is basically the second time derivative of the gravitaional wave) as functions of time. Both the amplitude and the frequency of the waves increase as time goes on, energy is given off, and the orbit of the binary becomes tighter.
Just for the fun of looking at it, let’s look at a movie: r·Ψ4 movie