ADM formalism: evolution of conjugate momentum

[Gregor Scholten]

Hi alltogether,

There are numerous articles about ADM formalism, e.g. this one:

http://www.math.toronto.edu/mccann/assignments/426/Tong.pdf

where they indicate expressions for the Hamiltonian (equation 20 in
upper article) and the time-derivative of the momentum \p^ij conjugated
to three-metric h_ij (equation 25). However, I didn't find any article
where the derivation of \dot \p^ij from the Hamiltonian is shown. In
every article I found, they only indicate the final result for the
time-evolution of p^ij (like in equation 25 in upper article), but not
the calculation that yields this result. Especially the terms with
second derivatives of the lapse function \alpha, like D^a D^b \alpha,
seem very strange to me. There are no terms in Hamiltonian where I could
imagine that those terms originate from them. The terms with
three-Ricci-tensor obviously originate from the three-Ricci-scalar, the
terms with p^ab from the p^ab terms in Hamiltonian, but I have no idea,
where the D^a D^b alpha occurs from.

Does anybody know about this?

Thanks in advance.

[Jonathan Thornburg [remove -animal to reply]]

I don't have a direct answer to your question handy, but here are
some references which might be useful:

You probably know that the name "ADM formalism" derives from the 1962
paper by Arnowitt, Deser, and Misner,

@incollection
{
ADM-1962,
author = "R. Arnowitt and S. Deser and Charles W. Misner",
title = "The Dynamics of General Relativity",
pages = "227--265",
editor = "L. Witten",
booktitle = "Gravitation: An Introduction to Current Research",
publisher = "Wiley",
address = "New York",
year = 1962,
}

This was republished as arXiv:gr-qc/0405109. The equations of motion
are their equations (3.15).

York has published a number of fairly-readable accounts of this material,
and many other researchers follow York's formalism. His 1979 paper
is the "classic" one, and was the key reference for my generation of
numerical relativists:

@incollection
{
York-1979-in-Yellow,
X-author = "James W. York, Jr.",
author = "York, Jr., James W.",
title = "Kinematics and Dynamics of General Relativity",
pages = "83--126",
editor = "Larry L. Smarr",
booktitle = "Sources of Gravitational Radiation",
publisher = "Cambridge University Press",
address = "Cambridge, UK",
year = 1979,
isbn = "0-521-22778-X",
snote = "Proceedings of the Battelle Seattle Workshop,
24 July -- 4 August, 1978",
}

There are also other papers by York with slightly different presentations
of essentially the same material:

@incollection
{
York-1983-in-Red,
X-author = "James W. York, Jr.",
author = "York, Jr., James W.",
title = "The Initial Value Problem and Dynamics",
pages = "175--201",
editor = "Nathalie Deruelle and Tsvi Piran",
booktitle = "Gravitational Radiation",
publisher = "North-Holland",
address = "Amsterdam",
year = 1983,
isbn = "0-444-86560-8",
snote = "Les Houches proceedings, 2--21 June 1982"
}

@incollection
{
York-1985-in-Centrella-LeBlank-Bowers,
X-author = "James W. York, Jr.",
author = "York, Jr., James W.",
title = "Spacetime Engineering",
pages = "176--189",
editor = "Joan M. Centrella and James M. LeBlanc
and Richard L. Bowers",
booktitle = "Numerical Astrophysics",
publisher = "Jones and Bartlett",
address = "Boston",
year = 1985,
isbn = "0-86720-048-0",
snote = "60th birthday festschrift for {J}ames {R}. {W}ilson",
}

@incollection
{
York-Piran-1982-in-Schild-lectures,
X-author = "James W. York, Jr. and Tsvi Piran",
author = "York, Jr., James W. and Tsvi Piran",
title = "The Initial Value Problem and Beyond",
pages = "147--176",
editor = "Richard A. Matzner and Lawrence C. Shepley",
booktitle = "Spacetime and Geometry: The {A}lfred {S}child Lectures",
publisher = "University of Texas Press",
address = "Austin (Texas)",
year = 1982,
isbn = "0-292-77567-9",
}

Finally, Misner, Thorne, & Wheeler present this material in section 21.7,
with the equations of motion appearing as their equation (21.115).

ciao,

--
-- "Jonathan Thornburg [remove -animal to reply]" <jth...@astro.indiana-zebra.edu>
Dept of Astronomy & IUCSS, Indiana University, Bloomington, Indiana, USA
on sabbatical in Canada starting August 2012
"Washing one's hands of the conflict between the powerful and the
powerless means to side with the powerful, not to be neutral."
-- quote by Freire / poster by Oxfam

[Jonathan Thornburg [remove -animal to reply]]

In article <kkkcg8$492$1...@news.albasani.net>,

Gregor Scholten <g.sch...@gmx.de> wrote:
> There are numerous articles about ADM formalism, e.g. this one:
>
> http://www.math.toronto.edu/mccann/assignments/426/Tong.pdf
>
> where they indicate expressions for the Hamiltonian (equation 20 in
> upper article) and the time-derivative of the momentum \p^ij conjugated
> to three-metric h_ij (equation 25). However, I didn't find any article
> where the derivation of \dot \p^ij from the Hamiltonian is shown. In
> every article I found, they only indicate the final result for the
> time-evolution of p^ij (like in equation 25 in upper article), but not
> the calculation that yields this result. Especially the terms with
> second derivatives of the lapse function \alpha, like D^a D^b \alpha,
> seem very strange to me. There are no terms in Hamiltonian where I could
> imagine that those terms originate from them. The terms with
> three-Ricci-tensor obviously originate from the three-Ricci-scalar, the
> terms with p^ab from the p^ab terms in Hamiltonian, but I have no idea,
> where the D^a D^b alpha occurs from.

In article <alpine.BSO.2.00.1...@cobalt.astro.indiana.edu>,
I replied with some general references.

I've just discovered another reference which works through the derivation
in detail, and thus should be more suitable for answering your question:

This reference is Matt Choptuik's notes "The 3+1 Einstein Equations"
from a course he taught in 1998. These notes are (still) online at

http://laplace.physics.ubc.ca/People/matt/Teaching/98Spring/Phy387N/Doc/3+1.ps

[Gregor Scholten]

Jonathan Thornburg [remove -animal to reply] wrote:

> I've just discovered another reference which works through the derivation
> in detail, and thus should be more suitable for answering your question:
>
> This reference is Matt Choptuik's notes "The 3+1 Einstein Equations"
> from a course he taught in 1998. These notes are (still) online at
>
> http://laplace.physics.ubc.ca/People/matt/Teaching/98Spring/Phy387N/Doc/3+1.ps

Thanks. However, this is not what I'm looking for. They derive the
equation of motion from the four-dimensional Einstein field equation,
but not from Lagrangian oder Hamiltonien density.