Dawood Kothawala

There is a considerable amount of evidence to suggest that the field equations of gravity have the same status as, say, the equations describing an emergent phenomenon like elasticity. In fact, it is possible to derive the field equations from a thermodynamic variational principle in which a set of normalized vector fields are varied rather than the metric. We show that this variational principle can arise as a low-energy [LP=(G/c3)1/2→0] relic of a plausible nonperturbative effect of quantum gravity, viz. the existence of a zero-point length in the spacetime. Our result is nonperturbative in the following sense: If we modify the geodesic distance in a spacetime by introducing a zero-point length, to incorporate some effects of quantum gravity, and take the limit LP→0 of the Ricci scalar of the modified metric, we end up getting a nontrivial, leading order (LP-independent) term. This term is identical to the expression for entropy density of spacetime used previously in the emergent gravity approach. This reconfirms the idea that the microscopic degrees of freedom of the spacetime, when properly described in the full theory, could lead to an effective description of geometry in terms of a thermodynamic variational principle. This is conceptually similar to the emergence of thermodynamics from the mechanics of, say, molecules. The approach also has important implications for the cosmological constant which are briefly discussed.

It is generally believed that any quantum theory of gravity should have a generic feature—a quantum of length. We provide a physical ansatz to obtain an effective non-local metric tensor starting from the standard metric tensor such that the spacetime acquires a zero-point-length (Formula presented.) of the order of the Planck length (Formula presented.). This prescription leads to several remarkable consequences. In particular, the Euclidean volume (Formula presented.) in a D-dimensional spacetime of a region of size (Formula presented.) scales as (Formula presented.) when (Formula presented.) , while it reduces to the standard result (Formula presented.) at large scales ((Formula presented.)). The appropriately defined effective dimension, (Formula presented.) , decreases continuously from (Formula presented.) (at (Formula presented.)) to (Formula presented.) (at (Formula presented.)). This suggests that the physical spacetime becomes essentially 2-dimensional near Planck scale.

Lanczos-Lovelockmodels of gravity represent a natural and elegant generalization of Einstein's theory of gravity to higher dimensions. They are characterized by the fact that the field equations only contain up to second derivatives of the metric even though the action functional can be a quadratic or higher degree polynomial in the curvature tensor. Because these models share several key properties of Einstein's theory they serve as a useful set of candidate models for testing the emergent paradigm for gravity. This review highlights several geometrical and thermodynamical aspects of Lanczos-Lovelockmodels which have attracted recent attention. © 2013 Elsevier B.V.

It is possible to obtain gravitational field equations in a large class of theories from a thermodynamic variational principle which uses the gravitational heat density Sg associated with null surfaces. This heat density is related to the structure of spacetime at Planck scale, LP2=(Għ/c3), which assigns A⊥/LP2 degrees of freedom to any area A⊥. On the other hand, it is also known that the surface term Kh in the gravitational action correctly reproduces the heat density of the null surfaces. We provide a link between these ideas by obtaining Sg, used in emergent gravity paradigm, from the surface term in the Einstein–Hilbert action. This is done using the notion of a nonlocal qmetric – introduced recently [arXiv:1307.5618, arXiv:1405.4967] – which allows us to study the effects of zero-point-length of spacetime at the transition scale between quantum and classical gravity. Computing Kh for the qmetric in the appropriate limit directly reproduces the entropy density Sg used in the emergent gravity paradigm.