Emergent gravity in spaces of constant curvature

Abstract

Abstract:In physical theories where the energy (action) is localized near a submanifold of a constant curvature space, there is a universal expression for the energy (or the action). We derive a multipole expansion for the energy that has a finite number of terms, and depends on intrinsic geometric invariants of the submanifold and extrinsic invariants of the embedding of the submanifold. This is the second of a pair of articles in which we try to develop a theory of emergent gravity arising from the embedding of a submanifold into an ambient space equipped with a quantum field theory. Our theoretical method requires a generalization of a formula due to by Hermann Weyl. While the first paper discussed the framework in Euclidean (Minkowski) space, here we discuss how this framework generalizes to spaces of constant sectional curvature. We focus primarily on anti de Sitter space. We then discuss how such a theory can give rise to a cosmological constant and Planck mass that are within reasonable bounds of the experimental values.

Emergent Gravity and Weyl's Volume Formula

“…Randall and Sundrum use the exponential change in the metric as you move from the hidden slice to the visible slice to generate an exponential hierarchy where the physical field theoretic parameters on the visible slice are TeV scale. An extension of the work presented here to the case of embedding in constant curvature spaces and the Randall-Sundrum scenario appears in a companion article [20]. There are also the models of emergent gravity motivated by the work of Dvali, Gabadadze and Porrati [21] where gravity on an infinite five dimensional spacetime reproduces the correct crossover to 4 dimensional behavior on a 4 dimensional submanifold.…”

“…The laplacian of the embedding map X is the mean curvature vector nj K a a j . The map X : Σ → E n is harmonic if and only if the submanifold X( Σ) = Σ → E n is minimal 20 , i.e., the mean curvature vector vanishes. By substituting (7.12) into eq.…”

“…More properly, the dalembertian in a Lorentzian framework 20. The Euler-Lagrange equation for Nambu action states that the map X : Σ → E n is harmonic with respect to the induced metric.…”

Some exact solutions for maximally symmetric topological defects in Anti de Sitter space

“…To work out the equations of motion for our maximally symmetric defect, it is convenient to use a coordinate system adapted to the geometry of the problem, i.e., an analog of spherical coordinates. The construction is based on the method of Cartan discussed in our previous paper [18]. Let Σ q be a Lorentzian q-dimensional submanifold of AdS n .…”

“…The volume element formula is a special case of a result from our previous paper [18]. We implicitly assumed spherical symmetry for the Lagrangian density to do the angular integrals.…”