Vacuum Energy-momentum Tensor Research Articles

Fermionic condensate and the vacuum energy-momentum tensor for planar fermions in homogeneous electric and magnetic fields

We consider a massive fermionic quantum field localized on a plane in external constant and homogeneous electric and magnetic fields. The magnetic field is perpendicular to the plane and the electric field is parallel. The complete set of solutions to the Dirac equation is presented. As important physical characteristics of the vacuum state, the fermion condensate and the expectation value of the energy-momentum tensor are investigated. The renormalization is performed using the Hurwitz function. The results are compared with those previously studied in the case of zero electric field. We discuss the behavior of the vacuum expectation values in different regions for the values of the problem parameters. Applications of the results include the electronic subsystem of graphene sheet described by the Dirac model in the long-wavelength approximation.

Induced energy-momentum tensor in de Sitter scalar QED and its implication for induced currents

The aim of this research is to investigate the vacuum energy-momentum tensor of a quantized, massive, nonminimally coupled scalar field induced by a uniform electric field background in a four-dimensional de Sitter spacetime (${\mathrm{dS}}_{4}$). We compute the expectation value of the energy-momentum tensor in the in-vacuum state and then regularize it using the adiabatic subtraction procedure. The correct trace anomaly of the induced energy-momentum tensor that confirmed our results is significant. The nonconservation equation for the induced energy-momentum tensor imposes the renormalization condition for the induced electric current of the scalar field. The findings of this research indicate that there are significant differences between the two induced currents which are regularized by this renormalization condition and the minimal subtraction condition.

Vacuum densities and the Casimir forces for branes orthogonal to the AdS boundary

For a massive scalar field with general curvature coupling we evaluate the Wightman function in the geometry of two parallel branes perpendicular to the AdS boundary. On the separate branes, the field operator is constrained by Robin boundary conditions, in general, with different coefficients. In the region between the branes their contribution to the Wightman function is explicitly separated. By using this decomposition, the brane-induced effects on the vacuum expectation values (VEVs) for the field squared and energy-momentum tensor are investigated. The behavior of those expectation values is studied in various asymptotic regions of the parameters. The vacuum energy-momentum tensor in addition to the diagonal components has a nonzero off-diagonal stress. Depending on the boundary conditions and also on the distance from the branes, the vacuum energy density can be either positive or negative. The Casimir forces acting on the branes have two components. The first one corresponds to the standard normal force and the second one is parallel to the branes and presents the vacuum shear force. Unlike to the problem of parallel plates in the Minkowski bulk, the normal Casimir forces acting on separate branes differ if the boundary conditions on the branes are different. They can be either repulsive or attractive. In a similar way, depending on the coefficients in the boundary conditions, the shear force is directed toward or from the AdS boundary. The separate components may also change their signs as functions of the interbrane separation. At large proper separations between the branes, compared to the AdS curvature radius, both the components of the Casimir forces exhibit a power-law decay. For a massive scalar field this behavior is in contrast to that for the Minkowski bulk, where the decrease is exponential.

Cosmic string and brane induced effects on the fermionic vacuum in AdS spacetime

We investigate the combined effects of a magnetic flux-carrying cosmic string and a brane on the fermionic condensate (FC) and on the vacuum expectation value (VEV) of the energy-momentum tensor for a massive charged fermionic field in background of 5-dimensional anti-de Sitter (AdS) spacetime. The brane is parallel to the AdS boundary and it divides the space into two regions with distinct properties of the fermionic vacuum. For two types of boundary conditions on the field operator and for the fields realizing two inequivalent representations of the Clifford algebra, the brane-induced contributions in VEVs are explicitly separated. The VEVs are even periodic functions of the magnetic flux, confined in the core, with the period of flux quantum. Near the horizon the FC and the vacuum energy-momentum tensor are dominated by the brane-free contribution, whereas the brane-induced part dominates in the region near the brane. Both the contributions vanish on the AdS boundary. At large distances from the cosmic string, the topological contributions in the VEVs, as functions of the proper distance, exhibit an inverse power-law decrease in the region between the brane and AdS horizon and an exponential decrease in the region between the brane and AdS boundary. We show that the FC and the vacuum energy density can be either positive or negative, depending on the distance from the brane. Applications are discussed in fermionic models invariant under the charge conjugation and parity transformation and also in Z2-symmetric braneworld models. By the limiting transition we derive the expressions of the FC and the vacuum energy-momentum tensor for a cosmic string on 5-dimensional Minkowski bulk in the presence of a boundary perpendicular to the string.

Fermionic vacuum polarization around a cosmic string in compactified AdS spacetime

We investigate topological effects of a cosmic string and compactificationof a spatial dimension on the vacuum expectation value (VEV) of theenergy-momentum tensor for a fermionic field in (4+1)-dimensional locallyAdS spacetime. The contribution induced by the compactification isexplicitly extracted by using the Abel-Plana summation formula. The meanenergy-momentum tensor is diagonal and the vacuum stresses along thedirection perpendicular to the AdS boundary and along the cosmic string areequal to the energy density. All the components are even periodic functionsof the magnetic fluxes inside the string core and enclosed by compactdimension, with the period equal to the flux quantum. The vacuum energydensity can be either positive or negative, depending on the values of theparameters and the distance from the string. The topological contributionsin the VEV of the energy-momentum tensor vanish on the AdS boundary. Nearthe string the effects of compactification and gravitational field are weakand the leading term in the asymptotic expansion coincides with thecorresponding VEV in (4+1)-dimensional Minkowski spacetime. At largedistances, the decay of the cosmic string induced contribution in the vacuumenergy-momentum tensor, as a function of the proper distance from thestring, follows a power law. For a cosmic string in the Minkowski bulk andfor massive fields the corresponding fall off is exponential. Within theframework of the AdS/CFT correspondence, the geometry for conformal fieldtheory on the AdS boundary corresponds to the standard cosmic string in(3+1)-dimensional Minkowski spacetime compactified along its axis.

Casimir densities induced by a sphere in the hyperbolic vacuum of de Sitter spacetime

Complete set of modes and the Hadamard function are constructed for a scalar field inside and outside a sphere in (D+1)-dimensional de Sitter spacetime foliated by negative constant curvature spaces. We assume that the field obeys Robin boundary condition on the sphere. The contributions in the Hadamard function induced by the sphere are explicitly separated and the vacuum expectation values (VEVs) of the field squared and energy-momentum tensor are investigated for the hyperbolic vacuum. In the flat spacetime limit the latter is reduced to the conformal vacuum in the Milne universe and is different from the maximally symmetric Bunch-Davies vacuum state. The vacuum energy-momentum tensor has a nonzero off-diagonal component that describes the energy flux in the radial direction. The latter is a purely sphere-induced effect and is absent in the boundary-free geometry. Depending on the constant in Robin boundary condition and also on the radial coordinate, the energy flux can be directed either from the sphere or towards the sphere. At early stages of the cosmological expansion the effects of the spacetime curvature on the sphere-induced VEVs are weak and the leading terms in the corresponding expansions coincide with those for a sphere in the Milne universe. The influence of the gravitational field is essential at late stages of the expansion. Depending on the field mass and the curvature coupling parameter, the decay of the sphere-induced VEVs, as functions of the time coordinate, is monotonic or damping oscillatory. At large distances from the sphere the fall-off of the sphere-induced VEVs, as functions of the geodesic distance, is exponential for both massless and massive fields.

Distance between collapsing matter and apparent horizon in evaporating black holes

Assuming that the vacuum energy-momentum tensor is not exceptionally large, we consider 4D evaporating black holes with spherical symmetry and evaluate the proper distance $\Delta L$ between the time-like apparent horizon and the surface of the collapsing matter after it has entered the apparent horizon. We show that $\Delta L$ can never be larger than $\mathcal{O}(n^{3/2}\ell_p)$ when the black hole is evaporated to $1/n$ of its initial mass, as long as $n \ll a^{2/3}/\ell_p^{2/3}$ (where $a$ is the Schwarzschild radius and $\ell_p$ is the Planck length). For example, the distance between the matter and the apparent horizon must be Planckian at the Page time.

Firewall from Effective Field Theory

For an effective field theory in the background of an evaporating black hole with spherical symmetry, we consider non-renormalizable interactions and their relevance to physical effects. The background geometry is determined by the semi-classical Einstein equation for an uneventful horizon where the vacuum energy–momentum tensor is small for freely falling observers. Surprisingly, after Hawking radiation appears, the transition amplitude from the Unruh vacuum to certain multi-particle states grows exponentially with time for a class of higher-derivative operators after the collapsing matter enters the near-horizon region, despite the absence of large curvature invariants. Within the scrambling time, the uneventful horizon transitions towards a firewall, and eventually the effective field theory breaks down.

The Casimir Densities for a Sphere in the Milne Universe

The influence of a spherical boundary on the vacuum fluctuations of a massive scalar field is investigated in the background of a ( D + 1 ) -dimensional Milne universe, assuming that the field obeys Robin boundary conditions on the sphere. The normalized mode functions are derived for the regions inside and outside the sphere and different vacuum states are discussed. For the conformal vacuum, the Hadamard function is decomposed into boundary-free and sphere-induced contributions and an integral representation is obtained for the latter in both the interior and exterior regions. As important local characteristics of the vacuum state, the vacuum expectation values (VEVs) of the field squared and of the energy-momentum tensor are investigated. It is shown that the vacuum energy-momentum tensor has an off-diagonal component that corresponds to the energy flux along the radial direction. Depending on the coefficient in Robin boundary conditions, the sphere-induced contribution to the vacuum energy and the energy flux can be either positive or negative. At late stages of the expansion and for a massive field the decay of the sphere-induced VEVs, as functions of time, is damping oscillatory. The geometry under consideration is conformally related to that for a static spacetime with negative constant curvature space and the sphere-induced contributions in the corresponding VEVs are compared.

Trapping horizon and negative energy

Assuming spherical symmetry and the semi-classical Einstein equation, we prove that, for the observers on top of the trapping horizon, the vacuum energy-momentum tensor is always that of an ingoing negative energy flux at the speed of light with a universal energy density ℰ ≃ − 1/(2κa2), (where a is the areal radius of the trapping horizon), which is responsible for the decrease in the black hole’s mass over time. This result is independent of the composition of the collapsing matter and the details of the vacuum energy-momentum tensor. The physics behind the universality of this quantity ℰ and its surprisingly large magnitude will be discussed.

Scalar Casimir effect in a linearly expanding universe

We investigate quantum vacuum effects for a massive scalar field, induced by two planar boundaries in background of a linearly expanding spatially flat Friedmann–Robertson–Walker spacetime for an arbitrary number of spatial dimensions. For the Robin boundary conditions and for general curvature coupling parameter, a complete set of mode functions is presented and the related Hadamard function is evaluated. The results are specified for the most important special cases of the adiabatic and conformal vacuum states. The vacuum expectation values of the field squared and of the energy–momentum tensor are investigated for a massive conformally coupled field. The vacuum energy–momentum tensor, in addition to the diagonal components, has nonzero off-diagonal component describing energy flux along the direction perpendicular to the plates. The influence of the gravitational field on the local characteristics of the vacuum state is essential at distances from the boundaries larger than the curvature radius of the background spacetime. In contrast to the Minkowskian bulk, at large distances the boundary-induced expectation values follow as power law for both massless and massive fields. Another difference is that the Casimir forces acting on the separate plates do not coincide if the corresponding Robin coefficients are different. At large separations between the plates the decay of the forces is power law. We show that during the cosmological expansion the forces may change the sign.

Scalar Casimir densities and forces for parallel plates in cosmic string spacetime

We analyze the Casimir densities and forces associated with a massive scalar quantum field confined between two parallel plates in a D-dimensional cosmic string spacetime. The plates are placed orthogonal to the string and the field obeys the Robin boundary conditions on them. The boundary-induced contributions are explicitly extracted in the vacuum expectation values (VEVs) of the field squared and energy-momentum tensor for both single and two plates. The VEV of the energy-momentum tensor, in additional to the diagonal components, contains an off-diagonal component corresponding to the shear stress. The latter vanishes on the plates in special cases of Dirichlet and Neumann boundary conditions. For points outside the string core the topological contributions in the VEVs are finite on the plates. Near the string the VEVs are dominated by the boundary-free part, whereas at large distances the boundary-induced contributions dominate. Due to the nonzero off-diagonal component of the vacuum energy-momentum tensor, in addition to the normal component, the Casimir forces have nonzero component parallel to the boundary. Unlike the problem on the Minkowski bulk, the normal forces acting on the separate plates, in general, do not coincide. Another difference is that in the presence of the cosmic string the Casimir forces for Dirichlet and Neumann boundary conditions differ. For Dirichlet case the normal Casimir force does not depend on the curvature coupling parameter. This is not the case for other boundary conditions. A new qualitative feature induced by the cosmic string is the appearance of the shear stress acting on the plates. The corresponding force is directed along the radial coordinate and vanishes for Dirichlet and Neumann cases. Depending on the parameters of the problem, the radial component of the shear force can be either positive or negative.

Static black hole and vacuum energy: thin shell and incompressible fluid

With the back reaction of the vacuum energy-momentum tensor consistently taken into account, we study static spherically symmetric black-hole-like solutions to the semi-classical Einstein equation. The vacuum energy is assumed to be given by that of 2-dimensional massless scalar fields, as a widely used model in the literature for black holes. The solutions have no horizon. Instead, there is a local minimum in the radius. We consider thin shells as well as incompressible fluid as the matter content of the black-hole-like geometry. The geometry has several interesting features due to the back reaction of vacuum energy. In particular, Buchdahl’s inequality can be violated without divergence in pressure, even if the surface is below the Schwarzschild radius. At the same time, the surface of the star can not be far below the Schwarzschild radius for a density not much higher than the Planck scale, and the proper distance from its surface to the origin can be very short even for very large Schwarzschild radius. The results also imply that, contrary to the folklore, in principle the Boulware vacuum can be physical for black holes.

Electromagnetic Casimir densities for a cylindrical shell on de Sitter space

Complete set of cylindrical modes is constructed for the electromagnetic field inside and outside a cylindrical shell in the background of [Formula: see text]-dimensional dS space–time. On the shell, the field obeys the generalized perfect conductor boundary condition. For the Bunch–Davies vacuum state, we evaluate the vacuum expectation values (VEVs) of the electric field squared and of the energy–momentum tensor. The shell-induced contributions are explicitly extracted. In this way, for points away from the shell, the renormalization is reduced to the one for the VEVs in the boundary-free dS bulk. As a special case, the VEVs are obtained for a cylindrical shell in the [Formula: see text]-dimensional Minkowski bulk. We show that the shell-induced contribution in the electric field squared is positive for both the interior and exterior regions. The corresponding Casimir–Polder forces are directed toward the shell. The vacuum energy–momentum tensor, in addition to the diagonal components, has a nonzero off-diagonal component corresponding to the energy flux along the direction normal to the shell. This flux is directed from the shell in both the exterior and interior regions. For points near the shell, the leading terms in the asymptotic expansions for the electric field squared and diagonal components of the energy–momentum tensor are obtained from the corresponding expressions in the Minkowski bulk replacing the distance from the shell by the proper distance in the dS bulk. The influence of the gravitational field on the local characteristics of the vacuum is essential at distances from the shell larger than the dS curvature radius. The results are extended for confining boundary conditions of flux tube models in QCD.

Scalar Casimir densities induced by a cylindrical shell in de Sitter spacetime

We evaluate the positive-frequency Wightman function, the vacuum expectation values (VEVs) of the field squared, and the energy–momentum tensor for a massive scalar field with general curvature coupling for a cylindrical shell in the background of de Sitter (dS) spacetime. The field is prepared in the Bunch–Davies vacuum state and on the shell, and the corresponding operator obeys the Robin boundary condition (BC). In the region inside the shell and for non-Neumann BC, the Bunch–Davies vacuum is a physically realizable state for all values of the mass and curvature coupling parameter. For both interior and exterior regions, the VEVs are decomposed into boundary-free dS and shell-induced parts. We show that the shell-induced part of the vacuum energy–momentum tensor has a nonzero off-diagonal component corresponding to the energy flux along the radial direction. Unlike in the case of a shell in Minkowski bulk, for the dS background, the axial stresses are not equal to the energy density. In dependence of the mass and the coefficient in the BC, the vacuum energy density and the energy flux can be either positive or negative. The influence of the background gravitational field on the boundary-induced effects is crucial at distances from the shell larger than the dS curvature scale. In particular, the decay of the VEVs with distance is power-law (monotonic or oscillatory with dependence of the mass) for both massless and massive fields. For the Neumann BC, the decay is faster than that for non-Neumann conditions.

Vacuum energy-momentum tensor in models with non-trivial topology in the presence of boundaries

The vacuum expectation value of the energy-momentum tensor for a scalar field in models with the compact subspace of an arbitrary geometry in the presence of parallel plates is investigated. The field operator on the plates obeys Robin boundary conditions with constant coefficients. Depending on the values of the coefficients, the vacuum energy density can be either positive or negative. As an example, the case of one-dimensional internal space is considered.

The large-scale structure of vacuum

The vacuum state in quantum field theory is known to exhibit an important number of fundamental physical features. In this work we explore the possibility that this state could also present a nontrivial spacetime structure on large scales. In particular, we will show that by imposing the renormalized vacuum energy–momentum tensor to be conserved and compatible with cosmological observations, the vacuum energy of sufficiently heavy fields behaves at late times as nonrelativistic matter rather than as a cosmological constant. In this limit, the vacuum state supports perturbations whose speed of sound is negligible and accordingly allows the growth of structures in the vacuum energy itself. This large-scale structure of vacuum could seed the formation of galaxies and clusters very much in the same way as cold dark matter does.

Casimir densities for a conducting plate in de Sitter spacetime

Two-point functions for the electromagnetic field in background of (D + 1)-dimensional dS spacetime are evaluated assuming that the field is prepared in the Bunch-Davies vacuum state. By using these functions, the vacuum expectation values (VEVs) of the field squared and the energy-momentum tensor are investigated in the geometry of a conducting plate. The VEVs are explicitly decomposed into the boundary-free and plate-induced parts. For points outside of the plate the renormalization is needed for the first parts only. Because of the maximal symmetry of the background spacetime and of the Bunch-Davies vacuum state, the boundary-free parts do not depend on spacetime coordinates, whereas the plate-induced parts are functions of the proper distance of the observation point from the plate. The plate-induced part in the VEV of the energy-momentum tensor vanishes for D = 3 which is a direct consequence of a conformal invariance of the electromagnetic field for this spatial dimension. For D > 3, in addition to the diagonal components, the vacuum energy-momentum tensor has nonzero off-diagonal component which describes energy flux along the direction normal to the plate.

Electromagnetic two-point functions and Casimir densities for a conducting plate in de Sitter spacetime

We evaluate the two-point function for the electromagnetic field tensor in (D+1)-dimensional de Sitter spacetime assuming that the field is prepared in Bunch–Davies vacuum state. This two-point function is used for the investigation of the vacuum expectation values (VEVs) of the field squared and the energy–momentum tensor in the presence of a conducting plate. The VEVs are decomposed into the boundary-free and plate-induced parts. For the latter, closed form analytical expressions are given in terms of the hypergeometric function. For 3≤D≤8 the plate-induced part in the VEV of the electric field squared is positive everywhere, whereas for D≥9 it is positive near the plate and negative at large distances. The VEV of the energy–momentum tensor, in addition to the diagonal components, contains an off-diagonal component which corresponds to the energy flux along the direction normal to the plate. Simple asymptotic expressions are provided at small and large distances from the plate compared with the de Sitter curvature scale. For D≥4, all the diagonal components of the plate-induced vacuum energy–momentum tensor are negative and the energy flux is directed from the plate.

Electromagnetic two-point functions and the Casimir effect in Friedmann-Robertson-Walker cosmologies

We evaluate the two-point functions of the electromagnetic field in (D+1) -dimensional spatially flat Friedmann-Robertson-Walker universes with a power-law scale factor, assuming that the field is prepared in the Bunch-Davies vacuum state. The range of powers are specified in which the two-point functions are infrared convergent and the Bunch-Davies vacuum for the electromagnetic field is a physically realizable state. The two-point functions are applied for the investigation of the vacuum expectation values of the field squared and the energy-momentum tensor, induced by a single and two parallel conducting plates. Unlike to the case of conducting plates in the Minkowski bulk, in the problem under consideration the stresses along the directions parallel to the plates are not equal to the energy density. We show that, in addition to the diagonal components, the vacuum energy-momentum tensor has a nonzero off-diagonal component which describes energy flux along the direction normal to the plates. For a single plate this flux is directed from the plate. The Casimir forces are investigated in the geometry of two plates. At separations between the plates smaller than the curvature radius of the background spacetime, to the leading order, we recover the corresponding result in the Minkowski spacetime and in this case the forces are attractive. At larger separations, the influence of the curvature on the Casimir forces is essential with different asymptotic behavior for decelerated and accelerated expansions. In particular, for the latter case there is a range of powers of the expansion law in which the forces become repulsive at large separations between the plates.