Dawood Kothawala

Given a Lorentzian spacetime (M, g) and a non-vanishing timelike vector field u(γ) with level surfaces σ, one can construct on M a Euclidean metric g-1E= g-1+ 2u ⊗u (Hawking and Ellis 1973 The Large Scale Structure of Space-Time (Cambridge: Cambridge University Press)). Motivated by this, we consider a class of metrics ∧g-1 = g-1-⊖ (γ) u⊗u with an arbitrary function ⊖that interpolates between the Euclidean ⊖= -2) and Lorentzian (⊖= 0) regimes, separated by the codimension one hypersurface ∑0 defined by ⊖= -1. Since ∧g can not, in general, be obtained from g by a diffeomorphism, its Euclidean regime is in general different from that obtained from Wick rotation t →-it. For example, if g is the k = 0 Lorentzian de Sitter metric corresponding to Λ> 0, the Euclidean regime of ∧g is the k = 0 Euclidean anti-de Sitter space with Λ < 0. We analyze the curvature tensors associated with ∧g for arbitrary Lorentzian metrics g and timelike geodesic fields u, and show that they have interesting and remarkable mathematical structures: (i) Additional terms arise in the Euclidean regime ⊖→-2 of ∧g g. (ii) For the simplest choice of a step-profile for ⊖, the Ricci scalar Ric[∧g ] of ∧g reduces, in the Lorentzian regime ⊖→ 0, to the complete EinsteinHilbert lagrangian with the correct GibbonsHawkingYork boundary term; the latter arises as a delta-function of strength 2K supported on sum;0. (iii) In the Euclidean regime ⊖→-2, Ric[∧g] also has an extra term 23R of the u-foliation. We highlight similar foliation dependent terms in the full Riemann tensor. We present some explicit examples for FLRW spacetimes in standard foliation and spherically symmetric spacetimes in the Painleve-Gullstrand foliation. We briefly discuss implications of the results for Euclidean quantum gravity and quantum cosmology.

We give a local description of the Euclidean regime (M,g,u) of Lorentzian spacetimes (M,g) based on timelike geodesics u passing through an arbitrary event p0M. We show that, to leading order, the Euclidean Einstein-Hilbert action IE is proportional to the Einstein tensor G[g](u,u). The positivity of IE follows if G[g](u,u)>0 holds. We suggest an interpretation of this result in terms of the amplitude A[Σ0]=exp[-IE] for a single spacelike hypersurface Σ0I+(p0) to emerge at a constant geodesic distance λ0 from p0. Implications for classical and quantum gravity are discussed.

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) reparameterization 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.

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.

I point out a simple expression for the 'Hubble' parameter defined by Borde, Guth and Vilenkin in their proof of past incompleteness of inflationary spacetimes. I show that the parameter which an observer O with four-velocity v will associate with a congruence u is equal to the fractional rate of change of the magnitude ξ of the Jacobi field ξ associated with u, measured along the points of intersection of O with u, with its direction determined by v . I then analyse the time dependence of 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.

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

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

There is a considerable amount of evidence to suggest that the field equations of gravity have the same status as, say, the equations describing an emergent phenomenon like elasticity. In fact, it is possible to derive the field equations from a thermodynamic variational principle in which a set of normalized vector fields are varied rather than the metric. We show that this variational principle can arise as a low-energy [LP=(G/c3)1/2→0] relic of a plausible nonperturbative effect of quantum gravity, viz. the existence of a zero-point length in the spacetime. Our result is nonperturbative in the following sense: If we modify the geodesic distance in a spacetime by introducing a zero-point length, to incorporate some effects of quantum gravity, and take the limit LP→0 of the Ricci scalar of the modified metric, we end up getting a nontrivial, leading order (LP-independent) term. This term is identical to the expression for entropy density of spacetime used previously in the emergent gravity approach. This reconfirms the idea that the microscopic degrees of freedom of the spacetime, when properly described in the full theory, could lead to an effective description of geometry in terms of a thermodynamic variational principle. This is conceptually similar to the emergence of thermodynamics from the mechanics of, say, molecules. The approach also has important implications for the cosmological constant which are briefly discussed.

It has recently been argued that if spacetime M possesses nontrivial structure at small scales, an appropriate semiclassical description of it should be based on nonlocal bitensors instead of local tensors such as the metric gab(p). Two most relevant bitensors in this context are Synge's world function Ω(p,p0) and the van Vleck determinant (VVD) Δ(p,p0), as they encode the metric properties of spacetime and (de)focusing behavior of geodesics. They also characterize the leading short distance behavior of two point functions of the d'Alembartian □p0p. We begin by discussing the intrinsic and extrinsic geometry of equigeodesic surfaces ΣG,p0≡{pM|Ω(p,p0)=constant} in a geodesically convex neighborhood of an event p0 and highlight some elementary identities relating the VVD with geometry of ΣG,p0. As an aside, we also comment on the contribution of ΣG,p0 to the surface term in the Einstein-Hilbert (EH) action and show that it can be written as a volume integral of □lnΔ. We then proceed to study the small scale structure of spacetime in presence of a Lorentz invariant short distance cutoff ℓ0 using Ω(p,p0) and Δ(p,p0), based on some recently developed ideas. We derive a second rank bitensor qab(p,p0;ℓ0)=qab[gab,Ω,Δ] which naturally yields geodesic intervals bounded from below and reduces to gab for Ωℓ02/2. We present a general and mathematically rigorous analysis of short distance structure of spacetime based on (a) geometry of equigeodesic surfaces ΣG,p0 of gab, (b) structure of the nonlocal d'Alembartian □p0p associated with qab, and (c) properties of VVD. In particular, we prove the following: (i) The Ricci biscalar Ric(p,p0) of qab is completely determined by ΣG,p0, the tidal tensor and first two derivatives of Δ(p,p0), and has a nontrivial classical limit (see text for details): limℓ0→0limΩ→0±Ric(p,p0)=±DRabqaqb (ii) The GHY term in EH action evaluated on equigeodesic surfaces straddling the causal boundaries of an event p0 acquires a nontrivial structure. These results strongly suggest that the mere existence of a Lorentz invariant minimal length ℓ0 can leave unsuppressed residues independent of ℓ0 and (surprisingly) independent of many precise details of quantum gravity. For example, the coincidence limit of Ric(p,p0) is finite as long as the modification of distances Sℓ0:2Ω2Ω satisfies (i) Sℓ0(0)=ℓ02 (the condition of minimal length), (ii) S0(x)=x, and (iii) [|Sℓ0|/Sℓ0′2](0)<∞. In particular, the function Sℓ0(x), which should eventually come from a complete framework of quantum gravity, need not admit a perturbative expansion in ℓ0. Finally, we elaborate on certain technical and conceptual aspects of our results in the context of entropy of spacetime and classical description of gravitational dynamics based on Noether charge of diffeomorphism invariance instead of the EH lagrangian.

It is generally believed that any quantum theory of gravity should have a generic feature—a quantum of length. We provide a physical ansatz to obtain an effective non-local metric tensor starting from the standard metric tensor such that the spacetime acquires a zero-point-length (Formula presented.) of the order of the Planck length (Formula presented.). This prescription leads to several remarkable consequences. In particular, the Euclidean volume (Formula presented.) in a D-dimensional spacetime of a region of size (Formula presented.) scales as (Formula presented.) when (Formula presented.) , while it reduces to the standard result (Formula presented.) at large scales ((Formula presented.)). The appropriately defined effective dimension, (Formula presented.) , decreases continuously from (Formula presented.) (at (Formula presented.)) to (Formula presented.) (at (Formula presented.)). This suggests that the physical spacetime becomes essentially 2-dimensional near Planck scale.