Talk:ADM formalism

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In Theoretical Physics, Hamiltonian formulation has succeeded to quantize field theory with canonical quantization method e.g., quantum electrodynamics and quantum chromodynamics. By taking the analogy of the above theories, Hamiltonian formulation can also be developed for Einstein gravity theory, which have been done by Arnowitt, Deser, and Misner (ADM) in 1962. ADM formalism is consistent with initial value formulation for general relativity. When general relativity can be cast into Hamiltonian form, one can attempt to quantize general relativity. However, a serious difficulty arises because of the presence of the constraint. Efforts to solve this constraint or to impose this constraint as an additional condition on state vector still have not been successful.


Benz Edy Kusuma, ITB

Reference: 1. Arnowitt, R., Deser, S., and Misner, C. W., The Dynamics of General Relativity, in Gravitation: an Introduction to Current Research, ed. Louis Witten, Wiley, New York, 1962, pp. 227-265. 2. Wald, R. M., General Relativity, The University of Chicago Press, Chicago,1984.

[edit] technical

I would rather not try and clean this up, when there are probably people who are probably more familiar with the formalism than I and can do it easier than I could. Salsb 16:00, 21 February 2006 (UTC)

[edit] Uh...Lemme Try Explaining

OK, so here's my shot at explaining the ADM formalism. Granted, I am pressed for time and I am referring to some lecture notes and collected notes that I've taken, but that's the best I've got at my disposal at the moment.

Initially, one starts with the Einstein–Hilbert action...The action S[g] which gives rise to the vacuum Einstein equations is given by the following integral of the Lagrangian

S[g]= \int kR \sqrt{-g} \, \mathrm{d}^4x

There may be additionally boundary terms depending on the boundary conditions, but they are ignored for the time being. Now, the ADM formalism basically takes this and uses the Legendre transform to yield a Hamiltonian formulation of General Relativity.

Because we are using canonical classical mechanics (i.e. Hamiltonian mechanics), we need to break spacetime up into space and time (see chapter 1.1 of [2]). This has caused some controversy and confusion (see chapter 1.4 of [2] for a thorough discussion of the problems). General Relativity is a generally covariant theory, meaning that we are working in 4 dimensions rather than 3 spatial dimensions plus some time parameter. However, if we keep things arbitrary and do not fix a coordinate system then there is no problem. The reason is that this arbitrariness exhausts the full diffeomorphism group.

A useful parametrization can be done with the aid of a deformation vector field

T^{\mu}(X) := \frac{\partial X^{\mu}(t, x)}{\partial t} =: N(X)n^{\mu}(X) + N^{\mu}(X)

where nμ(X) is a unit normal vector to the spatial hypersurface, i.e. gμνnμnν = − 1 in a (-+++) metric signature. Also Nμ is tangential which means g_{\mu\nu}n^{\mu}X^{\nu}_{,a}=0. Thus we have the vector field nμ determined solely by the metric tensor g and X by these two requirements. The coefficients of proportionality here (N and Nμ) are respectively called the lapse function and shift vector field. One moves "along" a spatial hypersurface by \vec{N}\delta t, and one moves from one spatial hypersurface (with time t) to another (with time t + δt) with the term Nnδt. It should be noted that each spatial hypersurface has the time constant.

I'm sorry but I'm short on time and cannot go any further, but I shall return!...tomorrow...and clarify what I have written and add more juicy details!

- pqnelson 1:43 (PST) 21 December 2007

References:

1. Arnowitt, R., Deser, S., and Misner, C. W., The Dynamics of General Relativity, in Gravitation: an Introduction to Current Research, ed. Louis Witten, Wiley, New York, 1962, pp. 227-265. Available online at Arxiv.org.

2. Thiemann, T. Modern Canonical Quantum General Relativity.

3. Hawking, S. and Ellis, G. The Large Scale Structure of Spacetime (Cambridge University Press, Cambridge, 1989).

4. Wald, R. M. General Relativity (University of Chicago Press, Chicago, 1989) (see especially chapter 10 and appendix E)

5. Misner, C., Thorne, K. and Wheeler, A. J. Gravitation (especially chapter 21 on the variational formalism of General Relativity which includes the construction of the Hamiltonian, i.e. ADM, formalism of general relativity)

6. Lee and Wald, "Local Symmetries and Constraints", Journal of Mathematical Physics 31 (1990) 725 (pdf)