Shyan said:
It is often ...
But if we say this isn't a conservation law for stress-energy tensor, we should ask what's its meaning?(Actually I see it being called a conservation law!)
I'm really confused about it and can't find a way out. I need urgent help!
Thanks
You can rewrite the equation \nabla_{ \mu } T^{ \mu \nu } = 0 in the form
\partial_{ \mu } \left( \sqrt{ - g } \ T^{ \mu }{}_{ \nu } \right) = \sqrt{ - g } \Gamma^{ \rho }_{ \nu \mu } \ T^{ \mu }{}_{ \rho } . \ \ \ \ (1)
Using the field equation, you can replace T^{ \mu }{}_{ \rho } in the RHS by ( - 1 / k \ G^{ \mu }{}_{ \rho } ). With some algebra, you can transform Eq(1) into a genuine conservation law
\partial_{ \mu } \left( \sqrt{ - g } \ ( T^{ \mu \nu } + \frac{ 1 }{ 2 k } t^{ \mu \nu } ) \right) = 0 , \ \ \ (2) where t^{ \mu \nu } is defined by \partial_{ \rho } ( \sqrt{ - g } \ t^{ \rho }{}_{ \sigma } ) \equiv \partial_{ \rho } \left( \frac{ \partial \mathcal{ L } }{ \partial ( \partial_{ \rho } g^{ \mu \nu } ) } \partial_{ \sigma } g^{ \mu \nu } - \delta^{ \rho }_{ \sigma } \mathcal{ L } \right) = - \sqrt{ - g } \ G_{ \mu \nu } \ \partial_{ \sigma } g^{ \mu \nu } , and \mathcal{ L } ( g , \partial g ) is the non-covariant part of the E-H scalar density ( \sqrt{ - g } R ). Explicitly, it is given by \mathcal{ L } = \sqrt{ - g } \ g^{ \mu \nu } \ ( \Gamma^{ \sigma }_{ \rho \sigma } \ \Gamma^{ \rho }_{ \mu \nu } - \Gamma^{ \sigma }_{ \mu \rho } \ \Gamma^{ \rho }_{ \nu \sigma } ) .
Notice that Eq(2) holds in all reference frames, i.e., it is a generally covariant statement, even though t^{ \mu \nu } is not a tensor. It only transforms as a tensor under linear transformations.
So, we can make the following observations: In a closed system of matter and gravitational field, the matter energy-momentum tensor is not conserved. On the other hand, while the object
\tau^{ \mu \nu } \equiv \sqrt{ - g } \ ( T^{ \mu \nu } + \frac{ 1 }{ 2 k } t^{ \mu \nu } ) , is conserved, it is not a tensor density.
Exactly the same thing occurs in Yang-Mills theories: the matter field current j^{ a }_{ \mu } is gauge-invariant, but it is not conserved. On the other hand, the total current (matter plus gauge fields)
J^{ a }_{ \mu } = j^{ a }_{ \mu } + C^{ a }{}_{ b c } \ F^{ b }_{ \mu \nu } \ A^{ \mu c } , is conserved but not gauge-invariant.
Ok, let us go back to GR and integrate Eq(2) with appropriate boundary conditions and obtain the following time-independent object
\mathcal{ P }_{ \mu } = \int d^{ 3 } x \sqrt{ - g } \ ( T^{ 0 }{}_{ \mu } + \frac{ 1 }{ 2 k } t^{ 0 }{}_{ \mu } ) = \mbox{ const. } \ \ \ (3)
Inspired by Eq(2) and Eq(3), Einstein called t_{ \mu \nu } the energy-momentum components of the gravitational field and \mathcal{ P }_{ \mu } the total energy and momentum of the closed system. Back then this interpretation was problematic at first sight. In the final analysis, all difficulties stem from the fact that t_{ \mu \nu } does not transform as a tensor. Since t_{ \mu \nu } contains only first derivatives of the metric tensor g_{ \mu \nu }, it can be made to vanish at an arbitrary point by a suitable choice of the coordinates. In other words, it is always possible to find a class of frames of reference relative to which the gravitational field vanishes locally and, therefore, the t_{ \mu \nu } vanishes locally too. On the other hand, in a perfectly flat spacetime it is possible to find a frame of reference relative to which we detect “inertial forces”. However, these inertial forces are locally equivalent to a gravitational field (principle of equivalence). Therefore, the components of t_{ \mu \nu } do not vanish in a non-inertial coordinate system. Also, since t_{ \mu \nu } is not symmetric, difficulties arise in defining conserved angular momentum. However, this deficiency is less disturbing. Indeed, dy adding appropriate “super-potential” an alternative symmetric pseudo-tensor can be defined (Landau & Lifshitz).
In spite of all these difficulties, it was hard to abandon the idea that an analogue to energy and momentum should exist. A final resolution was eventually made by Einstein (and subsequently completed by F. Klein). Einstein proved that the total 4-momentum \mathcal{ P }_{ \mu } of a closed system (matter plus field) is, to large extent, independent of the choice of coordinate system, although (in general) the localization of energy will be different for different coordinate systems.
The spirit of Einstein’s proof is, again, similar to the Yang-Mills case: While the total current, J^{ a }_{ \mu } = j^{ a }_{ \mu } + C^{ a }{}_{ b c } F^{ b }_{ \mu \nu } A^{ \nu c }, has no simple gauge transformation properties, the integrated time-independent charge,
Q^{ a } = \int d^{ 3 } x \ J^{ a }_{ 0 } ( x ) , is gauge covariant with respect to all gauge transformations which tend to a definite (angle-independent) limit at spatial infinity. Similarly, in GR, the quantity \mathcal{ P }_{ \mu } behaves as a 4-vector with respect to all coordinate transformations which approach Lorentz transformations at infinity.
Finally, I can summarize our current understanding as follow:
“It is impossible to localize energy and momentum in a gravitational field in a generally covariant and physically meaningful way, i.e., t_{ \mu \nu } has no physical meaning. However, the integral expression, Eq(3), for the total energy-momentum 4-vector, \mathcal{ P }_{ \mu }, has a definite physical meaning”.
Sam