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This paper explores modifications to General Relativity (GR) by considering higher-order curvature terms in the gravitational action, specifically focusing on the quadratic Ricci scalar and a particular cubic contraction of the Riemann tensor. These modifications introduce new interactions at short distances, potentially altering the dynamics of compact objects. We calculate the effective two-body binding potential energy for these modified theories to quantify these effects using the worldline effective field theory (WEFT) formalism. This approach allows us to systematically integrate out short-distance gravitational effects, capturing the modifications to the binding potential. Our results demonstrate how the quadratic Ricci scalar and cubic Riemann tensor terms contribute to the two-body interaction at the leading order, highlighting deviations from classical GR predictions. These findings provide insight into the possible observational signatures of such modified gravity theories in binary systems and other astrophysical settings., Comment: 22 pages

One version of the principle of equivalence, as originally formulated by Einstein, states that ``gravity" can be mimicked locally by going to an ``accelerated frame of reference". As highlighted by Synge, the physical content of this principle remains obscure in so far as it does not refer to the Riemann tensor $R_{abcd}$, which encodes the true effects of gravity. We here give the acceleration profile of a $Born$ rigid, Rindler$esque$, frame that can mimic a gravitational field corresponding to a given $R_{abcd}$. The generalised deviation equation that yields this result also has Centrifugal and Coriolis terms appearing in a purely relational context, yielding a similar connection between angular velocity of rotating, rigid inertial frames and the Riemann tensor. We comment briefly on implications for Mach principle., Comment: v1: 7 pages, 2 figures; v2: version accepted in PRD Letters

The workshop was organized at IUCAA on Oct 30 - Nov 3, 2023 as a compact discussion and discourse meeting with a threadbare exposition and discussion of the various aspects and the questions arising. It was occasioned by the visit of Professor Hakan Andreasson of the Gothenburg Technical University, Sweden. He has been exploring with his collaborators the Einstein - Vlasov system for over a decade and a half as a possible matter source for compact objects. This system characterizes itself by free particles in motion and interacting only through gravity. For a limiting compactness, this may be the most appropriate state. The main thrust of the workshop was to understand this new object, Buchdahl Star (BS), of limiting compactness without a horizon. It is almost as compact as a black hole (BH) and yet has no horizon and hence is open for interaction with the outside world. Ever since the proposal of the membrane paradigm envisaging a timelike fiducial surface near BH horizon, BS offers an excellent possibility of the existence of such a real astrophysical object. It could very well compete with BH as a mimicker for various physical and astrophysical phenomena. Thus, it opens up a new vista of study and investigation of all the questions that one asks for BH, for this new creature, BS. The workshop was intended to identify certain interesting questions as well as the people interested in studying them. On this count, the workshop has been a huge success as several interesting questions have been identified, a few groups have been formed to take up different problems, and the work has already started. Nothing more could one have asked from such an exercise. A brief summary of some of the talks is included, followed by a brief discussion of the projects identified as a result of the discussions during the workshop., Comment: 12 pages, Report of the "Workshop on the limiting compactness objects: Black holes and Buchdahl stars" at IUCAA on Oct 30 - Nov 3, 2023

We highlight some universal features concerning the role of spacetime curvature in the entanglement induced between quantum probes coupled to a quantum field in a suitable vacuum state. The probes are initially causally disconnected and non-entangled. We explore the parameter space $\{{\omega}, d_0, \boldsymbol{v}_0\}$ spanned by the energy gap $\omega$ of the detectors, and the initial values of separation distance $d_0$ and relative velocity $\boldsymbol{v}_0$, both covariantly defined in arbitrary curved spacetime. We also obtain numerical results in de Sitter spacetimes and use these to explore strong curvature regime, while also corroborating our perturbative results in arbitrary curved spacetime. Our analysis shows that curvature can induce entanglement features in certain regions of the above parameter space, in a manner which facilitates using entanglement as a probe of spacetime curvature., Comment: Published in PRD, 28 pages and 15 figures

We analyse several aspects of detectors with uniform acceleration $a$ and uniform rotation $\Omega$ in de Sitter ($\Lambda>0$) and anti-de Sitter ($\Lambda<0$) spacetimes, focusing particularly on the periodicity, in (Euclidean) proper time $\tau_{\rm traj}$, of geodesic interval $\tau_{\rm geod}$ between two events on the trajectory. For $\Lambda<0$, $\tau_{\rm geod}$ is periodic in ${\rm i} \tau_{\rm traj}$ for specific values of $a$ and $\Omega$. These results are used to obtain numerical plots for the response rate $\dot{\mathcal{F}}$ of Unruh-de Witt detectors, which display non-trivial combined effects of rotation and curvature through the dimensionless parameter $\Lambda c^2/\Omega^2$. In particular, periodicity does not imply thermality due to additional poles in the Wightman function away from the imaginary axis. We then present some results for stationary rotational motion in arbitrary curved spacetime, as a perturbative expansion in curvature., Comment: 10 pages, 3 figures, matches published version in PRD

A generic implication of incorporating gravitational effects in the analysis of quantum measurements is the existence of a zero-point length of spacetime. This requires an inherently non-local description of spacetime, beyond the usual one based on metric $g_{ab}(x)$ etc. The quantum spacetime should instead be reconstructed from non-local bi-tensors of the form $\mathscr{G}_{ab \ldots i'j' \ldots}(x,x')$. A deeper look then reveals a subtle interplay interplay between non-locality and the limit $G\hbar/c^3 \to 0$. In particular, the so called emergent gravity paradigm -- in which gravitational dynamics/action/spacetime are emergent and characterised by an *entropy functional* -- arises as the Cheshire grin of a fundamentally non-local quantum spacetime. This essay describes the flow of metric with respect to Planck length, and proposes a novel action for the same., Comment: Honorable Mention in the Gravity Research Foundation Essay Competition, 2023

All our observations that characterise space and time are expressed in terms of non-local, bi-tensorial objects such as geodesic intervals between events and two-point (Green) functions. In this contribution, I highlight the importance of characterising spacetime geometry in terms of such non-local objects, focusing particularly on two important bi-tensors that play a particular fundamental role -- Synge's World function and the van Vleck determinant. I will first discuss how these bi-tensors help capture information about spacetime geometry, and then describe their role in characterising quantum spacetime endowed with a lower bound, say $\ell_0$, on spacetime intervals. Incorporating such a length scale in a Lorentz covariant manner necessitates a description of spacetime geometry in terms of above bi-tensors, and naturally replaces the conventional description based on the metric tensor $g_{ab}(x)$ with a description in terms of a non-local bi-tensor $q_{ab}(x, y)$. The non-analytic structure of $q_{ab}(x, y)$ which renders a perturbative expansion in $\ell_0$ meaningless, also generically leaves a non-trivial ``relic" in the limit $\ell_0 \to 0$. I present some results where such a relic term is manifest; specifically, I will discuss how this: (i) suggests a description of gravitational dynamics different from the one based on Einstein-Hilbert lagrangian, (ii) implies dimensional reduction to $2$ at small scales, (iii) connects with the notion of cosmological constant itself being a non-local vestige of the small scale structure of spacetime, (iv) helps address the issues of spacetime singularities. I will conclude by discussing the ramifications of these ideas for quantum gravity., Comment: Based on the talk "Relics of the quantum spacetime: from Synge's world function as the fundamental probe of spacetime architecture to the emergent description of gravity", presented at DICE2022, Castiglioncello, Italy, Sep 19-23, 2022

There has been considerable interest over the past years in investigating the role of gravity in quantum phenomenon such as entanglement and decoherence. In particular, gravitational time dilation is believed to decohere superpositions of center of mass of composite quantum systems. Since true effects of gravity are encoded in the curvature of spacetime, the universality of such decoherence must be characterized through components of Riemann tensor $R_{abcd}$, with a clear separation from non-inertial kinematic effects. We obtain the reduced density matrix of a composite system in a generic curved spacetime and express the decoherence time scale explicitly in terms of curvature. The decoherence in an inertial frame is caused by tidal acceleration. We also analyze the effects of self-gravity and show that the coupling of gravitational interaction with external curvature can not be captured by the replacement $m \to m + H_{\rm int}/c^2$., Comment: 8 pages, 2 figures, typos fixed

We present a formulation of the generalised uncertainty principle based on commutator $\left[ {\hat x}^i, {\hat p}_j \right]$ between position and momentum operators defined in a covariant manner using normal coordinates. We show how any such commutator can acquire corrections if the momentum space is curved. The correction is completely determined by the extrinsic curvature of the surface $p^2=$ constant in the momentum space, and results in non-commutativity of normal position coordinates $\left[ {\hat x}^i, {\hat x}^j \right] \neq 0$. We then provide a construction for the momentum space geometry as a suitable four dimensional extension of a geometry conformal to the three dimensional relativistic velocity space - the Lobachevsky space - whose curvature is determined by the dispersion relation $F(p^2)=-m^2$, with $F(x)=x$ yielding the standard Heisenberg algebra., Comment: 6 pages, 3 figures, published as a Letter in Physical Review D

Thanu Padmanabhan was a renowned Indian theoretical physicist known for his research in general relativity, cosmology, and quantum gravity. In an extraordinary career spanning forty-two years, he published more than three hundred research articles, wrote ten highly successful technical and popular books, and mentored nearly thirty graduate students and post-doctoral fellows. He is best known for his deep work investigating gravitation as an emergent thermodynamic phenomenon. He was an outstanding teacher, and an indefatigable populariser of science, who travelled very widely to motivate and inspire young students. Paddy, as he was affectionately known, was also a close friend to his students and collaborators, treating them as part of his extended academic family. On September 17, 2021 Paddy passed away very unexpectedly, at the age of sixty-four and at the height of his research career, while serving as a Distinguished Professor at the Inter-University Centre for Astronomy and Astrophysics, Pune. His untimely demise has come as a shock to his family and friends and colleagues. In this article, several of them have come together to pay their tributes and share their fond memories of Paddy., Comment: In Memoriam of Prof. T. Padmanabhan (1957-2021)

We obtain a remarkable semi-analytic expression concerning the role of purely tidal curvature on accelerated probes, revealing some novel insights into the role of absolute vs. tidal acceleration in the response of such probes. The key quantity we evaluate is the relation between geodesic ($\tau_{\rm geod}$) and proper time ($\tau_{\rm acc}$) intervals between events on the probe trajectory. This is obtained as a covariant power series in curvature using a combination of analytical and numerical tools. A serendipitous observation then reveals that one can $exactly$ sum all terms involving the $purely\;tidal$ component ${\mathscr E}_n= R_{abcd} \varepsilon^{ab} \varepsilon^{cd}$ of curvature, with $\varepsilon^{ab}$ the bi-normal to the plane of motion: $$ \tau_{\rm geod} = \frac{2}{\sqrt{{-\mathscr E}_n}} \sinh ^{-1}\Biggl[\sqrt{\frac{-{\mathscr E}_n}{a^2-{\mathscr E}_n}} \sinh \left(\frac{1}{2} \sqrt{a^2-{\mathscr E}_n} \; \tau_{\rm acc} \right) \Biggl] $$ For classical clocks, the above result represents an interesting closed form contribution of tidal curvature to the differential ageing of twins in the classic $Twin\;paradox$. For quantum probes, it gives a thermal contribution to the $detector\;response$ with a modified $Unruh\;temperature$ $$ [k_{\rm B} T]_{{\mathscr E}_n} = \frac{\hbar \sqrt{a^2- {\mathscr E}_n }}{2 \pi} $$ As an operational tool, the computational framework we present and the corresponding results should find applications to a wide range of physical problems that involve measurements and observations by use of accelerated probes in curved spacetimes., Comment: 11 pages, 3 figures

We discuss the generic geometric properties of metrics $\widehat {g}_{ab}$ constructed from Lorentzian metric $g_{ab}$ and a nowhere vanishing, hypersurface orthogonal, timelike vector field $u^a$. The metric ${\widehat g}_{ab}$ has Euclidean signature in a certain domain, with the transition to Lorentzian signature occurring at some hypersurface $\Sigma$ orthogonal to $u^a$. Geometry associated with ${\widehat g}_{ab}$ has recently been shown to yield remarkable new insights for classical and quantum gravity. In this work, we prove several general results applicable in physically relevant spacetimes for congruences $u^i$ with non-zero acceleration $a^i$. We present as examples the cases of dynamical spherically symmetric spacetimes and spacetimes with maximal symmetry. We also investigate this formalism within the context of thermal effects in curved spacetimes with horizons. Specifically, we discuss: (i) the Holonomy of loops lying partially or wholly in the Euclidean regime. We show that the contribution of the Euclidean domain to holonomy is completely determined by extrinsic curvature $K_{ab}$ of $\Sigma$ and acceleration $a^i$. (ii) We also compute entropy using this formalism for simple field theories and obtain foliation dependent corrections for the Lanczos-Lovelock gravity, Bekenstein-Hawking entropy relation in four spacetime dimensions., Comment: 22 pages, 1 figure

The standard way of deriving Euler-Lagrange (EL) equations given a point particle action is to vary the trajectory and set the first variation of the action to zero. However, if the action is (i) reparameterisation invariant, and (ii) generally covariant, I show that one may derive the EL equations by suitably "nullifying" the variation through a judicious coordinate transformation. The net result of this is that the curve remains fixed, while all other geometrical objects in the action undergo a change, given precisely by the Lie derivatives along the variation vector field. This, then, is the most direct and transparent way to elucidate the connection between general covariance, diffeomorphism invariance, and Lie derivatives, without referring to covariant derivative. I highlight the geometric underpinnings and generality of above ideas by applying them to simplest of field theories, keeping the discussion at a level easily accessible to advanced undergraduates. As non-trivial applications of these ideas, I (i) derive the Geodesic Deviation Equation using first order diffeomorphisms, and (ii) demonstrate how they can highlight the connection between canonical and metric stress-energy tensors in field theories., Comment: v2: 32 pages, 4 figures, certain discussions expanded; v1: 26 pages, 4 figures

Riemann normal coordinates (RNC) at a regular event $p_0$ of a spacetime manifold $\mathcal{M}$ are constructed by imposing: (i) $g_{\textsf{ab}}|_{p_0}=\eta_{ab}$, and (ii) $\Gamma^\textsf{a}_{\phantom{\textsf a}\textsf{bc}}|_{p_0}=0$. There is, however, a third, $independent$, assumption in the definition of RNC which essentially fixes the $density$ $of$ $geodesics$ emanating from $p_0$ to its value in flat spacetime, viz.: (iii) the tangent space $\mathcal{T}_{p_0}(\mathcal{M})$ is $flat$. We relax (iii) and obtain the normal coordinates, along with the metric $g_{\textsf{ab}}$, when $\mathcal{T}_{p_0}(\mathcal{M})$ is a maximally symmetric manifold $\widetilde{\mathcal M}_{\Lambda}$ with curvature length $|\Lambda|^{-1/2}$. In general, the "rest" frame defined by these coordinates is non-inertial with an additional acceleration $\boldsymbol a = - ({\Lambda}/3) \, \boldsymbol x$ depending on the curvature of tangent space. Our geometric set-up provides a convenient probe of local physics in a universe with a cosmological constant $\Lambda$, now embedded into the local structure of spacetime as a fundamental constant associated with a curved tangent space. We discuss classical and quantum implications of the same., Comment: 13 pages, 4 figures, comments added and typos fixed, matches version accepted in Phys. Rev. D

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., Comment: Revised Version, Published, 16 pages, no figures

Locality plays a key role in the derivation of Hawking radiation. Could relaxing the locality criterion modify Hawking radiation? We consider Hawking radiation in a general class of non local quantum field theories. We prove that Hawking radiation is unmodified for these theories. Our result establishes the universality of Hawking radiation in a larger class of theories., Comment: 4 pages, no figures

I point out a simple expression for the "Hubble" parameter $\mathscr{H}$, defined by Borde, Guth and Vilenkin (BGV) in their proof of past incompleteness of inflationary spacetimes. I show that the parameter $\mathscr{H}$ which an observer $O$ with four-velocity $\bf v$ will associate with a congruence $\bf u$ is equal to the fractional rate of change of the magnitude $\xi$ of the Jacobi field associated with $\bf u$, measured along the points of intersection of $O$ with $\bf u$, with its direction determined by $\bf v$. I then analyse the time dependence of $\mathscr{H}$ and $\xi$ using the geodesic deviation equation, computing these exactly for some simple spacetimes, and perturbatively for spacetimes close to maximally symmetric ones. The perturbative solutions are used to characterise the rms fluctuations in these quantities arising due to possible fluctuations in the curvature tensor., Comment: 9 pages, 3 figures, mathematical details added in appendix, matches final published version

I give a local description of the Euclidean regime $(M, g_{ab}, u^a)$ of Lorentzian spacetimes $(M, g_{ab})$ based on timelike geodesics $u^a$ passing through an arbitrary event $p_0 \in M$. I show that, to leading order, the Euclidean Einstein-Hilbert action $I_E$ is proportional to the Einstein tensor $G_{ab}u^a u^b$ of $g_{ab}$. The positivity of $I_E$ follows if $G_{ab}u^a u^b>0$ holds. I suggest an interpretation of this result in terms of the amplitude $\mathcal{A}[\Sigma_0]=\exp[{-I_E}]$ for a single space-like hypersurface $\Sigma_0 \in I^{+}(p_0)$ to emerge at a constant geodesic distance $\lambda_0$ from $p_0$. Implications for classical and quantum gravity are discussed., Comment: 6 pages, 2 figures, added comments and expanded discussion of some implications, matches version accepted in Phys. Rev. D

Given a Lorentzian spacetime $(M, g)$ and a non-vanishing timelike vector field $u(\lambda)$ with level surfaces $\Sigma$, one can construct on $M$ a Euclidean metric $g_E^{ab} = g^{ab} + 2 u^a u^b$. Motivated by this, we consider a class of metrics $\hat{g}^{ab} = g^{ab} - \Theta(\lambda)\, u^a u^b$ with an arbitrary function $\Theta$ that interpolates between the Euclidean ($\Theta=-2$) and Lorentzian ($\Theta=0$) regimes. The Euclidean regime is in general different from that obtained from Wick rotation $t \rightarrow - i t$. For example, if $g_{ab}$ is the $k=0$ Lorentzian de Sitter metric corresponding to $\Lambda>0$, the Euclidean regime of $\hat{g}_{ab}$ is the $k=0$ Euclidean anti-de Sitter space with $\Lambda<0$. We analyze the curvature tensors associated with $\hat{g}$ for arbitrary Lorentzian metrics $g$ and timelike geodesic fields $u^a$, and show that they have interesting and remarkable mathematical structures: (i) Additional terms arise in the Euclidean regime $\Theta \to -2$ of $\hat{g}_{ab}$. (ii) For the simplest choice of a step profile for $\Theta$, the Ricci scalar Ric$[\widehat{g}]$ of $\hat{g}_{ab}$ reduces, in the Lorentzian regime $\Theta \to 0$, to the complete Einstein-Hilbert lagrangian with the correct Gibbons-Hawking-York boundary term; the latter arises as a delta-function of strength $2K$ supported on $\Sigma_0$. (iii) In the Euclidean regime $\Theta \to -2$, Ric$[\hat{g}]$ also has an extra term $2\, {}^3 R$ of the $u$-foliation. We highlight similar foliation dependent terms in the full Riemann tensor. We present some explicit examples and briefly discuss implications of the results for Euclidean quantum gravity and quantum cosmology., Comment: v2: discussions and references added; to appear in CQG Letters

The structure of quantum vacuum in presence of gravity, and the corresponding vacuum energy density $\rho_{v}$, is expected to depend on the coupling between the UV scale $\ell_\mathrm{\textsf{UV}}$ and spacetime curvature. We determine this coupling in an arbitrary freely falling frame characterised by it's geodesic tangent $U^i(\tau)$. We show that local vacuum modes within a small causal diamond based on $U^i(\tau)$, whose size is set by wavelength of the modes, generically give a contribution $\rho_{0}$ to $\rho_v$ which, to leading order, scales as: $\rho_{0} = \left( \pi \hbar c/2 \right) {\mathrm{\textsf R}} \, {\ell_\mathrm{\textsf{UV}}}^{-2}$, where the curvature term $\mathrm{\textsf R}=\alpha R_{ab} U^a U^b + \beta R$, and $(\alpha, \beta) \in \mathbb{R}$ are constants. The genericness of this result arises from the fact that, although the modes may reduce to Minkowski plane waves along $U^i(\tau)$, the stress-energy tensor $T_{ab}$, since it depends on derivatives of the modes, does not reduce to it's Minkowski value on $U^i(\tau)$. We discuss implications of our result for vacuum processes in freely falling frames, particularly in connection with certain aspects of the cosmological constant and horizon entropy., Comment: 6 pages, 1 figure

Assuming that an accelerated observer with four-velocity ${\bf u}_{\rm R}$ in a curved spacetime attributes the standard Bekenstein-Hawking entropy and Unruh temperature to his "local Rindler horizon", we show that the $\rm \it change$ in horizon area under parametric displacements of the horizon has a very specific thermodynamic structure. Specifically, it entails information about the time-time component of the Einstein tensor: $\bf G({\bf u}_{\rm R}, {\bf u}_{\rm R})$. Demanding that the result holds for all accelerated observers, this actually becomes a statement about the full Einstein tensor, $\rm \bf G$. We also present some perspectives on the free fall with four-velocity ${\bf u}_{\rm ff}$ across the horizon that leads to such a loss of entropy for an accelerated observer. Motivated by results for some simple quantum systems at finite temperature $T$, we conjecture that at high temperatures, there exists a universal, system-independent curvature correction to partition function and thermal entropy of $\rm \it any$ freely falling system, characterised by the dimensional quantity $\Delta = {\bf R({\bf u}_{\rm ff}, {\bf u}_{\rm ff})} \left(\hbar c/kT \right)^2$., Comment: Contribution to the Festschrift in honor of Professor T. Padmanabhan on the occassion of his 60th birthday

We investigate the random motion of a mirror in (1 + 1)-dimensions that is immersed in a thermal bath of massless scalar particles which are interacting with the mirror through a boundary condition. Imposing the Dirichlet or the Neumann boundary conditions on the moving mirror, we evaluate the mean radiation reaction force on the mirror and the correlation function describing the fluctuations in the force about the mean value. From the correlation function thus obtained, we explicitly establish the fluctuation-dissipation theorem governing the moving mirror. Using the fluctuation-dissipation theorem, we compute the mean-squared displacement of the mirror at finite and zero temperature. We clarify a few points concerning the various limiting behavior of the mean-squared displacement of the mirror. While we recover the standard result at finite temperature, we find that the mirror diffuses logarithmically at zero temperature, confirming similar conclusions that have been arrived at earlier in this context. We also comment on a subtlety concerning the comparison between zero temperature limit of the finite temperature result and the exact zero temperature result., Comment: v1: 14 pages, 1 figure; v2: 15 pages, 1 figure, discussions added, Phys. Rev. D version

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 $\ell _{0}$ of the order of the Planck length $L_{P}$. This prescription leads to several remarkable consequences. In particular, the Euclidean volume $V_D(\ell,\ell_{0})$ in a $D$-dimensional spacetime of a region of size $\ell $ scales as $V_D(\ell, \ell _{0}) \propto \ell_{0}^{D-2} \ell^2$ when $\ell \sim \ell_{0}$, while it reduces to the standard result $V_D(\ell,\ell_{0}) \propto \ell^D$ at large scales ($\ell \gg \ell_{0}$). The appropriately defined effective dimension, $D_{\rm eff} $, decreases continuously from $D_{\rm eff}=D$ (at $\ell \gg \ell_{0}$) to $D_{\rm eff}=2$ (at $\ell \sim \ell_{0}$). This suggests that the physical spacetime becomes essentially 2-dimensional near Planck scale., Comment: published version

It has recently been argued that if spacetime $\mathcal M$ 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 $g_{ab}$. Two most relevant bi-tensors in this context are Synge's World function $\Omega(p,p_0)$ and the van Vleck determinant (VVD) $\Delta(p,p_0)$, 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 $_{p_0} \square_p$. We begin by discussing the intrinsic and extrinsic geometry of equi-geodesic surfaces $\Sigma_{G,p_0}$ defined by $\Omega(p,p_0)=constant$ in a geodesically convex neighbourhood of an event $p_0$, and highlight some elementary identities relating the VVD with geometry of these surfaces. As an aside, we also comment on the contribution of $\Sigma_{G,p_0}$ to the surface term in the Einstein-Hilbert (EH) action and show that it can be written as a volume integral of $\square \ln \Delta$. We then study the small scale structure of spacetime in presence of a Lorentz invariant short distance cut-off $\ell_0$ using $\Omega$ and $\Delta$, based on some recently developed ideas. We derive a 2nd rank bi-tensor $q_{ab}$ which naturally yields geodesic intervals bounded from below, and present a general and mathematically rigorous analysis of short distance structure of spacetime based on (a) geometry of $\Sigma_{G,p_0}$, (b) structure of the non local d'Alembartian associated with $q_{ab}$, and (c) properties of the VVD. In particular, we show that the Ricci bi-scalar of $q_{ab}$ is completely determined by $\Sigma_{G,p_0}$, the tidal tensor, and first two derivatives of the van Vleck determinant, and has a non-trivial "classical" limit given by (constant) $R_{ab} q^a q^b$ (see text)., Comment: v1: 12 pages, 1 figure; abstract formatted (see PDF file for the complete abstract). v2: minor typos corrected, references added