Einstein's Theory Of Gravitation Research Articles

A Polyakov fermionic string as a quantum state of Einstein theory of gravitation

We show that a Polyakov fermionic string functional integral in the presence of the external spatial Ashtekar-Sen connection solves the Wheeler-De Witt equation of Einstein gravitation theory written in terms of the Ashtekar-Sen variables.

Freezing of the Affine Gravity and Singularity Free Cosmological Models

A modified version of the Einstein theory of gravity is proposed in which the constant G is dynamical, that is dependent on the energy stress tensor. This permits one to obtain a model in which the Planck's energy density is unbounded, meaning that gravity never reaches the condition ρ>ρ P . This permits one to separate the affine description from the quantum one, which has recently been observed not to be completely compatible. We describe moreover how this can bring us to violate the strong energy condition so as to avoid the constraints of singularity theorems.

Geometrization of linear perturbation theory for diffeomorphism-invariant covariant field equations. I. The notion of a gauge-invariant variable

Applying linear perturbation theory to the general-relativistic field equations, in a series of recent papers we have analyzed the gauge problem for an almost-Robertson-Walker universe. Mathematically, our analysis made use of a rather arbitrary choice of the background space-time geometry, and it turns out to possess the undersirable feature that the basic definitions and concepts are valid only for Einstein's gravity theory. The main purpose of this paper is to remedy all of the above deficiencies. Consequently, a new geometrical discussion of the notion of a gauge-invariant variable is presented with a view to demonstrating its usefulness in the context of an arbitrary diffeomorphism-invariant covariant field theory. Another welcome feature of this discussion is that, for linear perturbation theory, the proposed construction of gauge-invariant variables does not depend on the specific symmetry properties of the background “space-time” geometry chosen; in other words, it can be proven to hold for any possible choice of the background. In a companion paper, such an approach to the gauge problem will enable us to indicate in universal terms what geometrical objects are in fact essential if one is to obtain a fully satisfactory description of the equivalence classes of perturbations. A new example of the general structures, as compared with those already investigated for Einstein's gravity theory in the description of an almost-Robertson-Walker universe, is also given there. This example arises from consideration of the infinitesimal perturbation of the metric tensor itself (pure gravity) defined on a fixed background de Sitter space-time.

N=3 and N=4 Two-Form Supergravities

We construct the Lagrangians of N=3 and N=4 two-form supergravities. The two-form gravity theories are classically equivalent to the Einstein gravity theories and can be formulated as gauge theories. The gauge algebras used here can be identified with the subalgebra of N=3 superconformal algebra and SU (2)× SU (2)× U(1) -extended N=4 superconformal algebra.

Topological Symmetry Breaking on Einstein Manifolds

It is known that if gauge conditions have Gribov zero modes, then topological symmetry is broken. In this paper we apply it to the Witten type topological gravitational theory in dimension n ≥ 3. Our choice of the gauge condition for conformal invariance is R + α = 0, where R is the Ricci scalar curvature. We find when α ≠ 0, topological symmetry is not broken, but when α = 0 and solutions of the Einstein equations exist then topological symmetry is broken. These conditions connect to the Yamabe conjecture, namely negative constant scalar curvature exist on manifolds of any topology, but existence of nonnegative constant scalar curvature is restricted by topology. This fact is easily seen in this theory. Topological symmetry breaking means BRS symmetry breaking in cohomological field theory. But it is found that another BRS symmetry can be defined and physical states are redefined. The divergence due to the Gribov zero modes is regularized, and the theory after topological symmetry breaking becomes semiclassical Einstein gravitational theory under a special definition of observables.

Do Gravitational Waves Travel at Light Velocity?

We extend the standard Feynman–Deser approach of field theoretical derivation of Einstein's gravitational theory. We show that it is possible to obtain a theory that incorporates a great part of general relativity (GR) and can be interpreted in the standard geometrical way like GR, as far as the interaction of matter to gravity is concerned. The most important distinction of the new theory concerns the gravity-to-gravity interaction. This theory satisfies all standard tests of gravity and leads to new predictions about gravitational propagation. Since there is a strong expectation that the detection of gravitational waves will occur in the near future, the question of which theory describes nature better will probably be settled soon.

Dimensional reduction and renormalizability of the Wheeler-DeWitt equation: next-leading-order contribution

We solve the Wheeler-DeWitt equation for the wave function as an expansion in powers of the Planck mass by using heat kernel regularization. To solve the next-leading-order equation in this expansion, we introduce additional terms which are proportional to three-dimensional scalar curvature and Ricci tensor in the heat equation. We have an approximate wave function up to next-leading-order in this expansion. Expectation values computed with the leading-order approximation are reduced to the expectation value in three-dimensional Euclidean Einstein gravity theory in the region which is much smaller than the Planck scale. This means that the «New phase» (the dynamical system described by the three-dimensional quantum Einstein gravity) exists in the region beyond the Planck scale. We also discuss the renormalization group equation for the wave function of the Wheeler-DeWitt equation.

Unified analysis of cosmological perturbations in generalized gravity.

In a class of generalized Einstein's gravity theories we derive the equations and general asymptotic solutions describing the evolution of the perturbed universe in unified forms. Our gravity theory considers general couplings between the scalar field and the scalar curvature in the Lagrangian, thus includes broad classes of generalized gravity theories resulting from recent attempts for the unification. We analyze both the scalar-type mode and the gravitational wave in analogous ways. For both modes the large scale evolutions are characterized by the same conserved quantities which are valid in the Einstein's gravity. This unified and simple treatment is possible due to our proper choice of the gauges, or equivalently gauge invariant combinations.

Nonsingular Metric Elastic Universe

In the RTFD(Gusev (1986)) the conception of a Sakharov - Wheeler Metric Elasticity(SWME)(Sakharov (1967), Wheeler (1970)) had been worked out. On the basis of the exact solutions of Einstein equations and qualitative analysis RTFD the global evolution have been studied and the phase portraits of the early Universe is being constructed. An analysis of phase portraits show on the possibility description of spontaneous creation of Universe from an initial Minkowskian's vacuum to an inflationary de Sitter space-time in the frame of phenomenological non-quantum theory (Guth (1991)). During the past decade, a radically new picture of cosmology has emerged. The present homogeneous expanding Universe would have stated out with a de Sitter phase. The purpose of this paper is to shown that the geometry-dynamical approach to the Einstein's gravitation theory in the frame RTFD also is leaded to the nonsingular cosmological models (Brandenberger (1993)). Let us to propose that before the some moment of time the Universe is at the vacuum state and is described the geometry of Minkowskian's space. Deformations of vacuum state, identifying with empty Mikowskian's space are described by the deformations tensor, An arising of deformation ∊αβis leaded to appearance of the stress tensor ∊αβand the energy-momentumTαβ(∊γδ) which is connected with “creating” particles in the Universe. Here we are considered the deformations of Minkowskian's space (the initial vacuum state with∞αβ = 0) at the linear theory (~ ∊) of finite deformations. The final deformation stategαβare searched in the metric class of Friedmann's cosmological spaces. In the comoving reference systemUα(0, 0, 0, 1) the Friedmann's equations have form (Narlikar & Padmanabhan (1983), and Gusev (1989)):where R(t) is so called the expansion factor at the Robertson - Walker line element, k is the curvature parameter with the possible values −1, 0, + 1, P is pressure,k1,k2are the some combination from a Lame coefficients,l02is a “initial radius” Universe, a free parameter model. The phase space of this model is the two-dimensional (R,Ṙ) plane. We note that there is only two singular points (Ṙ= 0,Ṙ= 0) in the phase plane. The one of those points isR=l0,Ṙ= 0 and corresponds to Minkowski space - time. There are two classes trajectories which are asymptotically de Sitter. Those starting at large positive values ofṘgo off toṘ= + ∞, reaching their asymptotic value of H from above. Those starting with large negative values ofṘtend toR= + ∞ withṘ> 0. For small values ofṘand R we can see that there are periodic solutions about Minkowski space. The corresponding solutions oscillate with frequency given byH0(which is possible equal planck scale) about Minkowski space. Based on the preceding discussion of asymptotic solutions we see that there is a separatrix (Gusev, (1989)) in phase space dividing solutions which tend toR= + ∞ from those which oscillate or tend toR=l0. The above analyses of the phase portraits is an indication that in our theory Minkowski space may be unstable toward homogeneous deformations. We stress that all the general features of the phase portrait analyses are true for quadratic deformations of gravitational vacuum. Our model incorporates a very important feature: in the asymptotic de Sitter region, the quadratic deformations and temperature effects does not have an important effect on the geometry. The effective gravitational constant of coupling goes to zero as space - time approaches de Sitter space. In this sense the model is asymptotically free (gravitational confinement Linde, (1990)). At the late times the solutions are described a evolution of the de Sitter UniverseR~expHt(Hoyle et al. (1993)).

Gravitational field of a global cosmic string with non-linear Lagrangian

In this paper the metric outside a straight global string is derived, with the high-derivative gravitational theory which has a higher correction as compared with Einstein gravitational theory. The effect of accreting matter and the formation of the large-scale structure of the Universe may be very important.

Dimensional reduction and renormalizability of the Wheeler-DeWitt equation

We solve the Wheeler-DeWitt equation for the wave function as an expansion in powers of the Planck mass. Expectation values computed with the leading-order approximation are reduced to the expectation value in three-dimensional Euclidean Einstein gravity theory. This means that «new phase» (the dynamical system described by the three-dimensional quantum Einstein gravity) exists at the region beyond the Planck scale. We also briefly discuss about the renormalizability of the Wheeler-DeWitt equation for (N+1)-dimensional Lorentzian manifolds (N>3).

Celestial bodies and multidimensional Einstein theory of gravitation

Static spherically symmetric celestial bodies are investigated in multidimensional Einstein theory of gravitation. A model with compact Ricci-flat internal space ℝnis studied and confronted with observations. Integration constants in the exterior solution of field equations are determined using integrals over the distribution of matter inside the celestial body. Two independent formulas for the mass and mass squared of the celestial body are derived. The first one is similar to Tolman's formula for the mass in GRT. It is shown that integral parameters (mass, radius etc.) of the matter distribution inside the celestial body tend to definite limits in the case of large number (n → ∞) of extra dimensions.

ON THE WHEELER-DEWITT EQUATION FOR BLACK HOLES

Integration over the angular coordinates of the evaporating, four-dimensional Schwarzschild black hole leads to a two-dimensional action, for which the Wheeler-DeWitt equation has been found by Tomimatsu, on the apparent horizon, where the Vaidya metric is valid, using the Hamiltonian formalism of Hajicek. For the Einstein theory of gravity coupled to a massless scalar field ζ, the wave function Ψ obeys the Schrödinger equation [Formula: see text], where M is the mass of the hole. The solution is [Formula: see text], where k2 is the separation constant, and for k2>0 the hole evaporates at the rate Ṁ=−k2/4M2, in agreement with the result of Hawking. Here, this analysis is generalized to the two-dimensional theory [Formula: see text], which subsumes the spherical black holes formulated in D≥4 dimensions, when A = ½ (D - 2) (D - 3)ϕ2 (D - 4)/(D - 2), B=2(D−3)/(D−2), C=1, and also the twodimensional black hole identified by Witten and by Gautam et al., when A=4/α′, B=2, C=1/8π, c=+8/α′ being (minus) the central charge. In all cases an analogous Schrödinger equation is obtained. The evaporation rate is [Formula: see text] when D≥4 and [Formula: see text] when D=2. Since Ψ evolves without violation of unitarity, there is no loss of information during the evaporation process, in accord with the principle of black-hole complementarity introduced by Susskind et al. Finally, comparison with the four-dimensional, cosmological Schrödinger equation, obtained by reduction of the ten-dimensional heterotic superstring theory including terms [Formula: see text], shows in both cases that there is a positive semi-definite potential which evolves to zero, this corresponding to the ground state, which is Minkowski space.

An extension of Birkhoff's theorem to a class of 2-d gravity theories containing black holes

A class of 2-dimensional models including 2-d dilaton gravity, spherically symmetric reduction of d-dimensional Einstein gravity and other related theories are classically analyzed. The general analytic solutions in the absence of matter fields other than a U (1) gauge field are obtained under a new gauge choice and recast in the conventional conformal gauge. These solutions imply that Birkhoff's theorem, originally applied to spherically symmetric 4-d Einstein gravity, can be applied to all models we consider. Some issues related to the coupling of massless scalar fields and the quantization are briefly discussed.

General relativity from 'localization' of Killing vector fields

A new nonstandard method of constructing general relativity based on a principal analogical-to-local invariance is suggested. The internal symmetries of an action of any field theory given in a nondynamic spacetime having Killing vector fields are used as a starting point. The analogous properties for a required source action (at present without using generators of any group of motions) are required. The use of any fixed background spacetime (perhaps without any symmetries) is essential. In the result the gauge field appears. For it too the gauge (internal) transformation and the free field action are obtained. After including the connection between the gauge field and the background metric it becomes apparent that the constructed theory is the Einstein theory of gravity in the field (nongeometrical) formulation developed earlier.

The Clifford bundle and the nature of the gravitational field

In this paper we formulate Einstein's gravitational theory with the Clifford bundle formalism. The formalism suggests interpreting the gravitational field in the sense of Faraday, i.e., with the field residing in Minkowski spacetime. We succeeded in discovering the condition for this interpretation to hold. For the variables that play the role of the gravitational field in our theory, the Lagrangian density turns out to be of the Yang-Mills type (with an auto-interaction plus gauge-fixing terms). We give a brief comparison of our theory with other field theories of the gravitational field in the flat Minkowski spacetime.

Renormalization group for nonrenormalizable theories: Einstein gravity with a scalar field.

We develop a renormalization-group formalism for non-renormalizable theories and apply it to Einstein gravity theory coupled to a scalar field with the Lagrangian $L=\sqrt{g} [R U(\phi)-{1/2} G(\phi) g^{\mu\nu} \partial_{\mu}\phi \partial_{\nu}\phi- V(\phi)]$, where $U(\phi), G(\phi)$ and $V(\phi)$ are arbitrary functions of the scalar field. We calculate the one-loop counterterms of this theory and obtain a system of renormalization-group equations in partial derivatives for the functions $U, G$ and $V$ playing the role of generalized charges which substitute for the usual charges in multicharge theories. In the limit of a large but slowly varying scalar field and small spacetime curvature this system gives the asymptotic behaviour of the generalized charges compatible with the conventional choice of these functions in quantum cosmological applications. It also demonstrates in the over-Planckian domain the existence of the Weyl-invariant phase of gravity theory asymptotically free in gravitational and cosmological constants.

Quantum Einstein Gravity as a Differential Geometry: General Derivation of Gauge-Fixing Term and FP-Ghost Term Based on Fibre Bundle Theory

Intrinsic OSp(2D;2D)-symmetry (D≥3) is formally regarded as the intrinsic part of OSp(2D;2D)-symmetry which is derived from the quantum Einstein gravity theory. On the basis of the intrinsic OSp(2D;2D)-symmetry as the first principle, the geometrical structure of the theory is clarified and the gauge-fixing term plus the FP-ghost one in the quantum Einstein gravity is characterized as an orthosymplectic super-form on the totality of the cross-section of a tensor product bundle

Ashtekar's variables reexamined.

Ashtekar's variables for gravity are investigated in terms of exterior forms. The origin of the new variables is traced back to their role in eliminating the potential piece of the Lagrangian L=T-V which results in constraints polynomial in the canonically conjugate fields. The second-order general relativity and the real as well as complex (self-dual) Palatini first-order formalisms are collated with respect to this property. This analysis keeps track of the Hamiltonian through all these versions which are linked by canonical transformations; it presents a simple proof of the positive mass theorem, examines the possibility of scalar constraints linear in the momenta, and corrects a tetrad approach given by Ashtekar and collaborators. Also a new concise formulation of Einstein's gravity theory is proposed. To account for Cartan's calculus, a short but self-contained guide from Lagrangian to Hamiltonian formalisms in terms of forms is included.

Quantum gravity, the origin of time and time's arrow

The local Lorentz and diffeomorphism symmetries of Einstein's gravitational theory are spontaneously broken by a Higgs mechanism by invoking a phase transition in the early universe, at a critical temperature Tc below which the symmetry is restored. The spontaneous breakdown of the vacuum state generates an external time, and the wave function of the universe satisfies a time-dependent Schrödinger equation, which reduces to the Wheeler-deWitt equation in the classical regime for T<Tc, allowing a semiclassical WKB approximation to the wave function. The conservation of energy is spontaneously violated for T>Tc, and matter is created fractions of seconds after the big bang, generating the matter in the Universe. The time direction of the vacuum expectation value of the scalar Higgs field generates a time asymmetry, which defines the cosmological arrow of time and the direction of increasing entropy as the Lorentz symmetry is restored at low temperatures.