Talk:Stress–energy–momentum pseudotensor

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)

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)

The original work for the Einstein pseudotensor is:

• Einstein A. Das hamiltonisches Prinzip und allgemeine Relativitätstheorie. Sitzungsber. preuss. Acad. Wiss. 1916, 2, 1111-1116.

Additional work of Einstein defending its existence:

• Einstein A. Der Energiesatz in der allgemeinen Relativitätstheorie. Sitzungsber. preuss. Acad. Wiss. 1918, 1, 448-459. --Lantonov (talk) 07:03, 1 July 2008 (UTC)

Here [1] is the paper of Babak and Grishchuk, deriving their gravitational pseudotensor. I just found this very recent paper [2], 15 July 2008, which casts doubt on the physical significance of the Babak-Grishchuk gravitational stress-energy momentum tensor. --Lantonov (talk) 16:09, 24 July 2008 (UTC)

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 $G^{\mu \nu }$ 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 $((-g)(g^{\mu \nu }g^{\alpha \beta }-g^{\mu \alpha }g^{\nu \beta })),_{\alpha \beta }$ 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 $G^{\mu \nu }$ is defined, via Riemann's $R^{\mu \nu }$, in terms of Christoffel symbols, so $G^{\mu \nu }$ 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: $((-g)(g^{\mu \nu }g^{\alpha \beta }-g^{\mu \alpha }g^{\nu \beta })),_{\alpha \beta \mu }=0$ which algebraically holds in all frames due to antisymmetry.

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

1. Of course $G^{\mu \nu }$ 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, $G^{\mu \nu }=0$ 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, $((-g)(g^{\mu \nu }g^{\alpha \beta }-g^{\mu \alpha }g^{\nu \beta })),_{\alpha \beta \mu }=0$ 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 $G^{\mu \nu }$ 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 $G^{\mu \nu }$, 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)

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)

I thought that the "modern" view was that the LC connection is the "true" underlying dynamical object, while the metric pops out as a side effect. That's the viewpoint that the use of fibre bundles suggests. (in the same way that A is the dynamical object in E&M, not E+B). 67.198.37.16 (talk) 01:21, 11 April 2016 (UTC)

Also, in #96,8 $t^{ik}$ 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)

Wrong conservation law?

To MichaelCPrice: Gravitation (book) agrees with you that there are no square-roots there. What confused me was the supposed conservation law, $((-g)(T^{\mu \nu }+t_{LL}^{\mu \nu }))_{,\mu }=0$, which the LL pseudo-tensor satisfies. I believed that the correct conservation law would have to have the form $({\sqrt {-g}}(T^{\mu \nu }+t_{LL}^{\mu \nu }))_{,\mu }=0$. How do you justify using $(-g)T^{\mu \nu }\!$ rather than ${\sqrt {-g}}T^{\mu \nu }$ for the density of non-gravitational stress-energy? JRSpriggs (talk) 12:29, 27 June 2008 (UTC)

In Paul Dirac's book General Theory of Relativity (1975), he derives a pseudo-tensor for gravitational stress-energy in chapters 31 & 32 (pages 61 to 63). It satisfies

$[(t_{\mu }^{\nu }+Y_{\mu }^{\nu }){\sqrt {-g}}]_{,\nu }=0$

where $t_{\mu }^{\nu }$ is his pseudo-tensor for gravitational stress-energy and $Y_{\mu }^{\nu }$ is his non-gravitational stress-energy tensor which appears in the Einstein field equations, i.e. it is what we call $T_{\mu }^{\nu }$. In other words, Dirac's version works with the square-root in the conservation law. JRSpriggs (talk) 13:21, 27 June 2008 (UTC)

It is $-g$ and not ${\sqrt {-g}}$, and I am absolutely certain about this. In this formula, $-g$ has nothing to do with tensor density. $-g$ was obtained during the derivation of the formula 101,2 in LL by pulling it out of the derivative ${\frac {\partial }{\partial x^{l}}}$ to make ${\frac {\partial h^{ikl}}{\partial x^{l}}}$ symmetric. As LL write in the footnote: "This is why we pulled $-g$ from under the derivative by $x^{l}$ in the expression for $T^{ik}$ Otherwise ${\frac {\partial h^{ikl}}{\partial x^{l}}}$ and therefore, also $t^{ik}$ would be non-symmetric by i and k." LL thus derive the formula ${\frac {\partial h^{ikl}}{\partial x^{l}}}=(-g)T^{ik}$ because before this they had ${\frac {1}{(-g)}}$ in front of ${\frac {\partial h^{ikl}}{\partial x^{l}}}$ See also J. L. Synge (1960) General relativity, formula 6.148 where it is g instead of - g, $W^{ik}=\varkappa g\left(T^{ik}+t^{ik}\right)$ because Synge uses -+++ metric while LL use +--- metric. Dirac (formla 31.1) does another calculation in which he does not derive the LL pseudotensor, as it is given here. Dirac begins with $t_{\mu }^{\nu }{\sqrt {g}}$ (formula 31.1) but does not pass through the LL route. --Lantonov (talk) 14:01, 27 June 2008 (UTC)

In special relativity (i.e. the Minkowski metric) with arbitrary curvilinear coordinates, if $\xi ^{\mu }$ is a Killing vector of the metric, then we get that

$(\xi ^{\mu }T_{\mu }^{\nu }{\sqrt {-g}})_{,\nu }=0$

expresses the conservation law associated with the symmetry generated by the Killing vector field. For appropriate choices of that field, this includes conservation laws for: energy, linear momentum, and angular momentum. If we are going to generalize this to curved metrics so that we can include gravity, then the result should look something like this. In particular, the factor of ${\sqrt {-g}}$ seems unavoidable. Of course, there may not be any Killing vectors, but perhaps one could weaken that condition to a vector field which acts like a Killing vector in the limit at infinity, i.e. separate the metric into a symmetrical background metric and a perturbation. JRSpriggs (talk) 19:03, 27 June 2008 (UTC)

In the article this part of stress-energy-momentum is not treated at all and this is why the confusion. In LL, it is much better explained, and more logical. LL start with the conservation law in the absence of gravitational field which is ${\frac {\partial T^{ik}}{\partial x^{k}}}\equiv T_{,k}^{ik}=0$. When gravitational field is present, we have a curved space, and this means that the partial derivative "," must become covariant derivative ";" so instead of $T_{,k}^{ik}=0$ we have $T_{;k}^{ik}=0$. Because $T^{ik}$ is a symmetric tensor, using formula (86,11) from LL we have $T_{i;k}^{k}={\frac {1}{\sqrt {-g}}}{\frac {\partial \left(T_{i}^{k}{\sqrt {-g}}\right)}{\partial x^{k}}}-{\frac {1}{2}}{\frac {\partial g_{kl}}{\partial x^{i}}}T^{kl}=0$. Here the fly in the ointment is the term ${\frac {1}{2}}{\frac {\partial g_{kl}}{\partial x^{i}}}T^{kl}$. Were it not for this term, we would have $\left(T_{i}^{k}{\sqrt {-g}}\right)_{,k}=0$ and conservation of the integral $\int T_{i}^{k}{\sqrt {-g}}dS_{k}$. There is no way for this term to be avoided. LL chose a locality in which all first derivatives of the metric disappear (so getting rid of the objectionable term), and start their derivations from this point on. After this, they find a totally antisymmetric tensor whose second derivatives identically disappear, and so on and so on. There is just too much missing from the article. --Lantonov (talk) 10:36, 28 June 2008 (UTC)

To Lantonov: At the risk of re-stating the obvious, I agree that LL's pseudo-tensor satisfies the first three conditions. As the article states, they are easily checked. And I am willing to believe condition #4 holds in the absence of evidence to the contrary. My problem is that I think that condition #3 is the wrong criterion. Instead, the criterion that the pseudotensor should satisfy is

$3'\quad (\xi ^{\mu }(T_{\mu }^{\nu }+t_{\mu }^{\nu }){\sqrt {-g}})_{,\nu }=0$

for any $\xi ^{\mu }$ which is a Killing vector of the background metric, provided that the perturbation to the metric approaches zero at infinity sufficiently rapidly. I believe that this is different from the #3 in the article.

• Do you think that I should try to put Dirac's pseudotensor into the article in another section? JRSpriggs (talk) 13:13, 28 June 2008 (UTC)

Definitely yes. This would be a valuable addition. --Michael C. Price ^talk 22:08, 30 June 2008 (UTC)

Is Dirac's pseudotensor symmetric? --Michael C. Price ^talk 19:37, 1 July 2008 (UTC)

Yes, of course. Einstein pseudotensor (Dirac), and the other pseudotensors (SEM complexes), too. --Lantonov (talk) 05:41, 1 July 2008 (UTC)

P.S. With respect to Synge, the determinant of the metric, g, is negative regardless of whether you choose +--- or -+++ as the signature. Both have an odd number of negative numbers on the main diagonal, and hopefully zeros (or negligible values) off the diagonal in an appropriately chosen local coordinate system. JRSpriggs (talk) 13:55, 28 June 2008 (UTC)

The conserved quantity in the LL treatment is $h_{,l}^{ikl}=\left[(-g)\left(g^{ik}g^{lm}-g^{il}g^{km}\right)\right]_{,ml}$ where $h^{ikl}$ is a completely anti-symmetric tensor that identically disappears after second-differentiating it by k:$h_{,lk}^{ikl}=0$. Formula (96,5) in the English edition of LL book is: $(-g)\left(T^{ik}+t^{ik}\right)=h_{,l}^{ikl}$ so it comes out that $(-g)\left(T^{ik}+t^{ik}\right)$ is a conserved quantity, too. This is very important. To Michael C. Price in relation to discussion in Landau-Lifshitz pseudotensor:in words we have "matter/radiation tensor + gravitational pseudotensor = total stress-energy-momentum tensor". The total SEM is a tensor, and not pseudotensor because it is conserved regardless of any coordinate conditions. Invariance is a property of tensors, not of pseudotensors, especially in the loose sense that the term "pseudotensor" is used in GR. Also the total SEM is a tensor, and not its derivative. Its derivative by k is 0 which constitutes the conservation law. Back to present discussion: In the conservation law in GR we have no Killing vector. As far as I know, this is connected to the lack of symmetry in $t^{ik}$ but for more details I have to dig in the literature. Also, Killing vector cannot be put under integral in GR, because integration of vector in curved space depends on the path of integration, so the direction of any vector after integration will be different depending on the path. In other words, we have no integration operation which is the opposite of covariant differentiation. Absence of Killing field is very unfortunate for GR because it means that we do not have a way to preserve the metric. This is one of the big problems for GR. If you insist to work with tensor densities, it is still possible. For instance, dividing eq. (96,5) in LL by ${\sqrt {(}}-g)$ gives tensor densities for both $T^{ik}$ and $t^{ik}$. However, such division would be artificial and premature. It is better to be postponed until (96,11) where it is more natural: (96,11) $P^{i}={\frac {1}{c}}\int (-g)\left(T^{ik}+t^{ik}\right)dS_{k}$. Here $P^{i}$ is the 4-momentum (energy + 3-momentum) that is conserved, and the integral is taken over any infinite hypersurface $S_{k}$, including all of the 3d space. In order to express $P^{i}$ in tensor densities we can do the steps given in the problem to chapter 96 and derive a complicated expression in which all the tensors: Einstein, matter/radiation SEM, even the metric tensor are given in terms of tensor densities. However, the gravitational pseudotensor derived in this way is not symmetric for i and k so it does not conserve the angular momentum. For the conditions (conditions 3, 4 that you refer to), I must say that this article was not written by me so I don't understand what is the point of having points. If we take them as criteria, or conditions for something, we must have other things to compare, or relate to. We have only LL pseudotensor now. This brings me to the second topic. In his booklet Dirac derives the Einstein's gravitational pseudotensor. Einstein pseudotensor is listed along with other "stress-energy-momentum complexes" in the short list that I made above in this discussion. I do not allege that this list is complete but made this to suggest that this article be an umbrella article including as full as possible all the various gravitational pseudotensors, comparing their advantages and disadvantages. I have about a dozen papers with such comparisons stressing some more important criteria, as, for instance, what is conserved, how local is conservation, etc. The biggest advantage of LL pseudotensor I know of, is that it is the only possible pseudotensor that conserves angular momentum (proven by LL); otherwise, the 4-momentum is conserved by all listed pseudotensors. I would do this comparison myself but for the moment I have my hands full with other articles, most notably BKL singularity. For the metric determinant you are right, it is negative as long as signature has odd number of negative signs. This goes also for GR with non-diagonal metric. To quote LL: "By an appropriate choice of coordinates, we can, however, bring the quantities $g_{ik}$ to galilean form at any individual point of the non-galilean space-time: this amounts to the reduction to diagonal form of a quadratic form with constant coefficients (the values of $g_{ik}$ at the given point). Such a coordinate system is said to be galilean for the given point. We note that, after reduction to diagonal form at a given point the matrix of the quantities $g_{ik}$ has one positive and three negative principal values. From this it follows, in particular that the determinant g, formed from the quantities $g_{ik}$ is always negative for a real space-time: g < 0." --Lantonov (talk) 08:45, 30 June 2008 (UTC)

Proof

Here I prove $(\xi ^{\mu }g_{\mu \sigma }T^{\sigma \nu }{\sqrt {-g}})_{,\nu }=0$ in any coordinate system with any amount of curvature, provided ξ is a Killing vector. Let us begin by considering the covariant derivatives. According to Killing's equation:

$\xi _{\sigma ;\nu }+\xi _{\nu ;\sigma }=0\!$

as stated in the article on Killing vector field and in MTW page 651. In words, the covariant derivative of the covariant version $\xi _{\sigma }=\xi ^{\mu }g_{\mu \sigma }\!$ of a Killing vector field is anti-symmetric. Consequently,

$(\xi ^{\mu }g_{\mu \sigma })_{;\nu }T^{\sigma \nu }=0\!$

using the symmetry of the non-gravitational stress-energy. Thus

$(\xi ^{\mu }g_{\mu \sigma }T^{\sigma \nu }{\sqrt {-g}})_{;\nu }=(\xi ^{\mu }g_{\mu \sigma })_{;\nu }T^{\sigma \nu }{\sqrt {-g}}\,+\,\xi ^{\mu }g_{\mu \sigma }{T^{\sigma \nu }}_{;\nu }{\sqrt {-g}}\,+\,\xi ^{\mu }g_{\mu \sigma }T^{\sigma \nu }({\sqrt {-g}})_{;\nu }$ $=0+0+0=0.\!$

This is the covariant divergence of a contravariant vector density. Notice that the covariant divergence of a contravariant vector density is the same as its partial divergence. And so,

$(\xi ^{\mu }g_{\mu \sigma }T^{\sigma \nu }{\sqrt {-g}})_{,\nu }=(\xi ^{\mu }g_{\mu \sigma }T^{\sigma \nu }{\sqrt {-g}})_{;\nu }=0$

which is what was to be proved. As I said above, this law is integrable and includes the conservation of energy, linear momentum, and angular momentum in spacetimes with the corresponding symmetries. JRSpriggs (talk) 04:48, 3 July 2008 (UTC)

I don't understand the logic of

$(\xi ^{\mu }g_{\mu \sigma }T^{\sigma \nu }{\sqrt {-g}})_{;\nu }=(\xi ^{\mu }g_{\mu \sigma })_{;\nu }T^{\sigma \nu }{\sqrt {-g}}\,+\,\xi ^{\mu }g_{\mu \sigma }T_{;\nu }^{\sigma \nu }{\sqrt {-g}}\,+\,\xi ^{\mu }g_{\mu \sigma }T^{\sigma \nu }({\sqrt {-g}})_{;\nu }$ $=0+0+0=0.\!$

Agree that $(\xi ^{\mu }g_{\mu \sigma })_{;\nu }T^{\sigma \nu }{\sqrt {-g}}=0$ and $\xi ^{\mu }g_{\mu \sigma }T^{\sigma \nu }({\sqrt {-g}})_{;\nu }=0$

Why $\xi ^{\mu }g_{\mu \sigma }T_{;\nu }^{\sigma \nu }{\sqrt {-g}}=0$ ?

Also, $(\xi ^{\mu }g_{\mu \sigma }T^{\sigma \nu }{\sqrt {-g}})$ is a vector. In curved space, in general, the covariant derivative of a vector is different from its ordinary derivative.

--Lantonov (talk) 10:56, 3 July 2008 (UTC)

I am, of course, using the product rule for covariant differentiation. I am using $T^{\sigma \nu }\!$ for the stress-energy tensor of matter and radiation as I thought was our convention here. That its covariant divergence is zero, ${T^{\sigma \nu }}_{;\nu }=0\!$, is well known. Multiplying by the other factors (outside the derivative) should not affect that. Did you think that I was talking about the pseudotensor for gravitational stress-energy? My theorem is relevant to that only because it provides a model for what a more general expression of stress-energy (including both gravity and matter/radiation) should do.

This is not an ordinary vector, it is a contravariant vector density. The law for the covariant derivative of a vector density (of weight w) is

${{\mathfrak {A}}^{\alpha }}_{;\beta }={{\mathfrak {A}}^{\alpha }}_{,\beta }+{\mathfrak {A}}^{\gamma }{\Gamma ^{\alpha }}_{\gamma \beta }-w{\mathfrak {A}}^{\alpha }{\Gamma ^{\delta }}_{\delta \beta }$

Since we are taking a divergence, we sum over α = β from 0 to 3, getting

${{\mathfrak {A}}^{\beta }}_{;\beta }={{\mathfrak {A}}^{\beta }}_{,\beta }+{\mathfrak {A}}^{\gamma }{\Gamma ^{\beta }}_{\gamma \beta }-w{\mathfrak {A}}^{\beta }{\Gamma ^{\delta }}_{\delta \beta }$

When w=1 as in our case, the last two terms cancel each other yielding

${{\mathfrak {A}}^{\beta }}_{;\beta }={{\mathfrak {A}}^{\beta }}_{,\beta }$

Is that a satisfactory explanation? JRSpriggs (talk) 14:51, 3 July 2008 (UTC)

Yes, but would that not be true for any power of $(-g)^{m}$ since:

$(\xi ^{\mu }g_{\mu \sigma }T^{\sigma \nu }(-g)^{m})_{;\nu }=0\,$

by the product rule again?

Also a general space only admits a killing vector $\xi ^{\mu }=\delta x^{\mu }\,$ which, if true (?), means that we effectively only have:

$(\delta x_{\sigma }T^{\sigma \nu }(-g)^{m})_{;\nu }=0\,$

and I'm not sure I see the physical relevance of this. What is the Noether current here? --Michael C. Price ^talk 15:36, 3 July 2008 (UTC)

I was thinking that $T^{\sigma \nu }\,$ is the total SEM = matter/radiation + gravitational field. If it is only matter/radiation, no objection. Repeating myself, LL derive something very close to your final result in terms of variation in the action integral as conservation laws (LL eq. 94,6-94,7)

$\delta S=-{\frac {1}{c}}\int T_{i;k}^{k}\xi ^{i}{\sqrt {-g}}d\Omega =0$

where $T_{i}^{k}$ is the total SEM of a system and $\xi ^{i}$ is the usual Killing vector. However, they emphasize that this formula applies only to physical systems characterized by quantities different from $g_{ik}$; therefore it cannot be applied to the gravitational field which is determined by the quantities $g_{ik}$ themselves.

My second objection is answered, too, by showing that the Christoffel terms cancel in vector densities. Sorry, I didn't pay attention to your statement to that effect above. --Lantonov (talk) 15:59, 3 July 2008 (UTC)

To MichaelCPrice: No. The weight, w, is the sum of the weights of the factors. $\xi ^{\mu }$, $g_{\mu \sigma }$, and $T^{\sigma \nu }$ are ordinary tensors with weight 0. ${\sqrt {-g}}$ is the canonical scalar density of weight 1. In general, $(-g)^{m}$ being the product of 2m copies of the square-root has weight 2m. So unless m is half, the weight will not be 1 and the last line in my proof would fail because the two terms involving $\Gamma$ would no longer cancel in expression for the covariant derivative in terms of the partial derivative.

In a space with random curvature, there are no Killing vectors as far as I know. In Minkowski space (i.e. flat spacetime) with Cartesian coordinates, there are ten independent Killing vectors. For any K=0,1,2,3, we get that $\xi ^{\mu }=\delta _{K}^{\mu }$ (using Kronecker delta) are four of those Killing vectors corresponding to translations in the t, x, y, and z directions. Three of the vectors are rotations such as ${\vec {\xi }}={\hat {z}}\times {\vec {r}}$ for rotation around the z-axis. The last three generate Lorentz boosts. I do not know what you mean by your "Killing vector". JRSpriggs (talk) 18:50, 3 July 2008 (UTC)

I got my logic wrong about Killing vectors. I was thinking that $\delta g_{\mu \nu }=-x_{\mu ;\nu }-x_{\nu ;\mu }=0$, but of course the zero assumes that $\delta x_{\mu }$ is already a killing vector, which it isn't in general.

However I'm still confused about tensor densities; does the product rule apply to their covariant derivatives or not? And is $(g^{m})_{;\mu }=0$? My notes only give the case for m=1 and m=1/2. --Michael C. Price ^talk 08:06, 4 July 2008 (UTC)

Okay, I see why we are talking past each other. I am asking about the covariant divergence, which vanishes for :

$(\xi _{\sigma }T^{\sigma \nu }(-g)^{m})_{;\nu }=0\,$

whereas you are interested in the ordinary divergence, which only vanishes with m=1/2:

$(\xi _{\sigma }T^{\sigma \nu }{\sqrt {-g}})_{,\nu }=0\,$

--Michael C. Price ^talk 09:29, 4 July 2008 (UTC)

To Lantonov: I am pleased that you understand it now. Indeed, the theorem as it stands applies only to non-gravitational stress-energy. However, I consider it to be the proper expression of the pre-GTR conservation laws. A generalization to G≠0 and curved space should reduce to this expression in the limit of flat space. LL's formula does not do that. And neither does Dirac's. JRSpriggs (talk) 19:11, 3 July 2008 (UTC)

To MichaelCPrice: Yes, the product rule for covariant differentiation applies to tensor densities of any weight. And the covariant derivative of any power of -g is zero.

Indeed, I want the partial divergence to be zero because then we can integrate it to get

$0=\int _{\partial }\xi ^{\mu }g_{\mu \sigma }T^{\sigma \nu }{\sqrt {-g}}\,d^{3}s_{\nu }$

which expresses the idea that the net flux of momentum in the ξ direction across the boundary of the region is zero. In other words, that the momentum in the ξ direction is conserved globally.

If we are going to generalize this to curved spaces which do not admit Killing vectors, then the more general expression should reduce to this when there is a Killing vector. This is the correspondence principle -- the more general theory must reduce to the established & verified theory of the special case when that special case occurs. JRSpriggs (talk) 11:37, 4 July 2008 (UTC)

Total SEM a pseudotensor or tensor?

Hi Lantonov,

for readability I'm spinning this issue off into a new section. Our L-L equation numbers seem the same; the invariant total SEM entity is (95.10)

$((-g)\left(T^{ik}+t^{ik}\right)),k=0$

This equation, as you observe, is clearly a tensor since it is invariant, but I stress that this is only a property of the equation taken as a whole and not a property of each component term; since it is the ordinary derivative that appears here, not the covariant derivative, it does not follow that either $T^{ik}\,$ or $t^{ik}\,$ are also tensors -- although, as it happens, $T^{ik}\,$ is a tensor. If your argument were correct then this would imply that either $t^{ik}\,$ or $(-g)t^{ik}\,$ is a tensor (since a tensor + X = another tensor implies X is a tensor) and that isn't so (which is why we are forced to call $t^{ik}\,$ that ill-defined term, a pseudotensor).

PS I included point 4 about the lack/cancellation of the 2nd derivatives in the article since, combined with index symmetry, it appears to force the uniqueness of the form of the LL pseudotensor (see footnote between (96.12/13)). Perhaps this is what forces the appearance of (-g) and not its square root in the equation? -- I'm not sure.

--Michael C. Price ^talk 10:02, 30 June 2008 (UTC)

The whole point of the derivation of $t^{ik}$ is to make total SEM vanish on ordinary differentiation, as required by the conservation law, and thus to avoid the term appearing under covariant differentiation. That's why $t^{ik}$ contains all the first derivatives that must cancel to obtain the zero. The fact that we obtain zero means that $((-g)\left(T^{ik}+t^{ik}\right))$ is a constant in all the space irrespective of the coordinate system. It cannot be a pseudotensor if it behaves as such. If it is a pseudotensor in the mathematical sense, for instance, it must change sign when changing coordinate systems from left to right. In the loose sense of GR, it must change in a not easily predictable way when moving it in another locality. Only tensors can be constants in GR. The argument that it must disappear only by covariant differentiation to be a tensor is not valid because, for instance, for scalars (0-tensors) ordinary and covariant derivatives are the same. It all depends on whether the Christoffel terms in the covariant derivatives cancel or not. LL make it so that they cancel for the total SEM. The formula tensor + X = another tensor does not automatically imply that X is a tensor, because if X is specially chosen so that the sum disappears after ordinary differentiation, then X may be anything, not necessarily a tensor. If instead of ordinary differentiation in $((-g)\left(T^{ik}+t^{ik}\right)),k=0$ we had covariant differentiation, then the argument of $t^{ik}$ being a tensor would be valid. To complicate already complicated situation, we must differentiate also the determinant g which will spit a slew of metric tensors and their derivatives all over the place. This is my reasoning, at least, and I may well be wrong. Looking in different formulas I find con and pro. For instance, compare $T^{ik}=\eta _{,l}^{ikl}$ and $(-g)\left(T^{ik}+t^{ik}\right)=h_{,l}^{ikl}$. This supports $(-g)\left(T^{ik}+t^{ik}\right)$ being a tensor. On the other hand, $t^{ik}$ changing in response to metric while $T^{ik}$ remaining constant supports your thesis. This confounds me, so let's pass that on. --Lantonov (talk) 10:49, 30 June 2008 (UTC)

I cited above the footnote between 96.12/13 from the Russian edition (in various Russian editions the chapter on pseudotensors is 101-103 whereas in the most widespread English edition it is 96). For clearness, I paste it here from the English edition: "For just this purpose we took (-g) out from under the derivative sign in the expression for $T^{ik}$. If this had not been done, $h_{,l}^{ikl}$ and therefore also $t^{ik}$ would turn out not to be symmetric in i and k." The purpose LL refer to is for:"... the derivatives $h_{,l}^{ikl}$ [be] symmetric quantities." --Lantonov (talk) 11:32, 30 June 2008 (UTC)

Thanks Lantonov for drawing my attention to this footnote -- I was referring to the next footnote in my previous comment -- which explains the source of (-g) in the expression. It took awhile, but I'm beginning to see where they're coming from.--Michael C. Price ^talk 16:22, 30 June 2008 (UTC)

To Mike: Criterion #4 is essential because otherwise we could just let $t^{\mu \nu }=-{\frac {c^{4}}{8\pi G}}(G^{\mu \nu })$ which would result in $T^{\mu \nu }+t^{\mu \nu }=0$ which has lots of nice properties except that it is utterly useless.

To Lantonov: You said "The fact that we obtain zero means that $((-g)\left(T^{ik}+t^{ik}\right))$ is a constant in all the space irrespective of the coordinate system.". This is false. This is a divergence. To get that the function is constant everywhere you would need the gradient to be zero. JRSpriggs (talk) 14:44, 30 June 2008 (UTC)

Thanks JRSpriggs for the #4 observation. I had actually wondered why LL didn't define $t^{\mu \nu }=-{\frac {c^{4}}{8\pi G}}(G^{\mu \nu })$ -- after all triviallity is hardly a reason for rejection, but #4 provides the justification. --Michael C. Price ^talk 16:22, 30 June 2008 (UTC)

OK, this may probably explain some things. I will think about it. --Lantonov (talk) 14:51, 30 June 2008 (UTC)

Yes, it is divergence, and LL utilize (32,7) and from 32,3 and 32,4 they have $\psi _{,kl}^{ikl}=0$ so that is the primary source of their derivations. --Lantonov (talk) 15:23, 30 June 2008 (UTC)

LL derive something very close to what was proposed by JRSpriggs in terms of variation in the action integral as conservation laws (LL eq. 94,6-94,7)

$\delta S=-{\frac {1}{c}}\int T_{i;k}^{k}\xi ^{i}{\sqrt {-g}}d\Omega =0$

where $\xi ^{i}$ is the usual Killing vector. However, they emphasize that this formula applies only to physical systems characterized by quantities different from $g_{ik}$; therefore it cannot be applied to the gravitational field which is determined by the quantities $g_{ik}$ themselves. --Lantonov (talk) 16:00, 30 June 2008 (UTC)

Dirac's pseudotensor

Above MichaelCPrice (talk · contribs) asked whether Dirac's pseudotensor is symmetric. It is not manifestly symmetric. Whether there is a hidden symmetry, I do not know. Adjusting the notation (and raising the lower index) to agree with that used here, his formula becomes:

$t^{\mu \nu }={\frac {c^{4}}{16\pi G{\sqrt {-g}}}}(g^{\mu \rho }(g^{\alpha \beta }{\sqrt {-g}})_{,\rho }(\Gamma _{\alpha \beta }^{\nu }-\delta _{\beta }^{\nu }\Gamma _{\alpha \sigma }^{\sigma })-g^{\mu \nu }g^{\alpha \beta }(\Gamma _{\alpha \beta }^{\sigma }\Gamma _{\sigma \rho }^{\rho }-\Gamma _{\alpha \sigma }^{\rho }\Gamma _{\beta \rho }^{\sigma }){\sqrt {-g}}).$

OK? JRSpriggs (talk) 21:01, 1 July 2008 (UTC)

Trying to get rid of the square-root

$(g^{\alpha \beta }{\sqrt {-g}})_{,\rho }={g^{\alpha \beta }}_{,\rho }{\sqrt {-g}}+g^{\alpha \beta }({\sqrt {-g}})_{,\rho }=(-g^{\sigma \beta }\Gamma _{\sigma \rho }^{\alpha }-g^{\alpha \sigma }\Gamma _{\sigma \rho }^{\beta }+g^{\alpha \beta }\Gamma _{\rho \sigma }^{\sigma }){\sqrt {-g}}$

and thus

$t^{\mu \nu }={\frac {c^{4}}{16\pi G}}(g^{\mu \rho }(-g^{\sigma \beta }\Gamma _{\sigma \rho }^{\alpha }-g^{\alpha \sigma }\Gamma _{\sigma \rho }^{\beta }+g^{\alpha \beta }\Gamma _{\rho \sigma }^{\sigma })(\Gamma _{\alpha \beta }^{\nu }-\delta _{\beta }^{\nu }\Gamma _{\alpha \sigma }^{\sigma })-g^{\mu \nu }g^{\alpha \beta }(\Gamma _{\alpha \beta }^{\sigma }\Gamma _{\sigma \rho }^{\rho }-\Gamma _{\alpha \sigma }^{\rho }\Gamma _{\beta \rho }^{\sigma })).$

JRSpriggs (talk) 00:43, 2 July 2008 (UTC)

Excellent -- it if was symmetric then LL would have been incorrect in their uniqueness claim. The lack of symmetry means that it cannot be extended to include conservation of angular momentum. The Einstein SEM tensor, IIRC, also lacked symmetry. --Michael C. Price ^talk 05:21, 2 July 2008 (UTC)

This Dirac pseudotensor seems very similar, and maybe identical, to that given in LL book that I pointed to above. In the LL English edition, it is derived in the problem after section 96 (p. 285, just above section 97). It is:

$t_{i}^{k}={\frac {c^{4}}{16\pi k}}\left[G{\sqrt {-g}}\delta _{i}^{k}+\Gamma _{lm}^{k}\left(g^{lm}{\sqrt {-g}}\right)_{,i}-\Gamma _{ml}^{l}\left(g^{mk}{\sqrt {-g}}\right)_{,i}\right]$

especially when G is represented by the LL formula 93,3:

$G=g^{ik}\left(\Gamma _{il}^{m}\Gamma _{km}^{l}-\Gamma _{ik}^{l}\Gamma _{lm}^{m}\right)$

LL write about it: "[This] gives the four-momentum of the gravitational field in the absence of matter. The integrand is not symmetric in the indices i, k, so that one cannot formulate a law of conservation of angular momentum." --Lantonov (talk) 07:11, 2 July 2008 (UTC)

If you look more closely in the way that it is derived, you will see that this is, in fact, the Einstein pseudotensor. It is enough to compare Einstein (formula (20) in Einstein A. Das hamiltonisches Prinzip und allgemeine Relativitätstheorie. Sitzungsber. preuss. Acad. Wiss. 1916, 2, 1111-1116.):

$t_{\sigma }^{\nu }={\frac {1}{2}}\left({\mathfrak {G}}^{*}\delta _{\sigma }^{\nu }-{\frac {\partial {\mathfrak {G}}^{*}}{\partial g_{\nu }^{\mu \alpha }}}g_{\sigma }^{\mu \alpha }\right)$

and LL (p. 285, problem):

$t_{i}^{k}={\frac {c^{4}}{16\pi k}}\left[G{\sqrt {-g}}\delta _{i}^{k}-{\frac {\partial g^{lm}}{\partial x^{i}}}{\frac {\partial (G{\sqrt {-g}})}{\partial {\frac {\partial g^{lm}}{\partial x^{k}}}}}\right]$

--Lantonov (talk) 08:06, 2 July 2008 (UTC)

To MichaelCPrice: I am unclear on the uniqueness claim. Are they claiming uniqueness wrt the four conditions given here including the specific form of the divergence in $((-g)(T^{\mu \nu }+t_{LL}^{\mu \nu }))_{,\mu }=0$ with -g to the one-power? Or are they claiming uniqueness wrt any power of -g? If the former, then a different pseudotensor which was symmetric and used the half-power would be compatable.

To Lantonov: You might want to add credit to Einstein and another reference to the new section. Maybe even change the name to Einstein pseudotensor. JRSpriggs (talk) 09:01, 2 July 2008 (UTC)

Re name: I was going to suggest renaming it the Einstein-Dirac SEMP.

Re uniqueness: LL are claiming that the $(-g)^{1}\,$ factor is required for symmetry, so that any expression with just (say) $(-g)^{1/2}\,$ appearing would necessarily be unsymmetrical, which was why I asked about the symmetry of the Dirac expression. So, in answer, the LL uniqueness claim is wrt to all four conditions.

I hadn't realised that LL gave the Dirac expression in the problem on page 285 (my LL English edition seems identical to Lantonov's) -- do they credit it to Dirac (I guess not if Dirac's work is from 1975), or did Dirac rediscover it? If the latter it seems odd that Dirac was unfamiliar with LL's earlier work. (But then again, if it is identical to Einstein 1916/1918, this is rather moot...does Dirac credit Einstein?)

--Michael C. Price ^talk 09:29, 2 July 2008 (UTC)

The original Dirac book is earlier than LL, as far as I know (I don't have it here in the moment). 1975 is a date of re-publishing, otherwise it is from the 1930s or 1940s. I don't remember exactly if Dirac gave credit to Einstein. I guess he didn't. LL do not give any credit for this expression, neither to Einstein, nor Dirac. However, in the papers that I have (more than 100 of them) they all refer to it as Einstein pseudotensor. Some of the papers even cite Dirac's book for derivation of Einstein pseudotensor. As a simple test, google "Einstein pseudotensor" and "Dirac's pseudotensor" or "Dirac pseudotensor" and compare the results. --Lantonov (talk) 09:49, 2 July 2008 (UTC)

About uniqueness, see the footnote in LL p. 283: "It is necessary to note that the expression obtained by us for the four-momentum of matter plus field is by no means the only possible one. On the contrary; one can, in an infinity of ways (see, for example, the problem in this section [Einstein/Dirac pseudotensor - Lantonov (talk) 10:34, 2 July 2008 (UTC)]), form expressions which in the absence of a field reduce to $T^{i}k$, and which upon integration over $dS_{k}$, give conservation of some quantity. However, the choice made by us is the only one for which the energy-momentum pseudotensor of the field contains only first (and not higher) derivatives of $g_{ik}$ (a condition which is completely natural from the physical point of view), and is also symmetric, so that it is possible to formulate a conservation law for the angular momentum." And in the text further they derive the conservation law for the angular momentum. --Lantonov (talk) 10:34, 2 July 2008 (UTC)

I do not have LL. I have MTW; and I have Dirac's book. Dirac's book is very bare-bones. His preface is dated 1975 and says that the book is based on his lectures at Florida State University (Tallahassee) and that that GTR is Einstein's theory. Other than that he gives no credit to anyone, nor any history. He claims no originality for himself. The copyright is 1996 by Princeton University Press. JRSpriggs (talk) 04:07, 3 July 2008 (UTC)

I have LL and MTW, but not Dirac's book. It seems pretty clear that Einstein was the source of the pseudotensor and Dirac and LL merely described it. --Michael C. Price ^talk 06:17, 3 July 2008 (UTC)

Applying Noether's theorem to general relativity

Suppose the action is as seen at Einstein–Hilbert action, namely

$S_{\Omega }=\int _{\Omega }L\,d^{4}x$

with

$L={\frac {c^{4}}{16\pi G}}R{\sqrt {-g}}+L_{m}[\phi ^{A},\phi _{,\alpha }^{A},g_{\alpha \beta }]$

where $L_{m}\!$ is a scalar density, the Lagrangian density of matter and radiation, which is a function of the various material tensors (of unspecified type), $\phi ^{A}\!$, together with their first derivatives and the metric.

As you know, setting the variation of this integral to zero for infinitesimal changes in the fields which have compact support within the interior of the domain, $\Omega \!$, of integration yields the Euler–Lagrange equations. The Euler–Lagrange equations for the metric are the Einstein field equations.

To apply Noether's theorem to this, we follow essentially the same calculation, but instead of discarding the boundary term, we keep it and discard the bulk term instead (since it is zero by the Euler–Lagrange equations). The idea here is that we shift the values of the various fields (and the region they inhabit) by an infinitesimal amount in the direction of the flow generated by the contravariant vector field ξ which (by general covariance) should not affect the action, then differentiate with respect to that shift. Applying the Euler–Lagrange equations to remove the bulk terms provides the physical basis for the conservation law. This process yields

$\int _{\partial \Omega }L\,\xi ^{\mu }\,d^{3}s_{\mu }=\delta S_{\Omega }=\int _{\Omega }\left({\frac {c^{4}{\sqrt {-g}}}{16\pi G}}(g^{\alpha \beta }\delta \Gamma _{\alpha \beta }^{\mu }-g^{\alpha \mu }\delta \Gamma _{\rho \alpha }^{\rho })+{\frac {\partial L_{m}}{\partial \phi _{,\mu }^{A}}}{\mathcal {L}}_{\xi }\phi ^{A}\right)_{,\mu }\,d^{4}x=\int _{\partial \Omega }{\frac {c^{4}{\sqrt {-g}}}{16\pi G}}(g^{\alpha \beta }\delta \Gamma _{\alpha \beta }^{\mu }-g^{\alpha \mu }\delta \Gamma _{\rho \alpha }^{\rho })+{\frac {\partial L_{m}}{\partial \phi _{,\mu }^{A}}}{\mathcal {L}}_{\xi }\phi ^{A}\,d^{3}s_{\mu }$

where ${\mathcal {L}}_{\xi }\phi ^{A}$ is the Lie derivative of the tensors $\phi ^{A}\!$ in the $\xi ^{\nu }\!$ direction and

$\delta \Gamma _{\alpha \beta }^{\mu }={\frac {1}{2}}g^{\mu \gamma }(-2({\mathcal {L}}_{\xi }g_{\gamma \sigma })\Gamma _{\alpha \beta }^{\sigma }+({\mathcal {L}}_{\xi }g_{\beta \gamma })_{,\alpha }+({\mathcal {L}}_{\xi }g_{\gamma \alpha })_{,\beta }-({\mathcal {L}}_{\xi }g_{\alpha \beta })_{,\gamma })$

$={\frac {1}{2}}g^{\mu \gamma }(+({\mathcal {L}}_{\xi }g_{\beta \gamma })_{;\alpha }+({\mathcal {L}}_{\xi }g_{\gamma \alpha })_{;\beta }-({\mathcal {L}}_{\xi }g_{\alpha \beta })_{;\gamma })$

which (since it uses covariant derivatives) is a tensor.

Thus

$J^{\mu }={\frac {c^{4}{\sqrt {-g}}}{16\pi G}}(g^{\alpha \beta }\delta \Gamma _{\alpha \beta }^{\mu }-g^{\alpha \mu }\delta \Gamma _{\rho \alpha }^{\rho })+{\frac {\partial L_{m}}{\partial \phi _{,\mu }^{A}}}{\mathcal {L}}_{\xi }\phi ^{A}-L\xi ^{\mu }$

is the flux in the μ direction of the momentum in the $\xi ^{\nu }\!$ direction which is a conserved current, i.e.

${J^{\mu }}_{,\mu }=0.$

If we remove the matter terms,

$J_{matter}^{\mu }={\frac {\partial L_{m}}{\partial \phi _{,\mu }^{A}}}{\mathcal {L}}_{\xi }\phi ^{A}-L_{m}\xi ^{\mu }$

the gravitational part of this momentum remains

$J_{gravity}^{\mu }={\frac {c^{4}{\sqrt {-g}}}{16\pi G}}(g^{\alpha \beta }\delta \Gamma _{\alpha \beta }^{\mu }-g^{\alpha \mu }\delta \Gamma _{\rho \alpha }^{\rho }-R\xi ^{\mu }).$

Since

${\mathcal {L}}_{\xi }g_{\alpha \beta }=g_{\sigma \beta }\xi _{;\alpha }^{\sigma }+g_{\alpha \sigma }\xi _{;\beta }^{\sigma }$

we get

$\delta \Gamma _{\alpha \beta }^{\mu }={\frac {1}{2}}g^{\mu \gamma }(+g_{\sigma \gamma }\xi _{;\beta \alpha }^{\sigma }+g_{\beta \sigma }\xi _{;\gamma \alpha }^{\sigma }+g_{\sigma \alpha }\xi _{;\gamma \beta }^{\sigma }+g_{\gamma \sigma }\xi _{;\alpha \beta }^{\sigma }-g_{\sigma \beta }\xi _{;\alpha \gamma }^{\sigma }-g_{\alpha \sigma }\xi _{;\beta \gamma }^{\sigma })\,.$

OK? JRSpriggs (talk) 15:41, 17 July 2008 (UTC) [revised 19 March 2015]

At the moment I do not have time and resource to answer in substance. I just wanted to direct your attention to a paper I think is very relevant to the topic raised here:

• Leclerc, M. (2006), Noether’s theorem, the stress-energy tensor and Hamiltonian constraints, arXiv:gr-qc/0608096.

The paper you cite in the article is also very nice for a general treatment of the subject. --Lantonov (talk) 16:29, 17 July 2008 (UTC)

An ad hoc glance of what you wrote (take it as an out-of-hand comment, with great chances to be wrong) makes me think that one must be careful when introducing Killing vector for the metric field: The idea here is that we shift the values of the various fields (and the region they inhabit) by an infinitesimal amount in the direction of the flow generated by the contravariant vector field ξ. The problem is that a globally uniform shift in a particular direction is impossible in a general type non-symmetric spacetime. The same shift applied in different places directs the flow in different directions. From the above paper it is seen that symmetry plays a great role in applications of Noether's theorems. In particular, global symmetry leads to charge conservation (Noether's first theorem). Local symmetry allows for the expression of charge as two-dimensional surface integral (Noether's second theorem). The author shows that an intermediate linear symmetry transformation is important for applications in general relativity. Thus, Noether's theorems can be applied only in special spacetimes, such that possess a certain kind of symmetry.--Lantonov (talk) 16:44, 17 July 2008 (UTC)

To Lantonov: Although previously I had required ξ to be a Killing vector field, I am imposing no such requirement in this section — it is merely a contravariant vector field. Notice that I am making the metric flow along with all the other fields, so this flow can be reversed by a mere coordinate transformation. Essentially, I am subtracting the action at one place from the action at a neighboring place (compensating by introducing the end-points explicitly) and manipulating the difference (which is identically zero) mathematically with the Euler–Lagrange equations until I get a continuity equation.

If ξ were a Killing vector, then ${\mathcal {L}}_{\xi }g_{\beta \gamma }=0$ everywhere and thus $\delta \Gamma _{\alpha \beta }^{\mu }=0$ everywhere which would simplify the result significantly. JRSpriggs (talk) 03:58, 18 July 2008 (UTC)

Another paper on the topic is given as reference to Noether's theorem: Nina Byers. E. Noether's Discovery of the Deep Connection Between Symmetries and Conservation Laws. ISREAL MATHEMATICAL CONFERENCE PROCEEDINGS Vol. 12, 199. [3]. Especially the part around this quote (copy/paste):

"From (10) it is clear the ordinary divergence T mu nu, nu is in general different from zero in spacetime regions with gravitational fields. It is interesting to note, however, that at any given spacetime point one may choose a set of coordinates for which the gravitational fields vanish (gmu nu reduces to the flat spacetime Minkowski metric and the Christoffel symbols vanish). This is guaranteed by the equivalence principle which states that one can always choose a coordinate system such that spacetime in the neighborhood of a given point is Minkowski (flat). Thus one may see why it is not meaningful to speak of a localized energy density for gravitational fields. The physics of these relations is somewhat complicated. In regions of spacetime near gravitating sources, where the Riemann curvature is non-vanishing, there is failure of a principle of local energy conservation. The energy balance locally cannot be discussed independently of the coordinates one uses to calculate it, and consequently different results are obtained in various different coordinate frames — some being artifacts of the calculation itself. An amusing example of this is the case of an accelerating mass. For simplicity, let us take the mass and acceleration small, so that the deviations from Newtonian physics are small and may be treated perturbatively - the so-called post-Newtonian approximation. In this approximation the theory is similar to electromagnetic theory. In electromagnetic theory, an accelerating charge radiates and there is the radiation reaction force which tends to decelerate it; there is energy loss to the radiation field. In Einstein gravity, the accelerating mass is a source of gravitational radiation and one may calculate the radiation reaction force as one does in electromagnetic theory. A naive calculation along these lines yields a radiation reaction force term (å term in the equation of motion with a the acceleration) that has the opposite sign to the electromagnetic case; the particle appears to speed up gaining energy as it radiates! This is an unphysical radiation reaction force. The calculation is not gauge invariant and depends on the choice of coordinates used to describe the motion of the particle. However, though local energy conservation fails, there is a large scale principle of energy-momentum conservation in the general theory. The problem in principle was solved by Noether. She pointed out in I.V. that it is possible to construct a divergence-free quantity now generally referred to as the energy-momentum pseudotensor T mu nu eff. Jackiw et al. [21] have recently constructed such a pseudotensor using the methods of Noether introduced in her theorem II. Other constructions have been made by Weinberg [22] and Landau and Lifshitz [20]. Such pseudotensors have the form" ...--Lantonov (talk) 06:13, 18 July 2008 (UTC)

There are some differences between the approach I described in this section and the usual approach. What I have constructed above is not a pseudotensor; it is a tensor. However, the element of arbitrariness, which for pseudotensors lies in the choice of a coordinate system, remains; in my expression, the arbitrariness lies in the choice of the contravariant vector field ξ. Regarding the four criteria listed in this article:

(1) My expression is not constructed entirely from the metric tensor because it also contains ξ which unfortunately cannot be separated from the metric because of the way it enters in through the Lie derivative.

(2) It is not symmetric, if that question can even be asked of it. Nevertheless, by choosing an appropriate ξ, one gets a conserved angular momentum.

(3) It does combine the gravitational momentum with the Noether momentum of matter to make a conserved current as I explained above.

(4) It may fail to vanish locally in an inertial frame of reference because it contains the scalar of curvature, R; and thus it contains the second derivatives of the metric. Also, I cannot yet see whether the $\delta \Gamma _{\alpha \beta }^{\mu }$ can be made to vanish or not.

In view of these differences, the statements made in the usual references may not be applicable to this notion of gravitational momentum. JRSpriggs (talk) 13:04, 18 July 2008 (UTC)

We can change this into a pseudotensor by selecting a coordinate system and letting $\xi ^{\mu }=\delta _{\nu }^{\mu }\!$ for ν=0,1,2,3 in succession. This gets rid of ξ at the cost of losing the covariance of the formula. Also, these four specific choices may not include all the information available in the general form. In particular, one loses the angular momentum.

For these choices, ${\mathcal {L}}_{\xi }\phi ^{A}=\phi _{,\nu }^{A}$. So we get

$T_{\nu }^{\mu }{\sqrt {-g}}={\frac {\partial L_{m}}{\partial \phi _{,\mu }^{A}}}\phi _{,\nu }^{A}-L_{m}\delta _{\nu }^{\mu }$

and

$t_{\nu }^{\mu }{\sqrt {-g}}={\frac {c^{4}{\sqrt {-g}}}{16\pi G}}(g^{\alpha \beta }\Gamma _{\alpha \beta ,\nu }^{\mu }-g^{\alpha \mu }\Gamma _{\rho \alpha ,\nu }^{\rho }-R\delta _{\nu }^{\mu }).$

Be aware that you cannot reconstitute the original currents merely by multiplying these by ξ^ν except for linear combinations of the four specific choices made above. JRSpriggs (talk) 12:06, 20 July 2008 (UTC)

There is a similar calculation here: Covariant Energy-Momentum Conservation In General Relativity, 1997, arXiv:gr-qc/9701028. The expression above can be manipulated by replacing second derivatives of the vector field with expressions involving the curvature tensor. After simplifying using the EFE it reduces to the Komar superpotential.Weburbia (talk) 21:19, 23 August 2017 (UTC)

Landau and Lifshitz

I just bought "The Classical Theory of Fields" (fourth revised English edition) by L.D.Landau and E.M.Lifshitz. On page 303, the equation 96.8 gives their pseudotensor explicitly in terms of the Christoffel symbols. I think we should should include it in the article, if only to illustrate that their pseudotensor satisfies criterion 4. Unfortunately, equation 96.9 seems to have fallen prey to a typographical glitch which makes it impossible to distinguish $g^{\alpha \beta }\!$ from $g^{\alpha \beta }{\sqrt {-g}}$.

I do not see any place where they claim or justify the statement, made on this talk page, that their pseudotensor is the unique solution to those four criteria. JRSpriggs (talk) 20:45, 22 July 2008 (UTC)

I too have the 1975? fourth revised English edition (although our page numbers differ; mine has been been reprinted twice with corrections since 1975). I agree that the full Christoffel version should be included here -- laziness is my only excuse for not having already entered it (and that I haven't explicitly checked it -- i.e. laziness again!).

Re 96.9: I agree that it is quite hard to distinguish the terms, although I think I can detect a slight difference between the terms -- with the aid of a magnifying glass.

The uniqueness claim is found in the footnote raised between equations 96.12 and 96.13 (page 283 in my reprint version, probably pg 304 in yours?).

--Michael C. Price ^talk 07:48, 23 July 2008 (UTC)

Yes, I see that footnote now — "...the choice made by us is the only one for which the energy-momentum pseudotensor of the field contains only the first (and not higher) derivatives of g_ik ..., and is also symmetric, ...".

Mine says that it is copyright as of 1975. It is volume 2 in "Course of Theoretical Physics" published by Elsevier, Butterworth-Heinemann and translated from Russian by Morton Hamermesh.

Even with a magnifying glass the two "g"s (for the reciprocal of the metric and its product with the square-root of the negative of the determinate of the metric) are indistinguishable to me. However, I think I remember seeing something like this (where they are distinguishable) in MTW. JRSpriggs (talk) 09:20, 23 July 2008 (UTC)

I compared the formulae for the metric and connection LL pseudotensors in the 4 original Russian editions that I have (1962, 1966, 1975, and 1988) and they are correct with two exceptions: 1. LL use exclusively Latin symbols for 4-indices and Greek only for 3-indices, and 2. LL use the symbol ${\mathfrak {g}}^{ik}={\sqrt {-g}}g^{ik}$ for the density of the metric tensor. If you are interested, I made a Mathematica notebook with details of their calculation with the help of Tensorial package. Another interesting thing is that Babak and Grishchuk relatively recently (I don't have the paper at my disposal at the moment) derived a pseudotensor identical with LL metric formula with the only difference that they have metric instead of metric density. I'll place the Babak-Grishchuk reference here when I find it. --Lantonov (talk) 16:25, 23 July 2008 (UTC)

I presume then that the "Babak-Grishchuk pseudotensor" must fail one of the four LL requirements as listed in the article? Regarding the use of Greek vs Latin indices, it would be more in conformance with modern usage if we use Greek symbols in the expressions. Shall we change them? At the moments it's a bit of a muddle. I'll change them all to Greek if no one objects. --Michael C. Price ^talk 18:20, 23 July 2008 (UTC)

Responding to Mike's edit summary in this article: LL's equation 85.16 shows that they mean the connection to be symmetric. Both the Levi-Civita connection and Christoffel symbols articles say that they are usually taken as symmetric. But the former mentions the possibility of a torsion tensor, if one uses a basis for the tangent space which is not derived from a coordinate system. JRSpriggs (talk) 01:20, 24 July 2008 (UTC)

I'm not clear on the distinction between the different types of connection, nor am I fluent with the co-ordinate free representations; if it should be the Riemannian connection or whatever, please update accordingly. --Michael C. Price ^talk 05:27, 24 July 2008 (UTC)

You are right, Mike, most of the literature is opposite to LL: it uses Greek for 4D and Latin for 3D. So, for conformity, it will be ok to change here all indices to Greek (4D). --Lantonov (talk) 05:50, 24 July 2008 (UTC)

I checked the equations recently entered by MichaelCPrice (both latin and greek versions). They are correct (except for one index in the latin version which I fixed). It was rather tedious to do so, but I am sure that it was much more tedious for you to enter them. Thanks a lot.

I alway use coordinate systems rather than frame fields or any other such abstract garbage. And the only connection I use or know about is that derived at LL's 86.1 through 86.3. I do not regard a connection as a geometrical property of space-time, but merely as a way to cancel out the terms in the transformation law of the partial derivative of a tensor which would otherwise prevent the result from being a tensor. So, I despise the idea of torsion. JRSpriggs (talk) 07:13, 24 July 2008 (UTC)

Interjected comment about torsion (or rather contorsion, which is related): I too regard the connection as merely a field added to make the derivative covariant. But that doesn't mean that torsion is an abomination; quite the contrary.

By definition we have:

${T^{\mu }}_{;\nu }={T^{\mu }}_{,\nu }+\Gamma _{\alpha \nu }^{\mu }T^{\alpha }$

where $\Gamma _{\alpha \nu }^{\mu }$ is simply the connection (of whatever shade). Solving ${g^{\mu \nu }}_{;\lambda }=0$ gives

$\Gamma _{\alpha \nu }^{\mu }={\frac {1}{2}}g^{\mu \rho }(g_{\rho \alpha ,\nu }+g_{\rho \nu ,\alpha }-g_{\nu \alpha ,\rho })+{\frac {1}{2}}{K^{\mu }}_{\alpha \nu }\,$

where K is the contorsion tensor. The only constraint on K, at this stage, is that it is antisymmetric in the first two indices

${K^{\mu \nu }}_{\alpha }=-{K^{\nu \mu }}_{\alpha }\,$.

The factor of a 1/2 for K is historical (sometimes -1/2).

The presence of torsion is therefore almost demanded since it is permitted. --Michael C. Price ^talk 21:37, 24 July 2008 (UTC)

I have exactly the same attitude as JRSpriggs regarding frame fields, abstract geometry, and connection. I prefer to work with good old tensors, rather than say that something maps into another thing or define a manifold with associated metric, connection, describe vector bundles on it, etc. The latter makes the whole topic more intractable and unimaginable, hiding the individual trees in the forest. At some point, I considered replacing tensor algebra with Clifford algebra, as the latter generally gives more simple expressions, but gave up when I failed to apply Clifford algebra to GR energy conservation. --Lantonov (talk) 07:43, 24 July 2008 (UTC)

How heartening that you both -- who clearly know more about GR than me -- prefer the old style tensors (which is my preference also). I was going to try to master the more geometric or abstract approaches since so many people sing their praises, but now I'm not so sure. Also I understand that the abstract approach becomes ambiguous in some complex situations where you have to resort to loverly old tensors with all their wonderful indices. "Frame fields", though... are they the same as vierbein fields? I have to 'fess that I've fallen in love with the vierbein approach, which I think has much utility as well as being elegant. --Michael C. Price ^talk 08:32, 24 July 2008 (UTC)

Discussions with intelligent and knowledgeable people like you two are immensely satisfying and helpful for me too. No one is error-free and in the course of this discussion alone I was pointed to and corrected for 3 gross mistakes of mine. --Lantonov (talk) 10:15, 24 July 2008 (UTC)

Well, I learned tensors and general relativity around 1970 from one of Sir Arthur Stanley Eddington's books (unfortunately, a library book, so I do not have it). He took an approach which focuses on getting right to the physics without learning any more mathematics than necessary.

I recently learned about the Lie derivative in the course of trying to figure out how to apply Noether's theorem to tensors. It is very similar to the covariant derivative. Both modify the partial derivative to get something that is a tensor. Both are linear and satisfy the product rule.

The covariant derivative is defined to take the metric (and functions of it) to zero. You can transform to a coordinate system where the first derivative of the metric is zero, take a partial derivative there, and then transform back.

The Lie derivative takes a specified contravariant vector field to zero. If the vector is not zero at an event, you can transform to a coordinate system where the first derivative of that field is zero (or even make the field constant within a small ball). Then take the partial derivative and inner product with that vector field, and transform back.

One thing I tried to figure out a few times in the past but could not get was how to take the covariant derivative of a spinor which is needed to put the Dirac equation for electrons into general relativity. JRSpriggs (talk) 12:58, 24 July 2008 (UTC)

Which book by Eddington? I have the Russian translation of The Mathematical Theory of Relativity, 2nd ed. Cambridge University Press, 1924. --Lantonov (talk) 07:07, 25 July 2008 (UTC)

Thanks for the comments about the Lie derivative, which is one of those things I don't "get" -- the Lie derivative is often used a way of deriving the covariant derivative, whereas I prefer to start from the covariant derivative and just demand that it transform as a tensor. I shall dwell on your comments and see what develops.

Incidentally I found the discussion on tensor densities most illuminating; I had never "got" them either, although I could follow the formalism. Now I understand them as simply the product of a tensor and an arbitrary power of g and that $g_{;\mu }=0\,$ by definition; that was a eureka moment.

I believe I have the answer to the covariant derivative of spinor in my notes; let me double check it before I post it here.--Michael C. Price ^talk 20:54, 24 July 2008 (UTC)

I have, for a spinor field $\psi \,$:

$D_{\mu }\psi ^{j}=\psi ^{j},_{\mu }+w_{\mu }^{\ \ ab}(x){\frac {1}{4i}}\sigma _{ab\ \ \ k}^{\ \ \ j}\psi ^{k}$

where $\sigma _{ab\ \ k}^{\ \ j}={\frac {i}{2}}[\gamma _{a},\gamma _{b}]_{\ k}^{j}\,$ are the generators of the Lorentz group with $\gamma _{a}\,$ being the Dirac matrices. $w_{\mu }^{\ \ ab}(x)$ is a gauge field (from the local Lorentz freedom) known as the spin connection. --Michael C. Price ^talk 05:40, 25 July 2008 (UTC)

Mike said "The presence of torsion is therefore almost demanded since it is permitted.". On the contrary, when I took a course on elementary particles, I learned that each type of particle corresponds to an irreducible representation of the relevant symmetry group. Generalizing, each physically significant field should be as simple as it can be, that is, it should not be the direct sum (or other such combination) of two or more simpler fields. Adding an anti-symmetric connection to the symmetric connection which we must have is a clear violation of this principle.

I am even inclined to separate the metric itself into two fields: a conformal metric, $g^{\mu \nu }{\sqrt[{4}]{-g}}$, and a length scale, ${\sqrt[{8}]{-g}}$; except that I have never seen how this would simplify the equations. JRSpriggs (talk) 03:34, 25 July 2008 (UTC)

But at this stage we have not got to thinking about K as an irreducible representation. We are merely noting that the Christoffel symbol is a sufficient but not necessary solution for the connection, given the constraint: ${g^{\mu \nu }}_{;\lambda }=0$. --Michael C. Price ^talk 05:17, 25 July 2008 (UTC)

To Mike: Sorry I messed up your formula. I see now why you think it is anti-symmetric in the first two indices. However, since your contorsion tensor is the difference of two connections which are symmetric in the last two indices, it must be symmetric in the last two indices. So we get

$K_{cba}=K_{cab}=-K_{acb}=-K_{abc}=K_{bac}=K_{bca}=-K_{cba}\!$

which, being its own negative, must be zero. JRSpriggs (talk) 06:45, 25 July 2008 (UTC)

But I am not assuming a symmetric connection. Also I am using a different definition for K (which I probably shouldn't call contorsion); simply:

$K_{\ \alpha \nu }^{\mu }\equiv 2\Gamma _{\alpha \nu }^{\mu }-g^{\mu \rho }(g_{\rho \alpha ,\nu }+g_{\rho \nu ,\alpha }-g_{\nu \alpha ,\rho })$

It might be easier to stick with just torsion, which is defined as:

$S_{\ \mu \nu }^{\lambda }\equiv \Gamma _{\mu \nu }^{\lambda }-\Gamma _{\nu \mu }^{\lambda }$

Torsion free is simply the assumption that the connection be symmetric in its lower two indices. This assumption can be, and frequently is, dropped -- and I don't see that it has anything to do with the more abstract or geometric approaches (to get back to the original issue here). Einstein assumed a symmetric connection in his original formulation, which I imagine Eddington followed, but Cartan and Einstein generalised GR later to include torsion. --Michael C. Price ^talk 07:40, 25 July 2008 (UTC)

To Lantonov: It has been so long that I do not remember the title exactly. I thought it was something like "Spacetime and relativity", but I do not see exactly that on the list of his books. I assume it must be either

1. 1920. Space, Time and Gravitation: An Outline of the General Relativity Theory. Cambridge University Press. ISBN 0-521-33709-7; or
2. 1923, 1952. The Mathematical Theory of Relativity. Cambridge University Press.

JRSpriggs (talk) 07:22, 25 July 2008 (UTC)

To Mike: First, I must say that I find it rather irritating when people insert comments out of chronological order. It makes them harder to find and breaks up the flow. It also screws up the indentation.

LL justify the symmetry of Γ (which is equation 85.16), by saying

... by virtue of the equivalence principle there must be a "galilean" coordinate system in which the $\Gamma _{kl}^{i}$, and consequently also the $S_{kl}^{i}$, vanish at a given point. Since $S_{kl}^{i}$ is a tensor, if it vanishes in one coordinate system, it must vanish in all frames. This means the Christoffel symbols must be symmetric in their lower indices ...

which implies that torsion and contorsion are zero. Why do you not accept their argument? JRSpriggs (talk) 09:04, 25 July 2008 (UTC)

I find it convenient to group comments by subject within sections; I also find it clearer to read. Obviously a subjective thing.

As for the symmetric connection: my understanding is that the inclusion of fermion fields forces the existence of torsion, so the LL justification must be flawed (it looks suspiciously post-hoc to me; arguments based on the Equivalence principle are notoriously slippery, especially where they imply more than just general coordinate invariance and local lorentz invariance). My limited understanding of supersymmetry / supergravity -- not a concern of LL, Eddington etc -- is that it requires non-zero contorsion, perhaps purely from the inclusion of fermion fields.--Michael C. Price ^talk 10:06, 25 July 2008 (UTC)

There is, as of yet, no experimental evidence to support the supersymmetry hypothesis. How do fermionic fields force the existence of torsion? JRSpriggs (talk) 13:54, 25 July 2008 (UTC)

The lack of evidence is irrelevant: supersymmetric theories would not be developed if they violated the equivalence principle. As for the mechanism I believe that the it has something to do with the anticommutting nature of fermonic fields, but I'm hazy on the details.--Michael C. Price ^talk 15:16, 25 July 2008 (UTC)

So reality must adjust itself to the theory rather than the other way around?

Well, I will leave it to you to figure out how LL 96.8 should be adjusted to take torsion into consideration. Ha, Ha! JRSpriggs (talk) 20:44, 25 July 2008 (UTC)

See Einstein–Cartan theory "There is a qualitative theoretical proof showing that general relativity must be extended to Einstein-Cartan theory when matter with spin is present."

Probably no change required to LL 96.8 since the Einstein Field Equations are unaffected by torsion. Torsion generates a new set of equations alongside the EFEs. --Michael C. Price ^talk 20:48, 25 July 2008 (UTC)

6 conditions that an energy-momentum object must satisfy

I found the following in Babak-Grishchuk [4]:

"In traditional field theories, one arrives, after some work, at the energy-momentum object which is: 1) derivable from the Lagrangian in a regular prescribed way, 2) a tensor under arbitrary coordinate transformations, 3) symmetric in its components, 4) conserved due to the equations of motion obtained from the same Lagrangian, 5) free of the second (highest) derivatives of the field variables, and 6) is unique up to trivial modifications not containing the field variables. There is nothing else, in addition to these 6 conditions, that we could demand from an acceptable energy-momentum object, both on physical and mathematical grounds." --Lantonov (talk) 06:27, 25 July 2008 (UTC)

Removed a paragraph

I removed a paragraph from the intro that began "Some people object to this derivation on the grounds that pseudotensors are inappropriate objects in general relativity", because it doesn't make any sense:

• "... but the conservation law only requires the use of the 4-divergence of a pseudotensor which is, in this case, a tensor (which also vanishes)." Sure, but when one describes a vanishing divergence as a conservation law one is implying the existence of a conserved quantity. Anyone who objects to this construction is presumably objecting to the nature of the conserved quantity, not to the identically zero divergence.
• "Also, most pseudotensors are sections of jet bundles, which are perfectly valid objects in GR." When people complain about pseudotensors they're complaining about the breaking of general covariance. Nondynamical fields that explicitly break symmetries are of course valid mathematical objects; no one would argue otherwise, even the people who think that they're "inappropriate".

-- BenRG (talk) 05:42, 11 May 2010 (UTC)

...sure... presumably... Not good enough grounds for deletion.--Michael C. Price ^talk 05:52, 11 May 2010 (UTC)

An object does not have to be a tensor in order to participate in generally covariant formulas. Most notably the Christoffel symbols are widely used in general relativity (indeed they are the gravitational force field) and they are not tensors. Also tensor densities play a big role in Maxwell's equations in curved spacetime. You just need to make sure that you know what the transformation law of the object is and that the transformations combine to give an invariant formula. JRSpriggs (talk) 10:59, 11 May 2010 (UTC)

Can the Landau–Lifshitz pseudotensor be derived from the Lagrangian?

If so, and if it is not too complicated, I think it would make a nice addition to the article. Anyone have a source and/or the expertise for such a presentation? —Quantling (talk | contribs) 19:07, 18 November 2010 (UTC)

Being a pseudotensor I guess that it can't be derived by any variational technique. Just a guess, though. --Michael C. Price ^talk 00:09, 19 November 2010 (UTC)

Quadratic approximation to the total stress-energy

According to Landau and Lifshitz, the total stress-energy (both gravitational and non-gravitational, apart from a constant factor)

$((-g)(g^{\mu \nu }g^{\alpha \beta }-g^{\mu \alpha }g^{\nu \beta }))_{,\alpha \beta }\,$

is an exactly conserved quantity. If we let

${\mathfrak {g}}^{\alpha \beta }=g^{\alpha \beta }{\sqrt {-g}}\,$

and adopt the harmonic coordinate condition

${\mathfrak {g}}^{\alpha \beta }{}_{,\beta }=0\,,$

then the stress-energy can be rewritten as follows

$({\mathfrak {g}}^{\mu \nu }{\mathfrak {g}}^{\alpha \beta }-{\mathfrak {g}}^{\mu \alpha }{\mathfrak {g}}^{\nu \beta })_{,\alpha \beta }={\mathfrak {g}}^{\mu \nu }{}_{,\alpha \beta }\,{\mathfrak {g}}^{\alpha \beta }-{\mathfrak {g}}^{\mu \alpha }{}_{,\beta }\,{\mathfrak {g}}^{\nu \beta }{}_{,\alpha }\,.$

Assuming that spacetime is approximately Minkowskian, I will attempt to calculate the stress-energy to the quadratic order in

$h_{\alpha \beta }=g_{\alpha \beta }-\eta _{\alpha \beta }\,.$

Then

$g_{\alpha \beta }=\eta _{\alpha \beta }+h_{\alpha \beta }\,.$

$g^{\mu \nu }\approx \eta ^{\mu \nu }-\eta ^{\mu \alpha }h_{\alpha \beta }\eta ^{\beta \nu }+\eta ^{\mu \alpha }h_{\alpha \beta }\eta ^{\beta \gamma }h_{\gamma \delta }\eta ^{\delta \nu }\,.$

$g\approx \eta \left(1+h_{\alpha \beta }\eta ^{\beta \alpha }+{\tfrac {1}{2}}\left(h_{\alpha \beta }\eta ^{\beta \alpha }h_{\gamma \delta }\eta ^{\delta \gamma }-h_{\alpha \beta }\eta ^{\beta \gamma }h_{\gamma \delta }\eta ^{\delta \alpha }\right)\right)\,.$

${\sqrt {-g}}\approx 1+{\tfrac {1}{2}}h_{\alpha \beta }\eta ^{\beta \alpha }+{\tfrac {1}{8}}h_{\alpha \beta }\eta ^{\beta \alpha }h_{\gamma \delta }\eta ^{\delta \gamma }-{\tfrac {1}{4}}h_{\alpha \beta }\eta ^{\beta \gamma }h_{\gamma \delta }\eta ^{\delta \alpha }\,.$

${\mathfrak {g}}^{\mu \nu }\approx \eta ^{\mu \nu }-\eta ^{\mu \alpha }h_{\alpha \beta }\eta ^{\beta \nu }+\eta ^{\mu \alpha }h_{\alpha \beta }\eta ^{\beta \gamma }h_{\gamma \delta }\eta ^{\delta \nu }+{\tfrac {1}{2}}\eta ^{\mu \nu }h_{\alpha \beta }\eta ^{\beta \alpha }-{\tfrac {1}{2}}\eta ^{\mu \alpha }h_{\alpha \beta }\eta ^{\beta \nu }h_{\rho \sigma }\eta ^{\sigma \rho }+{\tfrac {1}{8}}\eta ^{\mu \nu }h_{\alpha \beta }\eta ^{\beta \alpha }h_{\gamma \delta }\eta ^{\delta \gamma }-{\tfrac {1}{4}}\eta ^{\mu \nu }h_{\alpha \beta }\eta ^{\beta \gamma }h_{\gamma \delta }\eta ^{\delta \alpha }\,.$

Since the derivatives wipe out the order zero terms, the stress-energy is zero to order zero. To order one, the only terms which survive are

$-\eta ^{\mu \alpha }h_{\alpha \beta ,\zeta \xi }\eta ^{\zeta \xi }\eta ^{\beta \nu }+{\tfrac {1}{2}}\eta ^{\mu \nu }h_{\alpha \beta ,\zeta \xi }\eta ^{\zeta \xi }\eta ^{\beta \alpha }\,$

which is the stress-energy of matter according to linearized gravity#With a coordinate condition.

<< still under construction >>

JRSpriggs (talk) 17:15, 24 June 2012 (UTC)

• Hmm, unless I did the algebra wrong, it seems that the quadratic term is:

$\eta ^{\mu \alpha }h_{\alpha \beta }{}_{,\zeta \xi }\eta ^{\zeta \xi }h^{\beta \nu }+h^{\mu \alpha }h_{\alpha \beta }{}_{,\zeta \xi }\eta ^{\zeta \xi }\eta ^{\beta \nu }-{\tfrac {1}{2}}h^{\mu \nu }h_{\alpha \beta }{}_{,\zeta \xi }\eta ^{\xi \zeta }\eta ^{\alpha \beta }-{\tfrac {1}{2}}\eta ^{\mu \nu }h_{\alpha \beta }{}_{,\zeta \xi }\eta ^{\zeta \xi }h^{\beta \alpha }-{\tfrac {1}{2}}\eta ^{\mu \alpha }h_{\alpha \beta }{}_{,\zeta \xi }\eta ^{\zeta \xi }\eta ^{\beta \nu }h_{\rho }^{\rho }+{\tfrac {1}{4}}\eta ^{\mu \nu }h_{\alpha \beta }{}_{,\zeta \xi }\eta ^{\zeta \xi }\eta ^{\beta \alpha }h_{\rho }^{\rho }$ $-\eta ^{\mu \alpha }h_{\alpha \beta }{}^{,\gamma }h_{\gamma \delta }{}^{,\beta }\eta ^{\delta \nu }+{\tfrac {1}{2}}h_{\alpha \beta }{}^{,\gamma }\eta ^{\beta \alpha }(h_{\gamma \delta }{}^{,\mu }\eta ^{\delta \nu }+h_{\gamma \delta }{}^{,\nu }\eta ^{\mu \delta })-{\tfrac {1}{4}}h_{\alpha \beta }{}^{,\nu }\eta ^{\beta \alpha }h_{\gamma \delta }{}^{,\mu }\eta ^{\delta \gamma }\,.$

Ben Standeven (talk) 07:51, 12 December 2014 (UTC)

To Ben: Thanks. I guess I burned out doing that calculation. That is, I felt that the effort required to finish it exceeded the potential benefit because there is still the issue of how one should interpret that expression. Can it be converted into a more intuitive notation as in classical physics? JRSpriggs (talk) 10:48, 12 December 2014 (UTC)

It occurs to me that perhaps one can separate the total stress-energy into two parts this way: Again assuming the harmonic coordinate condition and aside from a constant factor, the material stress-energy would be

${\mathfrak {g}}^{\mu \nu }{}_{,\alpha \beta }\,{\mathfrak {g}}^{\alpha \beta }\,$

and the gravitational stress-energy would be

$-{\mathfrak {g}}^{\mu \alpha }{}_{,\beta }\,{\mathfrak {g}}^{\nu \beta }{}_{,\alpha }\,.$

But this is just a guess. JRSpriggs (talk) 21:06, 19 March 2015 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Stress–energy–momentum pseudotensor/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Comment(s)

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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"

Last edited at 01:54, 1 January 2012 (UTC). Substituted at 07:09, 30 April 2016 (UTC)

Landau-Lifshitz pseudotensor blog

I made a blog page with a detailed derivation of of all forms of LL pseudotensor. Anyone can take from it what he/she deems helpful and put it in the wiki article. I prefer to write it in the blog and not here because in the blog I am not restricted by the conventions and conditionals of Wikipedia. Lantonov (talk) 14:35, 7 April 2021 (UTC)

An impressive achievement. Congratulations! cheers, Michael C. Price ^talk 09:09, 16 July 2021 (UTC)

LL should be just the gravitational energy

"The use of the Landau–Lifshitz pseudotensor, a stress–energy–momentum pseudotensor for combined matter (including photons and neutrinos) plus gravity" implies that the pseudotensor contains the energy of matter we well, which is not true. It represents purely the gravitational or metric components. cheers, Michael C. Price ^talk 18:29, 19 June 2023 (UTC)

Agreed. I made an edit. Feel free to edit further. —Quantling (talk | contribs) 22:36, 19 June 2023 (UTC)

Needs better explanation, supporting understanding of the mathematics.

I recognize this is a hard request, but much of this article could only be read by people with advanced degrees in mathematics. Those who have such, will already know a fair bit about it, so the article becomes a repository, rather than an explanation that invites the reader in. I have quite good mathematical skills, yet find this non-explanatory, and dense to the point of non-readable. Might someone provide an explanation of the meaning of the mathematics at each point, for someone who has perhaps a degree in physics, rather than a PhD level in the mathematics? Centroyd (talk) 10:55, 27 January 2024 (UTC)

It would be useful to list some of the things that were hard to understand, which parts seemed dense or opaque. It is equally difficult for the experts to guess what it is that is confusing to the newcomer. 67.198.37.16 (talk) 14:18, 4 June 2024 (UTC)

Weak field limit and relation to gravitational waves

It would be useful to add a section on what is the relation to the well known energy flux carried by gravitational waves in the weak-field limit, how that is related to the LL expression. Are these identical only in some particular gauge? Bkocsis (talk) 19:07, 21 February 2024 (UTC)

Yes, such a section would be useful. Weinberg's book does the weak-field limit, so he might present this. On the other hand, that book is old, maybe 50 years old now, and surely there are modern treatments. 67.198.37.16 (talk) 14:50, 4 June 2024 (UTC)