On effective spacetime dimension in the Hořava–Lifshitz gravity

Abstract

In this manuscript we explicitly compute the effective dimension of spacetime in some backgrounds of Hořava-Lifshitz (H-L) gravity. For all the cases considered, the results are compatible with a dimensional reduction of the spacetime to d + 1 = 2, at high energies (ultraviolet limit), which is confirmed by other quantum gravity approaches, as well as to d + 1 = 4, at low energies (infrared limit). This is obtained by computing the free energy of massless scalar and gauge fields. We find that the only effect of the background is to change the proportionality constant between the internal energy and temperature. Firstly, we consider both the non-perturbative and perturbative models involving the matter action, without gravitational sources but with manifest time and space symmetry breaking, in order to calculate modifications in the Stephan-Boltzmann law. When gravity is taken into account, we assume a scenario in which there is a spherical source with mass M and radius R in thermal equilibrium with radiation, and consider the static and spherically symmetric solution of the H-L theory found by Kehagias-Sfetsos (K-S), in the weak and strong field approximations. As byproducts, for the weak field regime, we used the current uncertainty of the solar radiance measurements to establish a constraint on the ω free parameter of the K-S solution. We also calculate the corrections, due to gravity, to the recently predicted attractive force that black bodies exert on nearby neutral atoms and molecules.

Dimension and dimensional reduction in quantum gravity

“…One popular example is 'Hořava-Lifshitz gravity' [128], a model that sacrifices Lorentz invariance in exchange for (possible) renormalizability. Here, a generalization of the spectral dimension flows from d S = 4 at low energies to d S = 2 at high energies [129], as do the thermodynamic dimensions d th1 and d th2 of section 2.2.1 [130]. Similarly, in curvature-squared models [131], which are renormalizable but very probably nonunitary, both the Green function dimension d G [132] and a generalized spectral dimension [133] fall to d = 2.…”

Dimension and Dimensional Reduction in Quantum Gravity

“…• Renormalizable modifications of general relativity: Another possible approach to quantum gravity is to modify the Lagrangian to make the theory renormalizable. In one such approach, Hořava-Lifshitz gravity [52], a generalized spectral dimension, flows to d = 2 at high energies [53], as do the thermodynamic dimensions [54]. In curvature-squared models, the Greens function dimension and a generalized spectral dimension exhibit a reduction to d = 2 [55,56].…”

Dependence of the black-body force on spacetime geometry and topology

“…Notice that by considering the Sun, this reduction in the temperature-independent part of the black-body potential is approximately 0.0007% at r = 1.5R. For a quantum gravity correction to this force, see [17].…”

“…It is worth noticing that, for the neutral hydrogen atom in the Minkowsky space (i.e., when ν = 1) we have the expression (17) again, and when one takes the limit R → 0 valid for an ideal cosmic string, the BBF vanishes. Then, an important conclusion is that ideal cosmic strings irradiating do not exert BBFs.…”