Five-dimensional Manifold Research Articles

Application of the Campbell–Magaard theorem to higher-dimensional physics

Stated succinctly, the original version of the Campbell–Magaard theorem says that it is always possible to locally embed any solution of four-dimensional general relativity in a five-dimensional Ricci-flat manifold. We discuss the proof of this theorem (and its variants) in n dimensions, and its application to current theories that postulate that our universe is a four-dimensional hypersurface Σ0 within a five-dimensional manifold, such as space–time–matter (STM) theory and the Randall and Sundrum (RS) braneworld scenario. In particular, we determine whether or not arbitrary spacetimes may be embedded in such theories, and demonstrate how these seemingly disparate models are interconnected. Special attention is given to the motion of test observers in five dimensions, and the circumstances under which they are confined to Σ0. For each five-dimensional scenario considered, the requirement that observers be confined to the embedded spacetime places restrictions on the 4-geometry. For example, we find that observers in the thin braneworld scenario can be localized around the brane if its total stress–energy tensor obeys the five-dimensional strong energy condition. As a concrete example of some of our technical results, we discuss a ℤ2 symmetric embedding of the standard radiation-dominated cosmology in a five-dimensional vacuum.

Nonrelativistic Wave Equations with Gauge Fields

We illustrate a metric formulation of Galilean invariance by constructing wave equations with gauge fields. It consists of expressing nonrelativistic equations in a covariant form, but with a five-dimensional Riemannian manifold. First we use the tensorial expressions of electromagnetism to obtain the two Galilean limits of electromagnetism found previously by Le Bellac and Levy-Leblond. Then we examine the nonrelativistic version of the linear Dirac wave equation. With an Abelian gauge field we find, in a weak field approximation, the Pauli equation as well as the spin—orbit interaction and a part reminiscent of the Darwin term. We also propose a generalized model involving the interaction of the Dirac field with a non-Abelian gauge field; the SU(2) Hamiltonian is given as an example.

Thick brane worlds arising from pure geometry

We study a non-Riemannian modification of 5-dimensional Kaluza–Klein theory. In our proposal the Riemannian structure of the five-dimensional manifold is replaced by a Weyl-integrable one. In this context a 4-dimensional Poincaré invariant solution is studied. Spacetime structures with thick smooth branes separated in the extra dimension arise. We focus our attention, mainly, in a case where the massless graviton is located in one of the thick branes at the origin, meanwhile the matter degrees of freedom are confined to the other brane. Due to the small overlap of the graviton's wave-function with the second thick brane, the model accounts for a resolution of the mass hierarchy problem a la Randall–Sundrum. Although, initially, no assumptions are made about the topology of the extra dimension, the solution found yields an extra space that is effectively compact and respects Z 2 symmetry. Unlike some other models with branes, a preliminary study (not including the effect of scalar perturbations) shows that the spectrum of massive Kaluza–Klein states is quantized and free of tachyonic modes.

Braneworld dynamics of inflationary cosmologies with exponential potentials

In this work we consider Randall-Sundrum brane-world type scenarios, in which the spacetime is described by a five-dimensional manifold with matter fields confined in a domain wall or three-brane. We present the results of a systematic analysis, using dynamical systems techniques, of the qualitative behavior of Friedmann-Lema\^{\i}tre-Robertson-Walker type models, whose matter is described by a scalar field with an exponential potential. We construct the state spaces for these models and discuss how their structure changes with respect to the general-relativistic case, and in particular, what new critical points appear and their nature and the occurrence of bifurcation.

Pin c and Lipschitz Structures on Products of Manifolds

The topological condition for the existence of a pin c structure on the product of two Riemannian manifolds is derived and applied to construct examples of manifolds having the weaker Lipschitz structure, but no pin c structure. An example of a five-dimensional manifold with this property is given; it is pointed out that there are no manifolds of lower dimension with this property.

SOLITONIC BRANE WORLD WITH COMPLETELY LOCALIZED (SUPER)GRAVITY

We construct a solitonic three-brane solution in the five-dimensional Einstein–Hilbert–Gauss–Bonnet theory. This solitonic brane is δ-function-like, and has the property that gravity is completely localized on the brane. That is, there are no propagating degrees of freedom in the bulk, while on the brane we have purely four-dimensional Einstein gravity. Thus, albeit the classical background is five-dimensional, the quantum theory (perturbatively) is four-dimensional. Our solution can be embedded in the supergravity context, where we have completely localized supergravity on the corresponding solitonic brane, which is a BPS object preserving 1/2 of the original supersymmetries. By including a scalar field, we also construct a smooth domain wall solution, which in a certain limit reduces to the δ-function-like solitonic brane solution (this is possible for the latter breaks diffeomorphisms only spontaneously). We then show that in the smooth domain wall background the only normalizable mode is the four-dimensional graviton zero mode, while all the other (including massive Kaluza–Klein) modes are not even plane-wave normalizable. Finally, we observe that in compactifications of Type IIB on five-dimensional Einstein manifolds other than a five-sphere the corresponding dual gauge theories on D3-branes are not conformal in the ultraviolet, and at the quantum level we expect the Einstein–Hilbert term to be generated in their world volumes. We conjecture that in full string theory on Type IIB side this is due to higher curvature terms, which cannot be ignored in such backgrounds. A stronger version of this conjecture also states that (at least in some cases) in such backgrounds D3-branes are solitonic objects with completely localized (super)gravity in their world volumes.

Mechanical Entropy and Its Implications

It is shown that the classical laws of thermodynamics require that mechanical systems must exhibit energy that becomes unavailable to do useful work. In thermodynamics, this type of energy is called entropy. It is further shown that these laws require two metrical manifolds, equations of motion, field equations, and Weyl's quantum principles. Weyl's quantum principle requires quantization of the electrostatic potential of a particle and that this potential be non-singular. The interactions of particles through these non-singular electrostatic potentials are analyzed in the low velocity limit and in the relativistic limit. It is shown that writing the two particle interactions for unlike particles allows an examination in two limiting cases: large and small separations. These limits are shown to have the limiting motions of: all motions are ABOUT the center of mass or all motion is OF the center of mass. The first limit leads to the standard Dirac equation. The second limit is shown to have equations of which the electroweak theory is a subset. An extension of the gauge principle into a five-dimensional manifold, then restricting the generality of the five-dimensional manifold by using the conservation principle, shows that the four-dimensional hypersurface that is embedded within the 5-D manifold is required to obey Einstein's field equations. The 5-D gravitational quantum equations of the solar system are presented.

Null Geodesics in Five-Dimensional Manifolds

We analyze a class of 5D non-compact warped-product spaces characterized by metrics that depend on the extra coordinate via a conformal factor. Our model is closely related to the so-called canonical coordinate gauge of Mashhoon et al. We confirm that if the 5D manifold in our model is Ricci-flat, then there is an induced cosmological constant in the 4D sub-manifold. We derive the general form of the 5D Killing vectors and relate them to the 4D Killing vectors of the embedded spacetime. We then study the 5D null geodesic paths and show that the 4D part of the motion can be timelike—that is, massless particles in 5D can be massive in 4D. We find that if the null trajectories are affinely parameterized in 5D, then the particle is subject to an anomalous acceleration or fifth force. However, this force may be removed by reparameterization, which brings the correct definition of the proper time into question. Physical properties of the geodesics—such as rest mass variations induced by a variable cosmological "constant," constants of the motion and 5D time-dilation effects—are discussed and are shown to be open to experimental or observational investigation.

Evolution of cosmological models in the brane-world scenario

In this work we consider Randall-Sundrum brane-world type scenarios, in which the spacetime is described by a five-dimensional manifold with matter fields confined in a domain wall or three-brane. We present the results of a systematic analysis, using dynamical systems techniques, of the qualitative behaviour of the Friedmann-Lemaitre-Robertson-Walker and the Bianchi I and V cosmological models in these scenarios. We construct the state spaces for these models and discuss how their structure changes with respect to the general-relativistic case, in particular, what new critical points appear and their nature, the occurrence of bifurcations and the dynamics of anisotropy.

Positive mass conjecture for five-dimensional Lorentzian manifolds

We extend Witten’s proof on positive mass conjecture to five-dimensional Lorentzian manifolds with a rigorous treatment in mathematics.

Geodesic equations for a charged particle in the unified theory of gravitational and electromagnetic interactions

A new model of gravitational and electromagnetic interactions is constructed as a version of the classical Kaluza-Klein theory based on a five-dimensional manifold as the physical space-time. The velocity space of moving particles in the model remains four-dimensional as in the standard relativity theory. The spaces of particle velocities constitute a four-dimensional distribution over a smooth five-dimensional manifold. This distribution depends only on the electromagnetic field and is independent of the metric tensor field. We prove that the equations for the geodesics whose velocity vectors always belong to this distribution are the same as the charged particle equations of motion in the general relativity theory. The gauge transformations are interpreted in geometric terms as a particular form of coordinate transformations on the five-dimensional manifold.

Waves and particles in Kaluza–Klein theory

We examine three overlapping problems in the application of five-dimensional (5D) manifolds to physics. First, we linearize the 5D theory along the lines of the four-dimensional (4D) theory, using the harmonic gauge condition. The resulting wave equations have sources, and can in principle describe gravitons and scalar particles with finite masses, but the natural choice of gauge parameters makes both massless. Second, we generalize the 5D metric by including separate conformal factors on its 4D and extra parts. Then the 5D harmonic gauge gives back the 4D harmonic gauge of gravitational waves and the Lorentz gauge of electromagnetic waves, but both particles are massless. Third, we again take a conformally rescaled metric, but rewrite the field equations in a novel manner. This allows us to interpret the finite masses of ordinary particles in terms of the wavelengths associated with 4D spaces embedded in a 5D space.

Five-Dimensional Bieberbach Groups with Holonomy Group Z2⊕Z2

In this paper we determine all five-dimensional compact flat Riemannian manifolds with holonomy group Z 2⊕Z 2. The classification is achieved by classifying their fundamental groups up to isomorphism. The Betti numbers of all these manifolds are also computed.

Einstein manifolds and conformal field theories

In light of the anti--de Sitter space conformal field theory correspondence, it is natural to try to define a conformal field theory in a large N, strong coupling limit via a supergravity compactification on the product of an Einstein manifold and anti--de Sitter space. We consider the five-dimensional manifolds ${T}^{\mathrm{pq}}$ which are coset spaces $[\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)]/\mathrm{U}(1).$ The central charge and a part of the chiral spectrum are calculated, respectively, from the volume of ${T}^{\mathrm{pq}}$ and the spectrum of the scalar Laplacian. Of the manifolds considered, only ${T}^{11}$ admits any supersymmetry: it is this manifold which characterizes the supergravity solution corresponding to a large number of $D3$-branes at a conifold singularity, discussed recently by Klebanov and Witten. Through a field theory analysis of anomalous three point functions we are able to reproduce the central charge predicted for the ${T}^{11}$ theory by supergravity: it is $\frac{27}{32}$ of the central charge of the $\mathcal{N}=2{\mathbf{Z}}_{2}$ orbifold theory from which it descends via a renormalization group flow.

Baryons and domain walls in an N=1 superconformal gauge theory

Coincident D3-branes placed at a conical singularity are related to string theory on $AdS_5\times X_5$, for a suitable five-dimensional Einstein manifold $X_5$. For the example of the conifold, which leads to $X_5=T^{1,1}=(SU(2)\times SU(2))/U(1)$, the infrared limit of the theory on $N$ D3-branes was constructed recently. This is ${\cal N}=1$ supersymmetric $SU(N)\times SU(N)$ gauge theory coupled to four bifundamental chiral superfields and supplemented by a quartic superpotential which becomes marginal in the infrared. In this paper we consider D3-branes wrapped over the 3-cycles of $T^{1,1}$ and identify them with baryon-like chiral operators built out of products of $N$ chiral superfields. The supergravity calculation of the dimensions of such operators agrees with field theory. We also study the D5-brane wrapped over a 2-cycle of $T^{1,1}$, which acts as a domain wall in $AdS_5$. We argue that upon crossing it the gauge group changes to $SU(N)\times SU(N+1)$. This suggests a construction of supergravity duals of ${\cal N}=1$ supersymmetric $SU(N_1)\times SU(N_2)$ gauge theories.

Cobordisms of embeddings of oriented five-dimensional manifolds in Euclidean space

The groups EΩ5,k of cobordism classes of embeddings of oriented five-dimensional manifolds in 5+ k-dimensional Euclidean space are studied. It is proved that EΩ5,3≅ℤ2 and EΩ5,4≅ℤ2⊕ℤ2 and specific generators are given. Bibliography: 5 titles.

Structures de contact sur les variétés algébriques de dimension 5

In this Note we study five-dimensional contact manifolds.

Generalized killing vectors in the five-dimensional lorentz manifold

Solutions of the generalized Killing equation are obtained for the metric with an initialL =t 2 ⊕so(3) symmetry in the five-dimensional manifold of signature −3.

Quantum phase transition in spin-3/2 systems on the hexagonal lattice — optimum ground state approach

Optimum ground states are constructed in two dimensions by using so called vertex state models. These models are graphical generalizations of the well-known matrix product ground states for spin chains. On the hexagonal lattice we obtain a one-parametric set of ground states for a five-dimensional manifold of S=3/2 Hamiltonians. Correlation functions within these ground states are calculated using Monte-Carlo simulations. In contrast to the one-dimensional situation, these states exhibit a parameter-induced second order phase transition. In the disordered phase, two-spin correlations decay exponentially, but in the Neel ordered phase alternating long-range correlations are dominant. We also show that ground state properties can be obtained from the exact solution of a corresponding free-fermion model for most values of the parameter.

Geometrical Models of Elementary Particles

A new concept of space and time, constructed from a de Sitter structured principal fibre bundle with a connection, is used to discuss a geometrical interpretation for the complex plane of the quantum theory and quantum behaviour of particles. In particular some features of a theory based on a torsion free metric linear connection in a five-dimensional base manifold are described.