Re: [Maxima-discuss] energy-momentum tensor

[Viktor T. T. <vt...@vt...>]

Dear Dimiter,

If L is a general field Lagrangian, all you know is that

T_{\mu\nu} = -2/\sqrt{-g} \partial (L\sqrt{-g})/\partial g^{\mu\nu}.

That's all you know. To work out T_{\mu\nu}, some specific form of L must be given.

The itensor package has the ability to compute functional derivatives with respect to fields, including g^{\mu\nu}. But depending on the specific form of the Lagrangian, it may take quite an effort to set up the problem and simplification rules in order to obtain a meaningful result.


Viktor




> -----Original Message-----
> From: Dimiter Prodanov [mailto:dim...@gm...]
> Sent: Monday, January 9, 2017 4:25 AM
> To: Viktor T. Toth <vt...@vt...>
> Cc: max...@li...
> Subject: Re: [Maxima-discuss] energy-momentum tensor
> 
> Hi Viktor,
> 
> Actually, I have in mind a general field lagrangian.
> So starting from an input lagrangian to compute the energy-momentum tensor according to the noether theorem.
> 
> best regards,
> 
> Dimiter
> 
> On Mon, Jan 9, 2017 at 7:39 AM, Viktor T. Toth <vt...@vt... <mailto:vt...@vt...> > wrote:
> 
> 
> 	Dear Dimiter,
> 
> 	No, I do not. The canonical energy-momentum depends on whatever fields we are dealing with. I am assuming
> that you have in mind the electromagnetic field and its energy-momentum tensor? In itensor, it would be something
> like
> 
> 	load(itensor);
> 	imetric(g);
> 	igeowedge_flag:true;
> 	components(F([i,j]),extdiff(A([i],[]),j));
> 	components(F([i,-k],[]), g([-j,-k])*F([i,j]));
> 	components(F([-l,-k],[]),g([-i,-l])*F([i,-k]));
> 	components(T([-m,-n]),F([-m,-a])*F([-n,a])-1/4*g([-m,-n])*F([a,b])*F([-a,-b]));
> 	ishow(T([-m,-n]))$
> 
> 	(Note the use of the negative sign for contravariant indices; this notation was implemented specifically to
> ensure that index ordering is preserved when indices are raised/lowered, as in this case.)
> 
> 
> 	Viktor
> 
> 
> 
> 
> 
> 
> 	> -----Original Message-----
> 	> From: Dimiter Prodanov [mailto:dim...@gm... <mailto:dim...@gm...> ]
> 	> Sent: Friday, January 6, 2017 11:45 AM
> 	> To: max...@li... <mailto:max...@li...>
> 	> Subject: [Maxima-discuss] energy-momentum tensor
> 	>
> 	> Dear Viktor,
> 	>
> 	> do you have code for compuation of the canonical energy-momentum tensor?
> 	> I would like to implement it in clifford so I want to compare it with reference implementation.
> 	>
> 	> best regards,
> 	>
> 	> Dimiter
> 
> 
>