1/<Emphasis Type="Italic">r</Emphasis> potential in higher dimensions

In Einstein gravity, gravitational potential goes as \(1/r^{d-3}\) in d non-compactified spacetime dimensions, which assumes the familiar 1 / r form in four dimensions. On the other hand, it goes as \(1/r^{\alpha }\), with \(\alpha =(d-2m-1)/m\), in pure Lovelock gravity involving only one mth order term of the Lovelock polynomial in the gravitational action. The latter offers a novel possibility of having 1 / r potential for the non-compactified dimension spectrum given by \(d=3m+1\). Thus it turns out that in the two prototype gravitational settings of isolated objects, like black holes and the universe as a whole – cosmological models, the Einstein gravity in four and mth order pure Lovelock gravity in \(3m+1\) dimensions behave in a similar fashion as far as gravitational interactions are considered. However propagation of gravitational waves (or the number of degrees of freedom) does indeed serve as a discriminator because it has two polarizations only in four dimensions.     

Gravity-Induced Four-Fermion Contact Interaction Implies Gravitational Intermediate W and Z Type Gauge Bosons

Coupling fermions to gravity necessarily leads to a non-renormalizable, gravitational four-fermion contact interaction. In this essay, we argue that augmenting the Einstein–Cartan Lagrangian with suitable kinetic terms quadratic in the gravitational gauge field strengths (torsion and curvature) gives rise to new, massive propagating gravitational degrees of freedom. This is to be seen in close analogy to Fermi’s effective four-fermion interaction and its emergent W and Z bosons.     

Dynamic compactification with stabilized extra dimensions in cubic Lovelock gravity

In this paper the dynamic compactification in Lovelock gravity with a cubic term is studied. The ansatz will be of space–time where the three dimensional space and the extra dimensions are constant curvature manifolds with independent scale factors. The numerical analysis shows that there exist a phenomenologically realistic compactification regime where the three dimensional hubble parameter and the extra dimensional scale factor tend to a constant. This result comes as surprise as in Einstein–Gauss–Bonnet gravity this regime exists only when the couplings of the theory are such that the theory does not admit a maximally symmetric solution (i.e. “geometric frustration”). In cubic Lovelock gravity however there always exists at least one maximally symmetric solution which makes it fundamentally different from the Einstein–Gauss–Bonnet case. Moreover, in opposition to Einstein–Gauss–Bonnet Gravity, it is also found that for some values of the couplings and initial conditions these compactification regimes can coexist with isotropizing solutions.     

Taub–NUT black holes in third order Lovelock gravity

We consider the existence of Taub–NUT solutions in third order Lovelock gravity with cosmological constant, and obtain the general form of these solutions in eight dimensions. We find that, as in the case of Gauss–Bonnet gravity and in contrast with the Taub–NUT solutions of Einstein gravity, the metric function depends on the specific form of the base factors on which one constructs the circle fibration. Thus, one may say that the independence of the NUT solutions on the geometry of the base space is not a robust feature of all generally covariant theories of gravity and is peculiar to Einstein gravity. We find that when Einstein gravity admits non-extremal NUT solutions with no curvature singularity at r=N$r=N$, then there exists a non-extremal NUT solution in third order Lovelock gravity. In 8-dimensional spacetime, this happens when the metric of the base space is chosen to be CP³${\mathbb{CP}}^3$. Indeed, third order Lovelock gravity does not admit non-extreme NUT solutions with any other base space. This is another property which is peculiar to Einstein gravity. We also find that the third order Lovelock gravity admits extremal NUT solution when the base space is T²×T²×T²$T^2\times T^2\times T^2$ or S²×T²×T²$S^2\times T^2\times T^2$. We have extended these observations to two conjectures about the existence of NUT solutions in Lovelock gravity in any even-dimensional spacetime.     

Thermodynamics of Third Order Lovelock Anti-de Sitter Black Holes Revisited

We compute the mass and temperature of third order Lovelock black holes with negative Gauss-Bonnet coefficient α₂<0 in anti-de Sitter space and perform the stability analysis of topological black holes. When k=-1, the third order Lovelock black holes are thermodynamically stable for the whole range r₊. When k=1, we found that the black hole has an intermediate unstable phase for D=7. In eight dimensional spacetimes, however, a new phase of thermodynamically unstable small black holes appears if the coefficient \tilde{\alpha} is under a critical value.For D≧ 9, black holes have similar the distributions of thermodynamically stable regions to the case where the coefficient \tilde{\alpha} is under a critical value for D=8. It is worth to mention that all the thermodynamic and conserved quantities of the black holes with flat horizon do not depend on the Lovelock coefficients and are the same as those of black holes in general gravity.     

Classical variational derivation and physical interpretation of Dirac's equation

A simple random walk model has been shown by Gaveauet al. to give rise to the Klein-Gordon equation under analytic continuation. This absolutely most probable path implies that the components of the Dirac wave function have a common phase; the influence of spin on the motion is neglected. There is a nonclassical path of relative maximum likelihood which satisfies the constraint that the probability density coincide with the quantum mechanical definition. In three space dimensions, and in the presence of electromagnetic interaction, the Lagrangian for this optimal, nonclassical path coincides with the Lagrangian of the Dirac particle. The nonrelativistic, or diffusion, limit is shown to be a formal consequence of Einstein's dynamical equilibrium condition; the continuity equation reduces to the same diffusion equation derived from Schrödinger's equation. The relativistic, massless limit, which would describe a neutrino, is comparable (in the sense of analytic continuation) to a nonviscous liquid whose molecules possess internal degrees of freedom.Dedicated to Professor Alfonso Maria Liquori on the occasion of his 60th birthday.     

Lovelock tensor as generalized Einstein tensor

We show that the splitting feature of the Einstein tensor, as the first term of the Lovelock tensor, into two parts, namely the Ricci tensor and the term proportional to the curvature scalar, with the trace relation between them is a common feature of any other homogeneous terms in the Lovelock tensor. Motivated by the principle of general invariance, we find that this property can be generalized, with the aid of a generalized trace operator which we define, for any inhomogeneous Euler–Lagrange expression that can be spanned linearly in terms of homogeneous tensors. Then, through an application of this generalized trace operator, we demonstrate that the Lovelock tensor analogizes the mathematical form of the Einstein tensor, hence, it represents a generalized Einstein tensor. Finally, we apply this technique to the scalar Gauss–Bonnet gravity as an another version of string–inspired gravity. This work was partially supported by a grant from the MSRT/Iran.     

Massive higher spin fields coupled to a scalar: Aspects of interaction and causality

We consider in detail the most general cubic Lagrangian which describes an interaction between two identical higher spin fields in a triplet formulation with a scalar field, all fields having the same values of the mass. After performing the gauge fixing procedure we find that for the case of massive fields the gauge invariance does not guarantee the preservation of the correct number of propagating physical degrees of freedom. In order to get the correct number of degrees of freedom for the massive higher spin field one should impose some additional conditions on parameters of the vertex. Further independent constraints are provided by the causality analysis, indicating that the requirement of causality should be imposed in addition to the requirement of gauge invariance in order to have a consistent propagation of massive higher spin fields.     

Thermodynamics of Third Order Lovelock-Born-Infeld Black Holes

We here explore black holes in the third order Lovelock gravity coupling with nonlinear Born-Infeld electromagnetic field. Considering special second and third order coefficients (\hat{\alpha}_2^2=3\hat{\alpha}_3=α), we analyze the thermodynamics of third order Lovelock-Born-Infeld black holes and, in 7-dimensional AdS space-time, discuss the stability of black holes in different event horizon structures. We find that the cosmological constantΛ plays an important role in the distribution of black hole stable regions.     

Restrictions on the interaction between massive vector mesons

A critical investigation of the most general Lagrangian describing any number of massive vector mesons coupled through parity conserving interactions with dimensionless coupling constants is made. The theory is required to satisfy the following three conditions which are necessary on physical grounds: the presence of the interaction must not change the number of degrees of freedom of the system, the equations must describe propagation, and the propagation must take place at a speed less than that of light. One finds that some of the coupling constants are zero and others are related. The Lagrangian density takes a very simple form which generalizes, apart from the mass term, the Yang-Mills Lagrangians.     

Dimensional regularization of gauge theories in spherical spacetime: Free field trace anomalies

The O(n + 1) covariant formulation of massless quantum electrodynamics in spherical spacetime is further developed to allow a calculation of the energy-momentum tensor trace anomalies for the free Dirac, electromagnetic, and SU(2) gauge fields. The principal technical development is the construction of the Faddeev-Popov ghosts for electrodynamics and SU(2) Yang-Mills theory. This construction is unconventional first in that the gauge fixing term in the Lagrangian is not a perfect square, and second because it is necessary to remove radial as well as gauge degrees of freedom from the measure of the functional integral. The ghost fields are shown to satisfy a minimal scalar field equation. The free field effective action is found to be divergent in four dimensions, and is renormalized by the inclusion in the Lagrangian of a counterterm local in the gravitational fields. The energy-momentum tensor calculated from this renormalized effective action is shown to have a trace anomaly.     

Lie Symmetries of Einstein’s Vacuum Equations in N Dimensions

Abstract

We investigate Lie symmetries of Einstein’s vacuum equations in N dimensions, with a cosmological term. For this purpose, we first write down the second prolongation of the symmetry generating vector fields, and compute its action on Einstein’s equations. Instead of setting to zero the coefficients of all independent partial derivatives (which involves a very complicated substitution of Einstein’s equations), we set to zero the coefficients of derivatives that do not appear in Einstein’s equations. This considerably constrains the coefficients of symmetry generating vector fields. Using the Lie algebra property of generators of symmetries and the fact that general coordinate transformations are symmetries of Einstein’s equations, we are then able to obtain all the Lie symmetries. The method we have used can likely be applied to other types of equations.     

Application of group representation theory to derive Hermite interpolation polynomials on a triangle

Current methods used to devise sets of Hermite interpolation polynomials of minimal order that ensure C⁽ⁿ⁾ continuity across triangular element boundaries in two dimensions are not readily extensible to higher dimensions. The extension of such methods is especially difficult when the number of degrees of freedom afforded by data at points is different from the number of degrees of freedom determined by the coefficients of a complete polynomial basis to a particular order. This work introduces a formalism based on group representation theory that can accomplish this task in general. The method is introduced through the derivation of C⁽¹⁾ continuous Hermite polynomials that interpolate data at the three vertices of an equilateral triangular element. These interpolation polynomials are reported here for the first time. The polynomials derived here are compared to the standard polynomials defined in a right triangle by using the two sets of polynomials to solve the Laplace equation over finite elements. The methodology presented here is of use in higher dimensional elements when the complete polynomial degrees of freedom exceed the total C⁽ⁿ⁾ degrees of freedom at the nodes.     

On the static Lovelock black holes

We consider static spherically symmetric Lovelock black holes and generalize the dimensionally continued black holes in such a way that they asymptotically for large $r$ go over to the d-dimensional Schwarzschild black hole in dS/AdS spacetime. This means that the master algebraic polynomial is not degenerate but instead its derivative is degenerate. This family of solutions contains an interesting class of pure Lovelock black holes which are the $N$ th order Lovelock $\Lambda $ -vacuum solutions having the remarkable property that their thermodynamical parameters have the universal character in terms of the event horizon radius. This is in fact a characterizing property of pure Lovelock theories. We also demonstrate the universality of the asymptotic Einstein limit for the Lovelock black holes in general.     

Quasinormal modes of the scalar field in five-dimensional Lovelock black hole spacetime

In this paper, using the third-order WKB approximation, we investigate the quasinormal frequencies of the scalar field in the background of a five-dimensional Lovelock black hole. We find that the ultraviolet correction to Einstein theory in the Lovelock theory makes the scalar field decay more slowly and oscillate more quickly, and the cosmological constant makes the scalar field decay more slowly and oscillate more slowly in the Lovelock black hole background.     

Lovelock gravity from entropic force

In this paper, we first generalize the formulation of entropic gravity to ( $n+1$ )-dimensional spacetime and derive Newton’s law of gravity and Friedmann equation in arbitrary dimensions. Then, we extend the discussion to higher order gravity theories and propose an entropic origin for Gauss–Bonnet gravity and more general Lovelock gravity in arbitrary dimensions. As a result, we are able to derive Newton’s law of gravitation as well as the corresponding Friedmann equations in these gravity theories. This procedure naturally leads to a derivation of the higher dimensional gravitational coupling constant of Friedmann/Einstein equation which is in complete agreement with the results obtained by comparing the weak field limit of Einstein equation with Poisson equation in higher dimensions. Our strategy is to start from first principles and assuming the entropy associated with the apparent horizon given by the expression previously known via black hole thermodynamics, but replacing the horizon radius $r_+$ with the apparent horizon radius $R$ . Our study shows that the approach presented here is powerful enough to derive the gravitational field equations in any gravity theory and further supports the viability of Verlinde’s proposal.     

A novel derivation of the boundary term for the action in Lanczos–Lovelock gravity

We present a novel derivation of the boundary term for the action in Lanczos–Lovelock gravity, starting from the boundary contribution in the variation of the Lanczos–Lovelock action. The derivation presented here is straightforward, i.e., one starts from the Lanczos–Lovelock action principle and the action itself dictates the boundary structure and hence the boundary term one needs to add to the action to make it well-posed. It also gives the full structure of the contribution at the boundary of the complete action, enabling us to read off the degrees of freedom to be fixed at the boundary, their corresponding conjugate momenta and the total derivative contribution on the boundary. We also provide a separate derivation of the Gauss–Bonnet case.     

Decoherence and Phase Transitions in Quantum Dynamics

A particle with internal degrees of freedom is in contact with a bath of photons (necessitating a relativistic treatment). The occurrence of decoherence is established and the density matrix is found to be diagonal in momentum space. In the case of non-trivial internal degrees of freedom and selection rules there is a first order phase transition separating those degrees of freedom. Finally, because probability amplitudes become probabilities, Einstein’s proposal that more than one detector could respond to a signal is answered.

     

Solution for static,spherically symmetric Lovelock gravity coupled with Yang–Mills hierarchy

The hierarchies of both Lovelock gravity and power-Yang–Mills field are combined through gravity in a single theory. In static, spherically symmetric ansatz exact particular integrals are obtained in all higher dimensions. The advantage of such hierarchies is the possibility of choosing coefficients, which are arbitrary otherwise, to cast solutions into tractable forms. To our knowledge the solutions constitute the most general spherically symmetric metrics that incorporate complexities both of Lovelock and Yang–Mills hierarchies within the common context. A large portion of our general class of solutions concerns and addresses to black holes for which specific examples are given. Thermodynamical behaviors of the system is briefly discussed in particular dimensions.     

On higher order gravities, their analogy to GR, and dimensional dependent version of Duff’s trace anomaly relation

An almost brief, though lengthy, review introduction about the long history of higher order gravities and their applications, as employed in the literature, is provided. We review the analogous procedure between higher order gravities and GR, as described in our previous works, in order to highlight/manipulate its important achievements. Amongst which are presentation of an easy classification of higher order Lagrangians and its employment as a criteria in order to distinguish correct metric theories of gravity. For example, it does not permit the inclusion of only one of the second order Lagrangians in isolation. But, it does allow the inclusion of the cosmological term. We also discuss on the compatibility of our procedure and the Mach idea. We derive a dimensional dependent version of Duff’s trace anomaly relation, which in four-dimension is the same as the usual Duff relation. The Lanczos Lagrangian satisfies this new constraint in any dimension. The square of the Weyl tensor identically satisfies it independent of dimension, however, this Lagrangian satisfies the previous relation only in three and four dimensions.