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GargleBlast42
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Having some generic curved spacetime, what are the Noether currents that are guaranteed to exist by diffeomorphism invariance? Is the energy-momentum tensor such a current?
DaleSpam said:What is the Noether current associated with symmetry under boosts?
GargleBlast42 said:Having some generic curved spacetime, what are the Noether currents that are guaranteed to exist by diffeomorphism invariance? Is the energy-momentum tensor such a current?
samalkhaiat said:Like local gauge invariance, diffeomorphism invariance of the action integral is the subject of the 2nd Noether theorem. The conservation statements of this theorem are nothing but the twice contracted Bianchi identities. However, one can use the (gravitational) field equations to obtain “genuine” conservation laws from the twice contracted Bianchi identities. That is;
[tex]\partial_{a}(T^{ab} + t^{ab}) = 0[/tex]
where [itex]t^{ab}[/itex] is the gravitational energy-momentum pseudotensor. It follows from this that (in curved spacetime) neither matter nor gravitational fields obey separate conservation laws. Also, it is a mistake to associate [itex]T^{ab}[/itex] solely with the matter field and [itex]t^{ab}[/itex] with the “pure” gravitational field, because the theory is highly nonlinear; the T depends on the metric (geometry) as well as the matter field quantities, and the t depends on the matter distribution through the metric. Further complications come from the fact that t is a frame dependent (non covariant) object.
An excellent, possibly the best (old) survey article discussing Noether theorems and conservation laws in curved spacetime is given by A.Trautman, “Foundations and Current Problems of General Relativity”, In Gravitation: An Introduction to Current Research, Ed, L. Witten, John Wiley & Sons Inc. N.Y.
Regards
sam
Noether currents associated with diffeomorphism invariance are mathematical quantities that arise in the study of field theories with diffeomorphism invariance. They are conserved quantities that are associated with symmetries of the theory.
Noether currents are closely related to diffeomorphism invariance because they are a consequence of the symmetry of the theory under diffeomorphisms, which are transformations that change the coordinates of the spacetime.
Noether currents play a crucial role in field theories because they represent the conservation laws of the theory. They provide a powerful tool for understanding the dynamics of the system and can reveal important symmetries and properties of the theory.
Noether currents can be calculated using Noether's theorem, which relates symmetries of a system to conserved quantities. In the case of diffeomorphism invariance, the Noether currents are calculated by taking variations of the action with respect to the diffeomorphism parameters.
Yes, Noether currents associated with diffeomorphism invariance have been used in a variety of practical applications, especially in the study of general relativity and other field theories. They have also been applied in cosmology, black hole physics, and other areas of theoretical physics.