arXiv subjects

Dawood Kothawala

Publications and source records attributed to Dawood Kothawala.

At least 37 records · Page 2Linked to original sources

arXiv2021

Geometric aspects of covariant Wick rotation

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 $Σ$ 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 $Σ$ 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.

gr-qc↗

arXiv2021

Varying without varying: Reparameterisations, Diffeomorphisms, General Covariance, Lie derivatives, and all that

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.

gr-qc↗

arXiv2021

BGV theorem, Geodesic deviation, and Quantum fluctuations

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 $ξ$ 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 $ξ$ 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.

gr-qc↗

arXiv2020

Normal coordinates based on curved tangent space

Riemann normal coordinates (RNC) at a regular event $p_0$ of a spacetime manifold $\mathcal{M}$ are constructed by imposing: (i) $g_{\textsf{ab}}|_{p_0}=η_{ab}$, and (ii) $Γ^\textsf{a}_{\phantom{\textsf a}\textsf{bc}}|_{p_0}=0$. 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 $\widetilde{\mathcal M}_Λ$ with curvature length $|Λ|^{-1/2}$. In general, the "rest" frame defined by these coordinates is non-inertial with an additional acceleration $\boldsymbol a = - (Λ/3) \, \boldsymbol x$ 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 $Λ$, 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.

gr-qc↗

arXiv2019

Raychaudhuri equation with zero point length

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.

gr-qc↗

arXiv2018

Hawking radiation in Non local field theories

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.

gr-qc↗

arXiv2018

Euclidean Action and the Einstein tensor

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 $\mathcal{A}[Σ_0]=\exp[{-I_E}]$ for a single space-like hypersurface $Σ_0 \in I^{+}(p_0)$ to emerge at a constant geodesic distance $λ_0$ from $p_0$. Implications for classical and quantum gravity are discussed.

gr-qc↗

arXiv2017

Action and Observer dependence in Euclidean quantum gravity

Given a Lorentzian spacetime $(M, g)$ and a non-vanishing timelike vector field $u(λ)$ with level surfaces $Σ$, 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 $\hat{g}^{ab} = g^{ab} - Θ(λ)\, u^a u^b$ with an arbitrary function $Θ$ that interpolates between the Euclidean ($Θ=-2$) and Lorentzian ($Θ=0$) 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 $Λ>0$, the Euclidean regime of $\hat{g}_{ab}$ is the $k=0$ Euclidean anti-de Sitter space with $Λ<0$. 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 $Θ\to -2$ of $\hat{g}_{ab}$. (ii) For the simplest choice of a step profile for $Θ$, the Ricci scalar Ric$[\widehat{g}]$ of $\hat{g}_{ab}$ reduces, in the Lorentzian regime $Θ\to 0$, 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 $Σ_0$. (iii) In the Euclidean regime $Θ\to -2$, 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.

gr-qc↗

arXiv2016

Vacuum energy in freely falling frames and spacetime curvature

The structure of quantum vacuum in presence of gravity, and the corresponding vacuum energy density $ρ_{v}$, 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 $U^i(τ)$. We show that local vacuum modes within a small causal diamond based on $U^i(τ)$, whose size is set by wavelength of the modes, generically give a contribution $ρ_{0}$ to $ρ_v$ which, to leading order, scales as: $ρ_{0} = \left( π\hbar c/2 \right) {\mathrm{\textsf R}} \, {\ell_\mathrm{\textsf{UV}}}^{-2}$, where the curvature term $\mathrm{\textsf R}=αR_{ab} U^a U^b + βR$, and $(α, β) \in \mathbb{R}$ are constants. The genericness of this result arises from the fact that, although the modes may reduce to Minkowski plane waves along $U^i(τ)$, the stress-energy tensor $T_{ab}$, since it depends on derivatives of the modes, does not reduce to it's Minkowski value on $U^i(τ)$. 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.

gr-qc↗

arXiv2016

Moving mirrors and the fluctuation-dissipation theorem

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.

hep-th↗

arXiv2016

Accelerated Observers, Thermal Entropy, and Spacetime Curvature

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 $Δ= {\bf R({\bf u}_{\rm ff}, {\bf u}_{\rm ff})} \left(\hbar c/kT \right)^2$.

gr-qc↗

arXiv2016

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

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.

gr-qc↗

arXiv2015

Small scale structure of spacetime: van Vleck determinant and equi-geodesic surfaces

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 $Ω(p,p_0)$ and the van Vleck determinant (VVD) $Δ(p,p_0)$, 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 $Σ_{G,p_0}$ defined by $Ω(p,p_0)=constant$ 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 $Σ_{G,p_0}$ to the surface term in the Einstein-Hilbert (EH) action and show that it can be written as a volume integral of $\square \ln Δ$. We then study the small scale structure of spacetime in presence of a Lorentz invariant short distance cut-off $\ell_0$ using $Ω$ and $Δ$, 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 $Σ_{G,p_0}$, (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 $Σ_{G,p_0}$, 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).

gr-qc↗

arXiv2014

Grin of the Cheshire cat: Entropy density of spacetime as a relic from quantum gravity

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.

gr-qc↗

arXiv2014

Intrinsic and Extrinsic curvatures in Finsler-esque spaces

We consider metrics related to each other by functionals of a scalar field $φ(x)$ and it's gradient $\nabla φ(x)$, 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 $φ$-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 $σ(p,P)$ from a fixed point $P$, and apply our results to a specific disformal geometry based on $σ(p,P)$ which was recently shown to arise in the context of spacetime with a minimal length.

gr-qc↗

arXiv2014

Entropy density of spacetime from the zero point length

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.

gr-qc↗

arXiv2013

Lanczos-Lovelock models of gravity

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.

gr-qc↗

arXiv2013

Minimal Length and Small Scale Structure of Spacetime

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 $Ω(p,P)$ 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 $Ω(p,P)$, which yields a geodesic interval of $L_0$ in the limit $p \rightarrow P$. Locality is recovered when $Ω(p,P) >> L_0^2/2$. I discuss several conceptual implications of the resultant small-scale structure of spacetime for QFT propagators as well as spacetime singularities.

gr-qc↗