Talk:ADM formalism

Untitled
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.

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)

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 $$n^{\mu}(X)$$ is a unit normal vector to the spatial hypersurface, i.e. $$g_{\mu\nu}n^{\mu}n^{\nu} = -1$$ in a (-+++) metric signature. Also $$N^{\mu}$$ is tangential which means $$g_{\mu\nu}n^{\mu}X^{\nu}_{,a}=0$$. Thus we have the vector field $$n^{\mu}$$ determined solely by the metric tensor g and X by these two requirements. The coefficients of proportionality here ($$N$$ and $$N^{\mu}$$) 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 $$t+\delta t$$) with the term $$Nn\delta t$$. 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)

Could use some historical perspective
Wouldn't this article (and others related) be better with a little historical perspective. Niczar ⏎ 18:22, 4 July 2008 (UTC)

Addition of a scalar field to Lagrangian density?
I wish this article was extended to include the addition of a scalar field as in the case of e.g. Brans-Dicke theory. I've seen the outcome cited in papers but I've never seen a detailed derivation. Can anyone point me in the right direction? —Preceding unsigned comment added by 208.65.181.179 (talk) 22:06, 8 July 2010 (UTC)


 * I think that the key aspect of the derivation - unless I am mistaken and if I am, can someone please correct me - is to use the Lagrangian to obtain the field equations and thereby isolate an expression for the Einstein tensor and then the Ricci tensor R_uv and then the Lagrangian being made of g^uv * R_uv provides a form for a transformed Lagrangian. This means taking on-shell results and plugging it back into the Lagrangian itself which could very well be a no-no unless you're careful.  At any rate, the on-shell/off-shell aspects are important.  I could use some help here TonyMath (talk) 22:05, 20 July 2010 (UTC)

Vacuum energy in non-stationary spacetime backgrounds
I find the sentence "Cosmic inflation in particular is able to produce energy (and mass) from 'nothing' because the vacuum energy density is roughly constant, but the volume of the Universe grows exponentially." from the section "ADM energy" doubtful as for a quantum field theory formulated on a non-stationary spacetime, there is no well defined vacuum state in the first place. There is a renormalization ambiguity in the calculation of the energy-momentum tensor for Hadamard states like discussed in Wald's "Quantum field theory in curved spacetime and black hole thermodynamics" but as far as I know this is not understood very well yet. I would therefore suggest removing it. Does anyone agree? — Preceding unsigned comment added by 139.18.9.66 (talk) 06:19, 18 October 2011 (UTC)

Far field limit?
Please correct me if I am wrong but isn't the last term in $$\partial_j$$ of the resulting Lagrangian a surface term since it's a derivative and consequently integration over $$x_j$$ would yield the function itself evaluated at $$x = +- \infty$$ and thus contribute nothing to Euler-Lagrange equations? Of course, I am assuming a Minkowski limit in the far field. TonyMath (talk) 05:10, 5 March 2012 (UTC)


 * It would nice to get some feedback on this. The section on canonical gravity mentions the elimination of total derivatives. It seems the only term left is $$ \sqrt{-g} ( R + K_{ij} K^{ij} - K^2) $$ where $$ K_{ij} $$ is the extrinsic curvature.  BTW, I do think the canonical gravity section and this section (clearly taken verbatim from the papers of Arnowitt Deser and Misner) need to be better integrated and related to each other.  The canonical gravity section uses $$ \pi_{ij} $$ which can be related to $$ K_{ij} $$ via the formula $$ \pi^{ij} =\sqrt{ + or -g} (K^{ij} - g^{ij} K) $$.  Of course surface terms can be important but then again you might just happy with the resulting Euler-Lagrange equations and ignore them.  TonyMath (talk) 09:00, 6 March 2012 (UTC)

Intro sucks
Intro should help the reader understand what ADM formalism is. It doesn't accomplish that atm. ScienceApe (talk) 18:09, 12 January 2013 (UTC)

agreed - one picture of a gravity well as if that is new! — Preceding unsigned comment added by Juror1 (talk • contribs) 19:14, 18 February 2016 (UTC)

ADM Mass
If ADM energy-momentum (or mass) doesn't have its own Wiki page then it really should be moved into the Mass in General Relativity article instead. Also I feel like that section needs a complete rewrite.

It should mention the energy-momentum 4 vector and give actual definitions. If it had its own article then a simple example using Schwarzschild could be included and reference to important results such as the positive mass theorem. Also some motivation for it - I like thinking about it as the boundary terms in the Regge Teitelboim Hamiltonian (http://adsabs.harvard.edu/abs/1974AnPhy..88..286R), but there are many ways of thinking about it.

I might have a look at typing something up myself soon but I'm pretty inexperienced with editing. Steve86au (talk) 19:26, 23 November 2013 (UTC)


 * at least ADM-mass exists in the German wiki. Ra-raisch (talk) 14:13, 13 September 2017 (UTC)
 * $$m_{ADM}(M,g):= \lim_{R\to\infty} \,\frac{1}{16\,\pi} \sum_{\mu,\nu=1,2,3}\,\,\int_{\partial K_R} \left(\frac{\partial}{\partial x_{\mu}}\, g_{\nu \nu}- \frac{\partial}{\partial x_{\nu}}\, g_{\nu\mu} \right)\,\mathrm{d}n^{\mu}$$

Controversy
I don't think there is a widely appreciated controversy. This paragraph cites a single paper by two authors from Ontario and it was added by an anonymous IP address from Ontario. This paragraph should either be substantiated with further references that show that there is really a controversy or be removed. Atdotde (talk) 10:00, 16 December 2020 (UTC)

I have removed this from the article. It is incorrect. — Preceding unsigned comment added by 2A00:23C5:C486:9701:641B:ED7A:311C:7225 (talk) 19:33, 21 February 2022 (UTC)