Topological effects in the polarization of the Fulling-Rindler vacuum

Technical Summary: Topological Effects in the Polarization of the Fulling-Rindler Vacuum

Problem Statement
This study investigates the vacuum polarization effects for a charged massive scalar field within a $(D+1)$-dimensional Rindler spacetime where a subset of spatial dimensions is compactified. Specifically, the authors analyze the vacuum expectation values (VEVs) of the field squared ($\langle \phi \phi^\dagger \rangle$) and the energy-momentum tensor ($\langle T_{ik} \rangle$) for a field prepared in the Fulling-Rindler (FR) vacuum state. The spatial topology is defined as $\mathbb{R}^p \times (S^1)^q$, where $p+q=D-1$. The analysis incorporates general quasiperiodicity conditions along the compact dimensions, which are interpreted physically in terms of magnetic fluxes enclosed by these dimensions, thereby inducing Aharonov-Bohm-type effects. The work aims to explicitly separate the contributions arising from the nontrivial topology from those inherent to the acceleration of the FR vacuum and the flat Minkowski background.

Methodology
The authors employ the canonical quantization of the scalar field in the Rindler wedge, utilizing mode functions that satisfy the Klein-Gordon equation with a constant gauge field and specific quasiperiodicity conditions. The core of the methodology relies on the evaluation of the Hadamard function, $G(x, x') = \langle 0 | \phi(x)\phi^\dagger(x') + \phi^\dagger(x')\phi(x) | 0 \rangle$, where $|0\rangle$ denotes the FR vacuum.

A key technical step involves the representation of the Hadamard function as a difference between the corresponding function in a locally Minkowskian spacetime with the same compact topology ($G_M$) and a finite remainder term. This decomposition allows for the renormalization of the Rindler VEVs to be reduced to the renormalization of the Minkowskian VEVs, which are calculated using zeta-function regularization (detailed in Appendix A).

The authors derive explicit integral representations for the VEVs by summing over the quantized momentum modes in the compact subspace. They utilize alternative representations of these sums (derived in Appendix B using the generalized Chowla-Selberg formula) to facilitate asymptotic analysis in two limiting regimes:

1. Near the Rindler horizon: Small values of the proper acceleration coordinate $\xi$ (large acceleration).
2. Far from the horizon: Large values of $\xi$ (small acceleration).

Key Contributions and Results

• Decomposition of VEVs: The VEVs are explicitly separated into three components:

  1. The Minkowski vacuum contribution ($\langle \cdot \rangle_M$), which depends on the compact topology and magnetic fluxes but is independent of acceleration.
  2. The contribution from the Rindler vacuum with trivial topology ($\langle \cdot \rangle_0$), representing the effect of acceleration in the absence of compact dimensions.
  3. The topological contribution ($\langle \cdot \rangle_t$), representing the specific interplay between the nontrivial topology and the acceleration.

• Behavior of the Field Squared:

  • Near the Rindler horizon ($\xi \to 0$), the leading term of the field squared coincides with the VEV in Rindler spacetime with trivial topology. This indicates that topological effects are suppressed near the horizon.
  • For small accelerations (large $\xi$), the difference between the FR and Minkowski VEVs is exponentially suppressed ($\sim e^{-2\xi\omega_0}$), where $\omega_0$ depends on the field mass and the phases of the periodicity conditions.
  • Exception: For a massless field with zero phases in the periodicity conditions, the decay is not exponential but follows a power law ($\sim \xi^{-p}$).

• Energy-Momentum Tensor (EMT):

  • Off-Diagonal Components: A significant finding is that for non-zero phases in the periodicity conditions, the vacuum EMT exhibits non-zero off-diagonal components ($\langle T_{il} \rangle$) where indices $i, l$ correspond to the compact subspace. These components vanish on the Rindler horizon and are odd periodic functions of the magnetic flux phases.
  • Diagonal Components: Near the horizon, the diagonal components behave as $1/\xi^{D+1}$, matching the trivial topology case. At large distances, the acceleration-induced corrections decay exponentially, except in the massless, zero-phase case where they decay via a power law.
  • Equation of State: The Minkowskian part of the EMT describes a gravitational source with a cosmological constant-type equation of state in the uncompact subspace.

• Instability Implications: The authors note that the VEV of the field squared can change sign depending on the acceleration and compactification parameters. In theories with self-interacting scalar fields (e.g., quartic interaction), a negative effective mass squared induced by this VEV could signal an instability or a phase transition.

Significance and Applications
The paper claims that these results provide a valuable insight into quantum phenomena in curved spacetime and black hole physics. By virtue of the equivalence principle, the Rindler geometry serves as the leading approximation for the near-horizon region of a broad class of black-hole spacetimes.

Consequently, the derived formulas are applied to the near-horizon geometry of:

1. Cylindrical Black Holes: Solutions in Anti-de Sitter (AdS) space relevant to supergravity and string theory compactifications.
2. Topological Black Holes: Specifically those with toroidal horizons.

The authors demonstrate that for black holes where the length scales of the mass and the cosmological constant are comparable, the simple asymptotic formulas derived for the Rindler wedge can be utilized to approximate vacuum densities near the horizon. The work also highlights the conformal relation between Rindler metrics and other physically relevant geometries, such as de Sitter spacetime and Friedmann-Robertson-Walker models with negative spatial curvature, suggesting that the results can be extended to these backgrounds via conformal transformations for conformally invariant fields.

The study concludes that while topological effects are generally weak near the horizon, they become significant at larger distances or for specific massless configurations, and the presence of magnetic fluxes fundamentally alters the structure of the vacuum energy-momentum tensor by introducing off-diagonal stresses.