Casimir Energy of a Long Wormhole Throat (1405.1283v1)

Abstract: We calculate the Casimir energy-momentum tensor induced in a scalar field by a macroscopic ultrastatic spherically-symmetric long-throated traversable wormhole, and examine whether this exotic matter is sufficient to stabilise the wormhole itself. The Casimir energy-momentum tensor is obtained (within the $\mathbb{R}\times S_2$ throat) by a mode sum approach, using a sharp energy cut-off and the Abel-Plana formula; Lorentz invariance is then restored by use of a Pauli-Villars regulator. The massless conformally-coupled case is found to have a logarithmic divergence (which we renormalise) and a conformal anomaly, the thermodynamic relevance of which is discussed. Provided the throat radius is above some fixed length, the renormalised Casimir energy-density is seen to be negative by all timelike observers, and almost all null rays; furthermore, it has sufficient magnitude to stabilise a long-throated wormhole far larger than the Planck scale, at least in principle. Unfortunately, the renormalised Casimir energy-density is zero for null rays directed exactly parallel to the throat, and this shortfall prevents us from stabilising the ultrastatic spherically-symmetric wormhole considered here. Nonetheless, the negative Casimir energy does allow the wormhole to collapse extremely slowly, its lifetime growing without bound as the throat-length is increased. We find that the throat closes slowly enough that its central region can be safely traversed by a pulse of light.

• The paper investigates whether Casimir energy, calculated using a mode sum approach with Abel-Plana and Pauli-Villars regulation, can stabilize a macroscopic long-throated traversable wormhole.
• For wormhole throats above a certain radius, the renormalized Casimir energy density is primarily negative, theoretically sufficient to stabilize large throats but insufficient along null rays parallel to the throat.
• While not achieving complete stabilization due to limitations along certain null geodesics, the negative Casimir energy can slow collapse enough to allow light pulses to traverse the wormhole throat.

An Analysis of Casimir Energy in Long-Wormhole Throats

The paper by Luke M. Butcher, "Casimir Energy of a Long Wormhole Throat," explores the generation of Casimir energy within the context of a macroscopic ultrastatic spherically-symmetric long-throated traversable wormhole. The paper investigates whether this exotic Casimir energy is adequate to stabilize the wormhole. This exploration considers a traversable wormhole framework, particularly assessing the negative energy attributes offered by the Casimir effect—a quantum phenomenon fundamentally connected to vacuum fluctuations.

The research adopts a mode sum approach, integrating a sharp energy cut-off with the Abel-Plana formula to determine the Casimir energy-momentum tensor in the wormhole throat. For restoring Lorentz invariance, a Pauli-Villars regulator is invoked, adjusting the quantum field computations by subtracting the contributions of a massive regulator field. This approach ensures that the physical results are consistent with relativistic principles.

Central to the paper's findings is that for wormhole throats with radii above a certain threshold, the renormalized Casimir energy-density is predominantly negative across all timelike observers and numerous null directions, but not all. While this negative energy is theoretically sufficient to stabilize substantial wormhole throats, particularly for those exceeding the Planck scale, the energy density presents inadequacies along null rays precisely aligned parallel to the throat. Hence, the overall stabilization required for complete wormhole support remains problematic, even as the negative Casimir energy is sufficient to enable an extremely slow collapse of the wormhole, facilitating light pulse traversal across its central region—a key observation in evaluating traversability.

The paper's results are methodically derived, highlighting the Casimir energy-momentum tensor's conformity and distinct deviations in various contexts, notably where energy conditions are violated. Through this process, the research illuminates how the exotic Casimir energy formed by a long wormhole could potentially be harnessed to support the stabilization of wormholes, albeit incompletely, due to limitations in energy-density along certain null geodesics.

Moreover, the investigation implies a rich connection between quantum field theory phenomena and classical spacetime constructs. While it stops short of declaring a stable wormhole regime solely reliant on Casimir energy, it paves the pathway for contemplating how quantum-induced macroscopic stability might be realized under certain symmetrical and topological conditions.

Future advancements might consider modifications to the throat geometry to resolve the parallel null ray energy-density issues or broader application of this quantum phenomenon across other traversable wormhole configurations. Importantly, Butcher’s insights encourage further exploration into employing Casimir energy as an intrinsic stabilizing agent within the context of general relativity and beyond, emphasizing the intersections of quantum field theory components and gravitational frameworks.

In conclusion, while the Casimir effect presents a tantalizing method for generating negative energy densities required for stabilizing traversable wormholes, this paper articulates its current limitations and prospects for future research into wormhole mechanics and the achievable manipulation of quantum fields in curvilinear spacetime arrangements.