The Raychaudhuri equation for a geodesic congruence in the presence of a zero-point length has been investigated. This is directly related to the small-scale structure of spacetime and possibly captures some quantum gravity effects. The existence of such a minimum distance between spacetime events modifies the associated metric structure and hence the expansion as well as its rate of change deviates from standard expectations. This holds true for any kind of geodesic congruences, including time-like and null geodesics. Interestingly, this construction works with generic spacetime geometry without any need of invoking any particular symmetry. In particular, inclusion of a zero-point length results into a non-vanishing cross-sectional area for the geodesic congruences even in the coincidence limit, thus avoiding formation of caustics. This will have implications for both time-like and null geodesic congruences, which may lead to avoidance of singularity formation in the quantum spacetime. © 2019 The Authors

Dawood Kothawala

Department Of Physics

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IIT Madras

Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics

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. © 2016, Springer Science+Business Media New York.

General Relativity and Gravitation

Spacetime with zero point length is two-dimensional at the Planck scale

It has recently been argued that if spacetime M possesses nontrivial structure at small scales, an appropriate semiclassical description of it should be based on nonlocal bitensors instead of local tensors such as the metric gab(p). Two most relevant bitensors in this context are Synge's world function Ω(p,p0) and the van Vleck determinant (VVD) Δ(p,p0), as they encode the metric properties of spacetime and (de)focusing behavior of geodesics. They also characterize the leading short distance behavior of two point functions of the d'Alembartian □p0p. We begin by discussing the intrinsic and extrinsic geometry of equigeodesic surfaces ΣG,p0≡{pM|Ω(p,p0)=constant} in a geodesically convex neighborhood of an event p0 and highlight some elementary identities relating the VVD with geometry of ΣG,p0. As an aside, we also comment on the contribution of ΣG,p0 to the surface term in the Einstein-Hilbert (EH) action and show that it can be written as a volume integral of □lnΔ. We then proceed to study the small scale structure of spacetime in presence of a Lorentz invariant short distance cutoff ℓ0 using Ω(p,p0) and Δ(p,p0), based on some recently developed ideas. We derive a second rank bitensor qab(p,p0;ℓ0)=qab[gab,Ω,Δ] which naturally yields geodesic intervals bounded from below and reduces to gab for Ωℓ02/2. We present a general and mathematically rigorous analysis of short distance structure of spacetime based on (a) geometry of equigeodesic surfaces ΣG,p0 of gab, (b) structure of the nonlocal d'Alembartian □p0p associated with qab, and (c) properties of VVD. In particular, we prove the following: (i) The Ricci biscalar Ric(p,p0) of qab is completely determined by ΣG,p0, the tidal tensor and first two derivatives of Δ(p,p0), and has a nontrivial classical limit (see text for details): limℓ0→0limΩ→0±Ric(p,p0)=±DRabqaqb (ii) The GHY term in EH action evaluated on equigeodesic surfaces straddling the causal boundaries of an event p0 acquires a nontrivial 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 example, the coincidence limit of Ric(p,p0) is finite as long as the modification of distances Sℓ0:2Ω2Ω satisfies (i) Sℓ0(0)=ℓ02 (the condition of minimal length), (ii) S0(x)=x, and (iii) [|Sℓ0|/Sℓ0′2](0)<∞. In particular, the function Sℓ0(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 diffeomorphism invariance instead of the EH lagrangian. © 2015 American Physical Society.

Physical Review D - Particles, Fields, Gravitation and Cosmology

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

We consider metrics related to each other by functionals of a scalar field (Formula presented) and it’s gradient (Formula presented), and give transformations of some key geometric quantities associated with such metrics. Our analysis provides useful and elegant geometric insights into the roles of conformal and non-conformal metric deformations in terms of intrinsic and extrinsic geometry of (Formula presented)-foliations. As a special case, we compare conformal and disformal transforms to highlight some non-trivial scaling differences. We also study the geometry of equi-geodesic surfaces formed by points (Formula presented) at constant geodesic distance (Formula presented) from a fixed point (Formula presented), and apply our results to a specific disformal geometry based on (Formula presented) which was recently shown to arise in the context of spacetime with a minimal length. © 2014, Springer Science+Business Media New York.

Intrinsic and extrinsic curvatures in Finsler esque spaces

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. © 2014 American Physical Society.

Entropy density of spacetime as a relic from quantum gravity

Many generic arguments support the existence of a minimum spacetime interval L0. Such a "zero-point" length can be naturally introduced in a locally Lorentz invariant manner via Synge's world function biscalar Ω(p,P) which measures squared geodesic interval between spacetime events p and P. I show that there exists a nonlocal deformation of spacetime geometry given by a disformal coupling of metric to the biscalar Ω(p,P), which yields a geodesic interval of L0 in the limit p→P. Locality is recovered when Ω(p,P) L02/2. I discuss several conceptual implications of the resultant small-scale structure of spacetime for QFT propagators as well as spacetime singularities. © 2013 American Physical Society.

Journal	Data powered by TypesetPhysics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics
Publisher	Data powered by TypesetElsevier B.V.
ISSN	03702693
Open Access	No