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

  1. Jun 9, 2007 #1
    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} -

    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)}


    \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.)

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    Last edited by a moderator: May 2, 2017
  2. jcsd
  3. Jun 13, 2007 #2
    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:


    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.
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