Dimensional Lorentzian Spacetimes Research Articles

An ideal conformally covariant characterization of the Kerr conformal structure

We present an ideal, conformally covariant characterization of the family of four dimensional Lorentzian spacetimes that are conformally related to the Kerr vacuum solution.

Hadamard renormalization of a two-dimensional Dirac field

The Hadamard renormalization procedure is applied to a free, massive Dirac field $\psi$ on a 2 dimensional Lorentzian spacetime. This yields the state-independent divergent terms in the Hadamard bispinor $G^{(1)}(x, x') = \frac{1}{2} \left\langle \left[ \bar{\psi}(x'), \psi(x) \right] \right\rangle$ as $x$ and $x'$ are brought together along the unique geodesic connecting them. Subtracting these divergent terms within the limit assigns $G^{(1)}(x, x')$, and thus any operator expressed in terms of it, a finite value at the coincident point $x' = x$. In this limit, one obtains a quadratic operator instead of a bispinor. The procedure is thus used to assign finite values to various quadratic operators, including the stress-energy tensor. Results are presented covariantly, in a conformally-flat coordinate chart at purely spatial separations, and in the Minkowski metric. These terms can be directly subtracted from combinations of $G^{(1)}(x, x')$ - themselves obtained, for example, from a numerical simulation - to obtain finite expectation values defined in the continuum.

Acoustic superradiance from a Bose–Einstein condensate vortex with a self-consistent background density profile

The propagation of acoustic perturbations in the velocity potential of a Bose–Einstein condensate (BEC) behave like minimally coupled massless scalar fields in a curved (1 + 1) dimensional Lorentzian space-time, where their propagation is governed by the Klein–Gordon wave equation. For linearized perturbations, this geometric picture can still apply in the presence of a single vortex state of the BEC. Thus far, the amplified scattering of axisymmetric perturbations from a vortex, as a manifestation of the acoustic superradiance, has been investigated by assuming a constant background density of the condensate. This paper addresses the validity of this approximation within the same theoretical workframe, by employing a self-consistent density profile that is obtained by solving the Gross–Pitaevskii equation for an unbound BEC and an approximation to the numerical solution is used throughout the paper. The resulting radial density profile around the vortex implies a radially varying speed of sound, which modifies the entire propagation dynamics as well as the loci of the event horizon and the ergosphere. We investigate the superradiance in temporal domain and, through an independent asymptotic formulation, in the spectral domain. The main conclusions are that the self-consistent density profile remedies the overshoot of the superradiance dynamics temporally and that the spectral profile of the superradiance differs significantly in the vortex-scaled low frequency regime between the constant density and self-consistent density formulations.

Objective Quantum Gravity, Its Possible Relation to Gauge Theories and Strings

In this paper the model of quantum gravity for the higher dimensional Lorentzian space-times, in the sense of the analogy with the Arnowitt–Deser–Misner decomposition well-known from General Relativity, is presented. The model is constructed through making use of the quantum geometrodynamics supplemented by the global onedimensionality conjecture, and considers the objective wave functionals. The framework of quantum field theory is applied in order to establish the phenomenological efficiency in accordance with high energy physics. The empirical deductions on the spatial dimensionality are presented as the relationship between the model and gauge theories, especially string theory.

Higher rank Killing tensors and Calogero model

(n+2)-dimensional Lorentzian spacetime which admits irreducible Killing tensors of rank up to n is constructed by applying the Eisenhart lift to the Calogero model.

Discriminating the Weyl type in higher dimensions using scalar curvature invariants

We consider higher dimensional Lorentzian spacetimes which are currently of interest in theoretical physics. It is possible to algebraically classify any tensor in a Lorentzian spacetime of arbitrary dimensions using alignment theory. In the case of the Weyl tensor, and using bivector theory, the associated Weyl curvature operator will have a restricted eigenvector structure for algebraic types II and D, which leads to necessary conditions on the discriminants of the associated characteristic equation which can be manifestly expressed in terms of polynomial scalar curvature invariants. The use of such necessary conditions in terms of scalar curvature invariants will be of great utility in the study and classification of black hole solutions in more than four dimensions.

Higher dimensional bivectors and classification of the Weyl operator

We develop the bivector formalism in higher dimensional Lorentzian spacetimes. We define the Weyl bivector operator in a manner consistent with its boost-weight decomposition. We then algebraically classify the Weyl tensor, which gives rise to a refinement in dimensions higher than four of the usual alignment (boost-weight) classification, in terms of the irreducible representations of the spins. We are consequently able to define a number of new algebraically special cases. In particular, the classification in five dimensions is discussed in some detail. In addition, utilizing the (refined) algebraic classification, we are able to prove some interesting results when the Weyl tensor has (additional) symmetries.

Vanishing scalar invariant spacetimes in higher dimensions

We study manifolds with Lorentzian signature and prove that all scalar curvature invariants of all orders vanish in a higher dimensional Lorentzian spacetime if and only if there exists an aligned non-expanding, non-twisting, geodesic null direction along which the Riemann tensor has negative boost order.

Acoustic black holes: horizons, ergospheres and Hawking radiation

It is a deceptively simple question to ask how acoustic disturbances propagate in a non-homogeneous flowing fluid. Subject to suitable restrictions, this question can be answered by invoking the language of Lorentzian differential geometry. This paper begins with a pedagogical derivation of the following result: if the fluid is barotropic and inviscid, and the flow is irrotational (though possibly time dependent), then the equation of motion for the velocity potential describing a sound wave is identical to that for a minimally coupled massless scalar field propagating in a (3 + 1)-dimensional Lorentzian geometry The acoustic metric governing the propagation of sound depends algebraically on the density, flow velocity, and local speed of sound. Even though the underlying fluid dynamics is Newtonian, non-relativistic, and takes place in flat space plus time, the fluctuations (sound waves) are governed by an effective (3 + 1)-dimensional Lorentzian spacetime geometry. This rather simple physical system exhibits a remarkable connection between classical Newtonian physics and the differential geometry of curved (3 + 1)-dimensional Lorentzian spacetimes, and is the basis underlying a deep and fruitful analogy between the black holes of Einstein gravity and supersonic fluid flows. Many results and definitions can be carried over directly from one system to another. For example, it will be shown how to define the ergosphere, trapped regions, acoustic apparent horizon, and acoustic event horizon for a supersonic fluid flow, and the close relationship between the acoustic metric for the fluid flow surrounding a point sink and the Painlevé-Gullstrand form of the Schwarzschild metric for a black hole will be exhibited. This analysis can be used either to provide a concrete non-relativistic analogy for black-hole physics, or to provide a framework for attacking acoustics problems with the full power of Lorentzian differential geometry.