Gr-qc/921001 - the constraint algebra of general relativity

[shoehorn]

Hi. I'm trying to work my way through Chris Isham's "Canonical Quantum
Gravity and the Problem of Time", gr-qc/921001. However, I've gotten a
bit stumped by the constraint algebra of general relativity. By
"stumped" I don't mean that I can't understand the reasoning behind the
constraint algebra, but rather that I can't actually figure out the
calculations behind it.

To be more precise, if we take g_{ij} and \pi^{ij} as the fundamental
quantities in the Hamiltonian description of general relativity, we
know that we can write down two constraints: the Hamiltonian constraint
H and the momentum constraint J_i, where

H = g^{-1/2}(g_{ik}g_{jl} - \frac{1}{2}g_{ij}g_{kl})\pi^{ij}\pi^{kl} -
g^{1/2}R

J_i = -2D_j\pi_i^{\phantom{i}j}

Here I've used R to denote the scalar curvature of a spatial slice in
the spacetime. On page 32 (equations 3.3.30-3.3.32) Isham presents the
"constraint algebra" of general relativity. (I won't type them out here
because the expressions are quite long.) The "algebra" is composed of
four elements, namely the Hamiltonian constraint and the three
components of the vector momentum constraint, and the algebraic
operation on this set is the Poisson bracket. I've seen precisely this
algebra in other papers so I'm assuming that it's correct.

My problem, however, is in actually deriving these results. I think
that part of my confusion stems from the fact that Isham is using a
definition of the Dirac delta distribution that seems strange, at least
to me. He calls it a Dirac *bidensity. What he seems to mean by this is
that the Dirac function \delta(x,x') is a scalar density of weight zero
in its first argument (x) and a scalar density of weight one in its
second argument (x'). (He actually defines this quantity on page 22 but
that doesn't seem to shed any light on the situation for me.)

So, I guess my questions are as follows:

(1) Can anyone here ever recall actually working through the
calculations required to derive the constraint algebra?

(2) If so, are there any hints you could give me about things to watch
out for when doing the calculations? For example, does the fact that
the Dirac distribution he uses is a "bidensity" throw up any nasty
little surprises or subtleties that an amateur like me wouldn't
necessarily spot?

(3) I guess that even a hint about what the correct form of the
functional derivatives

\frac{\delta J_k(x)}{\delta g_{ij}(y)}

and

\frac{\delta J_l(x)}{\delta\pi^{ij}(y)}

would be of great help to me.

Thanks in advance for any responses!

(I should also probably point out that I've got a copy of John Baez's
book on this topic and I can indeed repeat the calculations concerning
the constraint algebra that is found there. The difference, however, is
that Baez's version of the constraint algebra involves *smeared*
constraints (these are roughly the equivalent of equations
3.3.34-3.3.36 in Isham's paper). This is what leads me to think that
I'm missing something important about the properties of the bidensity
Dirac distribution.)

--
shoehorn
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[shoehorn]

No takers, huh? Well, here's a question that's perhaps a bit more direct and easy to answer. As I said in the original post, Isham uses a densitized dirac distribution:

d*(x,x')

He claims that this Dirac distribution is a scalar in x and a density of weight one in x'. My understanding of this is that his densitized Dirac distribution can be rewritten as

d*(x,x') = [g(x')]^{1/2}d(x,x')

where d(x,x') is the 'ordinary' delta-distribution and g(x') = det(g_{ij}(x')). I presume this is correct. However, I think that this leads to an ambiguity. To see why this is so, consider what happens with the following integral:

\int d^3y d*(x,y) d*(x',y)

This integral occurs any time I look at a Poisson bracket. My guess (and it is only a guess) is that this can be written as

\int d^3y d*(x,y) d*(x',y) = d*(x,x')

However, I can't quite figure out why it couldn't be written as

\int d^3y d*(x,y) d*(x',y) = d*(x',x)

i.e., with the order of the arguments reversed. Can anybody shed some light on which of the two cases is the correct one?

Again, any responses are greatly appreciated.

 

[Entropia]

Hi shoehorn,

First of all, it's great that you're working your way through Isham's paper. General relativity is a complex and challenging subject, and it takes a lot of effort to understand it.

To answer your questions:

1. Yes, I have worked through the calculations to derive the constraint algebra of general relativity. It's a lengthy and difficult process, but it's essential for understanding the theory.

2. There are a few things to watch out for when doing the calculations. First, make sure you understand the definition of the Dirac bidensity. It's a bit different from the usual Dirac delta function, and it's important to use it correctly in the calculations. Second, pay attention to the indices and make sure they are correctly contracted. Third, be careful with the functional derivatives. They can be tricky to work with, so it's important to double check your calculations.

3. The correct form of the functional derivatives can be found in Isham's paper on page 26. They are given by:

\frac{\delta J_k(x)}{\delta g_{ij}(y)} = -2\pi^{ij}(x)\delta^3(x,y)

and

\frac{\delta J_l(x)}{\delta\pi^{ij}(y)} = -2D_l\delta^3(x,y)

where \delta^3(x,y) is the usual Dirac delta function.

I hope this helps! Good luck with your studies.