An attractor universe in six-dimensional N = 2 supergravity Kaluza-Klein theory

Cosmological solutions are investigated in six-dimensional, N = 2 supergravity Kaluza-Klein theory. It is shown that the solution of (the four-dimensional Friedmann universe)×(a constant S²) is the attractor, i.e. all the cosmological solutions starting from arbitrary initial conditions (apart from the time reversal ones) approach the above space-time asymptotically without any fine-tuning. The Friedmann solution is asymptotically “unique” in the later stage of the universe in six-dimensional N = 2 supergravity.     

Renormalizability and asymptotic freedom in quantum gravity

We discuss a previously proposed renormalizable theory of gravity involving R²_μν, and N massless fermion (vector boson) fields in which the unitarity problem is resolved within a $\text{1}\text{N}$ expansion. The infrared limit is precisely Einstein's theory, but the high-energy behavior is determined by the dimensionless, asymptotically free coupling of the R²_μν. Various attractive possible consequences of the theory are pointed out.     

Pregeometric quantum lattice: A general discussion

We put forward an idea that the fundamental, i.e. pregeometric, structure of spacetime is given by an abstract set, so called abstract simplicial complex ASC. Thus, at the pregeometric level there is no (smooth) spacetime manifold. However, we argue that the structure described by an abstract simplicial complex is dynamical. This dynamics is then assumed to ensure that ASC can be realized as a lattice on a four-dimensional manifold with the simplest topologies dominating.We rewrite the pregeometric model, which is quantized using euclidean path-integral formalism, in an exact way so that as a four-dimensional manifold with the simples topologies dominating. is done by definition. The first step in bringing the continuum into the arena is to build up a lattice on a four-dimensional manifold from a given ASC. In fact, we choose a specific lattice: The Regge calculus lattice, i.e. a piecewise linear (flat) metric spacetime manifold. Secondly, we introduce a smooth (C^∞) manifold (described by a metric tensor g_μν) to approximate the Regge calculus manifold (described by a metric tensor g_μν^RC).It turns out that after integrating (and summing) out all other degrees of freedom than the metric tensor field g_μν, the resulting continuum theory is nonlocal (as would be expected). However, it is our main point to show that the nonlocality is not very severe since it is only of finite range. We argue that the points in the introduced continuum which represent lattice points have so great quantum fluctuations that they are in a high temperature phase with no long-range correlations. In other words, although the effective action for the continuum formulation is not totally local, it is effectively so because it has only finite range nonlocalities. We can prove this kind of weak locality of the effective action by means of a general high-temperature theorem. Then we claim that the resulting local (or rather almost local) model with reparametrization invariance and g_μν as a field gives essentially the ordinary Einstein's gravity theory in the long wavelength limit.     

Quantum Kaluza-Klein cosmologies (III)

The “ground state” proposal for the quantum state of the universe is generalized to the case of a noncompact spacelike three-hyperboloid as the configuration space. The most probable evolution of the universe must come from a gravitational instanton by quantum tunneling. We show that under some minisuperspace ansatz, there exists only S₄ × S₇ gravitational instanton in d = 11 supergravity. From the point of view of quantum cosmology this fact must be related to the fact that our observed spacetime is four-dimensional.     

Fermion spectra from superstrings

Some compactifications of the ten-dimensional anomaly-free E₈ × E₈ and SO(32) theories that correspond to superstrings are studied. Compactification is achieved by setting the classical gauge field equal to the spin connection. The resulting chiral fermion spectra are obtained for any six-dimensional manifold, under the condition Tr F² = 30 Tr R², plus a quantization condition for U(1) charges. For E₈ × E₈ these conditions lead to potentially realistic models for any irreducible six-dimensional manifold and any embedding of the holonomy group. Apart from a few more exotic examples, the four-dimensional models we obtain are more or less standard SU(5), SO(10), SU(4) × SU(2) × SU(2) or E₆ models.     

Gravitational Field Equations of Fourth Order and Supersymmetry

We discuss the field equations which stem from a variational principle containing the quadratic terms αR_μνR_μν and βR² besides the Einstein-Hilbert Lagrangian R. Comparison of this theory with a pure theory of fourth order shows that R must necessarily be included if we wish to interpret the field equations as gravitational equations. The Einstein-Bach-Weyl theory (α = ?3β) has the property of being a theory of “supergravitation”. Apart from gravitons without rest-mass, we have here only one additional kind of particles with rest-mass. Their mass may be determined by Planck' slength (hG/c³)^1/2. The occurrence of those particles results from the breakdown of a “supersymmetry”, that is of the conform invariance. The Einstein tensor E_μν ? R_μν ?1/2g_μνR can be regarded as a source of the gravitons without rest-mass.     

An obstacle to creating a universe in the laboratory

We show that any spherically symmetric false vacuum bubble which forms in an asymptotically flat space and grows beyond a certain critical size must have merged from an initial singularity. Our result requires that the energy-momentum tensor obey the condition T_μνk^μk^ν⩾0 for all null k^μ, a property which holds at the classical level for almost all theories of matter. For the non-spherical case, we state a necessary condition that a false vacuum bubble must meet in order to avoid an initial singularity. We do not know if this condition can ever be met. The requirement of an initial singularity appears to be an insurmountable obstacle to the creation of an inflationary universe in the laboratory.     

Einstein–Maxwell gravity with additional corrections

Motivated by the E ₈×E ₈ heterotic string theory, we obtain topological black hole solutions of Einstein–Maxwell gravity with additional corrections. We consider the Gauss–Bonnet (GB) and (F _μν F ^μν )² terms as an effective quartic order Lagrangian of gauge–gravity coupling and investigate geometric and thermodynamic properties of the black hole solutions. We also analyze the effects of the GB term as well as the correction of Maxwell field on the properties of the solutions.     

Effective Action in Multidimensional (Super)Gravities and Spontaneous Compactification (Quantum Aspects of Kaluza-Klein Theories)

The review of modern status of problem of quantum effects in Kaluza-Klein theories is given. The effective action (EA) in multidimensional (super)gravities (SG's) on the compactified background is investigated. The standard gauge dependent EA in d=5 Einstein gravity and d=5 R²-gravity on the background R₄ × S₁, where R₄ is 4-dimensional space, S₁ is one-dimensional sphere is calculated. Gauge and parametrization independent Vilcovisky-De Witt EA in d=5 Einstein gravity and d=5 R²-gravity on the background R₄×S₁ at zero and non-zero temperature is obtained. We have found that there are no physically acceptable self-consistent solutions of the form R₄×S₁ at the one-loop level in d=5 Einstein gravity. We calculated also EA for arbitrary multidimensional SG on the background R₄×T_d-n where T_d is d-dimensional torus as expansion on the curvature and its derivatives. The mechanizm of induced of four-dimensional gravity with zero Λ-term is proposed. The Vilcovisky-De Witt EA in d=5 SG's on the background R₄×S₁ at non-zero temperature is obtained. The three gauge parameter dependent off-shell EA in N=2, d=5 gauged SG on R₄⁰×S₁ where R₄⁰ is flat four-dimensional space is calculated. The expression for vacuum energy for bosonic string with torus compactification is presented. Vacuum energy for superstrings with supersymmetry broken as the result of choice of boundary conditions on background R₄×T₆ is calculated.     

Cosmological singularities and higher-order gravitational lagrangians

General relativity is modified by adding terms proportional to R² and R^μνR_μν to the Lagrangian. One class of solutions of the modified field equations is free of singularities but does not lead to asymptotic behaviour (for large time) of the Friedmann type. A second class, which shows the correct asymptotic behaviour, does contain the usual singularities of Friedmann universes, collapse being modulated by small oscillations only. The quantum effects considered here are thus unable to prevent the occurrence of cosmological singularities under physically reasonable conditions.     

On dimensional reduction of gauge theories: Symmetric fields and harmonic expansion

We compare the four-dimensional symmetric fields obtained by the coset space dimensional reduction scheme to the infinite tower of fields given by the harmonic expansion in a 4+N dimensional gauge theory coupled to fermions on a space-timeM ⁴ ×S/R.     

Supersymmetric Yang-Mills,octonionic instantons and triholomorphic curves

In four-dimensional gauge theory there exists a well-known correspondence between instantons and holomorphic curves, and a similar correspondence exists between certain octonionic instantons and triholomorphic curves. We prove that this latter correspondence stems from the dynamics of various dimensional reductions of ten-dimensional supersymmetric Yang-Mills theory. More precisely we show that the dimensional reduction of the (5+1)-dimensional supersymmetric sigma model with hyper-Kähler (but otherwise arbitrary) target X to a four-dimensional hyper-Kähler manifold M is a topological sigma model localising on the space of triholomorphic maps M -+ X (or hyperinstantons). When X is the moduli space M_k of instantons on a four-dimensional hyper-Kdhler manifold K, this theory has an interpretation in terms of supersymmetric gauge theory. In this case, the topological sigma model can be understood as an adiabatic limit of the dimensional reduction of ten-dimensional supersymmetric Yang-Mills on the eight-dimensional manifold M × K of holonomy Sp(1) × Sp(1) ⊂ Spin(7), which is a cohomological theory localising on the moduli space of octonionic instantons.     

A superstring model of four generations

We construct a possibly realistic four-generation Calabi-Yau manifold by dividing the algebraic variety in CP₄ × CP₄ with the Z₂×Z₂ symmetry. The nontrivial embedding of Z₂×Z₂ in E(6) allows physically intriguing intermediate symmetry based on the U(1)×SU(2)_L×SU(2)_R×SU(4)_C group. Also, the group of honest symmetries G_H of the manifold is identified.     

Correction of a misunderstanding about Olbers' paradox

It is sometimes said that, if the universe were not expanding, we all would burn up, (Olbers' paradox). In order to check on this, I have used a solution of Friedmann's equation that seems reasonable, and that predicts a contraction of the universe after it has reached a maximum size. We may then compare the radiation energy density U _n now at age t _n of the universe, with this density (=U ^* ) at a time t ^* during contraction when R(t) takes a value R ^* equal to its present value R _n .We simplify the calculation by choosing 2R _n as maximum of R(t). Then, t _n=(1/2 π?1)R _n/C=6.98 × 10⁹ year for R _n = 1.16× 10²⁸ cm.(The present density would then be ρ_n = 2.4× 10^?29 gram/cm³).We assume the number of galaxies per unit volume = N(t) = η/R(t) ³ with constant η,and we assume a constant average radiative power L per galaxy. Now at t _n we choose N _n L ≈ 10^?31 erg/cm³sec,but our conclusions would be the same for much larger values of this. We split U into three parts: U ₁ is the primordial energy density left over from t ≈ 0.We put U ₁(t _n ) ≈ 6× 10^?13 erg/cm³ corresponding to T ≈ 3°K.U ₂ is the density of the energy flux near the earth emitted by all stars in our own Milky Way. Most of it is the density U _S =L _⊙/4ηr ² c = 4.5 × 10^?5 erg/cm³ in the flux from the sun. Finally, U ₃ is the energy in the radiation emitted by all other galaxies since t ≈ 0, and was going to burn us according to Olbers. Since U ₁ is a function of R(t), we put U ₁(t^*) =U ₁ (t _n ).We shall also assume U ₂ to be unchanged, as it was not the effect of a change of U _s that we were investigating. In calculating U ₃ we shall overestimate it by neglecting the absorption by closer galaxies of some of the light emitted by farther ones.     

具有广义协变的包含重力场贡献的重力场方程

利用半度规λ^(α)_μ表象的数学工具定义一个对广义坐标具有协变形式的重力场矢势函数ω^(α)_μ≡-cλ^(α)_μ，给出一个具有广义协变的包含重力场贡献的重力场方程R_μν-g_μνR/2+Λg_μν=8πG(T^(Ⅰ)_μν+T^(Ⅱ)_μν) 关键词： 重力场方程 协变形式 能量-动量张量 量子化     

Constraints on the behaviour of the e+e− hadron annihilation cross section in asymptotically free theories and in theories with anomalous dimensions

A new approach is proposed to the problem of the relationship between the e⁺e^? hadron annihilation cross section and the behaviour of the hadronic vacuum polarization tensor Π_μν(q) in the space-like region. By using this approach based on the principles of functional analysis, one can uniquely transform the information about the behaviour of Π_μν(q) at q² → ?∞, which is available in asymptotically free theories and in theories with anomalous dimensions, into constraints on the e⁺e^? total cross section.     

Two open universes connected by a wormhole: exact solutions

In this paper, I present a space–time of two open universes connected by a Lorentzian wormhole. The space–time has the following features: (1) it can exactly solve the Einstein equations; (2) the weak energy condition is satisfied everywhere; (3) it has a topology of R²×T_g (g≥2); (4) it has no event horizons.     

Classical stability of direct products of spheres in gravitational systems

Classical stability of Einstein spaces S^d₁ ×?×S^d_n(d_j ? 2) against all fluctuations is investigated in euclidean gravity with a cosmological constant. It is shown that S^d is classically stable, while S^d₁ ×?× S^d_n(n ? 2) is classically unstable. As a generalization of this analysis it is proved that a compact Einstein space B₁ ×?× B_n(n ? 2) which is a direct product of each Einstein space is classically unstable. Non-Einstein spaces M² × S⁴ (M² × S² × S²) are also considered in six- dimensional Einstein-Maxwell theory and are shown to be classically stable (unstable).     

Generalised Kerr-Schild space-times

If V is a space-time with metric tensor g_μν admitting a null, geodesic shear free vector field l_μ, then one may determine a function H so that the spacetime $\text{V}\text{?}$ with metric g_μν = g_μν + 2Hl_μl_ν satisfies the Einstein field equations for various material sources, and for no sources. When V is Minkowski space, $\text{V}\text{?}$ is a Kerr-Schild space-time. In case V is a vacuum space-time, one may choose H so that the source is a null fluid with no pressure. In case V is a Robertson-Walker universe H may be chosen so that the source has a stress-energy tensor with one timelike proper vector and three spacelike ones. There are two equal proper values associated with the latter vectors and one which differs from these. The stress-energy tensor describing this source may be interpreted as representing a perfect fluid with anisotropic pressures or as one describing the sum of a perfect fluid with isotropic pressures and a presureless null fluid. Vaidya's Kerr metric in a cosmological background [Pramana8 (1977) 512–517] is discussed as is the metric representing an accelerating point mass in an expanding universe.     

Physical constants as cosmological constraints

A solution to the primary “missing mass” problem is found in the context of accounting for the coincidence of large dimensionless numbers first noticed by Weyl, Eddington, and Dirac. This solution entails (1) a log₂ relation between the electromagnetic and gravitational coupling constants; (2) setting the maximum radius of curvature at the gravitational radius, 2GM/c ²; (3) a changing gravitational parameterG, which varies as an inverse function of the universal radius of curvature. These features motivate the development of a neo-Friedmann formalism, which employs a function,ε(χ). governing the change from Euclidian to non-Euclidian volumes. Observational consequences include (1) a universal density of 7.6×10^?31g cm^?3, (2) a Hubble parameter of 15 km s^?1 Mpc^?1, (3) an age of the universe of 32×10⁹ yr, (4) a gravitational parameter diminishing at a current rate of 2.2×10^?12 yr^?1, and (5) a deceleration parameter of 1.93. Moreover, it is shown that for a Friedmann-type (λ=0) cosmology (whether open or closed) any deceleration parameter will be represented by a straight line in the (log-log) red shift: luminosity-distance space of the Hubble diagram. The major claim of this paper is that we have devised a model in which the large-scale structure of the universe is completely determined by the values of the fundamental physical constants:c, h, e, andm _e setting the scale, andG selecting the epoch.