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 $overline{L}_t (g) = mu( cdot , t) L_t (g)$ on $S^3$ (where $L_t$, $t in (-1,1)$, are the Rossi operators i.e., $L_t$ spans the globally nonembeddable CR structure $mathcal{H} (t)$ on $S^3$ discovered by H. Rossi) are derived such that to describe quasiconformal mappings $f: S^3 to N subset mathbb{C}^2$ from the Rossi sphere $(S^3 , mathcal{H} (t))$. 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 $g_t$ such that $g_t - v in W^{1,2}_F (S^3, theta)$ with $v in CR^infty (S^3 , mathcal{H} (0))$

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 $mathcal F$ of an indefinite locally conformal Kähler (l.c.K.) manifold $M$, given by the Pfaffian equation $omega = 0$, provided that $nabla omega = 0$ and $c = | omega | neq 0$ ($omega$ is the Lee form of $M$). If $M$ is conformally flat then every leaf of $mathcal F$ is shown to be a totally geodesic semi-Riemannian hypersurface in $M$, and a semi-Riemannian space form of sectional curvature $c/4$, 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 $2s$, 0 < s < n, is locally biholomorphically homothetic to an indefinite complex Hopf manifold ${mathbb C}H^n_s (lambda )$, 0 < lambda < 1, equipped with the indefinite Boothby metric $g_{s, n}$