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, field theory, and applications of geometry in physics.

Infrared finite scattering theory: Amplitudes and soft theorems

Kartik Prabhu, Gautam Satishchandran
Any non-trivial scattering with massless fields in four spacetime dimensions will generically produce an out-state with memory. Scattering with any massless fields violates the standard assumption of asymptotic completeness — that all “in” and “out” states lie in the standard (zero memory) Fock space — and therefore leads to infrared divergences in the standard \(S\)-matrix amplitudes. We define an infrared finite scattering theory valid for general quantum field theories and quantum gravity. The (infrared finite) “superscattering” map \(\$\) is defined as a map between “in” and “out” states which does not require any a priori choice of a preferred Hilbert space. We define a “generalized asymptotic completeness” which accommodates states with memory in the space of asymptotic states. We define a complete basis of improper states on any memory Fock space (called “BMS particle” states) which are eigenstates of the energy-momentum — or, more generally, the BMS supermomentum -- that generalize the usual \(n\)-particle momentum basis to account for states with memory. We then obtain infrared finite \(\$\)-amplitudes defined as matrix elements of \(\$\) in the BMS particle basis. This formulation of the scattering theory is a key step towards analyzing fine-grained details of the infrared finite scattering theory. In quantum gravity, invariance of \(\$\) under BMS supertranslations implies factorization of \(\$\)-amplitudes as the frequency of one of the BMS particles vanishes. This proves an infrared finite analog of the soft graviton theorem. Similarly, an infrared finite soft photon theorem in QED follows from invariance of \(\$\) under large gauge transformations. We comment on how one must generalize this framework to consider \(\$\)-amplitudes for theories with collinear divergences (e.g., massless QED and Yang-Mills theories).

Infrared finite scattering theory: Scattering states and representations of the BMS group

Kartik Prabhu, Gautam Satishchandran
Any non-trivial scattering with any massless fields in four spacetime dimensions will generically produce an "out" state with memory which gives rise to infrared divergences in the standard \(S\)-matrix. To obtain an infrared-finite scattering theory, one must suitably include states with memory. However, except in the case of QED with massive charged particles, asymptotic states with memory that have finite energy and angular momentum have not been constructed for more general theories (e.g. massless QED, Yang-Mills and quantum gravity). To this end, we construct direct-integral representations over the "Lorentz orbit" of a given memory and classify all "orbit space representations" that have well-defined energy and angular momentum. We thereby provide an explicit construction of a large supply of physical states with memory as well as the explicit action of the BMS charges all states. The construction of such states is a key step toward the formulation of an infrared-finite scattering theory. While we primarily focus on the quantum gravitational case, we outline how the methods presented in this paper can be applied to obtain representations of the Poincaré group with memory for more general quantum field theories.

BMS-supertranslation charges at the critical sets of null infinity

Mariem Magdy Ali Mohamed, Kartik Prabhu, Juan A. Valiente Kroon
For asymptotically flat spacetimes, a conjecture by Strominger states that asymptotic BMS-supertranslations and their associated charges at past null infinity \(\mathscr I^-\) can be related to those at future null infinity \(\mathscr I^+\) via an antipodal map at spatial infinity \(i^0\). We analyse the validity of this conjecture using Friedrich’s formulation of spatial infinity, which gives rise to a regular initial value problem for the conformal field equations at spatial infinity. A central structure in this analysis is the cylinder at spatial infinity representing a blow-up of the standard spatial infinity point \(i^0\) to a 2-sphere. The cylinder touches past and future null infinities \(\mathscr I^\pm\) at the critical sets. We show that for a generic class of asymptotically Euclidean and regular initial data, BMS-supertranslation charges are not well-defined at the critical sets unless the initial data satisfies an extra regularity condition. We also show that given initial data that satisfy the regularity condition, BMS-supertranslation charges at the critical sets are fully determined by the initial data and that the relation between the charges at past null infinity and those at future null infinity directly follows from our regularity condition.

Supergravity model of the Haldane-Rezayi fractional quantum Hall state

Dung Xuan Nguyen, Kartik Prabhu, Ajit C. Balram, Andrey Gromov
Supersymmetry and supergravity were invented in the 1970s to solve fundamental problems in high-energy physics. Even though neither of these ideas has yet been confirmed in high-energy and cosmology experiments, they have been beneficial in constructing numerous theoretical models, including superstring theory. Despite the absence of supersymmetry in particle physics, it can potentially emerge in exotic phases of strongly correlated condensed matter systems. In this paper, we propose a supergravity model that describes the low-energy physics of the Haldane-Rezayi state, a gapless quantum Hall state that occurs in a half-filled Landau level. We show that the corresponding edge modes of the Haldane-Rezayi state and the Girvin-MacDonald-Platzman algebra appear naturally in the supergravity model. Finally, we substantiate our theoretical findings with numerical exact diagonalization calculations that support the appearance of the emergent graviton and gravitino excitations in the Haldane-Rezayi state.

Infrared Finite Scattering Theory in Quantum Field Theory and Quantum Gravity

Kartik Prabhu, Gautam Satishchandran, Robert M. Wald
It has been known since the earliest days of quantum field theory (QFT) that infrared divergences arise in scattering theory with massless fields. These infrared divergences are manifestations of the memory effect: At order \(1/r\) a massless field generically will not return to the same value at late retarded times (\(u \to + \infty\)) as it had at early retarded times (\(u \to - \infty\)). There is nothing singular about states with memory, but they do not lie in the standard Fock space. Infrared divergences are merely artifacts of trying to represent states with memory in the standard Fock space. If one is interested only in quantities directly relevant to collider physics, infrared divergences can be successfully dealt with by imposing an infrared cutoff, calculating inclusive quantities, and then removing the cutoff. However, this approach does not allow one to treat memory as a quantum observable and is highly unsatisfactory if one wishes to view the \(S\)-matrix as a fundamental quantity in QFT and quantum gravity, since the \(S\)-matrix itself is undefined. In order to have a well-defined \(S\)-matrix, it is necessary to define “in” and “out” Hilbert spaces that incorporate memory in a satisfactory way. Such a construction was given by Faddeev and Kulish for quantum electrodynamics (QED) with a massive charged field. Their construction can be understood as pairing momentum eigenstates of the charged particles with corresponding memory representations of the electromagnetic field to produce states of vanishing large gauge charges at spatial infinity. (This procedure is usually referred to as “dressing” the charged particles.) We investigate this procedure for QED with massless charged particles and show that, as a consequence of collinear divergences, the required “dressing” in this case has an infinite total energy flux, so that the states obtained in the Faddeev-Kulish construction are unphysical. An additional difficulty arises in Yang-Mills theory, due to the fact that the “soft Yang-Mills particles” used for the “dressing” contribute to the Yang-Mills charge-current flux, thereby invalidating the procedure used to construct eigenstates of large gauge charges at spatial infinity. We show that there are insufficiently many charge eigenstates to accommodate scattering theory. In quantum gravity, the analog of the Faddeev-Kulish construction would attempt to produce a Hilbert space of eigenstates of supertranslation charges at spatial infinity. Again, the Faddeev-Kulish “dressing” procedure does not produce the desired eigenstates because the dressing contributes to the null memory flux. We prove that there are no eigenstates of supertranslation charges at spatial infinity apart from the vacuum. Thus, analogs of the Faddeev-Kulish construction fail catastrophically in quantum gravity. We investigate some alternatives to Faddeev-Kulish constructions but find that these also do not work. We believe that if one wishes to treat scattering at a fundamental level in quantum gravity — as well as in massless QED and Yang-Mills theory — it is necessary to approach it from an algebraic viewpoint on the “in” and “out” states, wherein one does not attempt to “shoehorn” these states into some pre-chosen “in” and “out” Hilbert spaces. We outline the framework of such a scattering theory, which would be manifestly infrared finite.

A novel supersymmetric extension of BMS symmetries at null infinity

Kartik Prabhu
We show that we can combine the (complex, self-dual) BMS vector fields with the recently defined BMS twistors to obtain a new supersymmetric extension of the BMS symmetries at null infinity. We compare our construction to other supersymmetric extensions of the BMS algebra proposed in the context of supergravity. Unlike the standard constructions the anticommutator in our superalgebra generates all the BMS vector fields including the Lorentz transformations. We also show that there exists a projection from our BMS Lie superalgebra to the global subalgebra of the Neveu-Schwarz supersymmetries on a 2-sphere, which are commonly considered in string theory and 2-dimensional conformal field theory.

A twistorial description of BMS symmetries at null infinity

Kartik Prabhu
We describe a novel twistorial construction of the asymptotic BMS symmetries at null infinity for asymptotically flat spacetimes. We define BMS twistors as spinor solutions to some set of components of the usual spacetime twistor equation restricted to null infinity. The space of BMS twistors is infinite-dimensional. We show that given two BMS twistors their symmetric tensor product can be used to generate (complex) vector fields which are the infinitesimal BMS symmetries of null infinity. In this sense BMS twistors are “square roots” of BMS symmetries. We also show that these BMS twistor equations can be written a pair of covariant spinor-valued equations which are completely determined by the intrinsic universal structure of null infinity.

Conservation of asymptotic charges from past to future null infinity: Lorentz charges in general relativity

Kartik Prabhu, Ibrahim Shehzad
We show that the asymptotic charges associated with Lorentz symmetries on past and future null infinity match in the limit to spatial infinity in a class of spacetimes that are asymptotically-flat in the sense of Ashtekar and Hansen. Combined with the results of \cite{KP-GR-match}, this shows that all Bondi-Metzner-Sachs (BMS) charges on past and future null infinity match in the limit to spatial infinity in this class of spacetimes, proving a relationship that was conjectured by Strominger. Assuming additional suitable conditions are satisfied at timelike infinities, this proves that the flux of all BMS charges is conserved in any classical gravitational scattering process.

The Wald-Zoupas prescription for asymptotic charges at null infinity in general relativity

Alexander M. Grant, Kartik Prabhu, Ibrahim Shehzad
We use the formalism developed by Wald and Zoupas to derive explicit covariant expressions for the charges and fluxes associated with the Bondi-Metzner-Sachs symmetries at null infinity in asymptotically flat spacetimes in vacuum general relativity. Our expressions hold in non-stationary regions of null infinity, are local and covariant, conformally-invariant, and are independent of the choice of foliation of null infinity and of the chosen extension of the symmetries away from null infinity. We also include detailed comparisons with other expressions for the charges and fluxes that have appeared in the literature, including expressions in conformal Bondi-Sachs coordinates, the Ashtekar-Streubel flux formula, the Komar formulae and the linkage and twistor charge formulae.

BMS-like symmetries in cosmology

Béatrice Bonga, Kartik Prabhu
Null infinity in asymptotically flat spacetimes posses a rich mathematical structure; including the BMS group and the Bondi news tensor that allow one to study gravitational radiation rigorously. However, FLRW spacetimes are not asymptotically flat because their stress-energy tensor does not decay sufficiently fast and in fact diverges at null infinity. This class includes matter- and radiation-dominated FLRW spacetimes. We define a class of spacetimes whose structure at null infinity is similar to FLRW spacetimes: the stress-energy tensor is allowed to diverge and the conformal factor is not smooth at null infinity. Interestingly, for this larger class of spacetimes, the asymptotic symmetry algebra is similar to the BMS algebra but not isomorphic to it. In particular, the symmetry algebra is the semi-direct sum of supertranslations and the Lorentz algebra, but it does not have any preferred translation subalgebra. Future applications include studying gravitational radiation in FLRW the full nonlinear theory, including the cosmological memory effect, and also asymptotic charges in this framework.

Asymptotic symmetries and charges at spatial infinity in general relativity

Kartik Prabhu, Ibrahim Shehzad
We analyze the asymptotic symmetries and their associated charges at spatial infinity in \(4\)-dimensional asymptotically-flat spacetimes. We use the covariant formalism of Ashtekar and Hansen where the asymptotic fields and symmetries live on the \(3\)-manifold of spatial directions at spatial infinity, represented by a timelike unit-hyperboloid (or de Sitter space). Using the covariant phase space formalism, we derive formulae for the charges corresponding to asymptotic supertranslations and Lorentz symmetries at spatial infinity. With the motivation of, eventually, proving that these charges match with those defined on null infinity — as has been conjectured by Strominger — we will not impose any restrictions on the choice of conformal factor in contrast to previous work on this problem. Since we work with a general conformal factor we expect that our charge expressions will be more suitable to prove the matching of the Lorentz charges at spatial infinity to those defined on null infinity, as has been recently shown for the supertranslation charges.

Angular momentum at null infinity in Einstein-Maxwell theory

Béatrice Bonga, Alexander M. Grant, Kartik Prabhu

On Minkowski spacetime, the angular momentum flux through null infinity of Maxwell fields, computed using the stress-energy tensor, depends not only on the radiative degrees of freedom, but also on the Coulombic parts. However, the angular momentum also can be computed using other conserved currents associated with a Killing field, such as the Noether current and the canonical current. The flux computed using these latter two currents are purely radiative. A priori, it is not clear which of these is to be considered the “true” flux of angular momentum for Maxwell fields. This situation carries over to Maxwell fields on non-dynamical, asymptotically flat spacetimes for fluxes associated with the Lorentz symmetries in the asymptotic BMS algebra.

We investigate this question of angular momentum flux in full Einstein-Maxwell theory. Using the prescription of Wald and Zoupas, we compute the charges associated with any BMS symmetry on cross-sections of null infinity. The change of these charges along null infinity then provides a flux. For Lorentz symmetries, the Maxwell fields contribute an additional term in the charge on a cross-section. With this additional term, the flux associated with Lorentz symmetries, e.g. the angular momentum flux, is purely determined by the radiative degrees of freedom of the gravitational and Maxwell fields. In fact, the contribution to this flux by the Maxwell fields is given by the radiative Noether current and not by the stress-energy flux.

Extensions of the asymptotic symmetry algebra of general relativity

Éanna É. Flanagan, Kartik Prabhu, Ibrahim Shehzad
We consider a recently proposed extension of the Bondi-Metzner-Sachs algebra to include arbitrary infinitesimal diffeomorphisms on a \(2\)-sphere. To realize this extended algebra as asymptotic symmetries, we work with an extended class of spacetimes in which the unphysical metric at null infinity is not universal. We show that the symplectic current evaluated on these extended symmetries is divergent in the limit to null infinity. We also show that this divergence cannot be removed by a local and covariant redefinition of the symplectic current. This suggests that such an extended symmetry algebra cannot be realized as symmetries on the phase space of vacuum general relativity at null infinity, and that the corresponding asymptotic charges are ill-defined. However, a possible loophole in the argument is the possibility that symplectic current may not need to be covariant in order to have a covariant symplectic form. We also show that the extended algebra does not have a preferred subalgebra of translations and therefore does not admit a universal definition of Bondi \(4\)-momentum.

Symmetries, charges and conservation laws at causal diamonds in general relativity

Venkatesa Chandrasekaran, Kartik Prabhu
We study the covariant phase space of vacuum general relativity at the null boundary of causal diamonds. The past and future components of such a null boundary each have an infinite-dimensional symmetry algebra consisting of diffeomorphisms of the \(2\)-sphere and boost supertranslations corresponding to angle-dependent rescalings of affine parameter along the null generators. Associated to these symmetries are charges and fluxes obtained from the covariant phase space formalism using the prescription of Wald and Zoupas. By analyzing the behavior of the spacetime metric near the corners of the causal diamond, we show that the fluxes are also Hamiltonian generators of the symmetries on the phase space. In particular, the supertranslation fluxes yield an infinite family of boost Hamiltonians acting on the gravitational data of causal diamonds. We show that the smoothness of the vector fields representing such symmetries at the bifurcation edge of the causal diamond implies suitable matching conditions between the symmetries on the past and future components of the null boundary. Similarly, the smoothness of the spacetime metric implies that the fluxes of all such symmetries is conserved between the past and future components of the null boundary. This establishes an infinite set of conservation laws for finite subregions in gravity analogous to those at null infinity. We also show that the symmetry algebra at the causal diamond has a non-trivial center corresponding to constant boosts. The central charges associated to these constant boosts are proportional to the area of the bifurcation edge, for any causal diamond, in analogy with the Wald entropy formula.

Conservation of asymptotic charges from past to future null infinity: Supermomentum in general relativity

Kartik Prabhu
We show that the BMS-supertranslations and their associated supermomenta on past null infinity can be related to those on future null infinity, proving the conjecture of Strominger for a class of spacetimes which are asymptotically-flat in the sense of Ashtekar and Hansen. Using a cylindrical 3-manifold of both null and spatial directions of approach towards spatial infinity, we impose appropriate regularity conditions on the Weyl tensor near spatial infinity along null directions. The asymptotic Einstein equations on this 3-manifold and the regularity conditions imply that the relevant Weyl tensor components on past null infinity are antipodally matched to those on future null infinity. The subalgebra of totally fluxless supertranslations near spatial infinity provides a natural isomorphism between the BMS-supertranslations on past and future null infinity. This proves that the flux of the supermomenta is conserved from past to future null infinity in a classical gravitational scattering process provided additional suitable conditions are satisfied at the timelike infinities.

Conservation of asymptotic charges from past to future null infinity: Maxwell fields

Kartik Prabhu
On any asymptotically-flat spacetime, we show that the asymptotic symmetries and charges of Maxwell fields on past null infinity can be related to those on future null infinity as recently proposed by Strominger. We extend the covariant formalism of Ashtekar and Hansen by constructing a \(3\)-manifold of both null and spatial directions of approach to spatial infinity. This allows us to systematically impose appropriate regularity conditions on the Maxwell fields near spatial infinity along null directions. The Maxwell equations on this \(3\)-manifold and the regularity conditions imply that the relevant field quantities on past null infinity are antipodally matched to those on future null infinity. Imposing the condition that in a scattering process the total flux of charges through spatial infinity vanishes, we isolate the subalgebra of totally fluxless symmetries near spatial infinity. This subalgebra provides a natural isomorphism between the asymptotic symmetry algebras on past and future null infinity, such that the corresponding charges are equal near spatial infinity. This proves that the flux of charges is conserved from past to future null infinity in a classical scattering process of Maxwell fields. We also comment on possible extensions of our method to scattering in general relativity.

Symmetries and charges of general relativity at null boundaries

Venkatesa Chandrasekaran, Éanna É. Flanagan, Kartik Prabhu
We study general relativity at a null boundary using the covariant phase space formalism. We define a covariant phase space and compute the algebra of symmetries at the null boundary by considering the boundary-preserving diffeomorphisms that preserve this phase space. This algebra is the semi-direct sum of diffeomorphisms on the two sphere and a nonabelian algebra of supertranslations that has some similarities to supertranslations at null infinity. By using the general prescription developed by Wald and Zoupas, we derive the localized charges of this algebra at cross sections of the null surface as well as the associated fluxes. Our analysis is covariant and applies to general non-stationary null surfaces. We also derive the global charges that generate the symmetries for event horizons, and show that these obey the same algebra as the linearized diffeomorphisms, without any central extension. Our results show that supertranslations play an important role not just at null infinity but at all null boundaries, including non-stationary event horizons. They should facilitate further investigations of whether horizon symmetries and conservation laws in black hole spacetimes play a role in the information loss problem, as suggested by Hawking, Perry, and Strominger.

Canonical Energy and Hertz Potentials for Perturbations of Schwarzschild Spacetime

Kartik Prabhu, Robert M. Wald
Canonical energy is a valuable tool for analyzing the linear stability of black hole spacetimes; positivity of canonical energy for all perturbations implies mode stability, whereas the failure of positivity for any perturbation implies instability. Nevertheless, even in the case of \(4\)-dimensional Schwarzschild spacetime — which is known to be stable — manifest positivity of the canonical energy is difficult to establish, due to the presence of constraints on the initial data as well as the gauge dependence of the canonical energy integrand. Consideration of perturbations generated by a Hertz potential would appear to be a promising way to improve this situation, since the constraints and gauge dependence are eliminated when the canonical energy is expressed in terms of the Hertz potential. We prove that the canonical energy of a metric perturbation of Schwarzschild that is generated by a Hertz potential is positive. We relate the energy quantity arising in the linear stability proof of Dafermos, Holzegel and Rodnianski (DHR) to the canonical energy of an associated metric perturbation generated by a Hertz potential. We also relate the Regge-Wheeler variable of DHR to the ordinary Regge-Wheeler and twist potential variables of the associated perturbation. Since the Hertz potential formalism can be generalized to a Kerr black hole, our results may be useful for the analysis of the linear stability of Kerr.

Black hole scalar charge from a topological horizon integral in Einstein-dilaton-Gauss-Bonnet gravity

Kartik Prabhu, Leo C. Stein
In theories of gravity that include a scalar field, a compact object’s scalar charge is a crucial quantity since it controls dipole radiation, which can be strongly constrained by pulsar timing and gravitational wave observations. However in most such theories, computing the scalar charge requires simultaneously solving the coupled, nonlinear metric and scalar field equations of motion. In this article we prove that in linearly-coupled Einstein-dilaton-Gauss-Bonnet gravity (which admits a shift symmetry of the dilaton), a black hole’s scalar charge is completely determined by the horizon surface gravity times the Euler characteristic of the bifurcation surface, without solving any equations of motion. Within this theory, black holes announce their horizon topology and surface gravity to the rest of the universe through the dilaton field. In our proof, a 4‐dimensional topological density descends to a 2‐dimensional topological density on the bifurcation surface of a Killing horizon. We also comment on how our proof can be generalised to other topological densities on general G‐bundles.

Electrons and composite Dirac fermions in the lowest Landau level

Kartik Prabhu, Matthew M. Roberts
We construct an action for the composite Dirac fermion consistent with symmetries of electrons projected to the lowest Landau level. First we construct a generalization of the \(g=2\) electron that gives a smooth massless limit on any curved background. Using the symmetries of the microscopic electron theory in this massless limit we find a number of constraints on any low-energy effective theory. We find that any low-energy description must couple to a geometry which exhibits nontrivial curvature even on flat space-times. Any composite fermion must have an electric dipole moment proportional and orthogonal to the composite fermion's wavevector. We construct the effective action for the composite Dirac fermion and calculate the physical stress tensor and current operators for this theory.

Stability of stationary-axisymmetric black holes in vacuum general relativity to axisymmetric electromagnetic perturbations

Kartik Prabhu, Robert M. Wald
We consider arbitrary stationary and axisymmetric black holes in general relativity in \((d +1)\) dimensions (with \(d \geq 3\)) that satisfy the vacuum Einstein equation and have a non-degenerate horizon. We prove that the canonical energy of axisymmetric electromagnetic perturbations is positive definite. This establishes that all vacuum black holes are stable to axisymmetric electromagnetic perturbations. Our results also hold for asymptotically deSitter black holes that satisfy the vacuum Einstein equation with a positive cosmological constant. Our results also apply to extremal black holes provided that the initial perturbation vanishes in a neighborhood of the horizon.

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

Michael Geracie, Kartik Prabhu, Matthew M. Roberts
For theories of relativistic matter fields with spin there exist two possible definitions of the stress‐energy tensor, one defined by a variation of the action with the coframes at fixed connection, and the other at fixed torsion. These two stress‐energy tensors do not necessarily coincide and it is the latter that corresponds to the Cauchy stress measured in the lab. In this note we discuss the corresponding issue for non‐relativistic matter theories. We point out that while the physical non‐relativistic stress, momentum, and mass currents are defined by a variation of the action at fixed torsion, the energy current does not admit such a description and is naturally defined at fixed connection. Any attempt to define an energy current at fixed torsion results in an ambiguity which cannot be resolved from the background spacetime data or conservation laws. We also provide computations of these quantities for some simple non‐relativistic actions.

A Variational Principle for the Axisymmetric Stability of Rotating Relativistic Stars

Kartik Prabhu, Joshua S. Schiffrin, Robert M. Wald
It is well known that all rotating perfect fluid stars in general relativity are unstable to certain non‐axisymmetric perturbations via the Chandrasekhar‐Friedman‐Schutz (CFS) instability. However, the mechanism of the CFS instability requires, in an essential way, the loss of angular momentum by gravitational radiation and, in many instances, it acts on too long a timescale to be physically/astrophysically relevant. It is therefore of interest to examine the stability of rotating, relativistic stars to axisymmetric perturbations, where the CFS instability does not occur. In this paper, we provide a Rayleigh‐Ritz type variational principle for testing the stability of perfect fluid stars to axisymmetric perturbations, which generalizes to axisymmetric perturbations of rotating stars a variational principle given by Chandrasekhar for spherical perturbations of static, spherical stars. Our variational principle provides a lower bound to the rate of exponential growth in the case of instability. The derivation closely parallels the derivation of a recently obtained variational principle for analyzing the axisymmetric stability of black holes.

Covariant effective action for a Galilean invariant quantum Hall system

Michael Geracie, Kartik Prabhu, Matthew M. Roberts
We construct effective field theories for gapped quantum Hall systems coupled to background geometries with local Galilean invariance i.e. Bargmann spacetimes. Along with an electromagnetic field, these backgrounds include the effects of curved Galilean spacetimes, including torsion and a gravitational field, allowing us to study charge, energy, stress and mass currents within a unified framework. A shift symmetry specific to single constituent theories constraints the effective action to couple to an effective background gauge field and spin connection that is solved for by a self‐consistent equation, providing a manifestly covariant extension of Hoyos and Son’s improvement terms to arbitrary order in \(m\).

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

Kartik Prabhu

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

Michael Geracie, Kartik Prabhu, Matthew M. Roberts

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

Michael Geracie, Kartik Prabhu, Matthew M. Roberts

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

Kartik Prabhu, Robert M. Wald

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

Naresh Dadhich, Josep M. Pons, Kartik Prabhu

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

Sanjit Das, Kartik Prabhu, Sayan Kar

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

Naresh Dadhich, Josep M. Pons, Kartik Prabhu

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

Kartik Prabhu, Sanjit Das, Sayan Kar

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

Kartik Prabhu, Naresh Dadhich

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

Sanjit Das, Kartik Prabhu, Sayan Kar

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.