Narrow Results

We discuss the relation between Noether (point) symmetries and discrete symmetries for a class of minisuperspace cosmological models. We show that when a Noether symmetry exists for the gravitational Lagrangian, then there exists a coordinate system in which a reversal symmetry exists. Moreover, as far as concerns, the scale-factor duality symmetry of the dilaton field, we show that it is related to the existence of a Noether symmetry for the field equations, and the reversal symmetry in the normal coordinates of the symmetry vector becomes scale-factor duality symmetry in the original coordinates. In particular, the same point symmetry as also the same reversal symmetry exists for the Brans–Dicke scalar field with linear potential while now the discrete symmetry in the original coordinates of the system depends on the Brans–Dicke parameter and it is a scale-factor duality when ${\omega }_{\mathrm{\text{BD}}}=1$. Furthermore, in the context of the O’Hanlon theory for f(R)-gravity, it is possible to show how a duality transformation in the minisuperspace can be used to relate different gravitational models.

We consider charged black holes with curved horizons, in five-dimensional dilaton gravity in the presence of Liouville-type potential for the dilaton field. We show how, by solving a pair of coupled differential equations, infinitesimally small angular momentum can be added to these static solutions to obtain charged rotating dilaton black hole solutions. In the absence of dilaton field, the nonrotating version of the solution reduces to the five-dimensional Reissner–Nordström black hole, and the rotating version reproduces the five-dimensional Kerr–Newman modification thereof for small rotation parameter. We also compute the angular momentum and the angular velocity of these rotating black holes which appear at the first order.

We investigate the space–time of a spinning cosmic string in conformal invariant gravity, where the interior consists of a gauged scalar field. We find exact solutions of the exterior of a stationary spinning cosmic string, where we write the metric as $g_{\mu \nu }={\omega }^2{\tilde{g}}_{\mu \nu }$, with $\omega$ a dilaton field which contains all the scale dependences. The “unphysical” metric ${\tilde{g}}_{\mu \nu }$ is related to the $(2+1)$-dimensional Kerr space–time. The equation for the angular momentum $J$ decouples, for the vacuum situation as well as for global strings, from the other field equations and delivers a kind of spin-mass relation. For the most realistic solution, $J$ falls off as $\sim \frac{1}{r}$ and ${\partial }_{r}J\to 0$ close to the core. The space–time is Ricci flat. The formation of closed timelike curves can be pushed to space infinity for suitable values of the parameters and the violation of the weak energy condition can be avoided. For the interior, a numerical solution is found. This solution can easily be matched at the boundary on the exterior exact solution by special choice of the parameters of the string. This example shows the power of conformal invariance to bridge the gap between general relativity and quantum field theory.

In this paper, we investigate the exact time-dependent black hole solution on a warped five-dimensional Randall–Sundrum space–time in conformal dilaton gravity. The zeroes of the model are described by a meromorphic quintic polynomial, which has no essential singularities. The quintic equation can be reduced to the Brioschi form by means of the Weierstrass elliptic curve over $\mathbb{Q}$. The model fits the antipodal boundary condition, i.e. antipodal points in the projected space are identified using the embedding of a Klein surface in ${ℂ}^2$, using the ${\mathbb{Z}}_2$ symmetry on the two sides of the brane. If one writes ${}^{(5)}{g}_{\mu \nu }={\omega }^{4∕3}{}^{(5)}{\tilde{g}}_{\mu \nu },{}^{(5)}{\tilde{g}}_{\mu \nu }={}^{(4)}{\tilde{g}}_{\mu \nu }+n_{\mu }{n}_{\nu }$, ${}^{(4)}{\tilde{g}}_{\mu \nu }={\bar{\omega }}^2{}^{(4)}{ḡ}_{\mu \nu }$, with $n_{\mu }$ the normal to the brane and $\omega$ the dilaton field, then ${}^{(4)}{ḡ}_{\mu \nu }$ is conformally flat. It is the contribution from the bulk which determines the real pole on the effective four-dimensional space–time. There is no objection applying ’t Hooft’s back reaction method in constructing the unitary S-matrix for the Hawking radiation. Again, there is no “inside” of the black hole. The zeroes can also be analyzed by the icosahedron equation and by the Hopf-fibration of the Klein surface.

The energy distribution of black holes with a dilaton and a pure monopole field is calculated by using Møller's energy–momentum complex. The four-dimensional space–times considered are static, spherically symmetric and asymptotically flat, exact solutions stemming from an action that besides gravity contains a dilaton field and a pure monopole field. The resulting "hairy" black holes have an essential singularity at the origin and two horizons. The energy obtained depends on the value of the dilaton field, the monopole charge and the ADM mass. All the momenta vanish for the space–time geometries considered.

We take a quick look at the different possible universally coupled scalar fields in nature. Then, we discuss how the gauging of the group of scale transformations (dilations), together with the Poincaré group, leads to a Weyl–Cartan spacetime structure. There the dilaton field finds a natural surrounding. Moreover, we describe shortly the phenomenology of the hypothetical axion field. In the second part of our essay, we consider a spacetime, the structure of which is exclusively specified by the premetric Maxwell equations and a fourth rank electromagnetic response tensor density ${\chi }^{ijkl}=-{\chi }^{jikl}=-{\chi }^{ijlk}$ with 36 independent components. This tensor density incorporates the permittivities, permeabilities and the magneto-electric moduli of spacetime. No metric, no connection, no further property is prescribed. If we forbid birefringence (double-refraction) in this model of spacetime, we eventually end up with the fields of an axion, a dilaton and the 10 components of a metric tensor with Lorentz signature. If the dilaton becomes a constant (the vacuum admittance) and the axion field vanishes, we recover the Riemannian spacetime of general relativity theory. Thus, the metric is encapsulated in ${\chi }^{ijkl}$, it can be derived from it.

Topological defects formed in the early stages of our universe can play a crucial role in understanding anisotropic deviations of the Friedmann Lemâitre Robertson Walker model we observe today. These defects are the result of phase transitions associated with spontaneous symmetry breaking in gauge theories at the grand unification energy scale. The most interesting defects are cosmic strings, vortex-like structures in the famous gauged U(1) abelian Higgs model with a “Mexican-hat” potential. Other defects, such as domain walls and monopoles are probably ruled out, because they should dominate otherwise the energy density of our universe. This local gauge model is the basis of the standard model of particle physics, where the Higgs-mechanism provides elementary particles with mass. It cannot be a coincidence that this model also explains the theory of superconductivity. The decay of the high multiplicity (n) super-conducting vortex into a lattice of n vortices of unit magnetic flux is energetically favourable and is experimentally confirmed. It explains the famous Meissner effect. This process could play an essential role by the entanglement of cosmic strings just after the symmetry breaking. The stability of the lattice depends critically on the parameters of the model, especially when gravity comes into play. The questions is how the imprint of the cosmic strings could be observed at present time. Up to now, no evidence is found. The recently found alignment of the spinning axes of quasars in large quasar groups on Mpc scales, could be a first indication of the existence of these cosmic strings. The temporarily broken axial symmetry will leave an imprint of a preferred azimuthal-angle on the lattice. This effect is only viable when a scaling factor is introduced. This can be realized in a warped five dimensional model. The warp factor plays the role of a dilaton field on an equal footing with the Higgs field. The resulting field equations can be obtained from a conformal invariant model. Conformal invariance, the missing symmetry in general relativity, will then spontaneously be broken, just as the Higgs field. The dilaton field, or equivalently, the warp factor, could also contribute to the expansion of the universe as it can act as a dark energy term coming from the bulk spacetime. It makes the cosmic string temporarily super-massive. This process could solve the cosmological constant and hierarchy problem. It is conjectured that the dilaton field has a dual meaning. At very early times, when the dilaton field approaches zero, it describes the small-distance limit of the model, while at later times it is a warp (or scale) factor that determines the dynamical evolution of the universe. When more data of quasars of high redshift will become available, one could prove that the alignment emerged after the symmetry breaking scale and must have a cosmological origin. The effect of the warp factor on the second-order perturbations could also be an indication of the existence of large extra dimensions.

An exact time-dependent solution of a black hole is found in a conformally invariant gravity model on a warped Randall-Sundrum spacetime, by writing the metric $g_{\mu \nu }={\omega }^{\frac{4}{n-2}}{\tilde{g}}_{\mu \nu }$. Here ${\tilde{g}}_{\mu \nu }$ represents the “un-physical” spacetime and ω the dilaton field, which will be treated on equal footing as any renormalizable scalar field. It is remarkable that the 5D and 4D effective field equations for the metric components and dilaton fields can be written in general dimension n = 4, 5. The location of the horizon(s) are determined by a quintic polynomial. This polynomial is related to the symmetry group of the icosahedron, isomorphic with the Galois group A₅. We applied the antipodal mapping on the axially symmetric black hole spacetime and make some connection with the information and firewall paradoxes. The dilaton field can be used to describe the different notion the in-going and outside observers have of the Hawking radiation by using different conformal gauge freedom. The disagreement about the interior of the black hole is explained by the antipodal map of points on the horizon. The free parameters of the solution can be chosen in such a way that ${\tilde{g}}_{\mu \nu }$ is singular-free and topologically regular, even for ω → 0.