Surface Energy-momentum Tensor Research Articles

Dynamical interiors of black-bounce spacetimes

Using the Israel-Darmois junction conditions, we obtain a class of regular dynamical interiors to the recently proposed black-bounce spacetimeswhich regularises the Schwarzschild singularity by introducing a regularisation parameter. We show that a regularised Friedmann-Lemaitre-Robertson-Walkerlike interior geometry can not be matched smoothly with the exterior black-bounce spacetime through a timelike hypersurface, as therealways exists a thin shell of non-zero energy-momentum tensor at the matching hypersurface. We obtain the expressions for the energy densityand pressure corresponding to the thin-shell surface energy-momentum tensor in terms of the regularisation parameter and derive an evolution equation for thescale factor of the interior geometry by imposing physical conditions on these components. We also discuss the formation of the event horizon inside the interior in the case when the initial conditions are such that thesituation describes a collapsing matter cloud. We elaborate upon the physical implications of these results.

Thin-shell wormholes with non-collapsing dark energy throats connecting flat Minkowski spacetimes

This research revisits the thin-shell wormhole connecting two Minkowski spacetimes — originally introduced by Matt Visser. Upon setting a particular timelike hypersurface with a de-sitter-induced line element, the corresponding thin-shell wormhole (TSW) possesses a minimum radius and a uniform surface energy–momentum tensor in the form of dark energy. The bounded from below radius for the throat implies that it does not collapse. The r−τ profile curve of the hypersurface is a Catenary in r−τ plane where r and τ are the radial coordinate and the proper time on the throat, respectively. It is also shown that while the throat is dynamic, the surface energy–momentum tensor of the dark energy is in the form of an effective cosmological constant in the spherically symmetric TSW. The 4-dimensional cylindrically symmetric TSW with a similar throat profile shares the same features as its spherical counterpart, however, its uniform surface energy–momentum tensor does not mimic an effective cosmological constant. The higher-dimensional generalization of the spherically symmetric TSW admits the same properties as in the 4-dimensional one.

Surface Casimir Densities on Branes Orthogonal to the Boundary of Anti-De Sitter Spacetime

The paper investigates the vacuum expectation value of the surface energy–momentum tensor (SEMT) for a scalar field with general curvature coupling in the geometry of two branes orthogonal to the boundary of anti-de Sitter (AdS) spacetime. For Robin boundary conditions on the branes, the SEMT is decomposed into the contributions corresponding to the self-energies of the branes and the parts induced by the presence of the second brane. The renormalization is required for the first parts only, and for the corresponding regularization the generalized zeta function method is employed. The induced SEMT is finite and is free from renormalization ambiguities. For an observer living on the brane, the corresponding equation of state is of the cosmological constant type. Depending on the boundary conditions and on the separation between the branes, the surface energy densities can be either positive or negative. The energy density induced on the brane vanishes in special cases of Dirichlet and Neumann boundary conditions on that brane. The effect of gravity on the induced SEMT is essential at separations between the branes of the order or larger than the curvature radius for AdS spacetime. In the considerably large separation limit, the decay of the SEMT, as a function of the proper separation, follows a power law for both massless and massive fields. For parallel plates in Minkowski bulk and for massive fields the fall-off of the corresponding expectation value is exponential.

A note on the stability of generic spherically symmetric thin-shell wormhole supported by a false vacuum

In the stability analysis of the Schwarzschild thin-shell wormhole (TSW) performed by Poisson and Visser in Poisson and Visser (1995) the equation of state (EoS) of the surface matter on the throat satisfies β2σ=∂p∂σ in which σ and p are the surface energy density and the surface transverse pressure. In this Letter, we show that β2σ=−1 has to be excluded as it results in the radius of the equilibrium a0=3M a radius which was problematic and later on was treated by Varela in Varela (2015). Furthermore, we present a formalism on the mechanical stability of generic spherically symmetric TSWs supported by a surface energy–momentum tensor of the form σ=−p corresponding to β2σ=−1. Such an EoS with σ>0 describes the so-called vacuum matter or dark energy, however, here it is called false vacuum because σ<0. We obtain the general conditions upon which such a TSW remains stable against a mechanical linear perturbation. In particular, we apply our formalism to first the TSW constructed in the Reissner–Nordström (RN) spacetime and then in the Bronnikov–Zaslavskii (BZ) closed black hole. We show that in both cases stable TSW exists, however, for the former TSW, it corresponds to the non-black hole RN solution.

Stability of thin-shell-wormhole in Robertson-Walker closed universe I: static universe

We construct a generic thin-shell wormhole (TSW) in the Robertson -Walker (RW) cosmological model of the Universe. The general formalism is presented and here in this paper—the first part of the research- we investigate the stability of the TSW constructed in the static RW universe. To do so, we perform two different stability analyses namely a mechanical linear radial perturbation and a small speed exact (not linearized) perturbation. It is shown that the corresponding TSW is stable against a linear radial perturbation provided that the variable equation of state of the surface energy-momentum tensor is finely tuned. On the other hand, the exact stability analysis in the context of small-speed perturbation reveals that the throat of the TSW for the closed static universe oscillates between the center of the Universe and the outer boundary.

Lightlike singular hypersurfaces in quadratic gravity

Using the principle of least action, the motion equations for a singular hypersurface of arbitrary type in quadratic gravity are derived. Equations containing the “external pressure” and the “external flow” components of the surface energy–momentum tensor together with the Lichnerowicz conditions serve to find the hypersurface itself, while the remaining ones define arbitrary functions that arise due to the implicit presence of the delta function derivative. It turns out that neither double layers nor thin shells exist for the quadratic Gauss–Bonnet term. It is shown that there is no “external pressure” for null singular hypersurfaces. The Lichnerowicz conditions imply the continuity of the scalar curvature in the case of spherically symmetric null singular hypersurfaces. These hypersurfaces must be thin shells if the Lichnerowicz conditions are necessary. It is shown that for this particular case the Lichnerowicz conditions can be completely removed therefore a spherically symmetric null double layer exists. Spherically symmetric null singular hypersurfaces in conformal gravity are explored as application.

Double layer from least action principle

We derived the equations for the double layers in quadratic gravity, using solely the least action principle. The advantage of our approach is that, in the process of calculation, the δ′-function does not appear at all, and the δ-functions appear for a moment and are mutually canceled prior to integration. We revealed the peculiar structure of the obtained equations, namely, that the surface energy–momentum tensor of the matter fields (constituents of the thin shells) does not play a role in the determination of the trajectory of the double layer. Also, we suggested that the space-like double layers may provide us with the adequate description of the creation of the Universe from the black hole singularity. The related topics, including the Gauss–Bonnet term and F(R)-theories, are shortly discussed.

Revisiting the Israel junction conditions in Einstein–Cartan gravity

The Israel junction conditions of a thin shell in the context of Einstein–Cartan gravity are revisited. It is shown that with a choice of the torsion discontinuity taken to be orthogonal to the hypersurface and consistent with the antisymmetric properties of torsion tensor and contorsion tensor, the generalized asymmetric surface energy–momentum tensor turns out to be tangent to the hypersurface while the resulting generalized Israel junction conditions are modified. The main differences to previous works are mentioned.

Induced Cosmological Constant in Brane Models with a Compact Dimension

The vacuum expectation value of the surface energy-momentum tensor of a charged scalar field on a flat brane in an anti-de Sitter space-time with a compact spatial dimension is studied. The existence of a constant gauge field is also assumed. Because of the nontrivial topology of the space, the latter leads to an Aharonov-Bohm type effect. A generalized zeta-function method is used for renormalizing the vacuum expectation value. The cosmological constant induced on the brane is a periodic function of the magnetic flux through the compact dimension and, depending on the parameters of the problem, can be either positive or negative.

ROTATING RELATIVISTIC THIN DISKS AS SOURCES OF CHARGED AND MAGNETIZED KERR–NUT SPACE–TIMES

A family of models of counterrotating and rotating relativistic thin disks of infinite extension based on a charged and magnetized Kerr–NUT metric are constructed using the well-known "displace, cut and reflect" method extended to solutions to vacuum Einstein–Maxwell equations. The metric considered has as limiting cases a charged and magnetized Taub–NUT solution and the well-known Kerr–Newman solutions. We show that for Kerr–Newman fields the eigenvalues of the energy–momentum tensor of the disk are for all the values of the parameters' real quantities so that these disks do not present heat flow in any case, whereas for charged and magnetized Kerr–NUT and Taub–NUT fields we always find regions with heat flow. We also find a general constraint on the counterrotating tangential velocities needed to cast the surface energy–momentum tensor of the disk as the superposition of two counterrotating charged dust fluids. We show that, in general, it is not possible to take the two counterrotating fluids as circulating along electrogeodesics or take the two counterrotating tangential velocities as equal and opposite.

SPHERICALLY SYMMETRIC GRAVITATIONAL COLLAPSE

In this paper, we discuss gravitational collapse of spherically symmetric spacetimes. We derive a general formalism by taking two arbitrary spherically symmetric spacetimes with g00 = 1. Israel's junction conditions are used to develop this formalism. The formulas for extrinsic curvature tensor are obtained. The general form of the surface energy–momentum tensor depending on extrinsic curvature tensor components is derived. This leads us to the surface energy density and the tangential pressure. The formalism is applied to two known spherically symmetric spacetimes. The results obtained show the regions for the collapse and expansion of the shell.

WORMHOLES OF K-ESSENCE IN ARBITRARY SPACE–TIME DIMENSIONS

We consider a K-essence involving a massless scalar field Φ minimally coupled to Einstein gravity in D ≥ 4 space–time dimensions. This theory admits a two-parameter family of spherical wormholes representing two asymptotically-flat universes connected via a (D-2)-dimensional spherical throat. The ADM masses of the two ends are unequal and of opposite sign except for a one-parameter family where both ends possess vanishing ADM masses. By cut and paste techniques, we construct a two-parameter family of wormholes where the ends possess equal and positive ADM masses but the throat is a (D-1)-dimensional thin-shell. The structure of the surface energy–momentum tensor is also analyzed.

Rotating and counterrotating relativistic thin disks as sources of stationary electrovacuum spacetimes

A detailed study is presented of the counterrotating model (CRM) for electrovacuum stationary axially symmetric relativistic thin disks of infinite extension without radial stress, in the case when the eigenvalues of the energy-momentum tensor of the disk are real quantities, so that there is not heat flow. We find a general constraint over the counterrotating tangential velocities needed to cast the surface energy-momentum tensor of the disk as the superposition of two counterrotating charged dust fluids. We then show that, in some cases, this constraint can be satisfied if we take the two counterrotating tangential velocities as equal and opposite or by taking the two counterrotating streams as circulating along electro-geodesics. However, we show that, in general, it is not possible to take the two counterrotating fluids as circulating along electro-geodesics nor take the two counterrotating tangential velocities as equal and opposite. A simple family of models of counterrotating charged disks based on the Kerr-Newman solution are considered where we obtain some disks with a CRM well behaved. We also show that the disks constructed from the Kerr-Newman solution can be interpreted, for all the values of parameters, as a matter distribution with currents and purely azimuthal pressure without heat flow. The models are constructed using the well-known "displace, cut and reflect" method extended to solutions of vacuum Einstein-Maxwell equations. We obtain, in all the cases, counterrotating Kerr-Newman disks that are in agreement with all the energy conditions.

Surface Casimir densities and induced cosmological constant in higher dimensional braneworlds

We investigate the vacuum expectation value of the surface energy-momentum tensor for a massive scalar field with general curvature coupling parameter obeying the Robin boundary conditions on two codimension one parallel branes in a $(D+1)$-dimensional background spacetime ${\mathrm{AdS}}_{{D}_{1}+1}\ifmmode\times\else\texttimes\fi{}\ensuremath{\Sigma}$ with a warped internal space $\ensuremath{\Sigma}$. These vacuum densities correspond to a gravitational source of the cosmological constant type for both subspaces of the branes. Using the generalized zeta function technique in combination with contour integral representations, the surface energies on the branes are presented in the form of the sum of single-brane and second-brane-induced parts. For the geometry of a single brane both regions, on the left and on the right of the brane, are considered. At the physical point the corresponding zeta functions contain pole and finite contributions. For an infinitely thin brane taking these regions together, in odd spatial dimensions the pole parts cancel and the total zeta function is finite. The renormalization procedure for the surface energies and the structure of the corresponding counterterms are discussed. The parts in the surface densities generated by the presence of the second brane are finite for all nonzero values of the interbrane separation and are investigated in various asymptotic regions of the parameters. In particular, it is shown that for large distances between the branes the induced surface densities give rise to an exponentially suppressed cosmological constant on the brane. The total energy of the vacuum including the bulk and boundary contributions is evaluated by the zeta function technique and the energy balance between separate parts is discussed.

A doubly covariant formula of deficit angle and its application to six-dimensional braneworld

We reformulate boundary conditions for axisymmetric codimension-2 braneworlds in a way which is applicable to linear perturbation with various gauge conditions. Our interest is in the thin brane limit and thus this scheme assumes that the perturbations are also axisymmetric and that the surface energy–momentum tensor of the brane is proportional to its induced metric. An advantage of our scheme is that it allows much more freedom for convenient coordinate choices than the other methods. This is because in our scheme, the coordinate system in the bulk and that on the brane are completely disentangled. Therefore, the latter does not need to be a subset of the former and the brane does not need to stay at a fixed bulk coordinate position. The boundary condition is manifestly doubly covariant: it is invariant under gauge transformations in the bulk and at the same time covariant under those on the brane. We take advantage of the double covariance when we analyse the linear perturbation of a particular model of a six-dimensional braneworld with warped flux compactification.

Surface Casimir densities on a spherical brane in Rindler-like spacetimes

Abstract The vacuum expectation value of the surface energy–momentum tensor is evaluated for a scalar field obeying Robin boundary condition on a spherical brane in ( D + 1 ) -dimensional spacetime Ri × S D − 1 , where Ri is a two-dimensional Rindler spacetime. The generalized zeta function technique is used in combination with the contour integral representation. The surface energies on separate sides of the brane contain pole and finite contributions. Analytic expressions for both these contributions are derived. For an infinitely thin brane in odd spatial dimensions, the pole parts cancel and the total surface energy, evaluated as the sum of the energies on separate sides, is finite. For a minimally coupled scalar field the surface energy–momentum tensor corresponds to the source of the cosmological constant type.

Surface vacuum energy and stresses for a brane in de Sitter spacetime

Vacuum expectation values of the surface energy–momentum tensor is investigated for a massless scalar field obeying mixed boundary condition on a brane in de Sitter bulk. To generate the corresponding vacuum surface densities we use the conformal relation between de Sitter and Rindler spacetimes.

Surface vacuum energy and stresses on a brane in AdS space with an application to the brane-world model

The vacuum expectations of the surface energy-momentum tensor generated on a brane in AdS space-time by quantum fluctuations of a scalar field with an arbitrary coupling parameter are studied. It is assumed that the field satisfies mixed boundary conditions on the brane. A generalized zeta function method is used as a regularization procedure. Two regions, located to the left (L-region) and right (R-region) of the brane, are considered. It is shown that the surface energies for both these regions contain pole and finite contributions. Analytic expressions are derived for both parts. When calculating the total surface energy including the contributions from the L- and R-regions, the pole terms add out in odd spatial dimensions. The surface energy-momentum tensor induced by vacuum quantum effects corresponds to the generation of a cosmological constant on the brane. These results are applied to the second Randall-Sundrum model.

Surface vacuum energy and stresses on a plate uniformly accelerated through the Fulling–Rindler vacuum

The vacuum expectation value of the surface energy–momentum tensor is evaluated for a massless scalar field obeying a mixed boundary condition on an infinite plate moving by uniform proper acceleration through the Fulling–Rindler vacuum. The generalized zeta function technique is used, in combination with the contour integral representation. The surface energies for separate regions on the left and on the right of the plate contain pole and finite contributions. Analytic expressions for both these contributions are derived. For a minimally coupled scalar, the surface energy–momentum tensor induced by vacuum quantum effects corresponds to a source of the cosmological constant type located on the plate and with the cosmological constant determined by the proper acceleration of the plate.

Surface Casimir densities and induced cosmological constant on parallel branes in AdS spacetime

Vacuum expectation value of the surface energy-momentum tensor is evaluated for a massive scalar field with general curvature coupling parameter subject to Robin boundary conditions on two parallel branes located on $(D+1)$-dimensional anti-de Sitter bulk. The general case of different Robin coefficients on separate branes is considered. As a regularization procedure the generalized zeta function technique is used, in combination with contour integral representations. The surface energies on the branes are presented in the form of the sums of single brane and second brane-induced parts. For the geometry of a single brane both regions, on the left (L-region) and on the right (R-region), of the brane are considered. The surface densities for separate L- and R-regions contain pole and finite contributions. For an infinitely thin brane taking these regions together, in odd spatial dimensions the pole parts cancel and the total surface energy is finite. The parts in the surface densities generated by the presence of the second brane are finite for all nonzero values of the interbrane separation. It is shown that for large distances between the branes the induced surface densities give rise to an exponentially suppressed cosmological constant on the brane. In the Randall-Sundrum braneworld model, for the interbrane distances solving the hierarchy problem between the gravitational and electroweak mass scales, the cosmological constant generated on the visible brane is of the right order of magnitude with the value suggested by the cosmological observations.