Stress-energy tensor for a quantised bulk scalar field in the Randall-Sundrum brane model

We calculate the vacuum expectation value of the stress-energy tensor for a quantised bulk scalar field in the Randall-Sundrum model, and discuss the consequences of its local behaviour for the self-consistency of the model. We find that, in general, the stress-energy tensor diverges in the vicinity of the branes. Our main conclusion is that the stress-energy tensor is sufficiently complicated that it has implications for the effective potential, or radion stabilisation, methods that have so far been used.Comment: 16 pages, 3 figures. Minor changes made and references added. To appear in Phys. Rev.

Wightman function and Casimir densities on AdS bulk with application to the Randall-Sundrum braneworld

Positive frequency Wightman function and vacuum expectation value of the energy-momentum tensor are computed for a massive scalar field with general curvature coupling parameter subject to Robin boundary conditions on two parallel plates located on $D+1$ - dimensional AdS background. The general case of different Robin coefficients on separate plates is considered. The mode summation method is used with a combination of a variant of the generalized Abel-Plana formula for the series over zeros of combinations of cylinder functions. This allows us to extract manifestly the parts due to the AdS spacetime without boundaries and boundary induced parts. The asymptotic behavior of the vacuum densities near the plates and at large distances is investigated. The vacuum forces acting on the boundaries are presented as a sum of the self-action and interaction forces. The first one contains well-known surface divergences and needs further regularization. The interaction forces between the plates are attractive for Dirichlet scalar. We show that threre is a region in the space of parameters defining the boundary conditions in which the interaction forces are repulsive for small distances and attractive for large distances. An application to the Randall-Sundrum braneworld with arbitrary mass terms on the branes is discussed.Comment: 26 pages, 6 figures, discussions and figure labels added, accepted for publication in Nuclear Physics

Casimir effect for scalar fields under Robin boundary conditions on plates

We study the Casimir effect for scalar fields with general curvature coupling subject to mixed boundary conditions $(1+\beta_{m}n^{\mu}\partial_{\mu})\phi =0$ at $x=a_{m}$ on one ($m=1$) and two ($m=1,2$) parallel plates at a distance $a\equiv a_{2}-a_{1}$ from each other. Making use of the generalized Abel-Plana formula previously established by one of the authors \cite{Sahrev}, the Casimir energy densities are obtained as functions of $\beta_{1}$ and of $\beta_{1}$,$\beta_{2}$,$a$, respectively. In the case of two parallel plates, a decomposition of the total Casimir energy into volumic and superficial contributions is provided. The possibility of finding a vanishing energy for particular parameter choices is shown, and the existence of a minimum to the surface part is also observed. We show that there is a region in the space of parameters defining the boundary conditions in which the Casimir forces are repulsive for small distances and attractive for large distances. This yields to an interesting possibility for stabilizing the distance between the plates by using the vacuum forces.Comment: 21 pages, 8 figures, consideration of the contribution from complex eigenmodes added, possibility for the stabilization of the distance between the plates is discussed; accepted for publication in J. Phys.

On the energy-momentum tensor for a scalar field on manifolds with boundaries

We argue that already at classical level the energy-momentum tensor for a scalar field on manifolds with boundaries in addition to the bulk part contains a contribution located on the boundary. Using the standard variational procedure for the action with the boundary term, the expression for the surface energy-momentum tensor is derived for arbitrary bulk and boundary geometries. Integral conservation laws are investigated. The corresponding conserved charges are constructed and their relation to the proper densities is discussed. Further we study the vacuum expectation value of the energy-momentum tensor in the corresponding quantum field theory. It is shown that the surface term in the energy-momentum tensor is essential to obtain the equality between the vacuum energy, evaluated as the sum of the zero-point energies for each normal mode of frequency, and the energy derived by the integration of the corresponding vacuum energy density. As an application, by using the zeta function technique, we evaluate the surface energy for a quantum scalar field confined inside a spherical shell.Comment: 25 pages, 2 figures, section and appendix on the surface energy for a spherical shell are added, references added, accepted for publication in Phys. Rev.

Wightman function and scalar Casimir densities for a wedge with two cylindrical boundaries

Wightman function, the vacuum expectation values of the field square and the energy-momentum tensor are investigated for a massive scalar field with general curvature coupling parameter inside a wedge with two coaxial cylindrical boundaries. It is assumed that the field obeys Dirichlet boundary condition on bounding surfaces. The application of a variant of the generalized Abel-Plana formula enables to extract from the expectation values the contribution corresponding to the geometry of a wedge with a single shell and to present the interference part in terms of exponentially convergent integrals. The local properties of the vacuum are investigated in various asymptotic regions of the parameters. The vacuum forces acting on the boundaries are presented as the sum of self-action and interaction terms. It is shown that the interaction forces between the separate parts of the boundary are always attractive. The generalization to the case of a scalar field with Neumann boundary condition is discussed.Comment: 19 pages, 3 figure

Calculating Casimir Energies in Renormalizable Quantum Field Theory

Quantum vacuum energy has been known to have observable consequences since 1948 when Casimir calculated the force of attraction between parallel uncharged plates, a phenomenon confirmed experimentally with ever increasing precision. Casimir himself suggested that a similar attractive self-stress existed for a conducting spherical shell, but Boyer obtained a repulsive stress. Other geometries and higher dimensions have been considered over the years. Local effects, and divergences associated with surfaces and edges have been studied by several authors. Quite recently, Graham et al. have re-examined such calculations, using conventional techniques of perturbative quantum field theory to remove divergences, and have suggested that previous self-stress results may be suspect. Here we show that the examples considered in their work are misleading; in particular, it is well-known that in two dimensions a circular boundary has a divergence in the Casimir energy for massless fields, while for general dimension $D$ not equal to an even integer the corresponding Casimir energy arising from massless fields interior and exterior to a hyperspherical shell is finite. It has also long been recognized that the Casimir energy for massive fields is divergent for $D\ne1$. These conclusions are reinforced by a calculation of the relevant leading Feynman diagram in $D$ and three dimensions. There is therefore no doubt of the validity of the conventional finite Casimir calculations.Comment: 25 pages, REVTeX4, 1 ps figure. Revision includes new subsection 4B and Appendix, and other minor correction

Local and Global Casimir Energies: Divergences, Renormalization, and the Coupling to Gravity

From the beginning of the subject, calculations of quantum vacuum energies or Casimir energies have been plagued with two types of divergences: The total energy, which may be thought of as some sort of regularization of the zero-point energy, $\sum\frac12\hbar\omega$, seems manifestly divergent. And local energy densities, obtained from the vacuum expectation value of the energy-momentum tensor, $\langle T_{00}\rangle$, typically diverge near boundaries. The energy of interaction between distinct rigid bodies of whatever type is finite, corresponding to observable forces and torques between the bodies, which can be unambiguously calculated. The self-energy of a body is less well-defined, and suffers divergences which may or may not be removable. Some examples where a unique total self-stress may be evaluated include the perfectly conducting spherical shell first considered by Boyer, a perfectly conducting cylindrical shell, and dilute dielectric balls and cylinders. In these cases the finite part is unique, yet there are divergent contributions which may be subsumed in some sort of renormalization of physical parameters. The divergences that occur in the local energy-momentum tensor near surfaces are distinct from the divergences in the total energy, which are often associated with energy located exactly on the surfaces. However, the local energy-momentum tensor couples to gravity, so what is the significance of infinite quantities here? For the classic situation of parallel plates there are indications that the divergences in the local energy density are consistent with divergences in Einstein's equations; correspondingly, it has been shown that divergences in the total Casimir energy serve to precisely renormalize the masses of the plates, in accordance with the equivalence principle.Comment: 53 pages, 1 figure, invited review paper to Lecture Notes in Physics volume in Casimir physics edited by Diego Dalvit, Peter Milonni, David Roberts, and Felipe da Ros

Systematics of the Relationship between Vacuum Energy Calculations and Heat Kernel Coefficients

Casimir energy is a nonlocal effect; its magnitude cannot be deduced from heat kernel expansions, even those including the integrated boundary terms. On the other hand, it is known that the divergent terms in the regularized (but not yet renormalized) total vacuum energy are associated with the heat kernel coefficients. Here a recent study of the relations among the eigenvalue density, the heat kernel, and the integral kernel of the operator $e^{-t\sqrt{H}}$ is exploited to characterize this association completely. Various previously isolated observations about the structure of the regularized energy emerge naturally. For over 20 years controversies have persisted stemming from the fact that certain (presumably physically meaningful) terms in the renormalized vacuum energy density in the interior of a cavity become singular at the boundary and correlate to certain divergent terms in the regularized total energy. The point of view of the present paper promises to help resolve these issues.Comment: 19 pages, RevTeX; Discussion section rewritten in response to referees' comments, references added, minor typos correcte

p90 ribosomal S6 kinases play a significant role in early gene regulation in the cardiomyocyte response to Gq protein-coupled receptor stimuli, endothelin-1 and α1-adrenergic receptor agonists

Extracellular signal-regulated kinases 1/2 (ERK1/2) and their substrates, p90 ribosomal S6 kinases (RSKs), phosphorylate different transcription factors, contributing differentially to transcriptomic profiles. In cardiomyocytes, ERK1/2 are required for >70% of the transcriptomic response to endothelin-1. Here, we investigated the role of RSKs in the transcriptomic responses to Gq protein-coupled receptor agonists, endothelin-1, phenylephrine (generic α1-adrenergic receptor agonist) and A61603 (α1A-adrenergic receptor selective). Phospho-ERK1/2 and phospho-RSKs appeared in cardiomyocyte nuclei within 2-3 min of stimulation (endothelin-1>a61603≈phenylephrine). All agonists increased nuclear RSK2, but only endothelin-1 increased nuclear RSK1 content. PD184352 (inhibits ERK1/2 activation) and BI-D1870 (inhibits RSKs) were used to dissect the contribution of RSKs to the endothelin-1-responsive transcriptome. Of 213 RNAs upregulated at 1 h, 51% required RSKs for upregulation whereas 29% required ERK1/2 but not RSKs. The transcriptomic response to phenylephrine overlapped with, but was not identical to, endothelin-1. As with endothelin-1, PD184352 inhibited upregulation of most phenylephrine-responsive transcripts, but the greater variation in effects of BI-D1870 suggests that differential RSK signalling influences global gene expression. A61603 induced similar changes in RNA expression in cardiomyocytes as phenylephrine, indicating that the signal was mediated largely through α1A-adrenergic receptors. A61603 also increased expression of immediate early genes in perfused adult rat hearts and, as in cardiomyocytes, upregulation of the majority of genes was inhibited by PD184352. PD184352 or BI-D1870 prevented the increased surface area induced by endothelin-1 in cardiomyocytes. Thus, RSKs play a significant role in regulating cardiomyocyte gene expression and hypertrophy in response to Gq protein-coupled receptor stimulation