Background Metric Research Articles

Directζ-function approach and renormalization of one-loop stress tensors in curved spacetimes

A method which uses a generalized tensorial $\zeta$-function to compute the renormalized stress tensor of a quantum field propagating in a (static) curved background is presented. The starting point of the method is the direct computation of the functional derivatives of the Euclidean one-loop effective action with respect to the background metric. This method, when available, gives rise to a conserved stress tensor and produces the conformal anomaly formula directly. It is proven that the obtained stress tensor agrees with statistical mechanics in the case of a finite temperature theory. The renormalization procedure is controlled by the structure of the poles of the stress-tensor $\zeta$ function. The infinite renormalization is automatic and is due to a ``magic'' cancellation of two poles. The remaining finite renormalization involves conserved geometrical terms arising by a certain residue. Such terms renormalize coupling constants of the geometric part of Einstein's equations (customary generalized through high-order curvature terms). The method is checked on particular cases (closed and open Einstein`s universe) finding agreement with other approaches. The method is also checked considering a massless scalar field in the presence of a conical singularity in the Euclidean manifold (i.e. Rindler spacetimes/large mass black hole manifold/cosmic string manifold). There, the method gives rise to the stress tensor already got by the point-splitting approach for every coupling with the curvature regardless of the presence of the singular curvature. Comments on the measure employed in the path integral, the use of the optical manifold and different approaches to renormalize the Hamiltonian are made.

Massive scalar field near a cosmic string

The -function of a massive scalar field near a cosmic string is computed and then employed to find the vacuum fluctuation of the field. The vacuum expectation value of the energy - momentum tensor is also computed using a point-splitting approach and employed to find the back-reaction on the background metric. The results obtained could also be useful for the finite-temperature Rindler space theory.

Proof of Gauge Independence of the Conformal Anomaly of Bosonic String in the Sense of Kraemmer and Rebhan

Kraemmer and Rebhan claimed the gauge independence of the conformal anomaly of bosonic string for various gauge fixings in the framework of the perturbation theory of two-dimensional quantum gravity. It is pointed out that their proof is wrong. The gauge independence is proved for the gauge-fixings which reduce to the linearized de Donder gauge in the flat limit of the background metric. Similar remarks are made also for the Rebhan-Kraemmer current anomaly.

Black holes and critical points in moduli space

We study the stabilization of scalars near a supersymmetric black hole horizon using the equation of motion of a particle moving in a potential and background metric. When the relevant 4-dimensional theory is described by special geometry, the generic properties of the critical points of this potential can be studied. We find that the extremal value of the central charge provides the minimal value of the BPS mass and of the potential under the condition that the moduli space metric is positive at the critical point. This is a property of a regular special geometry. We also study the critical points in all N ⩾ 2 supersymmetric theories. We relate these ideas to the Weinhold and Ruppeiner metrics introduced in the geometric approach to thermodynamics and used for the study of critical phenomena.

Nonrelativistic scattering problem by a global monopole

We study the nonrelativistic quantum scattering problem of a charged or massive particle by the global monopole background metric. In addition to the purely gravitational effects, we consider the electrostatic or gravitational self-interaction.

Mass for the Graviton

Can we give the graviton a mass? Does it even make sense to speak of a massive graviton? In this essay I shall answer these questions in the affirmative. I shall outline an alternative to Einstein Gravity that satisfies the Equivalence Principle and automatically passes all classical weak-field tests (GM/r approx 10^{-6}). It also passes medium-field tests (GM/r approx 1/5), but exhibits radically different strong-field behaviour (GM/r approx 1). Black holes in the usual sense do not exist in this theory, and large-scale cosmology is divorced from the distribution of matter. To do all this we have to sacrifice something: the theory exhibits {*prior geometry*}, and depends on a non-dynamical background metric.

Semiclassical effects induced by Aharonov-Bohm interaction between a cosmic string and a scalar field

In the context of the vacuum polarization effect, we consider the back-reaction of the energy-momentum tensor of a charged scalar field on the background metric of a cosmic string carrying a magnetic flux F. Working within the semiclassical approach to the Einstein equations we find the first-order (in ħ) metric associated to the magnetic flux cosmic string. We show that the contribution to the vacuum polarization effect coming from the Aharonov-Bohm interaction is larger than the one coming from the non-trivial gravitational interaction.

Quantum theory of geometry II: Volume operators

A functional calculus on the space of (generalized) connections was recently introduced without any reference to a background metric. It is used to continue the exploration of the quantum Riemannian geometry. Operators corresponding to volume of three-dimensional regions are regularized rigorously. It is shown that there are two natural regularization schemes, each of which leads to a well-defined operator. Both operators can be completely specified by giving their action on states labelled by graphs. The two final results are closely related but differ from one another in that one of the operators is sensitive to the differential structure of graphs at their vertices while the second is sensitive only to the topological characteristics. (The second operator was first introduced by Rovelli and Smolin and De Pietri and Rovelli using a somewhat different framework.) The difference between the two operators can be attributed directly to the standard quantization ambiguity. Underlying assumptions and subtleties of regularization procedures are discussed in detail in both cases because volume operators play an important role in the current discussions of quantum dynamics.

Conserved currents and the energy-momentum tensor in conformally invariant theories for general dimensions

The implications of conformal invariance, as relevant in quantum field theories at a renormalization group fixed point, are analysed with particular reference to results for correlation functions involving conserved currents and the energy-momentum tensor. Ward identities resulting from conformal invariance are discussed. Explicit expressions for two and three-point functions, which are essentially determined by conformal invariance, are obtained. As special cases we consider the three-point functions for two vector and an axial current in four dimensions, which realises the usual anomaly simply and unambiguously, and also for the energy-momentum tensor in general dimension d. The latter is shown to have two linearly independent forms in which the Ward identities are realised trivially, except if d = 4, when the two forms become degenerate. This is necessary in order to accommodate the two independent forms present in the trace of the energy-momentum tensor on curved space backgrounds for conformal field theories in four dimensions. The coefficients of the two trace anomaly terms are related to the three parameters describing the general energy-momentum tensor three-point function. The connections with gravitational effective actions depending on a background metric are described. A particular form due to Riegert is shown to be unacceptable. Conformally invariant expressions for the effective action in four dimensions are obtained using the Green function for a differential operator which has simple properties under local rescalings of the metric.

Conformal Perturbation of Heat Kernels and Their Traces

On a compactn-dimensional Riemannian manifoldMwe consider self-adjoint operators of typew−1Δacting onL2(M,wdx), wheredxis the volume element ofMandw>0 smooth. Forn=2 this setup arises naturally from a conformal change of metric. We derive perturbation series for the heat kernels of these operators and their traces, in terms ofwand of objects in the given background metric. The method involves Gaussian estimates on space and time derivatives of the complex time heat kernel.

Absorption versus decay of black holes in string theory and T-symmetry

Classically a black hole can absorb but not emit energy. We discuss how this T-asymmetric property of black holes arises in the recently proposed (T-symmetric) microscopic models of black holes based on bound states of D-branes. In these string theory based models, the nonvanishing classical absorption is made possible essentially by the exponentially increasing degeneracy of quantum states with mass of the black hole. The classical limit of the absorption cross section computed in the microscopic model agrees with the result obtained from a classical analysis of a wave propagating in the background metric of the corresponding black hole (upto a numerical factor).

NEWMAN–PENROSE FORMALISM FOR GRAVITATIONAL SHOCK WAVES

First-order perturbations for the fields with spin on the background metric of the gravitational shock waves are discussed. Applying the Newman–Penrose formalism, exact solutions of the perturbation equations are obtained. For particle physics, this would be an approach to the problem of scattering particle at Planck energy.

Differential geometry on the space of connections via graphs and projective limits

In a quantum mechanical treatment of gauge theories (including general relativity), one is led to consider a certain completion A G of the space A G of guage equivalent connections. This space serves as the quantum configuration space, or, as the space of all Euclidean histories over which one must integrate in the quantum theory A G is a very large is a very large space and serves as a “universal home” for measures in theories in which the Wilson loop observables are well defined. In this paper, A G is considered as the projective limit of a projective family of compact Hausdorff manifolds, labelled by graphs (which can be regarded as “floating lattices” in the physics terminology). Using this characterization, differential geometry is developed through algebraic methods. In particular, we are able to introduce the following notions on A G : differential forms, exterio derivatives, volume forms, vector fields and Lie brackets between them, divergence of a vector field with respect to a volume form, Laplacians and associated heat kernels and heat kernel measures. Thus, although A G is very large, it is small enough to be mathematically interesting and physically useful. A key feature of this approach is that it does not require a background metric. The geometrical framework is therefore well suited for diffeomorphism invariant theories such as quantum general relativity.

Viscosity of quantum Hall fluids.

The viscosity of quantum fluids with an energy gap at zero temperature is non-dissipative and is related to the adiabatic curvature on the space of flat background metrics (which plays the role of the parameter space). For a quantum Hall fluid on two dimensional tori this viscosity is computed. In this case the average viscosity is quantized and is proportional to the total magnetic flux through the torus.

On spherically symmetric string solutions in four dimensions

We reconsider here the problem of finding the general 4D spherically symmetric, asymptotically flat and time-independent solutions to the lowest-order string equations in the $\ap$ expansion. Our construction includes earlier work, but differs from it in three ways. (1) We work with general background metric, dilaton, axion and $U(1)$ gauge fields. (2) Much of the original solutions were required to be nonsingular at the apparent horizon, motivated by an interest in finding string corrections to black hole spacetimes. We relax this condition throughout, motivated by the realization that string theory has a less restrictive notion of what constitutes a singular field configuration than do point particle theories. (3) We can construct the general solution from a particularly simple one, by generating it from successive applications of the {\it noncommuting} \sltwor\ and \ooneone\ symmetries of the low-energy string equations containing $S$ and target--space dualities respectively. This allows its construction using relatively simple, purely algebraic, techniques. The general solution is determined by the asymptotic behaviour of the various fields: \ie\ by the mass, dilaton charge, axion charge, electric charge, magnetic charge, and Taub-NUT parameter.

Theory of linear cosmological perturbations in flat space-time theory of gravitation

Covariant linear cosmological perturbations are considered in flat space-time theory of gravitation. The background metric is not altered. The perturbed energy-momentum is given. The basic equations for the propagation of the perturbations are presented. The perturbed equations for a homogeneous, isotropic universe are stated.

The bimetric theory of gravitation with a dynamic background metric

We study the bimetric theory of gravitation with background metric γik. In contrast to the accepted point of view, in which γik, is a metric given a priori, we assume that γik is a dynamic variable determined from the condition that the total action of the gravitating system must be an extremum. As a result it turns out that (1) γik can be described by the Einstein equation in space-time with the metric γik and (2) the energy-momentum tensor of the graviational field γik, is the source of γik. In this sense γik can be considered a secondary field in relation to gik. We determine the conditions for existence of integral covariant conservation laws. Two of the latter have no analogs in the theory with the background metric given a priori.

Evolution of scalar field cosmological perturbations

view Abstract Citations (41) References (22) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Evolution of Scalar Field Cosmological Perturbations Hwang, Jai-Chan Abstract A method of dealing with cosmological perturbations was suggested recently which allows a simple way of choosing the gauge appropiate to each separate problem. Using this method, if we have one solution in a particular gauge we can easily derive the rest of the solutions in all the other gauges. In this paper we apply the method to a medium dominated by a minimally coupled scalar field. By considering full metric perturbations we are doing scalar field quantum field theory in a spatially flat homogeneous and isotropic background spacetime, now including the metric perturbations as additonal quantum fields consistently excited from the scalar field perturbations. Asymptotic solutions in a general potential case, and the exact solutions in an exponential potential case -- thus a power-law expanding background case -- are presented in tabular forms. Perturbations in de Sitter space -- thus an exact exponentially expanding background case -- need a separate treatment which is also presented. It is interesting to note that it is in the case that takes into account the full metric perturbations which allows more self-consistent analysis, compared to the one based on pure background metric. We identify the uniform-curvature gauge as providing the simple analysis in the scalar field perturbations. The full solutions may provide a wider perspective on the subject. Publication: The Astrophysical Journal Pub Date: June 1994 DOI: 10.1086/174166 Bibcode: 1994ApJ...427..542H Keywords: Astrophysics; Cosmology; Field Theory (Physics); Gauge Theory; Perturbation; Perturbation Theory; Universe; Curvature; Expansion; Inflating; Newton Theory; Quantum Theory; Astrophysics; COSMOLOGY: LARGE-SCALE STRUCTURE OF UNIVERSE; COSMOLOGY: EARLY UNIVERSE; COSMOLOGY: THEORY full text sources ADS |

Fermions in a global monopole background

We study the scattering of fermions in a “global monopole” background metric. This is the four-dimensional analogue of the scattering on a cone in three dimensions. The scattering amplitude is exactly obtained. We then study massless fermion-dyon systems in such a background metric. The density of the S-wave fermion condensate is found to be given by a constant times the flat space value of Callan and Rubakov.

COORDINATE-INDEPENDENT DEFINITION OF TIME AND VACUUM IN CURVED SPACE–TIME

We propose a definition of time and of the vacuum such that they are intrinsic to a given globally hyperbolic 1 + (n − 1)-dimensional space–time geometry and independent of the choice of coordinates. To arrive at this definition we use the new physical principle that a 1 + 1-dimensional Poincaré algebra, including Killing conditions on the generators, should be valid on the hypersurface of instantaneity. Given a timelike vector at a point (an observer's velocity) we define "an instant of time" to be the spacelike surface of geodesics which pass through that point and are orthogonal to that timelike vector. Gaussian coordinates erected on this surface yield 1 + 1-dimensional subspaces with Poincaré symmetry valid on that surface. The generator associated with time translation now uniquely picks out the direction of time on that surface. This fact permits unambiguous quantization on the surface of a field evolving in this background metric. For flat space–time the corresponding vacuum is always the Minkowski vacuum. We also consider in detail the case of static and Robertson–Walker metrics in 1 + 1 dimensions and find our vacuum to be different from those given before. The vacuum for the de Sitter metric in 1 + 1 dimensions is compared with the results in the literature and found to be different. Our definition of particles, and hence particle production, is consequently different also.