Einstein Spacetimes Research Articles

Exact solutions of embedding the four-dimensional perfect fluid in a five- or higher-dimensional Einstein spacetime and the cosmological interpretations

We investigate an exact solution that describes the embedding of the four-dimensional (4D) perfect fluid in a five-dimensional (5D) Einstein spacetime. The effective metric of the 4D perfect fluid as a hypersurface with induced matter is equivalent to the Robertson–Walker metric of cosmology. This general solution shows interconnections among many 5D solutions, such as the solution in the braneworld scenario and the topological black hole with cosmological constant. If the 5D cosmological constant is positive, the metric periodically depends on the extra dimension. Thus we can compactify the extra dimension on S1 and study the phenomenological issues. We also generalize the metric ansatz to the higher-dimensional case, in which the 4D part of the Einstein equations can be reduced to a linear equation.

BARYOGENESIS BY QUANTUM GRAVITY

A novel mechanism of baryogenesis is proposed on the basis of the phase transition from the conformal invariant spacetime to the Einstein spacetime in quantum gravity. Strong-coupling gravitational excitations with dynamical mass about 1017 GeV are generated at the transition. They eventually decay into ordinary matters. We show that the low energy effective interactions between the gravitational potential describing the excitation and the non-conserving matter currents by the axial anomalies can explain matter asymmetry out of thermal equilibrium.

Einstein's space-time: an introduction to special and general relativity

Space and Time Before Einstein.- In Search of the Ether.- Space and Time in Special Relativity.- Geometric Structure of Space-Time.- Transformation of the Electromagnetic Field.- Energy and Momentum.- Covariant Formulation.- Inertia and Gravity.- Results of General Relativity.

Type D Einstein spacetimes in higher dimensions

We show that all static spacetimes in higher dimensions n > 4 are necessarily of Weyl types G, Ii, D or O. This also applies to stationary spacetimes provided additional conditions are fulfilled, as for most known black hole/ring solutions. (The conclusions change when the Killing generator becomes null, such as at Killing horizons, on which we briefly comment.) Next we demonstrate that the same Weyl types characterize warped product spacetimes with a one-dimensional Lorentzian (timelike) factor, whereas warped spacetimes with a two-dimensional Lorentzian factor are restricted to the types D or O. By exploring algebraic consequences of the Bianchi identities, we then analyze the simplest non-trivial case from the above classes—type D vacuum spacetimes, possibly with a cosmological constant, dropping, however, the assumptions that the spacetime is static, stationary or warped. It is shown that for ‘generic’ type D vacuum spacetimes (as defined in the text) the corresponding principal null directions are geodetic in arbitrary dimension (this in fact also applies to type II spacetimes). For n ⩾ 5, however, there may exist particular cases of type D vacuum spacetimes which admit non-geodetic multiple principal null directions and we explicitly present such examples in any n ⩾ 7. Further studies are restricted to five dimensions, where the type D Weyl tensor is fully described by a 3 × 3 real matrix Φij. In the case with ‘twistfree’ (Aij = 0) principal null geodesics we show that in a ‘generic’ case Φij is symmetric and eigenvectors of Φij coincide with eigenvectors of the expansion matrix Sij providing us thus in general with three preferred spacelike directions of the spacetime. Similar results are also obtained when relaxing the twistfree condition and assuming instead that Φij is symmetric. The five-dimensional Myers–Perry black hole and Kerr–NUT–AdS metrics in arbitrary dimension are also briefly studied as specific illustrative examples of type D vacuum spacetimes.

Stochastic quantization of scalar fields in Einstein and Rindler spacetime

We consider the stochastic quantization method for scalar fields defined in a curved manifold and also in a flat spacetime with event horizon. The two-point function associated with a massive self-interacting scalar field is evaluated, up to the first-order level in the coupling constant λ, for the case of an Einstein and also a Rindler Euclidean metric, respectively. Its value for the asymptotic limit of the Markov parameter τ → ∞ is exhibited. The divergences therein are taken care of by employing a covariant stochastic regularization, where all the symmetries of the original theory are preserved.

THE CHRONO-GEOMETRICAL STRUCTURE OF SPECIAL AND GENERAL RELATIVITY: A RE-VISITATION OF CANONICAL GEOMETRODYNAMICS

A modern re-visitation of the consequences of the lack of an intrinsic notion of instantaneous 3-space in relativistic theories leads to a reformulation of their kinematical basis emphasizing the role of non-inertial frames centered on an arbitrary accelerated observer. In special relativity the exigence of predictability implies the adoption of the 3 + 1 point of view, which leads to a well posed initial value problem for field equations in a framework where the change of the convention of synchronization of distant clocks is realized by means of a gauge transformation. This point of view is also at the heart of the canonical approach to metric and tetrad gravity in globally hyperbolic asymptotically flat space-times, where the use of Shanmugadhasan canonical transformations allows the separation of the physical degrees of freedom of the gravitational field (the tidal effects) from the arbitrary gauge variables. Since a global vision of the equivalence principle implies that only global non-inertial frames can exist in general relativity, the gauge variables are naturally interpreted as generalized relativistic inertial effects, which have to be fixed to get a deterministic evolution in a given non-inertial frame. As a consequence, in each Einstein's space-time in this class the whole chrono-geometrical structure, including also the clock synchronization convention, is dynamically determined and a new approach to the Hole Argument leads to the conclusion that "gravitational field" and "space-time" are two faces of the same entity. This view allows to get a classical scenario for the unification of the four interactions in a scheme suited to the description of the solar system or our galaxy with a deparametrization to special relativity and the subsequent possibility to take the non-relativistic limit.

Space-time evolution and CMB anisotropies from quantum gravity

We propose an evolutional scenario of the universe which starts from quantum states with conformal invariance, passing through the inflationary era, and then makes a transition to the conventional Einstein space-time. The space-time dynamics is derived from the renormalizable higher-derivative quantum gravity on the basis of a conformal gravity in four dimensions. Based on the linear perturbation theory in the inflationary background, we simulate evolutions of gravitational scalar, vector, and tensor modes, and evaluate the spectra at the transition point located at the beginning of the big bang. The obtained spectra cover the range of the primordial spectra for explaining the anisotropies in the homogeneous cosmic microwave background.

HERMITIAN VECTOR FIELDS AND SPECIAL PHASE FUNCTIONS

We start by analyzing the Lie algebra of Hermitian vector fields of a Hermitian line bundle. Then, we specify the base space of the above bundle by considering a Galilei, or an Einstein spacetime. Namely, in the first case, we consider, a fibred manifold over absolute time equipped with a spacelike Riemannian metric, a spacetime connection (preserving the time fibring and the spacelike metric) and an electromagnetic field. In the second case, we consider a spacetime equipped with a Lorentzian metric and an electromagnetic field. In both cases, we exhibit a natural Lie algebra of special phase functions and show that the Lie algebra of Hermitian vector fields turns out to be naturally isomorphic to the Lie algebra of special phase functions. Eventually, we compare the Galilei and Einstein cases.

CANONICAL FOLIATION ON A NULL HYPERSURFACE

We prove the existence of a "canonical foliation" on a null incoming hypersurface. Here "canonical" denotes a foliation whose level function is a solution of a given nonlinear system of equations. The existence of this foliation is an important ingredient for the construction of asymptotically flat global Einstein spacetimes, with appropriate initial conditions.

Review: Hamiltonian Linearization of the Rest-Frame Instant Form of Tetrad Gravity in a Completely Fixed 3-Orthogonal Gauge: A Radiation Gauge for Background-Independent Gravitational Waves in a Post-Minkowskian Einstein Spacetime

In the framework of the rest-frame instant form of tetrad gravity, where the Hamiltonian is the weak ADM energy ${\hat E}_{ADM}$, we define a special completely fixed 3-orthogonal Hamiltonian gauge, corresponding to a choice of {\it non-harmonic} 4-coordinates, in which the independent degrees of freedom of the gravitational field are described by two pairs of canonically conjugate Dirac observables (DO) $r_{\bar a}(\tau ,\vec \sigma)$, $\pi_{\bar a}(\tau ,\vec \sigma)$, $\bar a = 1,2$. We define a Hamiltonian linearization of the theory, i.e. gravitational waves, {\it without introducing any background 4-metric}, by retaining only the linear terms in the DO's in the super-hamiltonian constraint (the Lichnerowicz equation for the conformal factor of the 3-metric) and the quadratic terms in the DO's in ${\hat E}_{ADM}$. {\it We solve all the constraints} of the linearized theory: this amounts to work in a well defined post-Minkowskian Christodoulou-Klainermann space-time. The Hamilton equations imply the wave equation for the DO's $r_{\bar a}(\tau ,\vec \sigma)$, which replace the two polarizations of the TT harmonic gauge, and that {\it linearized Einstein's equations are satisfied} . Finally we study the geodesic equation, both for time-like and null geodesics, and the geodesic deviation equation.

Conformal Einstein equations and Cartan conformal connection

Necessary and sufficient conditions for a spacetime to be conformal to an Einstein spacetime are interpreted in terms of curvature restrictions for the corresponding Cartan conformal connection.

Randall–Sundrum zero mode as a Penrose limit

A generalization of the limiting procedure of Penrose, which allows non-zero cosmological constants and takes into account metrics that contain homogeneous functions of degree zero, is presented. It is shown that any spacetime which admits a spacelike conformal Killing vector has a limit which is conformal to plane waves. If the spacetime is an Einstein space, its limit exists only if the cosmological constant is negative or zero. When the conformal Killing vector is hypersurface orthogonal, the limits of Einstein spacetimes are certain AdS plane waves. In this case the non-linear version of the Randall–Sundrum zero mode is obtained as the limit of the brane world scenarios.

Future Global in Time Einsteinian Spacetimes with U(1) Isometry Group

We prove that spacetimes satisfying the vacuum Einstein equations on a manifold of the form $\Sigma \times U(1)\times R$ where $\Sigma $ is a compact surface of genus $G>1$ and where the Cauchy data is invariant with respect to U(1) and sufficiently small exist for an infinite proper time in the expanding direction.

Huygens' principle for theself-adjoint scalar wave equation in Einstein spacetimes

A necessary condition that the self-adjoint scalar waveequation satisfies Huygens' principle is that the spacetime shouldbe Einstein.

Existence of Lanczos potentialsand superpotentials for theWeyl spinor/tensor

A new and concise proof ofexistence - emphasizing the very naturaland simple structure - is given for theLanczos spinor potential LABCA' of anarbitrary symmetric spinor WABCDdefined by WABCD = 2∇(AA'LBCD)A'; this proof is easily translated into tensorsin such a way that it is valid in four-dimensional spaces of any signature.In particular, this means that the Weyl spinor ΨABCD has Lanczospotentials in all spacetimes, and furthermore that the Weyl tensor hasLanczos potentials on all four-dimensional spaces, irrespective ofsignature. In addition, two superpotentials for WABCD are identified:the first TABCD ( = T(ABC)D) isgiven by LABCA' = ∇A'DTABCD, while the secondHABA'B' ( = H(AB)(A'B')) (which isrestricted to Einstein spacetimes) is givenbyLABCA' = ∇(AB'HBC)A'B'. The superpotential TABCD is used to describe the gaugefreedom in the Lanczos potential.

Local existence of symmetric spinor potentials for symmetric (3,1)-spinors in Einstein space–times

We investigate the possibility of existence of a symmetric potential H ABA′ B′ = H ( AB)( A′ B′) for a symmetric (3,1)-spinor L ABCA′ , e.g., a Lanczos potential of the Weyl spinor, as defined by the equation L ABCA′ =∇ ( A B′ H BC) A′ B′ . We prove that in all Einstein space–times such a symmetric potential H ABA′ B′ exists. Potentials of this type have been found earlier in investigations of some very special spinors in restricted classes of space–times. A tensor version of this result is also given. We apply similar ideas and results by Illge to Maxwell’s equations in a curved space–time.

Petrov type D Einstein space–times of embedding class two

A space–time is of embedding class 2 if its curvature tensor can be written in terms of two symmetric tensors aαβ and bαβ which satisfy the Gauss, Codazzi, and Ricci equations. For an Einstein space–time the Ricci tensor is proportional to the metric tensor. Using the Petrov classification of the curvature tensor and canonical forms of aαβ and bαβ, this paper produces a list of the embedding class 2 Petrov type D Einstein space–times. After a summary of the basic equations and the previous work on embedding class 2 space–times, the field equations for Einstein space–times are expressed in a canonical form and the type D Petrov conditions applied. The resulting equations are solved and give two sets of metrics. The first are known metrics which are products of two 2-spaces of constant curvature and the second set is a generalization of the Kottler solution. The form of the second set of solutions depends upon whether the metric depends upon one or two of the four coordinates.

Extensions of bundles of null directions

The geometry of , the bundle of null directions over an Einstein spacetime, is studied. The full set of invariants of the natural G-structure on is constructed using the Cartan method of equivalence. This leads to an extension of which is an elliptic fibration over the spacetime. Examples are given which show that such an extension, although natural, is not unique. A reinterpretation of the Petrov classification in terms of the fibres of an extension of is presented.

Elliptic fibrations associated with the Einstein space–times

Given a conformally nonflat Einstein space–time we define a fibration P̃ over it. The fibers of this fibration are elliptic curves (two-dimensional tori) or their degenerate counterparts. Their topology depends on the algebraic type of the Weyl tensor of the Einstein metric. The fibration P̃ is a double branched cover of the bundle 𝒫 of null direction over the space–time and is equipped with six linearly independent one-forms which satisfy a certain relatively simple system of equations.

Cosmological solution in multidimensional gravitation based on principal layering

Multidimensional gravitation based on principal layering, the basis of which is four-dimensional Einsteinian space-time while the layers are the SU(2) gage group, is considered. In the spirit of ideas in generalized Kaluza-Klein theories, gage fields correspond to the nondiagonal components of the multidimensional metric in this theory. A cosmological solution is obtained on the basis of an instant-like ansatz for nondiagonal components of the multidimensional metric. It is shown that, at least locally, this solution has a bounce; its behavior near the singularity is investigated.