Nondiagonal Metrics Research Articles

On geodesics in spherical Rindler space

The geodesics in various spherical Rindler frames are investigated. A display of some kinematical quantities of the spacetime is given. The constant acceleration from the metric acts as the surface gravity of the horizon [Formula: see text]. The radial geodesics are computed both for the Balasubramanian et al. form of the spherical Rindler space and for the non-diagonal metric of Huang and Sun.

Spinor domain wall and test fermions on an arbitrary domain wall

We consider a spinor domain wall embedded in a five-dimensional spacetime with a nondiagonal metric. The corresponding plane symmetric solutions for linear and nonlinear spinor fields with different parameters are obtained. It is shown that in the general case the metric functions and spinor fields do not possess Z_2 symmetry with respect to the domain wall. We study the angular momentum density of the domain wall arising because of the presence of the spinor field creating the wall. The properties of test fermions located on an arbitrary domain wall are considered. The concepts of the “second spin” (arising due to the properties of the Lorentz group generators in a five-dimensional spacetime) and of the “second magnetic field” (representing the components F_{i 5} of the electromagnetic field five-tensor) are introduced. We find eigenspinors of the “second spin” and show that some of them represent the Bell states. In the nonrelativistic limit we derive the Pauli equation for the test fermions on the domain wall which contains an extra term describing the interaction of a spin-1/2 particle with the “second magnetic field”; this allows the possibility of an experimental verification of the existence of extra dimensions.

Relativistic star solutions in mass-varying massive gravity with a diagonal metric

We investigate relativistic star solutions in Mass-Varying Massive Gravity (MVMG) with a diagonal metric. Contrary to the intuition that there is no fundamental difference between diagonal metric and non-diagonal metric solutions regarding relativistic stars, we find that with a diagonal metric, well-behaved relativistic star solutions may not exist except for trivial ones in which the graviton mass is a constant, whereas non-trivial relativistic star solutions had been found in MVMG with a non-diagonal metric. The reason is that with a diagonal metric, the field equations constitute a system of differential-algebraic equations of differential index-2 with two extra constraints that have a significant influence on the system, rendering the relativistic star solution with a non-trivial graviton mass configuration impossible in most cases.

Riemannian Geometry of Ising Model in the Bethe Approximation

A method combining statistical equilibrium theory and metric geometry is used to study thermodynamic scalar curvature in the neighborhood of the Curie critical temperature for an Ising model of ferromagnetism. Using a Bethe type free energy expression, a non-diagonal metric is introduced on the two-dimensional phase space of long-range and short-range order parameters. Based on the metric elements Christoffel symbols, curvature tensor and Ricci tensor are found. An expression (containing equilibrium order parameters) is derived for Riemann scalar curvature (R). Its behavior near the critical temperature is examined analytically. We find that R tends toward plus infinity while approaching the critical point. This result fits well with those in the exact one-dimensional chain and mean-field Ising model in the lowest order approximation.

The open string membrane paradigm with external electromagnetic fields

We study the effective geometry felt by the fluctuations of open strings living on the worldvolume of probe D-branes in the presence of background electromagnetic fields. This is captured by an effective action consisting of a Maxwell term and a topological term, with the role of the metric played by the open string metric. Studying generalized Eddington-Finkelstein coordinates for stationary but non-static manifolds, we consider an open string membrane paradigm to obtain a generic formula for the DC transport coefficients, including the effect of external electromagnetic fields present on the world volume of the probe branes. We show that the previously studied singular shell, present when a critical electric field strength is turned on, behaves as a horizon for the open string degrees of freedom. The results of this analysis can be used to define a membrane paradigm for a very general class of spacetimes with non-diagonal metrics.

Poisson-Lie T-plurality of three-dimensional conformally invariant sigma models II: Nondiagonal metrics and dilaton puzzle

We look for 3-dimensional Poisson-Lie dualizable sigma models that satisfy the vanishing beta-function equations with constant dilaton field. Using the Poisson-Lie T-plurality we then construct 3-dimensional sigma models that correspond to various decompositions of Drinfeld double. Models with nontrivial dilaton field may appear. It turns out that for ``traceless'' dual algebras they satisfy the vanishing beta-function equations as well. In certain cases the dilaton cannot be defined in some of the dual models. We provide an explanation why this happens and give criteria predicting when it happens.

Perfect-fluid cosmologies with a proper conformal Killing vector

We study the Einstein field equations for spacetimes admitting a maximal two-dimensional Abelian group of isometries acting orthogonally transitively on spacelike surfaces and, in addition, with at least one conformal Killing vector. The three-dimensional conformal group is restricted to the case when the two-dimensional Abelian isometry subalgebra is an ideal and it is also assumed to act on non-null hypersurfaces (both spacelike and timelike cases are studied). We consider both diagonal and non-diagonal metrics, and find all the perfect-fluid solutions under these assumptions (except those already known). We find four families of solutions, each one containing arbitrary parameters for which no differential equations remain to be integrated. We write the line-elements in a simplified form and perform a detailed study for each of these solutions, giving the kinematical quantities of the fluid velocity vector, the energy density and pressure, values of the parameters for which the energy conditions are fulfilled everywhere, the Petrov type, the singularities in the spacetimes and the Friedmann - Lemaître - Robertson - Walker metrics contained in each family.

Particle creation in a noninertial frame associated to a nondiagonal metric

We study the problem of particle creation for massless scalar and spin-(1/2) particles in a two-dimensional space-time associated to a metric with one null coordinate and constant acceleration. We find that the associated thermal distributions are those corresponding to bosons and fermions.

Separation of variables in the Dirac equation for one class of non-diagonal metrics

The problem of separation of variables in the Dirac equation for one class of non-diagonal metrics is investigated by means of an algebraic method. Such an approach allows four types of non-diagonal metrics to be marked out where separation of variables is possible. As partial cases the authors reproduce here some results of other authors, in particular for Kerr and Kerr-Newman metrics.

Dirac equation in external fields: Separation of variables in nondiagonal metrics

The algebraic method of separation of variables in the Dirac equation proposed in earlier works by one of the present authors [Theor. Math. Phys. 70, 204 (1987); J. Math. Phys. 30, 2132 (1989)] is developed for the space-time with nondiagonal metrics. The essence of the method consists of the separation of the first-order matricial differential operators that define the dependence of the Dirac’s bispinor on the related variables. In contrast to some other authors the pairs of operators are commuted on each step of separation including the variables mixed by nondiagonal elements of fundamental tensor of space-time. There are reasons to believe that it must be some local similarity transformation connected these commuted operators with noncommuted corresponding operators of other authors, although such transformation in view of mathematical difficulties of problem, in general, were not successfully found.

On Kruskal-Novikov co-ordinate systems

We rederive the transformation from Schwarzschild coordinates to Novikov co-ordinates, which appear to be equivalent to Kruskal co-ordinates. Our derivation takes place in two steps. We first replace the Schwarzschild time co-ordinateT with a new time coordinate τ measured by radially moving geodesic clocks, but keep the Schwarzschild radial co-ordinateR. The transformation fromT to τ results in a nondiagonal metric which is regular at the Schwarzschild radius,R=2M, and geodesics can be followed across the Schwarzschild radius in terms of the new (R, τ) co-ordinates. However, the metric does contain an expected co-ordinate singularity, arising because of the limitation of the reference system that one cannot have a geodesic clock with a turning point smaller than the Schwarzschild radius. Because of this, the co-ordinate system is incomplete in the sense one can find geodesic trajectories that cannot be followed to the intrinsic singularity atR=0. In our second step, the Schwarzschild radial co-ordinateR is replaced with a new spatial co-ordinateRi equal to the maximum Schwarzschild radius of each geodesic clock forming the reference system, a constant uniquely associated with the world-line of each co-ordinate clock. In terms of (Ri, τ) co-ordinates the metric assumes a diagonal form, but still maintains the previous co-ordinate singularity and is still geodesically incomplete. It is shown that the co-ordinate singularity in the metric can be removed by the mathematical procedure of replacing the Schwarzschild valueRi with a derived co-ordinateR* monotonically related toRi byR*=(Ri/2M−1)1/2. Because the metric written in terms of (R*, τ) now contains no appearance of co-ordinate singularities, it is possible to consider lettingR* take on negative values. However, arguments are given for excluding both negative values ofR* and the corresponding left-hand side of a Novikov diagram that would be generated by these negative values. Similar arguments for excluding the left-hand side of Kruskal diagrams are also given.