World sheet EM tensor (Hilbert principle?)

[da_willem]

In my ST book the world sheet EM tensor for the Polyakov action is given by

$$T_{\alpha \beta} = -\frac{2}{T} \frac{1}{\sqrt{-h}} \frac{\delta S_{\sigma}}{\delta h^{\alpha \beta}}$$

Why is this?

(Searching through the internet I found clues to what is called Hilbert's prescription, namely that the EM tensor in GR is given by

$$T^{\mu \nu} = -2 \frac{\delta L}{\delta g^{\mu \nu}}$$

and that to obtain the conserved current one should calculate the functional derivative of the Lagrangian L wrt the gauge field. Why is this all so?!)

 

[nonplus]

In my ST book the world sheet EM tensor for the Polyakov action is given by

$$T_{\alpha \beta} = -\frac{2}{T} \frac{1}{\sqrt{-h}} \frac{\delta S_{\sigma}}{\delta h^{\alpha \beta}}$$

Why is this?

(Searching through the internet I found clues to what is called Hilbert's prescription, namely that the EM tensor in GR is given by

$$T^{\mu \nu} = -2 \frac{\delta L}{\delta g^{\mu \nu}}$$

and that to obtain the conserved current one should calculate the functional derivative of the Lagrangian L wrt the gauge field. Why is this all so?!)

Hmm...it's been a while since I took GR... flipping through some books I see that General Relativity by Wald seems to have a nice discussion of this in appendix E. And so I will simply refer you to that

Let me however comment that
$$T^{\mu \nu} = -2 \frac{\delta L}{\delta g^{\mu \nu}}$$
is usually taken as the definition of the EM tensor in relativity (and it works pretty good to, try finding the EM tensor for electrodynamics and you'll see!)
And it makes quite a bit of sense actually: think of some field theory with gravity. The action for this theory will have two parts:
S=S_{matter}+S_{GR},
where S_{GR} is the ordinary Einstein-Hilbert action. If you try to find the equation of motion for the metric you go through the standard prescription: vary the action with respect to g_{\mu\nu}. Doing that you get two parts; the variation of S_{GR} will become the standard Einstein tensor, and the variation of S_{matter} part is...wait for it...the EM tensor! And so you get the Einstein field equation.
G=T
Which is pretty cool. Of course there is a bit more to this, the standard EM tensor in flat space is of course related to all that Noether stuff. I can't remember the details at all, I simply take the GR definition as *the* definition.
Anyway: Take a look in Walds book and it should become clearer.
(If you have a hard time getting this book, this is a subjects that should be discussed in most GR books I think)

nonplus

 

[da_willem]

Ah, I see. Thanks very much for that, now I know at least that there is quite something to it. I just never got acquainted with 'the other definition' of the energy momentum tensor. I read the text of Wald you referred to, but it didn't give me an answer to the relation with 'my definition', the one arising from spacetime translations as a Noether current

$$T^{\mu} _{\nu} = \frac{\partial L}{\partial (\partial _{\mu} \phi)}\partial _{\nu} \phi - L\delta ^{\mu} _{\nu}$$

Using google I found the following:

I think it is reasonable to take the following viewpoint:
the traditional method (with the asymmetric T) uses
Noether's first theorem with respect to translational
symmetry (when it exists!). The general relativity inspired
method uses Noether's second theorem! Recall that Noether's
second theorem says that if you have a symmetry of the
Lagrangian that involves arbitrary functions (of all
spacetime coords) then there is a differential identity
satisfied by the field equations. By using a general
metric and allowing it to be a variable, you can use the
resulting diffeo symmetry (which certainly involves
arbitrary functions) to get an identity, which leads to the
GR version of T.

But another search on Noether's second theorem leaves me clueless as to how the relation between the two definitions goes...

 

[nonplus]

Ah, I see. Thanks very much for that, now I know at least that there is quite something to it. I just never got acquainted with 'the other definition' of the energy momentum tensor. I read the text of Wald you referred to, but it didn't give me an answer to the relation with 'my definition', the one arising from spacetime translations as a Noether current

$$T^{\mu} _{\nu} = \frac{\partial L}{\partial (\partial _{\mu} \phi)}\partial _{\nu} \phi - L\delta ^{\mu} _{\nu}$$

Using google I found the following:



But another search on Noether's second theorem leaves me clueless as to how the relation between the two definitions goes...

I also had problems finding out more. Unfortunately I don't have time to think much about this now...but I had a look in the "phonebook": Gravitation by MTW. At page 504 there is a discussion about the problems of defining the EM tensor. Its a bit short, but it is interesting... it turns out that there are some technical problems in defining the EM tensor in general. One of these is related to the localization of energy (I just remembered that my field theory prof. made a joke about refusing to pay his electricity bill since he could show from the EM tensor that no energy went into his house )

Anyway: A long trip to the library should help

nonplus

 

[Demystifier]

The book of Steven Weinberg "Gravitation and Cosmology" also gives a good derivation of such a definition of the energy-momentum.

See also pages 116 and 139 of S. M. Carroll, gr-qc/9712019

 

[da_willem]

Luckily I have an (electronic) library in my room, so not such a long trip...:rofl:.

So, I guess my investigations are not yet over. I don't mind, it's quite an interesting issue. Thanks for your help!