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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: 7 pages, 2 figures

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

It is possible to obtain the gravitational field equations in a large class of theories from a thermodynamic variational principle which uses the gravitational heat density $\mathcal{S}_g$ associated with null surfaces. This heat density is related to the discreteness of spacetime at Planck scale, $L_P^2 = (G\hbar / c^3)$, which assigns $A_{\perp}/L_P^2$ degrees of freedom to any area $A_{\perp}$. On the other hand, it is also known that the surface term $K\sqrt{h}$ in the gravitational action principle correctly reproduces the heat density of the null surfaces. We provide a link between these ideas by obtaining $\mathcal{S}_g$, used in emergent gravity paradigm, from the surface term in the action in Einstein's gravity. This is done using the notion of a nonlocal qmetric -- introduced recently [arXiv:1307.5618, arXiv:1405.4967] -- which allows us to study the effects of zero-point-length of spacetime at the transition scale between quantum and classical gravity. Computing $K\sqrt{h}$ for the qmetric in the appropriate limit directly reproduces the entropy density $\mathcal{S}_g$ used in the emergent gravity paradigm., Comment: 8 pages; no figures

We consider metrics related to each other by functionals of a scalar field $\varphi(x)$ and it's gradient $\nabla \varphi(x)$, and give transformations of some key geometric quantities associated with such metrics. Our analysis provides useful and elegant geometric insights into the roles of {\it conformal} and {\it non-conformal} metric deformations in terms of intrinsic and extrinsic geometry of $\varphi$-foliations. As a special case, we compare {\it conformal} and {\it disformal} transforms to highlight some non-trivial scaling differences. We also study the geometry of {\it equi-geodesic} surfaces formed by points $p$ at constant geodesic distance $\sigma(p,P)$ from a fixed point $P$, and apply our results to a specific disformal geometry based on $\sigma(p,P)$ which was recently shown to arise in the context of spacetime with a minimal length., Comment: 8 pages, 1 figure; Sec IV now includes a more refined expression for Ricci scalar completely in terms of conformal (3-)geometry, and discusses a couple of applications; to appear in Gen. Rel. Grav._

There is considerable 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 ($L_P = (G\hbar/c^3)^{1/2} \to 0$) 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 $L_P \to 0$ of the Ricci scalar of the modified metric, we end up getting a nontrivial, leading order ($L_P$ - independent) term. \textit{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 mechanics of, say, molecules. The approach also has important implications for cosmological constant which are briefly discussed., Comment: v3: minor corrections; matches version accepted in Phys. Rev. D

Many generic arguments support the existence of a minimum spacetime interval $L_0$. Such a "zero-point" length can be naturally introduced in a locally Lorentz invariant manner via Synge's world function bi-scalar $\Omega(p,P)$ which measures squared geodesic interval between spacetime events $p$ and $P$. I show that there exists a \emph{non-local} deformation of spacetime geometry given by a \emph{disformal} coupling of metric to the bi-scalar $\Omega(p,P)$, which yields a geodesic interval of $L_0$ in the limit $p \rightarrow P$. Locality is recovered when $\Omega(p,P) >> L_0^2/2$. I discuss several conceptual implications of the resultant small-scale structure of spacetime for QFT propagators as well as spacetime singularities., Comment: v3: 6 pages; added details and a discussion on the relationship between minimal length and maximal acceleration; final version to appear in Phys. Rev. D

Lanczos-Lovelock models of gravity represent a natural and elegant generalization of Einstein's theory of gravity to higher dimensions. They are characterized by the fact that the field equations only contain up to second derivatives of the metric even though the action functional can be a quadratic or higher degree polynomial in the curvature tensor. Because these models share several key properties of Einstein's theory they serve as a useful set of candidate models for testing the emergent paradigm for gravity. This review highlights several geometrical and thermodynamical aspects of Lanczos-Lovelock models which have attracted recent attention., Comment: 95 pages; final version published in Physics Reports

Quantum mechanics with a generalized uncertainty principle arises through a representation of the commutator $[\hat{x}, \hat{p}] = i f(\hat{p})$. We apply this deformed quantization to free scalar field theory for $f_\pm =1\pm \beta p^2$. The resulting quantum field theories have a rich fine scale structure. For small wavelength modes, the Green's function for $f_+$ exhibits a remarkable transition from Lorentz to Galilean invariance, whereas for $f_-$ such modes effectively do not propagate. For both cases Lorentz invariance is recovered at long wavelengths., Comment: 5 pages, 5 figures

We show that for asymptotically anti-deSitter backgrounds with negative energy, such as the AdS soliton and regulated negative mass AdS-Schwarzshild metrics, the Wilson loop expectation value in the AdS/CFT conjecture exhibits a Coulomb to confinement transition. We also show that the quark-antiquark ($q \bar q$) potential can be interpreted as affine time along null geodesics on the minimal string world sheet,and that its intrinsic curvature provides a signature of transition to confinement phase. The result demonstrates a UV/IR relation in that the boundary separation of the $q \bar{q}$ pair exhibits an inverse relationship with the radial descent of the world sheet into the bulk. Our results suggest a generic (holographic) relationship between confinement in gauge theory and repulsive gravity, which in turn is connected with singularity avoidance in quantum gravity., Comment: 8 pages, 4 figures

We study the membrane paradigm for horizons in Lanczos-Lovelock models of gravity in arbitrary D dimensions and find compact expressions for the pressure p and viscosity coefficients \eta and \zeta of the membrane fluid. We show that the membrane pressure is intimately connected with the Noether charge entropy S_Wald of the horizon when we consider a specific m-th order Lanczos-Lovelock model, through the relation pA/T=(D-2m)/(D-2)S_Wald, where T is the temperature and A is the area of the horizon. Similarly, the viscosity coefficients are expressible in terms of entropy and quasi-local energy associated with the horizons. The bulk and shear viscosity coefficients are found to obey the relation \zeta=-2(D-3)/(D-2)\eta., Comment: v1: 13 pages, no figure. (v2): refs added, typos corrected, new subsection added on the ratio \eta/s. (v3): some clarification added, typos corrected, to appear in JHEP

We consider two specific approaches to evaluate the black hole entropy which are known to produce correct results in the case of Einstein's theory and generalize them to Lanczos-Lovelock models. In the first approach (which could be called extrinsic) we use a procedure motivated by earlier work by Pretorius, Vollick and Israel, and by Oppenheim, and evaluate the entropy of a configuration of densely packed gravitating shells on the verge of forming a black hole in Lanczos-Lovelock theories of gravity. We find that this matter entropy is not equal to (it is less than) Wald entropy, except in the case of Einstein theory, where they are equal. The matter entropy is proportional to the Wald entropy if we consider a specific m-th order Lanczos-Lovelock model, with the proportionality constant depending on the spacetime dimensions D and the order m of the Lanczos-Lovelock theory as (D-2m)/(D-2). Since the proportionality constant depends on m, the proportionality between matter entropy and Wald entropy breaks down when we consider a sum of Lanczos-Lovelock actions involving different m. In the second approach (which could be called intrinsic) we generalize a procedure, previously introduced by Padmanabhan in the context of GR, to study off-shell entropy of a class of metrics with horizon using a path integral method. We consider the Euclidean action of Lanczos-Lovelock models for a class of metrics off-shell and interpret it as a partition function. We show that in the case of spherically symmetric metrics, one can interpret the Euclidean action as the free energy and read off both the entropy and energy of a black hole spacetime. Surprisingly enough, this leads to exactly the Wald entropy and the energy of the spacetime in Lanczos-Lovelock models obtained by other methods. We comment on possible implications of the result., Comment: v1: 20 pages, no figures v2: added some discussion, to appear in Phys. Rev. D

We study the \textit{quantum} partition function of non-relativistic, ideal gas in a (non-cubical) box falling freely in arbitrary curved spacetime with centre 4-velocity u^a. When perturbed energy eigenvalues are properly taken into account, we find that corrections to various thermodynamic quantities include a very specific, sub-dominant term which is independent of \textit{kinematic} details such as box dimensions and mass of particles. This term is characterized by the dimensionless quantity, \Xi=R_00 \Lambda^2, where R_00=R_ab u^a u^b and \Lambda=\beta \hbar c, and, quite intriguingly, produces Euler relation of homogeneity two between entropy and energy -- a relation familiar from black hole thermodynamics., Comment: 5 pages, no figures; abstract abridged and an appendix added outlining some relevant mathematical steps; accepted in Phys. Lett. B

We analyze the generic structure of Einstein tensor projected onto a 2-D spacelike surface S defined by unit timelike and spacelike vectors u_i and n_i respectively, which describe an accelerated observer (see text). Assuming that flow along u_i defines an approximate Killing vector X_i, we then show that near the corresponding Rindler horizon, the flux j_a=G_ab X^b along the ingoing null geodesics k_i normalised to have unit Killing energy, given by j . k, has a natural thermodynamic interpretation. Moreover, change in cross-sectional area of the k_i congruence yields the required change in area of S under virtual displacements \emph{normal} to it. The main aim of this note is to clearly demonstrate how, and why, the content of Einstein equations under such horizon deformations, originally pointed out by Padmanabhan, is essentially different from the result of Jacobson, who employed the so called Clausius relation in an attempt to derive Einstein equations from such a Clausius relation. More specifically, we show how a \emph{very specific geometric term} [reminiscent of Hawking's quasi-local expression for energy of spheres] corresponding to change in \emph{gravitational energy} arises inevitably in the first law: dE_G/d{\lambda} \alpha \int_{H} dA R_(2) (see text) -- the contribution of this purely geometric term would be missed in attempts to obtain area (and hence entropy) change by integrating the Raychaudhuri equation., Comment: added comments and references; matches final version accepted in Phys. Rev. D

As was first noted by Isaac Newton, the two most famous ellipses of classical mechanics, arising out of the force laws F~r and F~1/r^2, can be mapped onto each other by changing the location of center-of-force. What is perhaps less well known is that this mapping can also be achieved by the complex transformation, z -> z^2. We give a simple derivation of this result (and its generalization) by writing the Gaussian curvature in its "covariant" form, and then changing the \emph{metric} by a conformal transformation which "mimics" this mapping of the curves. The final result also yields a relationship between Newton's constant G, mass M of the central attracting body in Newton's law, the energy E of the Hooke's law orbit, and the angular momenta of the two orbits. We also indicate how the conserved Laplace-Runge-Lenz vector for the 1/r^2 force law transforms under this transformation, and compare it with the corresponding quantities for the linear force law. Our main aim is to present this duality in a geometric fashion, by introducing elementary notions from differential geometry., Comment: 8 pages, 1 figure; typos fixed, references added, matches version accepted in Am. J. Phys

The existence of a minimal and fundamental length scale, say, the Planck length, is a characteristic feature of almost all the models of quantum gravity. The presence of the fundamental length is expected to lead to an improved ultra-violet behavior of the semi-classical propagators. The hypothesis of path integral duality provides a prescription to evaluate the modified propagator of a free, quantum scalar field in a given spacetime, taking into account the existence of the fundamental length in a locally Lorentz invariant manner. We use this prescription to compute the quantum gravitational modifications to the propagators in spacetimes with constant curvature, and show that: (i) the modified propagators are ultra-violet finite, and (ii) the modifications are non-perturbative in the Planck length. We discuss the implications of our results., Comment: 3 Pages, To appear in the proceedings of the 12th Marcel Grossman Meeting

It is well known that a detector, coupled linearly to a quantum field and accelerating through the inertial vacuum with a constant acceleration $g$, will behave as though it is immersed in a radiation field with temperature $T=(g/2\pi)$. We study a generalization of this result for detectors moving with a time-dependent acceleration $g(\tau)$ along a given direction. After defining the rate of excitation of the detector appropriately, we evaluate this rate for time-dependent acceleration, $g(\tau)$, to linear order in the parameter $\eta = \dot g / g^2$. In this case, we have three length scales in the problem: $g^{-1}, (\dot g/g)^{-1}$ and $\omega^{-1}$ where $\omega$ is the energy difference between the two levels of the detector at which the spectrum is probed. We show that: (a) When $\omega^{-1} \ll g^{-1} \ll (\dot g/g)^{-1}$, the rate of transition of the detector corresponds to a slowly varying temperature $T(\tau) = g(\tau)/2 \pi $, as one would have expected. (b) However, when $ g^{-1}\ll \omega^{-1} \ll (\dot g/g)^{-1}$, we find that the spectrum is modified \textit{even at the order $\mathcal{O}(\eta)$}. This is counter-intuitive because, in this case, the relevant frequency does not probe the rate of change of the acceleration since $(\dot g/g) \ll \omega$ and we certainly do not have deviation from the thermal spectrum when $\dot g =0$. This result shows that there is a subtle discontinuity in the behaviour of detectors with $\dot g = 0$ and $\dot g/g^2$ being arbitrarily small. We corroborate this result by evaluating the detector response for a particular trajectory which admits an analytic expression for the poles of the Wightman function., Comment: v1, 7 pages, no figures; v2, an Acknowledgment and some clarifying comments added, matches version accepted for publication in Physics Letters B

The hypothesis of path integral duality provides a prescription to evaluate the propagator of a free, quantum scalar field in a given classical background, taking into account the existence of a fundamental length, say, the Planck length, $\lp$, in a {\it locally Lorentz invariant manner}. We use this prescription to evaluate the duality modified propagators in spacetimes with {\it constant curvature} (exactly in the case of one spacetime, and in the Gaussian approximation for another two), and show that: (i) the modified propagators are ultra violet finite, (ii) the modifications are {\it non-perturbative} in $\lp$, and (iii) $\lp$ seems to behave like a `zero point length' of spacetime intervals such that $\l< \sigma^2(x,x')\r> = \l[\sigma^{2}(x,x')+ {\cal O}(1) \lp^2 \r]$, where $\sigma(x,x')$ is the geodesic distance between the two spacetime points $x$ and $x'$, and the angular brackets denote (a suitable) average over the quantum gravitational fluctuations. We briefly discuss the implications of our results., Comment: v1. 10 pages, no figures; v2. 11 pages, acknowledgments added

It is well known that, for a wide class of spacetimes with horizons, Einstein equations near the horizon can be written as a thermodynamic identity. It is also known that the Einstein tensor acquires a highly symmetric form near static, as well as stationary, horizons. We show that, for generic static spacetimes, this highly symmetric form of the Einstein tensor leads quite naturally and generically to the interpretation of the near-horizon field equations as a thermodynamic identity. We further extend this result to generic static spacetimes in Lanczos-Lovelock gravity, and show that the near-horizon field equations again represent a thermodynamic identity in all these models. These results confirm the conjecture that this thermodynamic perspective of gravity extends far beyond Einstein's theory., Comment: RevTeX 4; 10 pages; no figures

In Einstein's gravity, the entropy of horizons is proportional to their area. Several arguments given in the literature suggest that, in this context, both area and entropy should be quantized with an equally spaced spectrum for large quantum numbers. But in more general theories (like, for e.g, in the black hole solutions of Gauss-Bonnet or Lanczos-Lovelock gravity) the horizon entropy is \emph{not} proportional to area and the question arises as to which of the two (if at all) will have this property. We give a general argument that in all Lanczos-Lovelock theories of gravity, it is the \emph{entropy} that has equally spaced spectrum. In the case of Gauss-Bonnet gravity, we use the asymptotic form of quasi normal mode frequencies to explicitly demonstrate this result. Hence, the concept of a quantum of area in Einstein Hilbert (EH) gravity needs to be replaced by a concept of \emph{quantum of entropy} in a more general context., Comment: RevTeX 4; 4 pages; no figures

Modes emerging out of a collapsing black hole are red-shifted to such an extent that Hawking radiation at future null infinity consists of modes that have energies beyond the Planck scale at past null infinity. This indicates that physics at the Planck scale may modify the spectrum of Hawking radiation and the associated stress-energy tensor of the quantum field. Recently, it has been shown that, the T-duality symmetry of string fluctuations along compact extra dimensions leads to a modification of the standard propagator of point particles in quantum field theory. At low energies (when compared to the string scale), the modified propagator is found to behave as though the spacetime possesses a minimal length, say, $\lp$, which we shall assume to be of the order of the Planck length. We utilize the duality approach to evaluate the modified propagator around the rotating Banados-Teitelboim-Zanelli black hole and show that the propagator is finite in the coincident limit. We compute the stress-energy tensor associated with the modified Green's function and illustrate graphically that the quantum gravitational corrections turn out to be negligibly small. We conclude by briefly commenting on the results we have obtained., Comment: v1. 7 pages, 2 figures; v2. 11 pages, 4 figures, discussion extended to the case of the rotating BTZ black hole, figures improved

We present a derivation of Hawking radiation through tunneling mechanism for a general class of asymptotically flat, spherically symmetric spacetimes. The tunneling rate $\Gamma \sim \exp{(\Delta S)}$ arises as a consequence of the first law of thermodynamics, TdS=dE + PdV. Therefore, this approach demonstrates how tunneling is intimately connected with the first law of thermodynamics through the principle of conservation of energy. The analysis is also generally applicable to any reasonable theory of gravity so long as the first law of thermodynamics for horizons holds in the form, TdS=dE + PdV., Comment: RevTeX 4; 11 pages; no figures

There is an intriguing analogy between the gravitational dynamics of the horizons and thermodynamics. In case of general relativity, as well as for a wider class of Lanczos-Lovelock theories of gravity, it is possible to interpret the field equations near any spherically symmetric horizon as a thermodynamic identity TdS = dE + PdV. We study this approach further and generalize the results to two more generic cases within the context of general relativity: (i) stationary axis-symmetric horizons and (ii) time dependent evolving horizons. In both the cases, the near horizon structure of Einstein equations can be expressed as a thermodynamic identity under the virtual displacement of the horizon. This result demonstrates the fact that the thermodynamic interpretation of gravitational dynamics is not restricted to spherically symmetric or static horizons but is quite generic in nature and indicates a deeper connection between gravity and thermodynamics., Comment: revtex; 6 pages; no figures