Small scale structure of spacetime: The van Vleck determinant and equigeodesic surfaces

Abstract

It has recently been argued that if spacetime M possesses non-trivial structure at small scales, an appropriate semi-classical description of it should be based on non-local bi-tensors instead of local tensors such as the metric g ab (p). Two most relevant bi-tensors in this context are Synge's World function Ω(p, p0) and the van Vleck determinant (VVD) ∆(p, p0), as they encode the metric properties of spacetime and (de)focussing behaviour of geodesics. They also characterize the leading short distance behavior of two point functions of the d'Alembartian p 0 p.We begin by discussing the intrinsic and extrinsic geometry of equi-geodesic surfaces ΣG,p 0 ≡ {p ∈ M|Ω(p, p0) = constant} in a geodesically convex neighbourhood of an event p0, and highlight some elementary identities relating the VVD with geometry of ΣG,p 0 . As an aside, we also comment on the contribution of ΣG,p 0 to the surface term in the Einstein-Hilbert (EH) action and show that it can be written as a volume integral of ln ∆.We then proceed to study the small scale structure of spacetime in presence of a Lorentz invariant short distance cut-off ℓ0 using Ω(p, p0) and ∆(p, p0), based on some recently developed ideas. We derive a 2nd rank bi-tensor q ab (p, p0; ℓ0) = q ab [g ab , Ω, ∆] which naturally yields geodesic intervals bounded from below and reduces to g ab for Ω ≫ ℓ 2 0 /2. We present a general and mathematically rigorous analysis of short distance structure of spacetime based on (a) geometry of equi-geodesic surfaces ΣG,p 0 of g ab , (b) structure of the non local d'Alembartian p 0 p associated with q ab , and (c) properties of VVD. In particular, we prove the following: (i) The Ricci bi-scalar Ric(p, p0) of q ab is completely determined by ΣG,p 0 , the tidal tensor and first two derivatives of ∆(p, p0), and has a non-trivial classical limit (see text for details):(ii) The GHY term in EH action evaluated on equi-geodesic surfaces straddling the causal boundaries of an event p0 acquires a non-trivial structure.These results strongly suggest that the mere existence of a Lorentz invariant minimal length ℓ0 can leave unsuppressed residues independent of ℓ0 and (surprisingly) independent of many precise details of quantum gravity. For e.g., the coincidence limit of Ric(p, p0) is finite as long as the modification of distances S ℓ 0 : 2Ω → 2 Ω satisfies (i) S ℓ 0 (0) = ℓ 2 0 (the condition of minimal length), (ii) S0(x) = x, and (iii) |S ℓ 0 |/S ′2 ℓ 0 (0) < ∞. In particular, the function S ℓ 0 (x), which should eventually come from a complete framework of quantum gravity, need not admit a perturbative expansion in ℓ0.Finally, we elaborate on certain technical and conceptual aspects of our results in the context of entropy of spacetime and classical description of gravitational dynamics based on Noether charge of Diff invariance instead of the EH lagrangian.

Gravity and quantum theory: Domains of conflict and contact

“…(182), just by the simple replacement of σ 2 → (σ 2 + L 2 0 ). This just happens to be true in this case; but it turns out that this replacement does not work for the volume element √ q which actually vanishes [77,78,86] when σ → 0. As a result, each event has zero volume, but a finite area, associated with it!…”

“…We first introduce the notion of an equi-geodesic surface, which can be done either in the Euclidean sector or in the Lorentzian sector; let us work in the Euclidean sector. An equi-geodesic surface S is made of the set of all points located at the same geodesic distance σ from some specific point P , which we take to be the origin [75,[77][78][79]. We next "associate" an area element with a point P in a fairly natural way by the following limiting procedure: (i) Construct an equi-geodesic surface S around a point P at some geodesic distance σ.…”

“…The q ab (x, x ′ ) will be a bitensor, which must be singular at all events in the coincidence limit x → x ′ . It turns out that one can determine [77,78,86] the form of q ab just from the demand that the pair (q ab , S(σ 2 )) should satisfy the same relationships as (g ab , σ 2 ). Once we have determined q ab we can compute the area element ( √ h d 3 x) of an equi-geodesic surface using the qmetric.…”

Raychaudhuri equation with zero point length

“…As a consequence the classical metric g ab gets modified to an effective metric q ab (which we will call the qmetric). The qmetric provides a squared geodesic interval between two events P and p which approximates to that provided by g ab in the limit of large geodesic distances, while at the same time approaches a finite value different from zero in the coincidence limit, i.e., as p → P [3][4][5]. Note that the above approach incorporates some relics of quantum gravity irrespective of any specific theory of gravitational interaction.…”

“…Removal of these curvature singularities has remained a puzzle for decades. In this work, we will present a novel approach where formation of caustics can be avoided which possibly will lead to avoidance of curvature singularities as well [3][4][5].…”