Talk:Stress-energy-momentum pseudotensor

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Help with this template Comments: edit – history – watch – refresh The more well-known stress energy pseudotensors are: Einstein [1], Papapetrou [2], Bergmann [3], Landau and Lifshitz [4], Moeller [5], and Weinberg [6]. - Lyudmil Antonov, Bulgaria [1] See, e.g., A. Trautman, in Gravitation: an Introduction to Current Research, ed. L. Witten (Wiley, New York, 1962), 169-198. [2] A. Papapetrou, Proc. Roy. Irish Acad. A 52, 11-23 (1948); S. N. Gupta, Phys. Rev. 96, 1683-1685 (1954); this pseudotensor has more recently been rediscovered by D. Bak, D. Cangemi, and R. Jackiw, Phys. Rev. D 49, 5173-5181 (1994). [3] P. G. Bergmann and R. Thompson, Phys. Rev. 89, 400- 407 (1953). [4] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, 2nd ed. (Reading, Mass.: Addison-Wesley, 1962). [5] C. Møller, Ann. Phys. 4, 347-371 (1958). [6] S. Weinberg, Gravitation and Cosmology (Wiley, New York, 1972); the same energy-momentum density is used in [7], §20.3. [7] C.W. Misner, K. Thorne, and J. A.Wheeler, Gravitation (Freeman, San Francisco, 1973). Retrieved from "http://en.wikipedia.org/wiki/Talk:Stress-energy-momentum_pseudotensor"

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1 Move
2 Stress energy momentum complexes (pseudotensors)
3 Definition of LL pseudotensor
4 Levi-Civita connection vs. Christoffel symbols

[edit] Move

Right now, Landau-Lifshitz pseudotensor is a redirect to this article. But this article should be named Landau-Lifshitz pseudotensor (and expanded) since there are many others like Møller complex which probably need their own articles, since they have somewhat different motivations and uses. An article called Stress-energy-momentum pseudotensors should be an umbrella article putting them in context and comparing them.

Just to add to the confusion, there are several variant spellings of "Lifshitz", although this is most common and should be preferred.---CH 22:50, 13 June 2006 (UTC)

I hadn't realised there were so many stress-energy-momentum pseudotensors! Yes, we need an umbrella page to list them. You seem to be suggesting that "Møller complex" is one; "Einstein" and "Landau-Lifshitz" makes three. Any more? --Michael C Price 11:03, 14 June 2006 (UTC)

Moving LL pseudo in a separate article and SEM pseudotensors as umbrella is a good idea. However, we first have to take a good look at the SEM pseudotensors from those listed below to see what is common and what is different and put that unifying material in SEM pseudotensors rather than in the articles on individual pseudotensors. BTW, Michael, good work on putting meat in Landau-Lifshitz pseudotensor. What remains is giving the more usual definitions of LL pseudo in terms of 'metric densities' and as a combination of Christoffel symbols, as given in more recent editions of the LL book (1965 and later) and also uses of LL tensor, eg in gravitational waves research. --Lantonov 06:53, 13 April 2007 (UTC)

I agree. I have LL to hand and will add the Christoffel description soon. I am not familar with the other pseudotensors you listed, so am probably not the best judge of what should go in the SEM pseudo article and what belongs in the LL pseudo article.--Michael C. Price ^talk 07:52, 13 April 2007 (UTC)

[edit] Stress energy momentum complexes (pseudotensors)

The more well-known stress energy pseudotensors are: Einstein [1], Papapetrou [2], Bergmann [3], Landau and Lifshitz [4], Møller [5], and Weinberg [6].Lantonov 06:45, 5 April 2007 (UTC)

[1] See, e.g., A. Trautman, in Gravitation: an Introduction to Current Research, ed. L. Witten (Wiley, New York, 1962), 169-198.

[2] A. Papapetrou, Proc. Roy. Irish Acad. A 52, 11-23 (1948); S. N. Gupta, Phys. Rev. 96, 1683-1685 (1954); this pseudotensor has more recently been rediscovered by D. Bak, D. Cangemi, and R. Jackiw, Phys. Rev. D 49, 5173-5181 (1994).

[3] P. G. Bergmann and R. Thompson, Phys. Rev. 89, 400- 407 (1953).

[4] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, 2nd ed. (Reading, Mass.: Addison-Wesley, 1962).

[5] C. Møller, Ann. Phys. 4, 347-371 (1958).

[6] S. Weinberg, Gravitation and Cosmology (Wiley, New York, 1972); the same energy-momentum density is used in [7], §20.3.

[7] C.W. Misner, K. Thorne, and J. A.Wheeler, Gravitation (Freeman, San Francisco, 1973).

Retrieved from "http://en.wikipedia.org/wiki/Talk:Stress-energy-momentum_pseudotensor"

Note [7] is the same as [4], is the same as [6]. --Michael C. Price ^talk 08:28, 23 July 2007 (UTC)

[edit] Definition of LL pseudotensor

On further reflection, I feel a growing unease about the definition $t_{LL}^{\mu \nu} = - \frac{1}{8\pi G}(G^{\mu \nu}) + \frac{1}{16\pi G (-g)}((-g)(g^{\mu \nu}g^{\alpha \beta} - g^{\mu \alpha}g^{\nu \beta})),_{\alpha \beta}$

Two points for discussion spring immediately:

1. Inclusion of the Einstein tensor $G μν$ in the formula for $t_{LL}^{\mu \nu}$ is questionable on the ground that $\frac{1}{8\pi G}(G^{\mu \nu}) = T^{\mu \nu}$ (the Einstein eqs), the rhs being the stress-energy-momentum tensor for matter. Since $t_{LL}^{\mu \nu}$ is supposed to represent the 'pure' gravitational stress-energy-momentum, ie that only of the gravitational field without matter, inclusion of matter-related quantities in its definition is disconcerting.

2. LL make up the quantitity $(( - g)(g μν g αβ - g μα g νβ)), αβ$ in such a way as to vanish upon differentiation in a special reference frame, chosen such that all first derivatives of the metric are zero. In the general case, the first derivatives of the metric are not zero, and this necessitated the long calculation, at the end of which LL found the expression of $t_{LL}^{\mu \nu}$ in terms of Christoffel symbols (ie first derivatives of the metric). After another long calculation, they found the expression in first derivatives of 'metric densities' which is the starting point for most applications of the LL pseudotensor. --Lantonov 11:01, 13 April 2007 (UTC)

Re 1. I understand your unease, but

G μν

is defined, via Riemann's

R μν

, in terms of Christoffel symbols, so

G μν

is a purely geometric object. Assuming that the Christoffel expression LL give matches (which I haven't checked yet) does that allay your concerns?

Re 2. Yes, LL do most of their calculation in a frame where the first derivatives of the metric vanish; but the result they get hinges upon:

(( - g)(g μν g αβ - g μα g νβ)), αβμ = 0

which algebraically holds in all frames due to antisymmetry.

--Michael C. Price ^talk 11:56, 13 April 2007 (UTC)

1. Of course $G μν$ is purely geometric object, but its magnitude depends on matter since 'matter curves space'. This is the whole meaning of the Einstein eqs. Barring a possible cosmological constant, $G μν = 0$ in the absence of matter. I didn't understand the part about 'matches'. If you have in mind doing the calculation of (96,8) and (96,9), I have done it in several ways, by hand and with the help of 'Mathematica' with 'Tensorial' package and they match the expressions in the LL book. BTW, Synge gives a much easier algorithm for calculating (96,8) with the help of definitions for the covariant derivative. 2. This is right, $(( - g)(g μν g αβ - g μα g νβ)), αβμ = 0$ which algebraically holds in all frames due to antisymmetry. I knew this but I had in mind something else. Nevermind, disregard this question until I formulate it better. --Lantonov 15:55, 13 April 2007 (UTC)

1. (Note that I've corrected a typo in my previous post.) Yes, by "matches" I meant checking that the expression for $t_{LL}^{\mu \nu}$ in terms of

G μν

is equivalent to #96.8 and #96.9. Your point about the geometric terms having the same magnitude as the matter tensor still holds, no matter whether we express the result in terms of

G μν

, metric tensor densities or the connection/Christoffel symbols. It really comes down to taste, I think; and I intend to add the other expressions as well. Thanks for the Synge tip.--Michael C. Price ^talk 20:30, 13 April 2007 (UTC)

[edit] Levi-Civita connection vs. Christoffel symbols

Just another nag before you continue: LLP is expressed by Christoffel symbols which are components of the Levi-Civita connection, not by the Levi-Civita connection in its coordinate-independent sense. --Lantonov 09:41, 18 April 2007 (UTC)

Thanks, I'll look further into this: my understanding was that Christoffel symbols are defined in terms of the metric, whilst the LC connection is defined in terms of the covariant derivative. In the absence of torsion tensor they become identical. The coordinate independent approach is an area I am currently weak on.--Michael C. Price ^talk 09:56, 18 April 2007 (UTC)

Also, in #96,8 $t i k$ is in the terms of $\mathfrak{g}^{ik} = \sqrt{-g} g^{ik}$ which is a metric density in accordance with the convention that $\sqrt{-g}A^{ik}$ is called a density of a tensor (see LL footnote in ch. 83). Well, it is true that something very similar to #96,8 was found by Babak and Grishchuk which is in terms of pure metric, not of metric density. However, the approach there is different and the relationship of BG and LL entities is not straightforward --Lantonov 09:41, 18 April 2007 (UTC)