# Papers

Oh, he seems like an okay person, except for being a little strange in some ways. All day he sits at his desk and scribbles, scribbles, scribbles. Then, at the end of the day, he takes the sheets of paper he’s scribbled on, scrunges them all up, and throws them in the trash can.

A list of scientific papers I have written. They mainly concern black holes, spacetimes, and applications of geometry in physics.

## Physical stress, mass, and energy for non‐relativistic spinful matter

## A Variational Principle for the Axisymmetric Stability of Rotating Relativistic Stars

## Covariant effective action for a Galilean invariant quantum Hall system

## The First Law of Black Hole Mechanics for Fields with Internal Gauge Freedom

We derive the first law of black hole mechanics for physical theories based on a local, covariant and gauge‐invariant Lagrangian where the dynamical fields transform non‐trivially under the action of some internal gauge transformations. The theories of interest include General Relativity formulated in terms of tetrads, Einstein‐Yang‐Mills theory and Einstein‐Dirac theory. Since the dynamical fields of these theories have some internal gauge freedom, we argue that there is no natural group action of diffeomorphisms of spacetime on such dynamical fields. In general, such fields cannot even be represented as smooth, globally well‐defined tensor fields on spacetime. Consequently the derivation of the first law by Iyer and Wald cannot be used directly. Nevertheless, we show how such theories can be formulated on a principal bundle and that there is a natural action of automorphisms of the bundle on the fields. These bundle automorphisms encode both spacetime diffeomorphisms and internal gauge transformations. Using this reformulation we define the Noether charge associated to an infinitesimal automorphism and the corresponding notion of stationarity and axisymmetry of the dynamical fields. We first show that we can define certain potentials and charges at the horizon of a black hole so that the potentials are constant on the bifurcate Killing horizon, giving a generalised zeroth law for bifurcate Killing horizons. We further identify the gravitational potential and perturbed charge as the temperature and perturbed entropy of the black hole which gives an explicit formula for the perturbed entropy analogous to the Wald entropy formula. We then obtain a general first law of black hole mechanics for such theories. The first law relates the perturbed Hamiltonians at spatial infinity and the horizon, and the horizon contributions take the form of a “potential times perturbed charge” term. We also comment on the ambiguities in defining a prescription for the total entropy for black holes.

## Fields and fluids on curved non‐relativistic spacetimes

We consider non-relativistic curved geometries and argue that the background structure should be generalized from that considered in previous works. In this approach the derivative operator is defined by a Galilean spin connection valued in the Lie algebra of the Galilean group. This includes the usual spin connection plus an additional “boost connection” which parameterizes the freedom in the derivative operator not fixed by torsion or metric compatibility. As an example of this approach we develop the theory of non‐relativistic dissipative fluids and find significant differences in both equations of motion and allowed transport coefficients from those found previously. Our approach also immediately generalizes to systems with independent mass and charge currents as would arise in multicomponent fluids. Along the way we also discuss how to write general locally Galilean invariant non‐relativistic actions for multiple particle species at any order in derivatives. A detailed review of the geometry and its relation to non‐relativistic limits may be found in a companion paper [arXiv:1503.02682].

## Curved non‐relativistic spacetimes, Newtonian gravitation and massive matter

There is significant recent work on coupling matter to Newton‐Cartan spacetimes with the aim of investigating certain condensed matter phenomena. To this end, one needs to have a completely general spacetime consistent with local non‐relativisitic symmetries which supports massive matter fields. In particular, one can not impose a priori restrictions on the geometric data if one wants to analyze matter response to a perturbed geometry. In this paper we construct such a Bargmann spacetime in complete generality without any prior restrictions on the fields specifying the geometry. The resulting spacetime structure includes the familiar Newton‐Cartan structure with an additional gauge field which couples to mass. We illustrate the matter coupling with a few examples. The general spacetime we construct also includes as a special case the covariant description of Newtonian gravity, which has been thoroughly investigated in previous works. We also show how our Bargmann spacetimes arise from a suitable non‐relativistic limit of Lorentzian spacetimes. In a companion paper [arXiv:1503.02680] we use this Bargmann spacetime structure to investigate the details of matter couplings, including the Noether‐Ward identities, and transport phenomena and thermodynamics of non‐relativistic fluids.

## Black Hole Instabilities and Exponential Growth

Recently, a general analysis has been given of the stability with respect to axisymmetric perturbations of stationary‐axisymmetric black holes and black branes in vacuum general relativity in arbitrary dimensions. It was shown that positivity of canonical energy on an appropriate space of perturbations is necessary and sufficient for stability. However, the notions of both “stability” and “instability” in this result are significantly weaker than one would like to obtain. In this paper, we prove that if a perturbation of the form \(£_t \delta g\)—with \(\delta g\) a solution to the linearized Einstein equation—has negative canonical energy, then that perturbation must, in fact, grow exponentially in time. The key idea is to make use of the \(t\)‐ or (\(t\)‐\(\phi\))‐reflection isometry, \(i\), of the background spacetime and decompose the initial data for perturbations into their odd and even parts under \(i\). We then write the canonical energy as \(\mathscr E = \mathscr K + \mathscr U\), where \(\mathscr K\) and \(\mathscr U\), respectively, denote the canonical energy of the odd part (kinetic energy) and even part (potential energy). One of the main results of this paper is the proof that \(\mathscr K\) is positive definite for any black hole background. We use \(\mathscr K\) to construct a Hilbert space \(\mathscr H\) on which time evolution is given in terms of a self‐adjoint operator \(\tilde {\mathcal A}\), whose spectrum includes negative values if and only if \(\mathscr U\) fails to be positive. Negative spectrum of \(\tilde{\mathcal A}\) implies exponential growth of the perturbations in \(\mathscr H\) that have nontrivial projection into the negative spectral subspace. This includes all perturbations of the form \(£_t \delta g\) with negative canonical energy. A “Rayleigh‐Ritz” type of variational principle is derived, which can be used to obtain lower bounds on the rate of exponential growth.

## On the static Lovelock black holes

We consider static spherically symmetric Lovelock black holes and generalize the dimensionally continued black holes in such a way that they asymptotically for large r go over to the d‐dimensional Schwarzschild black hole in dS/AdS spacetime. This means that the master algebraic polynomial is not degenerate but instead its derivative is degenerate. This family of solutions contains an interesting class of pure Lovelock black holes which are the Nth order Lovelock \(\Lambda\)‐vacuum solutions having the remarkable property that their thermodynamical parameters have the universal character in terms of the event horizon radius. This is in fact a characterizing property of pure Lovelock theories. We also demonstrate the universality of the asymptotic Einstein limit for the Lovelock black holes in general.

## Higher order geometric flows on three dimensional locally homogeneous spaces

We analyse second order (in Riemann curvature) geometric flows (un‐normalised) on locally homogeneous three manifolds and look for specific features through the solutions (analytic whereever possible, otherwise numerical) of the evolution equations. Several novelties appear in the context of scale factor evolution, fixed curves, phase portraits, approaches to singular metrics, isotropisation and curvature scalar evolution. The distinguishing features linked to the presence of the second order term in the flow equation are pointed out. Throughout the article, we compare the results obtained, with the corresponding results for un‐normalized Ricci flows.

## Thermodynamical universality of the Lovelock black holes

The necessary and sufficient condition for the thermodynamical universality of the static spherically symmetric Lovelock black hole is that it is the pure Lovelock \(\Lambda\)‐vacuum solution. By universality we mean the thermodynamical parameters: temperature and entropy always bear the same relationship to the horizon radius irrespective of the Lovelock order and the spacetime dimension. For instance, the entropy always goes in terms of the horizon radius as \(r_h\) and \(r_h^2\) respectively for odd and even dimensions. This universality uniquely identifies the pure Lovelock black hole with \(\Lambda\).

## On higher order geometric and renormalisation group flows

Renormalisation group flows of the bosonic nonlinear \(\sigma\)‐model are governed, perturbatively, at different orders of \(\alpha'\), by the perturbatively evaluated \(\beta\)‐functions. In regions where \(\frac{\alpha'}{R_c^2} << 1\) the flow equations at various orders in \(\alpha'\) can be thought of as *approximating* the full, non‐perturbative RG flow. On the other hand, taking a different viewpoint, we may consider the abovementioned RG flow equations as viable *geometric* flows in their own right and without any reference to the RG aspect. Looked at as purely geometric flows where higher order terms appear, we no longer have the perturbative restrictions. In this paper, we perform our analysis from both these perspectives using specific target manifolds such as \(S^2\), \(H^2\), unwarped \(S^2 \times H^2\) and simple warped products. We analyze and solve the higher order RG flow equations within the appropriate perturbative domains and find the *corrections* arising due to the inclusion of higher order terms. Such corrections, within the perturbative regime, are shown to be small and they provide an estimate of the error which arises when higher orders are ignored.

We also investigate the higher order geometric flows on the same manifolds and figure out generic features of geometric evolution, the appearance of singularities and solitons. The aim, in this context, is to demonstrate the role of the higher order terms in modifying the flow. One interesting aspect of our analysis is that, separable solutions of the higher order flow equations for simple warped spacetimes, correspond to constant curvature Anti‐de Sitter (AdS) spacetime, modulo an overall flow‐parameter dependent scale factor. The functional form of this scale factor (which we obtain) changes on the inclusion of successive higher order terms in the flow.

## Energetics of a rotating charged black hole in 5‐dimensional supergravity

We investigate the properties of the event horizon and static limit for a charged rotating black hole solution of minimal supergravity theory in \(1 + 4\) dimension. Unlike the four‐dimensional case, there are in general two rotations, and they couple to both mass and charge. This gives rise to much richer structure to ergosphere leading to energy extraction even for axial fall. Another interesting feature is that the metric in this case is sensitive to the sign of the Maxwell charge.

## Ricci flow of unwarped and warped product manifolds

We analyse Ricci flow (normalised/un‐normalised) of product manifolds — unwarped as well as warped, through a study of generic examples. First, we investigate such flows for the unwarped scenario with manifolds of the type \(\mathbb S^n\times \mathbb S^m\), \(\mathbb S^n\times \mathbb H^m\), \(\mathbb H^m\times \mathbb H^n\) and also, similar multiple products. We are able to single out generic features such as singularity formation, isotropisation at particular values of the flow parameter and evolution characteristics. Subsequently, motivated by warped braneworlds and extra dimensions, we look at Ricci flows of warped spacetimes. Here, we are able to find analytic solutions for a special case by variable separation. For others we numerically solve the equations (for both the forward and backward flow) and draw certain useful inferences about the evolution of the warp factor, the scalar curvature as well the occurence of singularities at finite values of the flow parameter. We also investigate the dependence of the singularities of the flow on the inital conditions. We expect our results to be useful in any physical/mathematical context where such product manifolds may arise.