Vacuum energy density for massless scalar fields in flat homogeneous spacetime manifolds with nontrivial topology

TL;DR

The paper computes the renormalized vacuum stress-energy tensor for a massless scalar field in all 17 multiply-connected flat homogeneous spatial manifolds with nontrivial topology. It finds the vacuum energy density is negative relative to Minkowski in every case and becomes position-dependent when reflections or rotations are present.

Key findings

• Every calculated vacuum energy density is negative relative to the simply-connected Minkowski vacuum; the stress-energy is proportional to the inverse-fourth power of the Fundamental Polyhedron length, so the density approaches zero as the polyhedron grows.
• Table I gives constant, position-independent shifts for \(L_x=L_y=L_z=L=1\): 1-Torus \(-0.11\), 2-Torus \(-0.31\), rectangular 3-Torus \(-0.83\), hexagonal prism \(-0.99\), Klein space \(-2.39\), and Hantzsche-Wendt Space \(-0.32\).
• Closing more dimensions lowers the vacuum energy density further; the cross-section of the Fundamental Polyhedron strongly affects the relative density, while flips and turns introduce only minor position-dependent modulations.
• Position dependence appears with reflections or rotations: a flip in a direction makes \(T_{\mu\nu}\) depend on that direction, and a turn makes it depend on coordinates in the plane perpendicular to the rotational axis and produces off-diagonal elements. For example, the 1-Torus with Flip has \(T_{\mu\nu}^{E17}=\frac{1}{16}T_{\mu\nu}^{E16}+T_{\mu\nu}^{\mathrm{flip}}(y)\), which is \(y\)-dependent.

Methods

• Assumes a static, flat, homogeneous spacetime \(\mathbb{R}\times M^3\), where \(M^3\) has Euclidean \(\mathbb{R}^3\) as universal covering space; treats a free massless scalar field with conformal coupling \(\xi=1/6\), natural units, and metric signature \(+2\).
• Uses the method of images: writes the half-squared interval \(\sigma\) for each manifold, sums over image points while excluding the simply-connected \(n=0\) term to form the renormalized Hadamard function, applies the stress-energy derivative operator, takes the coincidence limit, and subtracts the Minkowski vacuum expectation value.
• Applies this procedure to all 17 multiply-connected flat homogeneous spacetimes, including non-orientable manifolds; some results are given analytically, while others are presented numerically in plots.

Limitations

• The paper states assumptions rather than limitations: the universe is taken to be static, flat, and homogeneous; the field is massless; topology refers to the spatial manifold \(M^3\), not spacetime; natural units are used.
• Evident from the text: the calculations are for a massless scalar field only, with massive scalar cases cited elsewhere; some spaces, such as the Quarter-Turn Space, omit an explicit analytical expression in favor of a numerical plot.
• Evident from the text: Table I reports only the constant, position-independent terms for \(L_x=L_y=L_z=L=1\), and the results depend on the chosen Fundamental Polyhedron size and cross-section.