Four-dimensional Case Research Articles

Asymptotic structure of radiation in higher dimensions

We characterize a general gravitational field near conformal infinity (null, spacelike or timelike) in spacetimes of any dimension. This is based on an explicit evaluation of the dependence of the radiative component of the Weyl tensor on the null direction from which infinity is approached. Behaviour similar to a peeling property is recovered, and it is shown that the directional structure of radiation has a universal character that is determined by the algebraic type of the spacetime. This is a natural generalization of analogous results obtained previously in the four-dimensional case.

Some remarks on curvature tensor integrability in spacetimes

If a vector field is covariantly constant then the Ricci identity shows that it annihilates the curvature tensor and all its covariant derivatives. This paper attempts a ‘best possible’ converse to this result in the four-dimensional Lorentz case (the spacetime of general relativity theory) by supposing a vector field annihilates the curvature tensor and a certain number of its derivatives and showing how this leads to a covariantly constant vector field. A direct proof of these results, together with a brief description of how the proof can be obtained using holonomy theory, is presented. The two- and three-dimensional cases are also remarked upon.

Motion of four-dimensional rigid body around a fixed point: an elementary approach I

The goal of this note is to give the explicit solution of Euler-Frahm equations for the Manakov four-dimensional case by elementary means. For this, we use some results from the original papers by Schottky [Sch 1891], Koetter [Koe 1892], Weber [We 1878], and Caspary [Ca 1893]. We hope that such approach will be useful for the solution of the problem of $n$-dimensional top.

Invariant resolutions for several Fueter operators

A proper generalization of complex function theory to higher dimension is Clifford analysis and an analogue of holomorphic functions of several complex variables were recently described as the space of solutions of several Dirac equations. The four-dimensional case has special features and is closely connected to functions of quaternionic variables. In this paper we present an approach to the Dolbeault sequence for several quaternionic variables based on symmetries and representation theory. In particular we prove that the resolution of the Cauchy–Fueter system obtained algebraically, via Gröbner bases techniques, is equivalent to the one obtained by R.J. Baston (J. Geom. Phys. 1992).

Gravitational Collapse and Black Hole Evolution: -- Do Holographic Black Holes Eventually "Anti-Evaporate"? --

We study the gravitational collapse of compact objects in the Brane-World. We begin by arguing that the regularity of the five-dimensional geodesics does not allow the energy-momentum tensor of matter on the brane to have (step-like) discontinuities, which are instead admitted in the four-dimensional General Relativistic case, and compact sources must therefore have an atmosphere. Under the simplifying assumption that matter is a spherically symmetric cloud of dust without dissipation, we can find the conditions for which the collapsing star generically ``evaporates'' and approaches the Hawking behavior as the (apparent) horizon is being formed. Subsequently, the apparent horizon evolves into the atmosphere and the back-reaction on the brane metric reduces the evaporation, which continues until the effective energy of the star vanishes. This occurs at a finite radius, and the star afterwards re-expands and ``anti-evaporates''. We clarify that the Israel junction conditions across the brane (holographically related to the matter trace anomaly) and the projection of the Weyl tensor on the brane (holographically interpreted as the quantum back-reaction on the brane metric) contribute to the total energy as, respectively, an ``anti-evaporation'' and an ``evaporation'' term. Concluding, we comment on the possible effects of dissipation and obtain a new stringent bound for the brane tension.

Quasi-local rotating black holes in higher dimension: geometry

With the help of a generalized Raychaudhuri equation non-expanding null surfaces are studied in an arbitrary dimensional case. The definition and basic properties of non-expanding and isolated horizons known in the literature in the four- and three-dimensional cases are generalized. A local description of the horizon's geometry is provided. The zeroth law of black-hole thermodynamics is derived. The constraints have a similar structure to that of the four-dimensional spacetime case. The geometry of a vacuum isolated horizon is determined by the induced metric and the rotation 1-form potential, local generalizations of the area and the angular momentum typically used in the stationary black-hole solutions case.

Lightlike limit of the boosted Kerr black holes in higher-dimensional spacetimes

Deriving the gravitational field of the high-energy objects is an important theme because the black holes might be produced in particle collisions in the brane world scenario. In this paper, we develop a method for boosting the metric of the stationary objects and taking the lightlike limit in higher-dimensional spacetimes, using the Kerr black hole as one example. We analytically construct the metric of lightlike Kerr black holes and find that this method contains no ambiguity in contrast to the four-dimensional case. We discuss the possible implications of our results in the brane world context.

Renormalization properties of the mass operator [formula omitted] in three-dimensional Yang-Mills theories in the Landau gauge

Massive renormalizable Yang-Mills theories in three dimensions are analyzed within the algebraic renormalization in the Landau gauge. In analogy with the four-dimensional case, the renormalization of the mass operator A μ a A μ a turns out to be expressed in terms of the fields and coupling constant renormalization factors. We verify the relation we obtain for the operator anomalous dimension by explicit calculations in the large N f expansion. The generalization to other gauges such as the non-linear Curci–Ferrari gauge is briefly outlined.

Necessary and sufficient conditions for n-dimensional conformal Einstein spaces via dimensionally dependent identities

Listing has recently extended results of Kozameh, Newman, and Tod for four-dimensional space–times and presented a set of necessary and sufficient conditions for a metric to be locally conformally equivalent to an Einstein metric in all semi-Riemannian spaces of dimension n⩾4—subject to a nondegeneracy restriction on the Weyl tensor. By exploiting dimensionally dependent identities we demonstrate how to construct two alternative versions of these necessary and sufficient conditions which we believe will be useful in applications. The four-dimensional case is discussed in detail and examples are also given in five and six dimensions.

Gauged Nambu-Jona-Lasinio model with extra dimensions: Phase structure and renormalizability

We investigate phase structure of the D (> 4)-dimensional gauged Nambu-Jona-Lasinio (NJL) model with $\delta(=D-4)$ extra dimensions compactified on TeV scale, based on the improved ladder Schwinger-Dyson (SD) equation in the bulk. We assume that the bulk running gauge coupling in the SD equation for the SU(N_c) gauge theory with N_f massless flavors is given by the truncated Kaluza-Klein effective theory and hence has a nontrivial ultraviolet fixed point (UVFP). We find the critical line in the parameter space of two couplings, the gauge coupling and the four-fermion coupling, which is similar to that of the gauged NJL model with fixed (walking) gauge coupling in four dimensions. It is shown that in the presence of such walking gauge interactions the four-fermion interactions become ``nontrivial'' even in higher dimensions, similarly to the four-dimensional gauged NJL model. Such a nontriviality holds only in the restricted region of the critical line (``nontrivial window'') with the gauge coupling larger than a non-vanishing value (``marginal triviality (MT)'' point), in contrast to the four-dimensional case where such a nontriviality holds for all regions of the critical line except for the pure NJL point. In the nontrivial window the renormalized effective potential yields a nontrivial interaction which is conformal invariant. The exisitence of the nontrivial window implies ``cutoff insensitivity'' of the physics prediction in spite of the ultraviolet dominance of the dynamics. In the formal limit D -> 4, the nontrivial window coincides with the known condition of the nontriviality of the four-dimensional gauged NJL model, $9/(2N_c) < N_f - N_c < 9/2 N_c$.

Killing Symmetries of Generalized Minkowski Spaces, 3: Spacetime Translations in Four Dimensions

In this paper, we continue the study of the Killing symmetries of a N-dimensional generalized Minkowski space, i.e. a space endowed with a (in general non-diagonal) metric tensor, whose coefficients do depend on a set of non-metrical coodinates. We discuss here the translations in such spaces, by confining ourselves (without loss of generality) to the four-dimensional case. In particular, the results obtained are specialized to the case of a ''deformed'' Minkowski space $\widetilde{M_{4}}$ (i.e. a pseudoeuclidean space with metric coefficients depending on energy).

Letter: Constraint Propagation in (N + 1)-Dimensional Space-Time

Higher dimensional space-time models provide us an alternative interpretation of nature, and give us different dynamical aspects than the traditional four-dimensional space-time models. Motivated by such recent interests, especially for future numerical research of higher-dimensional space-time, we study the dimensional dependence of constraint propagation behavior. The $N+1$ Arnowitt-Deser-Misner evolution equation has matter terms which depend on $N$, but the constraints and constraint propagation equations remain the same. This indicates that there would be problems with accuracy and stability when we directly apply the $N+1$ ADM formulation to numerical simulations as we have experienced in four-dimensional cases. However, we also conclude that previous efforts in re-formulating the Einstein equations can be applied if they are based on constraint propagation analysis.

Killing Symmetries of Generalized Minkowski Spaces. Part 2: Finite Structure of Space–Time Rotation Groups in Four Dimensions

In this paper, we continue the study of the Killing symmetries of an N-dimensional generalized Minkowski space, i.e., a space endowed with a (in general non-diagonal) metric tensor, whose coefficients do depend on a set of non-metrical coordinates. We discuss here the finite structure of the space–time rotations in such spaces, by confining ourselves (without loss of generality) to the four-dimensional case. In particular, the results obtained are specialized to the case of a “deformed” Minkowski space M_4 (i.e., a pseudoeuclidean space with metric coefficients depending on energy), for which we derive the explicit general form of the finite rotations and boosts in different parametric bases.

Quasinormal modes of Schwarzschild black holes in four and higher dimensions

We make a thorough investigation of the asymptotic quasinormal modes of the four- and five-dimensional Schwarzschild black holes for scalar, electromagnetic, and gravitational perturbations. Our numerical results give full support to all the analytical predictions by Motl and Neitzke for the leading term. We also compute the first-order corrections analytically, by extending to higher dimensions, previous work of Musiri and Siopsis and find excellent agreement with the numerical results. For a generic spacetime dimension number D the first-order corrections go as ${1/n}^{(D\ensuremath{-}3)/(D\ensuremath{-}2)}.$ This means that there is a more rapid convergence to the asymptotic value for the five-dimensional case than for the four-dimensional case, as we also show numerically.

A solution to Lipsman's conjecture for ℝ4

We consider the affine transformation of a nilpotent Lie group on ℝ n . Lipsman (1995) conjectured that such an action is proper in the sense of Palais if and only if the action is compact isotropy (CI) in the sense of Kobayashi. This paper gives an affirmative solution to Lipsman's conjecture in the four-dimensional case.

Convolution of n-Dimensional Tempered Ultradistributions and Field Theory

In this work, a general definition of convolution between two arbitrary Tempered Ultradistributions is given. When one of the Tempered Ultradistributions is rapidly decreasing this definition coincides with the definition of J. Sebastiao e Silva. In the four-dimensional case, when the Tempered Ultradistributions are even in the variables $k^0$ and $\rho$ (see Section 5) we obtain an expression for the convolution, which is more suitable for practical applications. The product of two arbitrary even (in the variables $x^0$ and $r$) four dimensional distributions of exponential type is defined via the convolution of its corresponding Fourier Transforms. With this definition of convolution, we treat the problem of singular products of Green Functions in Quantum Field Theory. (For Renormalizable as well as for Nonrenormalizable Theories). Several examples of convolution of two Tempered Ultradistributions are given. In particular we calculate the convolution of two massless Wheeeler's propagators and the convolution of two complex mass Wheeler's propagators.

Constraints on the mass spectrum of primordial black holes and braneworld parameters from the high-energy diffuse photon background

We investigate the spectral shape of a high-energy diffuse photon emitted by evaporating primordial black holes (PBHs) in the Randall-Sundrum type II (RS2) braneworld. In their braneworld scenario, the nature of small PBHs is drastically modified from the ordinary four-dimensional case for the following two reasons. (i) dropping Hawking temperature, which equivalently lengthens the lifetime of the individual PBH due to the change of space-time topology and (ii) the effective increase of the total amount of PBHs caused by accretion during the earliest part of the radiation-dominated epoch, the brane high-energy phase. From studies of the expected spectral shape and its dependence on braneworld parameters, we obtain two qualitatively distinctive possibilities of constraints on the braneworld PBHs from the observations of diffuse high-energy photon background. If the efficiency of accretion in the high-energy phase exceeds a critical value, the existence of the extra dimension gives a more stringent upper bound on the abundance of PBHs than the 4D case and a small length scale for the extra dimension is favored. On the contrary, in the case below the critical accretion efficiency, we find that the constraint on the PBH abundance can be relaxed by a few orders of magnitude in exchange for the existence of the large extra dimension; its size may be even bounded in the region above 10^{19} times 4D Planck length scale provided the rest mass energy density of the PBHs relative to energy density of radiation is actually larger than 10^{-27} (4D upper bound) at their formation time. The above analytical studies are also confirmed numerically, and an allowed region for braneworld parameters and PBH abundance is clearly obtained.

Echoing the extra dimension

We study the propagating gravitational waves as a tool to probe the extra dimensions. In the set-up with one compact extra dimension and non-gravitational physics resigning on the 4-dimensional subspace (brane) of 5-dimensional spacetime we find the Green's function describing the propagation of 5-dimensional signal along the brane. The Green's function has a form of the sum of contributions from large number of images due to the compactness of the fifth dimension. Additionally, a peculiar feature of the causal wave propagation in five dimensions (making a five-dimensional spacetime very much different from the familiar four-dimensional case) is that the entire region inside the past light-cone contributes to the signal at the observation point. The 4-dimensional propagation law is nevertheless reproduced at large (compared to the size of extra dimension) intervals from the source as a superposition of signals from large number of images. The fifth dimension however shows up in the form of corrections to the purely 4-dimensional picture. We find three interesting effects: a tail effect for a signal of finite duration, screening at the forefront of this signal and a frequency-dependent amplification for a periodic signal. We discuss implications of these effects in the gravitational wave astronomy and estimate the sensitivity of gravitational antenna needed for detecting the extra dimension.

Solvable models of strings in homogeneous plane-wave backgrounds

We solve closed string theory in all regular homogeneous plane-wave backgrounds with homogeneous NS three-form field strength and a dilaton. The parameters of the model are constant symmetric and anti-symmetric matrices k ij and f ij associated with the metric, and a constant anti-symmetric matrix h ij associated with the NS field strength. In the light-cone gauge the rotation parameters f ij have a natural interpretation as a constant magnetic field. This is a generalisation of the standard Landau problem with oscillator energies now being non-trivial functions of the parameters f ij and k ij . We develop a general procedure for solving linear but non-diagonal equations for string coordinates, and determine the corresponding oscillator frequencies, the light-cone Hamiltonian and level matching condition. We investigate the resulting string spectrum in detail in the four-dimensional case and compare the results with previously studied examples. Throughout we will find that the presence of the rotation parameter f ij can lead to certain unusual and unexpected features of the string spectrum like new massless states at non-zero string levels, stabilisation of otherwise unstable (tachyonic) modes, and discrete but not positive definite string oscillator spectra.

Comment on “Time-dependent solution for a star immersed in a background radiation” [H. Liu and Q. Zhang, J. Math. Phys. 43, 4904 (2002)

In a recent paper [J. Math. Phys. 43, 4904 (2002)] Liu and Zhang present a time-dependent solution for a star which is immersed in a background radiation. This solution was obtained by analyzing equations arising from the analysis of a five-dimensional solution that contained their solution. We show that, while their analysis and conclusions are valid, their equations are unnecessarily restrictive when applied to the four-dimensional case.