We consider two non-statistical definitions of entropy for dynamic (non-stationary) black holes in spherical symmetry. The first is analogous to the original Clausius definition of thermodynamic entropy: there is a first law containing an energy-supply term which equals surface gravity times a total differential. The second is Wald’s Noether-charge method, adapted to dynamic black holes by usin...

We prove that the geodesic flow of a pseudo-Riemannian metric g that admits a ”nontrivial” projective symmetry X is completely integrable. Nontriviality condition of the projective symmetry is expressed in the terms of the invariants of the pair forms g and LXg, where LX denotes the Lie derivative with respect to the vector field X. The theorem we propose can be considered as a ”commutative” an...

Quantum cosmology with quintom dark energy model has been investigated in the present work using symmetry analysis of underlying physical system. In background flat FLRW cosmological studied Noether and appropriate conserved charge is obtained. The Wheeler-DeWitt (WD) equation constructed on minisuperspace solutions are obtained charge.

(a) What symmetries do these charges generate (i.e. how do the fields transform)? Show that they are symmetries of S. (b) If you want to, show that [Q, H] = 0, where H is the Hamiltonian. (c) Evaluate [Q, Q]. Hence, non-Abelian. (d) To complete the circle, find the Noether currents J i μ associated to the symmetry transformations you found in part 1a. (e) Generalize to the case of N scalar fields.

We introduce a method to construct conservation laws for a large class of linear partial differential equations. In contrast to the classical result of Noether, the conserved currents are generated by any symmetry of the operator, including those of the non-Lie type. An explicit example is made of the Dirac equation were we use our construction to find a class of conservation laws associated wi...

We present the discussion of the energy-momentum tensor of the scalar φ-theory on a noncommutative space. The Noether procedure is performed at the operator level. Additionally, the broken dilatation symmetry will be considered in a Moyal-Weyl deformed scalar field theory at the classical level. Work supported by The Danish Research Agency. Work supported in part by “Fonds zur Förderung der Wis...

We demonstrate that the existence of a Noether symmetry in f (R) theories of gravity gives rise to an additional gravitational radius, besides the standard Schwarzschild one, determining the dynamics at galactic scales. By this feature, it is possible to explain the baryonic Tully-Fisher relation and the rotation curve of gas-rich galaxies without the dark matter hypothesis. Furthermore, under ...

Usually we consider the symmetry of action as the symmetry of the theory, however, in the Keplar problem the scaling symmetry existing in equation of motion is not the ones for action. It changes the multiplicative constant of action and the time boundary. In such a case that the scale transformation does not leave the action invariant but keeping the equation invariant, the following statement...