Dimensional Gravity Theory Research Articles

Universality of Final State in Two Dimensional Dilaton Cosmology

Two dimensional dilaton gravity theories provide a class of cosmological toy models in which the semiclassical quantum correction is analytically solvable. The most general semiclassical solution describing inhomogeneous universe is discussed, and all the big bang type universe is shown to end in a flat universe with a universally created matter, irrespective of arty irregularities initially present. It is also pointed out that the same universal behavior towards the final stage of universe remains valid to the leading 'Ii order in the nonintegrable model.

Wormhole and Hawking Radiation

It is shown in a variant of two dimensional dilaton gravity theories that an arbitrary, localized massive source put in an initially regular spacetime gives rise to formation of the wormhole classically, without accompanying the curvature singularity. The semiclassical quantum correction under this wormhole spacetime yields Hawking radiation. It is expected, with the quantum back reaction added to the classical equation, that the information loss paradox may be resolved in this model.

Calculation of Knot Polynomials for Unknotted Knots

Recently, we have succeeded in constructing algebraic equations among knot and link polynomials. In this paper we report that the polynomials for unknot are really solved exactly from the equations, in the case of SU(N) and E 6• Such data are expected to give us a new insight of the Chern· Simons theory because knot polynomials can be regarded as vacuum expectation values of Wilson loop operators in the Chern· Simons theory. Knot polynomials 1H ) can be regarded as vacuum expectation values of Wilson loop operators in 2+ 1 dimensional Chern-Simons theories. 5H9 ) Interestingly enough, we could evaluate them exactly by using the relationship5),20)-22) between the Chern­ Simons theories and 1 + 1 dimensional rational conformal field theories. 23 )-29) Recently, we have shown that vacuum expectation values of Wilson loop opera­ tors in the Chern-Simons theory satisfy algebraic equations if the gauge group is compact and simple. 30 ) In this paper we report briefly typical examples. Algebraic equations for vacuum expectation values of unknotted Wilson loop operators are derived and solved exactly, in the case of SU(N) and E 6 • Because this paper is devoted to rapid communications, we have omitted explanations of many interesting topics. They are discussed in detail in Ref. 30). Our calculations show that vacuum expectation value of unknotted Wilson loop operator in p representation of gauge group G, which is denoted by Eo(p), is nothing but a q-dimensions for Uq(G).**) Consequently, it has close relations between char­ acter formulas of affine Lie algebra, S-matrix of modular transformation, Verlinde algebra, and so on. Although no argument about these properties of Eo(p) is included in this paper, they are discussed in Ref. 30). Here we stress one point. Explicit evaluation of Eo(p) must be useful to eluci­ date some features of the Chern-Simons theory. For example, our computation here is regarded as the manifestations of quantum group symmetry which is hidd~n in the Chern-Simons theory, although it is rather indirect. We think, however, such an observation is important because it is almost impossible to foresee the existence of the quantum group symmetry at the level of Chern-Simons action. As is well known, 2+ 1 dimensional gravity theories are also formulated as the Chern-Simons theories, though, in these cases, the gauge group must be non-compact. Our method might be applicable also for such cases. It means that our computations

EDGE CURRENTS AND VERTEX OPERATORS FOR CHERN-SIMONS GRAVITY

We apply elementary canonical methods for the quantization of 2+1 dimensional gravity, where the dynamics is given by E. Witten’s ISO(2, 1) Chern-Simons action. As in a previous work, our approach does not involve choice of gauge or clever manipulations of functional integrals. Instead, we just require the Gauss law constraint for gravity to be first class and also to be everywhere differentiable. When the spatial slice is a disc, the gravitational fields can either be unconstrained or constrained at the boundary of the disc. The unconstrained fields correspond to edge currents which carry a representation of the ISO(2, 1) Kac-Moody algebra. Unitary representations for such an algebra have been found using the method of induced representations. In the case of constrained fields, we can classify all possible boundary conditions. For several different boundary conditions, the field content of the theory reduces precisely to that of 1+1 dimensional gravity theories. We extend the above formalism to include sources. The sources take into account self-interactions. This is done by punching holes in the disc, and erecting an ISO(2, 1) Kac–Moody algebra on the boundary of each hole. If the hole is originally sourceless, a source can be created via the action of a vertex operator V. We give an explicit expression for V. We shall show that when acting on the vacuum state, it creates particles with a discrete mass spectrum. The lowest mass particle induces a cylindrical space-time geometry, while higher mass particles give an n fold covering of the cylinder. The vertex operator therefore creates cylindrical space-time geometries from the vacuum.

Two dimensional quantum gravity coupled to matter

A classical two dimensional theory of gravity which has a number of interesting features (including a newtonian limit, black holes and gravitational collapse) is quantized using conformal field theoretic techniques. The critical dimension depends upon Newton's constant, permitting models with d = 4. The constraint algebra and scaling properties of the model are computed.

Causality in (2+1)-dimensional gravity

A method is presented to characterize fully the evolution of an arbitrary set of spinless particles in (unquantized) (2+1)-dimensional gravity theory. The method produces a complete series of time ordered Cauchy surfaces, which are being triangulated. By construction, closed timelike curves never arise, even if the initial conditions contain a Gott pair. In particular the author's construction shows that the configuration proposed by Carrol et al. (1992) in which a Gott pair is formed in a closed universe, nevertheless does not admit closed timelike curves; this universe has a finite lifetime, ending in a 'big crunch'.

Lower dimensional black holes

A survey of black hole physics in two spacetime dimensions is presented. Basic properties, specific solutions and quantum aspects are considered in turn. The relationship between string theoretic black holes and those arising in other (1+l)-dimensional theories of gravity is discussed.

Regularization of Quantum Gravity in the Matrix Model Approach

We study divergence problem of the partition function in the matrix model approach for two· dimensional quantum gravity. We propose a new model V(¢)=1/2Tr¢2+g/NTr¢4+dN 4 (Tr¢4)2 and show that in the sphere case it has no divergence problem and the critical exponent is of pure gravity. 23 Recently quantization of two dimensional surfaces has been studied extensively by many people. It is generally believed that in doing so, we can treat a theory of two dimensional quantum gravity and string theory in a non-perturbative manner. To study two dimensional surfaces the idea of N x N hermitean matrix model was brought about in the large N limit, but some difficulties exist in this approach; the most serious problem appears when calculating the partition function of pure gravity. In fact for the simplest ¢4 matrix model, the partition function diverges. To avoid this difficulty we may change the model into the ¢4+¢6 model/) but the latter also has the same divergence. What is the action of the model that does not have this difficulty? We find that ¢4 + (¢4? model has good property in the sphere case by means of saddle point approximation. We consider a matrix model which has the following action,

Regularization of Quantum Gravity in the Matrix Model Approach

We study divergence problem of the partition function in the matrix model approach for two· dimensional quantum gravity. We propose a new model V(¢)=1/2Tr¢2+g/NTr¢4+dN 4 (Tr¢4)2 and show that in the sphere case it has no divergence problem and the critical exponent is of pure gravity. 23 Recently quantization of two dimensional surfaces has been studied extensively by many people. It is generally believed that in doing so, we can treat a theory of two dimensional quantum gravity and string theory in a non-perturbative manner. To study two dimensional surfaces the idea of N x N hermitean matrix model was brought about in the large N limit, but some difficulties exist in this approach; the most serious problem appears when calculating the partition function of pure gravity. In fact for the simplest ¢4 matrix model, the partition function diverges. To avoid this difficulty we may change the model into the ¢4+¢6 model/) but the latter also has the same divergence. What is the action of the model that does not have this difficulty? We find that ¢4 + (¢4? model has good property in the sphere case by means of saddle point approximation. We consider a matrix model which has the following action,

Matrix Models as Superstrings

We propose an extension of Marinari and Parisi's supersymmetric one dimensional strings. This model is expected to describe superstring in less than one dimension. The target space of these string theories is a finite lattice with supersymmetry. The supersymmetry guarantees that the free energy of the matrix model, i.e., the vacuum amplitude of the corresponding string theory, vanishes in any order of the perturbation. When the number of matrices is odd, the supersymmetry breaks down in general. On the other hand, the supersymmetry never breaks down when the number is even. We analyze the correlation functions in one matrix model which will be equivalent to zero dimensional superstring theory in the large N limit. This model is closely related to the bosonic two dimensional gravity coupled with c = -2 conformal matter (-2 dimensional string theory). When Nicolai's mapping is degenerate, there exists a critical point and the continuum limit can be defined to all orders in the genus expansion. We consider the string susceptibility and compare it with those of the known two dimensional gravity theories. Furthermore we analyze two matrix model and we show that there exists a multi critical point.

DYNAMICAL TIME VARIABLE IN COSMOLOGY

Dynamical time variables are studied in two dimensional gravity theory. Dynamical time and space variables exchange their role at the maximum radius (amax) like a black hole at event horizon. Dynamical arrows of time are directed towards amax in both expanding and contracting phases. Both time flows cannot go beyond amax, and the universe becomes static at amax.

Torsion in a gravity theory with SO(k) × SO(d − k) as tangent group

Abstract We consider a d -dimensional theory of gravity where the tangent group is SO( k ) × SO( d − k ) rather than SO( d ) as in riemannian theories. This theory has nonvanishing torsion (which is required if the theory is to yield gauge fields). The torsion is determined consistently in terms of vielbein derivatives.

Positive energy and the scalar potential in higher dimensional (super) gravity theories

We consider, for spacetimes of arbitrary dimension, the general two-derivative gravity plus scalar field theory allowing anti-de Sitter space as a solution. We study the restrictions on the scalar field potential required by positivity of the energy. In particular, we rederive the perturbative stability criterion. Applied to the d = 5 gauged Maxwell/Einstein supergravity theory our results demonstrate the stability of the supersymmetric (adS) 5 ground state.

Dimensionality

We examine the role played by the dimensions of space and space-time in determining the form of various physical laws and constants of Nature. Low dimensional manifolds are also seen to possess special mathematical properties. The concept of fractal dimension is introduced and we discuss the recent renaissance of Kaluza-Klein theories obtained by dimensional reduction from higher dimensional gravity or supergravity theories. A formulation of the anthropic principle is suggested.

Extra space-time dimensions: Towards a solution to the cosmological constant problem

We discuss the possibility that the cosmological constant problem is solved by raising the number of spatial dimensions a la Kaluza-Klein. In (4+2)-dimensional pure gravity theory with the explicit Λ-term we find classical solutions with vanishing physical cosmological constant and compact 2-dimensions. However, there also exist solutions with the non-vanishing physical Λ-term, and the case Λ phys=) is not preferred at the classical level. We conjecture that quantum corrections and/ or additional interactions single out the solution with vanishing physical cosmological constant.