Casimir Energy of a Relativistic Perfect Fluid Confined to a D-dimensional Hypercube

Abstract

Compact formulas are obtained for the Casimir energy of a relativistic perfect fluid confined to a $D$-dimensional hypercube with von Neumann or Dirichlet boundary conditions. The formulas are conveniently expressed as a finite sum of the well-known gamma and Riemann zeta functions. Emphasis is placed on the mathematical technique used to extract the Casimir energy from a $D$-dimensional infinite sum regularized with an exponential cut-off. Numerical calculations show that initially the Dirichlet energy decreases rapidly in magnitude and oscillates in sign, being positive for even $D$ and negative for odd $D$. This oscillating pattern stops abruptly at the critical dimension of D=36 after which the energy remains negative and the magnitude increases. We show that numerical calculations performed with 16-digit precision are inaccurate at higher values of $D$.