Higher Dimensional Gravity Research Articles

Description of the Riemannian geometry in the presence of conical defects.

A consistent approach to the description of integral coordinate invariant functionals of the metric on manifolds ${\cal M}_{\alpha}$ with conical defects (or singularities) of the topology $C_{\alpha}\times\Sigma$ is developed. According to the proposed prescription ${\cal M}_{\alpha}$ are considered as limits of the converging sequences of smooth spaces. This enables one to give a strict mathematical meaning to a number of invariant integral quantities on ${\cal M}_{\alpha}$ and make use of them in applications. In particular, an explicit representation for the Euler numbers and Hirtzebruch signature in the presence of conical singularities is found. Also, higher dimensional Lovelock gravity on ${\cal M}_{\alpha}$ is shown to be well-defined and the gravitational action in this theory is evaluated. Other series of applications is related to computation of black hole entropy in the higher derivative gravity and in quantum 2-dimensional models. This is based on its direct statistical-mechanical derivation in the Gibbons-Hawking approach, generalized to the singular manifolds ${\cal M}_{\alpha}$, and gives the same results as in the other methods.

Recent progress in the theory of random surfaces and simplicial quantum gravity

Some of the recent developments in the theory of random surfaces and simplicial quantum gravity is reviewed. For 2d quantum gravity this includes the failure of Regge calculus, our improved understanding of the c > 1 regime, some surprises for q-state Potts models with q > 4, attempts to use renormalization group techniques, new critical behavior of random surface models with extrinsic curvature and improved algorithms. For simplicial quantum gravity in higher dimensions it includes a discussion of the exponential entropy bound needed for the models to be well defined, the question of “computational ergodicity” and the question of how to extract continuum behavior from the lattice simulations.

A family of cosmological solutions in higher-dimensional Einstein gravity

A two-parameter family of solutions to Einstein’s vacuum field equations in six dimensions is presented. The solutions are a natural extension of a one-parameter family of five-dimensional vacuum solutions found by Ponce de Leon [Gen. Relativ. Gravit. 20, 539 (1988)] and are closely related to the generalized Kasner metrics. The solution is generalized to 4+n dimensions and is briefly discussed.

Random surfaces and lattice quantum gravity

Recent developments in the study of the crystalline and fluid random surface models are discussed. Concerning two dimensional gravity theories, a new method for measuring the string susceptibility exponent is described and so are new results on the distribution of boundary lengths in, and the fractal dimension of, the mainfolds which dominate the partition function. Possible lattice models for higher dimensional quantum gravity are described together with the results from recent simulations.

Quantum Gravity with Dynamical Torsion in Two Dimensions

One of the main reasons for considering two-dimensional gravity is an expectation that it may provide a conceptual laboratory for investigation of quantum gravity in higher dimensions. In particular, dynamical coupling of quantum matter fields seems manageable only in two spacetime dimensions. However, as long as pure gravity is concerned, it is so special in two dimensions that it bears little resemblance to pure gravity in three or more dimensions. Naive counting gives a negative degree of freedom for the metric field gPJJ with general covariance. Indeed, in two dimensions, the Einstein tensor GpJJ=RpJJ-(1/2)gpJJR vanishes identically, and hence there is no sensible Einstein equation at all. The introduction of an additional scalar field (J makes it possible to define a nontrivial theoryl)

Two-dimensional quantum cosmology

Two-dimensional quantum gravity coupled to conformally invariant matter with central charge c > 25 has been proposed as a toy model for quantum gravity in higher dimensions. The associated “Wheeler-DeWitt equation” is non-linear and unstable to forming a condensate of baby universes. This will occur even in the classical c → ∞ limit. Small fluctuations about this background describe the propagation of single universes and satisfy a more conventional linear Wheeler-DeWitt equation. The resulting two-dimensional cosmology depends on details of the non-linear dynamics. In particular the existence of a large-scale cosmological constant is determined by the behavior of a string theoretic tachyon potential near its minimum.

Higher dimensional simplical quantum gravity

The generalization of the dynamically triangulated quantum gravity (DT) model to more than two-dimensions is discussed. In three-dimensions two updates are described which appear to allow for ergodic sampling of states, and an updating algorithm which obeys detailed balance is presented. The tensor model generalization of the matrix model formulation of higher dimensional gravity is discussed.

Simplicial quantum gravity in more than two dimensions.

The dynamical triangulation (DT) model of two-dimensional quantum gravity is generalized to higher dimensions. In three dimensions two updates are described which appear to allow for ergodic sampling of states, and an updating procedure which obeys detailed balance is given. The tensor-model generalization of the matrix-model formulation of higher-dimensional gravity is discussed and the relations between DT-, continuum-, and tensor-model couplings are given.

Extended inflation from higher-dimensional theories.

We consider the possibility that higher-dimensional theories may, upon reduction to four dimensions, allow extended inflation to occur. We analyze two separate models. One is a very simple toy model consisting of higher-dimensional gravity coupled to a scalar field whose potential allows for a first-order phase transition. The other is a more sophisticated model incorporating the effects of nontrivial field configurations (monopole, Casimir, and fermion bilinear condensate effects) that yield a nontrivial potential for the radius of the internal space. We find that extended inflation does not occur in these models. We also find that the bubble nucleation rate in these theories is time dependent unlike the case in the original version of extended inflation.

State vectors in higher-dimensional gravity with kinematic quantum numbers of quarks and leptons

We examine the question of whether fundamental fermions can be regarded as topological solitons (geons) of a theory of pure quantum gravity in n + 4 dimensions. In particular, we consider the quantum gravitational field for ( n + 4)-dimensional space-times which have asymptotic Kaluza-Klein behavior: asymptotic topology of the form R 4 × G/H and a metric that is asymptotically the direct sum of the Minkowski metric on R 4 and the natural symmetric metric on G/H. We argue that if there is a nonsingular theory of pure quantum gravity (a theory that, below Planck energy, involves only the space-time metric), then there are stable ground states with the kinematical quantum numbers of fundamental fermions: that is, states which have spin 1 2 and belong to the fundamental representation of G . Multi-valued representations of the asymptotic symmetry group can arise from diffeomorphisms (diffeos) of the ( n + 3)-dimensional space that are not isotopic to the identity. In a path integral approach, the creation of fundamental fermions (topological geons) requires a space-time in which the topology of space-like hypersurfaces changes and in which the relevant diffeos on the final ( n + 3)-space (containing the geons) are not extendible to diffeos of the ( n + 4)-space-time. We prove that such pair-creation geometries occur in a lorentzian as well as euclidean framework and that the diffeos are not extendible. Although zero-mass modes of finite-size fermions need not belong to a real representation of the internal group, it is not difficult to prove that, independent of the fermion size, one cannot avoid mirror fermions. There is, however, an interesting loophole: A nonzero vacuum expectation value of a Higgs-like vector constructed from the metric can break chiral invariance without changing the lowest-order Einstein-Yang-Mills form of the gravitational lagrangian.

Electroweak boson masses from gravity-coupled sigmas

It is shown that a non-linear sigma model coupled to gravity in higher dimensions can induce gauge-symmetry breaking in the effective theory. This result is achieved without loss of the compactification and the vanishing of the cosmological constant which the same sigma field induces in the classical vacuum. In the explicitly studied SU (2) × U (1) case, the construction predicts the massless photon as well as the value ϱ = 1 for the electroweak bosons. The ordinary Higgs mechanism as well as the renormalizability of the electroweak sector may be recovered, following an appropriate linearization of the theory.

Fibre bundle generated theory of higher dimensional gravity

With the help of Weil polynomials on the Lie algebra of the Poincare group in D dimensions, we construct a Lagrangian form for higher dimensional gravity on a principal fibre bundle whose base space is an even-dimensional Riemann-Cartan spacetime and whose structure group is the Poincare group.

On higher dimensional gravity: the Lagrangian, its dimensional reduction and a cosmological model

The author motivates the study of, and gives a brief introduction to, the theory of higher dimensional gravity described by the Lovelock Lagrangian. After studying its dimensional reduction, a cosmological model is discussed where the internal space is the Euclidian n-torus.

A two-dimensional model for quantum gravity

Two-dimensional quantum gravity coupled to conformally invariant matter is a good model for the physics of higher-dimensional gravity, provided the central charge of the matter theory is greater than 25. In particular, the theory has a euclidean saddle point of large negative action, analogous to that considered by Baum, Hawking and Coleman in d = 4. Further, the theory is identical to string theory in an appropriate background, making it a promising laboratory for the study of topology change. Some preliminary applications are made to cosmological solutions, the Hartle-Hawking wavefunction, wormholes, and the classical field equation for the baby universe background.

Electromagnetic solutions of higher-dimensional gravity

We investigate the field equations of the higher-dimensional extensions of the Taub-NUT-de Sitter space-times. Exact solutions are given both in the vacuum as well as in the dust case, in general with a non-vanishing cosmological constant and an additional electromagnetic field.

Gravitational theta states and the wave function of the universe.

The naive no-boundary wave function of the universe is shown to be invariant under diffeomorphisms only for the simplest spacetime topologies. A more general construction which does give an invariant wave function of the universe is exhibited. Similar problems, some familiar, some not, are encountered in a wide range of theories whose physical configuration space is topologically nontrivial. These include the theory of identical particles, Yang-Mills theory, higher-dimensional gravity, and membrane theories. The sum-over-histories formulation of quantum mechanics provides a unified approach to these problems and their resolution.

Consistent interaction for infinitely many massless spin-two fields by dimensional reduction

We present a higher-dimensional gravity model realizing the recently discovered new gauge symmetries for a collection of massless spin-two fields. The commutative associative algebra characterizing these theories is obtained in a Kac-Moody-like way by harmonic analysis on internal space. The dynamics is governed by a pure dimensionally continued Euler form. More general dimensionally reduced gravity theories can be interpreted as spontaneous symmetry-breaking of an infinite-dimensional gauge symmetry.

Spectrum degeneracy and new symmetries for generalized Euler form actions

Higher-dimensional gravity theories based on dimensionally continued Euler forms lead to a degenerate spectrum. It contains fewer excitations than the one of a genetic theory with arbitrary coefficients of the higher derivative terms. This can be traced back to a residual topological invariance of the action. The symmetry of the theory is enhanced. As a consequence the stability group of the vacuum can be larger than in the generic case. At the four-dimensional level this can even lead to infinite dimensional (Kac-Moody type) gauge algebras. A simple six-dimensional example leads to infinite towers of massless gravitons and scalars, but no spin-one bosons (gauge bosons) are present. We describe how modes decouple when the action approaches the Euler form. Concerning classical stability the Euler form corresponds to a singular point in the space of six-dimensional gravity actions.

Soliton transformations for axially symmetric higher-dimensional gravity. I. Generalized Neugebauer–Kramer transformations

The field equations for axially symmetric gravitational fields in 4+K dimensions with K-dimensional toroidal compatification can be reduced to those of a generalized nonlinear sigma model. The dual symmetry of this model is considered first. Then the soliton transformations for these equations are derived using the method of Belinskii and Zakharov consequently generalizing their results to the higher-dimensional case. From the result of one-soliton transformation, a series of discrete symmetries generalizing the Neugebauer–Kramer transformation in the four-dimensional case are obtained. The basic properties of these new transformations are studied.

On energy in 5-dimensional gravity and the mass of the Kaluza-Klein monopole

We discuss the concept of energy in higher-dimensional gravity, with special attention given to the problem of the choice of a background. Three different approaches to the calculation of energy for solutions of the 5-dimensional Einstein equation are considered. They are then shown to be equivalent and applied to the calculation of the mass of the Kaluza-Klein monopole. The question of the reduction of a 5-dimensional theory to an effective 4-dimensional one in the presence of a Killing vector field along the compact dimension is then discussed, in particular with regard to the conformal ambiguity in the definition of the metric in the reduced theory. We show that there exists an essentially unique choice of reduction for which the 4-dimensional energy has the usual general relativistic form, and that for the Kaluza-Klein monopole this is the reduction that will make the “naive” 4-dimensional energy additive.