181 research outputs found
On Lagrangian and Hamiltonian systems with homogeneous trajectories
Motivated by various results on homogeneous geodesics of Riemannian spaces,
we study homogeneous trajectories, i.e. trajectories which are orbits of a
one-parameter symmetry group, of Lagrangian and Hamiltonian systems. We present
criteria under which an orbit of a one-parameter subgroup of a symmetry group G
is a solution of the Euler-Lagrange or Hamiltonian equations. In particular, we
generalize the `geodesic lemma' known in Riemannian geometry to Lagrangian and
Hamiltonian systems. We present results on the existence of homogeneous
trajectories of Lagrangian systems. We study Hamiltonian and Lagrangian g.o.
spaces, i.e. homogeneous spaces G/H with G-invariant Lagrangian or Hamiltonian
functions on which every solution of the equations of motion is homogeneous. We
show that the Hamiltonian g.o. spaces are related to the functions that are
invariant under the coadjoint action of G. Riemannian g.o. spaces thus
correspond to special Ad*(G)-invariant functions. An Ad*(G)-invariant function
that is related to a g.o. space also serves as a potential for the mapping
called `geodesic graph'. As illustration we discuss the Riemannian g.o. metrics
on SU(3)/SU(2).Comment: v3: some misprints correcte
Weak cosmic censorship, dyonic Kerr-Newman black holes and Dirac fields
It was investigated recently, with the aim of testing the weak cosmic
censorship conjecture, whether an extremal Kerr black hole can be converted
into a naked singularity by interaction with a massless classical Dirac test
field, and it was found that this is possible. We generalize this result to
electrically and magnetically charged rotating extremal black holes (i.e.
extremal dyonic Kerr-Newman black holes) and massive Dirac test fields,
allowing magnetically or electrically uncharged or nonrotating black holes and
the massless Dirac field as special cases. We show that the possibility of the
conversion is a direct consequence of the fact that the Einstein-Hilbert
energy-momentum tensor of the classical Dirac field does not satisfy the null
energy condition, and is therefore not in contradiction with the weak cosmic
censorship conjecture. We give a derivation of the absence of superradiance of
the Dirac field without making use of the complete separability of the Dirac
equation in dyonic Kerr-Newman background, and we determine the range of
superradiant frequencies of the scalar field. The range of frequencies of the
Dirac field that can be used to convert a black hole into a naked singularity
partially coincides with the superradiant range of the scalar field. We apply
horizon-penetrating coordinates, as our arguments involve calculating
quantities at the event horizon. We describe the separation of variables for
the Dirac equation in these coordinates, although we mostly avoid using it.Comment: 28 pages, LaTeX, sections 2, 3 and appendix A shortened, appendix C
omitted, subsection 4.1 and references added, results unchange
Noether currents for the Teukolsky Master Equation
Conserved currents associated with the time translation and axial symmetries
of the Kerr spacetime and with scaling symmetry are constructed for the
Teukolsky Master Equation (TME). Three partly different approaches are taken,
of which the third one applies only to the spacetime symmetries. The results
yielded by the three approaches, which correspond to three variants of
Noether's theorem, are essentially the same, nevertheless. The construction
includes the embedding of the TME into a larger system of equations, which
admits a Lagrangian and turns out to consist of two TMEs with opposite spin
weight. The currents thus involve two independent solutions of the TME with
opposite spin weights. The first approach provides an example of the
application of an extension of Noether's theorem to nonvariational differential
equations. This extension is also reviewed in general form. The variant of
Noether's theorem applied in the third approach is a generalization of the
standard construction of conserved currents associated with spacetime
symmetries in general relativity, in which the currents are obtained by the
contraction of the symmetric energy-momentum tensor with the relevant Killing
vector fields. Symmetries and conserved currents related to boundary conditions
are introduced as well, and Noether's theorem and its variant for
nonvariational differential equations are extended to them. The extension of
the latter variant is used to construct conserved currents related to the
Sommerfeld boundary condition.Comment: 19 pages, revised manuscript, new subsections on boundary condition
On the mass-coupling relation of multi-scale quantum integrable models
We determine exactly the mass-coupling relation for the simplest multi-scale
quantum integrable model, the homogenous sine-Gordon model with two independent
mass-scales. We first reformulate its perturbed coset CFT description in terms
of the perturbation of a projected product of minimal models. This
representation enables us to identify conserved tensor currents on the UV side.
These UV operators are then mapped via form factor perturbation theory to
operators on the IR side, which are characterized by their form factors. The
relation between the UV and IR operators is given in terms of the sought-for
mass-coupling relation. By generalizing the sum rule Ward identity we
are able to derive differential equations for the mass-coupling relation, which
we solve in terms of hypergeometric functions. We check these results against
the data obtained by numerically solving the thermodynamic Bethe Ansatz
equations, and find a complete agreement.Comment: 55 pages, 9 figures, reference added, minor changes, published
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