877 research outputs found
One-loop analysis with nonlocal boundary conditions
In the eighties, Schroder studied a quantum mechanical model where the
stationary states of Schrodinger's equation obey nonlocal boundary conditions
on a circle in the plane. For such a problem, we perform a detailed one-loop
calculation for three choices of the kernel characterizing the nonlocal
boundary conditions. In such cases, the zeta(0) value is found to coincide with
the one resulting from Robin boundary conditions. The detailed technique here
developed may be useful for studying one-loop properties of quantum field
theory and quantum gravity if nonlocal boundary conditions are imposed.Comment: 17 pages, Revtex4. In the final version, the presentation in section
5 has been improved, and important References have been adde
The scalar wave equation in a non-commutative spherically symmetric space-time
Recent work in the literature has studied a version of non-commutative
Schwarzschild black holes where the effects of non-commutativity are described
by a mass function depending on both the radial variable r and a
non-commutativity parameter theta. The present paper studies the asymptotic
behaviour of solutions of the zero-rest-mass scalar wave equation in such a
modified Schwarzschild space-time in a neighbourhood of spatial infinity. The
analysis is eventually reduced to finding solutions of an inhomogeneous
Euler--Poisson--Darboux equation, where the parameter theta affects explicitly
the functional form of the source term. Interestingly, for finite values of
theta, there is full qualitative agreement with general relativity: the
conformal singularity at spacelike infinity reduces in a considerable way the
differentiability class of scalar fields at future null infinity. In the
physical space-time, this means that the scalar field has an asymptotic
behaviour with a fall-off going on rather more slowly than in flat space-time.Comment: 19 pages, Revtex4, 7 figure
Self-dual road to noncommutative gravity with twist: a new analysis
The field equations of noncommutative gravity can be obtained by replacing
all exterior products by twist-deformed exterior products in the action
functional of general relativity, and are here studied by requiring that the
torsion 2-form should vanish, and that the Lorentz-Lie-algebra- valued part of
the full connection 1-form should be self-dual. Other two conditions,
expressing self-duality of a pair 2-forms occurring in the full curvature
2-form, are also imposed. This leads to a systematic solution strategy, here
displayed for the first time, where all parts of the connection 1-form are
first evaluated, hence the full curvature 2-form, and eventually all parts of
the tetrad 1-form, when expanded on the basis of {\gamma}-matrices. By assuming
asymptotic expansions which hold up to first order in the noncommutativity
matrix in the neighbourhood of the vanishing value for noncommutativity, we
find a family of self-dual solutions of the field equations. This is generated
by solving first a inhomogeneous wave equation on 1-forms in a classical curved
spacetime (which is itself self-dual and solves the vacuum Einstein equations),
subject to the Lorenz gauge condition. In particular, when the classical
undeformed geometry is Kasner spacetime, the above scheme is fully computable
out of solutions of the scalar wave equation in such a Kasner model.Comment: 37 pages, Revtex. Appendix A is a recollection of mathematical tools
used in the paper. In the final version, Appendix C and some valuable
References have been added. arXiv admin note: text overlap with
arXiv:hep-th/0703014 by other authors. Misprints in Eq. (10.23) and (10.25)
have been amended, as well as their propagation in Sec.
Non-commutative Einstein equations and Seiberg-Witten map
The Seiberg--Witten map is a powerful tool in non-commutative field theory,
and it has been recently obtained in the literature for gravity itself, to
first order in non-commutativity. This paper, relying upon the pure-gravity
form of the action functional considered in Ref. 2, studies the expansion to
first order of the non-commutative Einstein equations, and whether the
Seiberg--Witten map can lead to a solution of such equations when the
underlying classical geometry is Schwarzschild.Comment: 6 and 1/2 pages, based on talk prepared for the Friedmann Seminar,
May-June 2011. In the final version, the presentation has been improved,
including a better notatio
The Seiberg-Witten map for non-commutative pure gravity and vacuum Maxwell theory
In this paper the Seiberg-Witten map is first analyzed for non-commutative
Yang-Mills theories with the related methods, developed in the literature, for
its explicit construction, that hold for any gauge group. These are exploited
to write down the second-order Seiberg-Witten map for pure gravity with a
constant non-commutativity tensor. In the analysis of pure gravity when the
classical space-time solves the vacuum Einstein equations, we find for three
distinct vacuum solutions that the corresponding non-commutative field
equations do not have solution to first order in non-commutativity, when the
Seiberg-Witten map is eventually inserted. In the attempt of understanding
whether or not this is a peculiar property of gravity, in the second part of
the paper, the Seiberg-Witten map is considered in the simpler case of Maxwell
theory in vacuum in the absence of charges and currents. Once more, no obvious
solution of the non-commutative field equations is found, unless the
electromagnetic potential depends in a very special way on the wave vector.Comment: Misprints corrected. References adde
Beltrami equations on Rossi spheres
Beltrami equation on (where , , are the Rossi operators i.e., spans the globally nonembeddable CR structure on discovered by H. Rossi) are derived such that to describe quasiconformal mappings from the Rossi sphere . Using the Greiner-Kohn-Stein solution to the Lewy equation and the Bargmann representations of the Heisenberg group, we solve the Beltrami equations for Sobolev-type solutions such that with
On Schwarzschild's interior solution and perfect fluid star model
We solve the boundary value problem for Einstein’s gravitational field equations in the presence of matter in the form of an incompressible perfect fluid of density rho and pressure field p(r) located in a ball r leq r_0. We find a 1-parameter family of time-independent and radially symmetric solutions {(g_a, rho_a, p_a) : -2m < a < 9 kappa M/(4c^2) identifies the “physical” (i.e., such that p_a(r) geq 0 and p_a(r) is bounded in 0 leq r leq r_0) solutions {p_a : a in mathcal{U}_0} for some neighbourhood mathcal{U}_0 subset (-2m , +infty) of a = 0. For every star model {g_a : a_0 < a < a_1}, we compute the volume V(a) of the region r leq r_0 in terms of abelian integrals of the first, second, and third kind in Legendre form
On the canonical foliation of an indefinite locally conformal Kähler manifold with a parallel Lee form
We study the semi-Riemannian geometry of the foliation of an indefinite locally conformal Kähler (l.c.K.) manifold , given by the Pfaffian equation , provided that and ( is the Lee form of ). If is conformally flat then every leaf of is shown to be a totally geodesic semi-Riemannian hypersurface in , and a semi-Riemannian space form of sectional curvature , carrying an indefinite c-Sasakian structure (in the sense of T. Takahasi). As a corollary of the result together with a semi-Riemannian version of the de Rham decomposition theorem (due to H. Wu) any geodesically complete, conformally flat, indefinite Vaisman manifold of index , 0 < s < n, is locally biholomorphically homothetic to an indefinite complex Hopf manifold , 0 < lambda < 1, equipped with the indefinite Boothby metric
Convection of physical quantities of random density
We study the random flow, through a thin cylindrical tube, of a physical quantity of
random density, in the presence of random sinks and sources. We model convection in terms of the expectations of the flux and density and solve the initial value problem for the resulting convection equation. We propose a difference scheme for the convection equation, that is both stable and satisfies the Courant–Friedrichs–Lewy test, and estimate the difference between the exact and approximate solutions
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