Structured objects in quantum gravity. The external field approximation

Abstract

In the external field approximation (EFA) gravity and inertia are represented by a two-point vector that is the byproduct of symmetry breaking. The vector is accompanied by the appearance of classical, vortical structures. Its interaction range is, in general, that of the metric tensor, but, in the context of a simple symmetry breaking model, the range can be made finite by the presence of massive scalar particles. Vortices can then be produced that conceal matter making it effectively "dark". In EFA fermion relativistic vortices can be induced, in particular, by rotation. THE EXTERNAL FIELD APPROXIMATIONSymmetry breaking in quantum many-body systems gives rise to macroscopic objects like vortices in superconductors, dislocations in crystals and domain walls in ferromagnets. These structures normally appear in a quantum context, but behave classically. They are properties of matter, in the form other than particles, that emerge from a quantum background when quantum fluctuations become negligible. Here we consider the possibility that similar phenomena occur in quantum gravity, still far from the essential quantum regime that is supposed to take over at Planck's length. The suggestion comes from studies of covariant wave equations [1-5] that can be solved exactly to first order in the metric deviation γ µν = g µν − η µν , where η µν is the Minkowski metric and whose solutions are a useful tool in the study of the interaction of gravity with quantum systems [6][7][8][9][10][11][12].In the EFA context, gravity is represented exclusively by a two-point vector K λ (z, x) [13,14] that is known only if γ µν and its derivatives are known. In what follows the coordinates z µ refer to a frame that is moved by parallel displacement along a particle path and x µ to a particle local inertial frame. Though the essential steps of the discussion apply to any wave equation, spin is an unnecessary complication and is momentarily ignored. A brief discussion of fermions is given in section 4. Without loss of generality, we can therefore consider the Klein-Gordon equation that, in its minimal coupling form and after applying the Lanczos-DeDonder condition γ αν , ν −1/2γ σ σ , α = 0, becomes(1)We use unitsh = c = 1 and the notations are as in [12]. In particular, ∇ µ is the covariant derivative and partial derivatives with respect to a variable y µ are interchangeably indicated by ∂ µ , or by a comma followed by µ. The first order solution of (1) iswhereΦ G is the operatorΦwhere P is an arbitrary point, henceforth dropped, and φ 0 (x) is a wave packet solution of the free Klein-Gordon equationThe transformation (2) that makes the ground state of the system space-time dependent, results in a breakdown of symmetry. This is essentially produced by EFA because it is this approximation that generates the solution (2) and preserves its structure even at higher order iterations according to the relation φ(x) = Σ n φ (n) (x) = Σ n e −iΦG φ (n−1) . For simplicity we choose a plane wave for φ 0 . We also writeΦ G (x)φ 0 (x) ≡ Φ G (x)φ 0 (x) wh...

Long-range order in gravity

“…in agreement with the equation of geodesic deviation [32,33]. From ( 4) and φ 0 one can derive the important relation…”

“…Consider scalar particles first. It has already been shown [32,33] that the covariant Klein-Gordon (KG) equation gives rise to classical objects that have a vortical structure. This result can be rapidly derived.…”

“…x dz λ K λ (z, x) and is obviously classical. By differentiating (6) with respect to z α , one finds [33] Fµλ (z,…”

Gravitational Qubits

“…In what follows, use is made of the external field approximation [1,2] that treats gravity as a classical theory when it interacts with quantum particles. This approximation can be applied successfully to all those problems involving gravitational sources of weak to intermediate strength for which the full-fledged use of general relativity is not required [3][4][5][6][7][8][9][10][11], it is encountered in the solution of relativistic wave equations and takes different forms according to the statistics obeyed by the particles [2,6,[12][13][14]. The approximation can also be applied to theories in which acceleration has an upper limit [15][16][17][18][19][20][21][22][23][24] and that allow for the resolution of astrophysical and cosmological singularities in quantum gravity [25,26].…”

Condensation phenomena in gravity

“…where R αβστ is the linearized Riemann tensor [25,26]. It follows from ( 8) that Φ G is not single-valued and that, after a gauge transformation, K α satisfies the equations…”