Penrose Limit Research Articles

Moving holographic boundaries

In this paper, we show that for one sign of the deformation coupling single-trace TT‾ deformation moves the holographic screen in Gödel universe radially inward. For the other sign of the coupling it moves the holographic screen radially outward. We (thus) argue, on general grounds, that in holography (single-trace) TT‾ deformation can be generally thought of as either moving the holographic boundary into the bulk or washing it away to infinity. We explain in what sense. In Anti-de Sitter this breaks the spacetime conformal symmetry. We further note that moving timelike holographic boundary into bulk creates (at onset) a curvature singularity. In the boundary the singularity is understood by states with imaginary energies. To define or make the bulk theory sensible we introduce an ultraviolet cutoff and thereby move the boundary into the bulk. However, it is not clear whether suitable boundary conditions that lead to a consistent string theory exist. In this paper we first obtain the Penrose limit of the single-trace TT‾ deformed string background and then perform T-duality along a space-like isometry to obtain a class of deformed Gödel universes. The string background we consider is AdS3×S3×M4. The single-trace TT‾ deformation is a particular example of the more general O(d,d) transformations.

Quasinormal modes and universality of the Penrose limit of black hole photon rings

We study the physics of photon rings in a wide range of axisymmetric black holes admitting a separable Hamilton-Jacobi equation for the geodesics. Utilizing the Killing-Yano tensor, we derive the Penrose limit of the black holes, which describes the physics near the photon ring. The obtained plane wave geometry is directly linked to the frequency matrix of the massless wave equation, as well as the instabilities and Lyapunov exponents of the null geodesics. Consequently, the Lyapunov exponents and frequencies of the photon geodesics, along with the quasinormal modes, can be all extracted from a Hamiltonian in the Penrose limit plane wave metric. Additionally, we explore potential bounds on the Lyapunov exponent, the orbital and precession frequencies, in connection with the corresponding inverted harmonic oscillators and we discuss the possibility of photon rings serving as effective holographic horizons in a holographic duality framework for astrophysical black holes. Our formalism is applicable to spacetimes encompassing various types of black holes, including stationary ones like Kerr, Kerr-Newman, as well as static black holes such as Schwarzschild, Reissner-Nordström, among others.

Penrose limits of I-branes, twist-compactified D5-branes, and spin chains

Penrose limits of I-branes, twist-compactified D5-branes, and spin chains

Semi-Riemannian manifolds with linear differential conditions on the curvature

Semi-Riemannian manifolds that satisfy (homogeneous) linear differential conditions of arbitrary order r on the curvature are analyzed. They include, in particular, the spaces with (rth-order) recurrent curvature, (rth-order) symmetric spaces, as well as entire new families of semi-Riemannian manifolds rarely, or never, considered before in the literature—such as the spaces whose derivative of the Riemann tensor field is recurrent, among many others. Definite proof that all types of such spaces do exist is provided by exhibiting explicit examples of all possibilities in all signatures, except in the Riemannian case with a positive definite metric. Several techniques of independent interest are collected and presented. Of special relevance is the case of Lorentzian manifolds, due to its connection to the physics of the gravitational field. This connection is discussed with particular emphasis on Gauss–Bonnet gravity and in relation with Penrose limits. Many new lines of research open up and a handful of conjectures, based on the results found hitherto, is put forward.

Nonlinear gravitational waves in Horndeski gravity: scalar pulse and memories

We present and analyze a new non-perturbative radiative solution of Horndeski gravity. This exact solution is constructed by a disformal mapping of a seed solution of the shift-symmetric Einstein-Scalar system belonging to the Robinson-Trautman geometry describing the gravitational radiation emitted by a time-dependent scalar monopole. After analyzing in detail the properties of the seed, we show that while the general relativity solution allows for shear-free parallel transported null frames, the disformed solution can only admit parallel transported null frames with a non-vanishing shear. This result shows that, at the nonlinear level, the scalar-tensor mixing descending from the higher-order terms in Horndeski dynamics can generate shear out of a pure scalar monopole. We further confirm this analysis by identifying the spin-0 and spin-2 polarizations in the disformed solution using the Penrose limit of our radiative solution. Finally, we compute the geodesic motion and the memory effects experienced by two null test particles with vanishing initial relative velocity after the passage of the pulse. This exact radiative solution offers a simple framework to witness nonlinear consequences of the scalar-tensor mixing in higher-order scalar-tensor theories.

Supersymmetric backgrounds from λ-deformations

We provide the first supersymmetric embedding of an integrable λ-deformation to type-II supergravity. Specifically, that of the near horizon of the NS1-NS5 brane intersection, geometrically corresponding to AdS3 × S3 × T4. We show that the deformed background preserves 1/4 of the maximal supersymmetry. In the Penrose limit we show that it preserves no-more than one half of the maximal supersymmetry.

Superstrings on Penrose limits of AdS7 solutions

We consider the Penrose limit of AdS7 solutions of massive Type IIA supergravity. The resulting pp-wave geometry is supported by RR and NSNS fields. We quantize the Green-Schwarz superstring on the obtained pp-wave background, in the light-cone gauge.

Quasinormal modes from Penrose limits

We use Penrose limits to approximate quasinormal modes (QNMs) with large real frequencies. The Penrose limit associates a plane wave to a region of spacetime near a null geodesic. This plane wave can be argued to geometrically realize the geometrical optics approximation. Therefore, when applied to the bound null orbits around black holes, the Penrose limit can be used to study QNMs. For instance, this Penrose limit point of view makes manifest the symmetry that emerges in the geometrical optics approximation of QNMs in terms of isometries of the resulting plane wave spacetime. We apply the procedure to warped , Schwarzschild black holes and Kerr black holes. In the former we show explicitly how the symmetry algebra contracts to that of the limiting plane wave while in the latter we find the expected agreement with numerically computed QNMs for large (real) frequencies.

Penrose limits of inhomogeneous spacetimes, their diagonalizability, and twistors

Penrose limits of inhomogeneous spacetimes, their diagonalizability, and twistors

The Maslov index and some applications to dispersion relations in curved space times

The aim of the present work is to generalize the results given in Osorio Morales and Santillán [Eur. Phys. J. C 82, 353 (2022)] to a generic situation for causal geodesics. It is argued that these results may be of interest for causality issues. Recall that the presence of superluminal signals in a generic space time (M, gμν) does not necessarily imply violations of the principle of causality {[G. M. Shore, Nucl. Phys. B 778, 219 (2007)] and [T. J. Hollowood and G. M. Shore, Phys. Lett. B 655, 67 (2007)]}. In flat spaces, global Lorenz invariance leads to the conclusion that closed time-like curves appear if these signals are present. In a curved space instead, there is only local Poincare invariance, and the presence of closed causal curves may be avoided even in the presence of a superluminal mode, especially when terms violating the strong equivalence principle appear in the action. This implies that the standard analytic properties of the spectral components of these functions are therefore modified, and in particular, the refraction index n(ω) is not analytic in the upper complex ω plane. The emergence of these singularities may also take place for non-superluminal signals due to the breaking of global Lorenz invariance in a generic space time. In the present work, it is argued that the homotopy properties of the Maslov index are useful for studying how the singularities of n(ω) vary when moving along a geodesic congruence. In addition, several conclusions obtained in Shore [Nucl. Phys. B 778, 219 (2007)] and Hollowood and Shore [Phys. Lett. B 655, 67 (2007)] are based on the Penrose limit along a null geodesic, and they are restricted to GR with matter satisfying strong energy conditions. The use of the Maslov index may allow a more intrinsic description of singularities, not relying on that limit, and a generalization of these results about non-analyticity to generic gravity models with general matter content.

Integrable models based on non-semi-simple groups and plane wave target spacetimes

We initiate the construction of integrable λ-deformed WZW models based on non-semisimple groups. We focus on the four-dimensional case whose underlying symmetries are based on the non-semisimple group {E}_2^c . The corresponding gravitational backgrounds of Lorentzian signature are plane waves which can be obtained as Penrose limits of the λ-deformed SU(2) background times a timelike coordinate for appropriate choices of the λ-matrix. We construct two such deformations which we demonstrate to be integrable. They both deform the Nappi-Witten plane wave and are inequivalent. Nevertheless, they have the same underlying symmetry algebra which is a Saletan-type contraction of that for the λ-deformed SU(2) background with a timelike direction. We also construct a plane wave from the Penrose limit of the λ-deformation of the frac{textrm{SU}(2)}{textrm{U}(1)} coset CFT times a timelike coordinate which represents the deformation of a logarithmic CFT constructed in the past. Finally, we briefly consider contractions based on the simplest Yang-baxter σ-models.

Finsler pp-waves and the Penrose limit

We extend the notion of a Lorentzian pp-wave to that of Finsler spacetimes by providing a coordinate-independent definition of a Finsler pp-wave with respect to the Chern connection; our definition also includes the special case of a plane wave. This treatment introduces suitable lightlike coordinates, in analogy with the Lorentzian case, and utilizes the anisotropic calculus recently developed by one of the authors. We then extend Penrose’s “plane wave limit” to the setting of Finsler spacetimes. New examples of such Finsler pp-waves are also presented.

Marginally deformed Schrödinger/dipole CFT correspondence

We construct and thoroughly study a new integrable example of the AdS/CFT correspondence with Schrödinger symmetry. On the gravity side, the supergravity solution depends on two parameters and is obtained by marginally deforming the internal space of the Schrödinger background through a series of TsT transformations. On the field theory side, we identify the dual field theory which also depends on two parameters.We find a point-like string solution and derive its dispersion relation. By using the Landau-Lifshitz coherent state Lagrangian, which originates from field theory, we reproduce the leading, in the deformation parameters, terms of the string theory prediction. This constitutes a non-trivial test of the correspondence. Then, we calculate the Wilson loop, describing the quark/anti-quark potential at strong coupling. It exhibits confining behaviour when the separation length is much less than the Schrödinger parameter. When the separation length is much greater than the Schrödinger parameter the behaviour is that of a conformal theory. Subsequently, we take the Penrose limit along a certain null geodesic of the constructed background and calculate the bosonic spectrum. Based on that spectrum, we make an educated guess for the exact, in the ’t Hooft coupling, dispersion relation of the magnon excitations in the original doubly deformed background. This provides us with an exact prediction for the dimensions of the dual field theory operators. This applies to operators of large length, for which finite size corrections are suppressed.

Gravitational lensing, memory, and the Penrose limit

In this paper, we discuss the gravitational field of ultrarelativistic extended spinning objects. For this purpose, we use a solution of the linearized gravitational equations obtained in the frame where such an object is translationally at rest, and boost this solution close to the speed of light. In order to obtain a regular limiting metric for non-spinning matter, it is sufficient to keep the energy of the boosted body fixed. This process is known as the Penrose limit. We demonstrate that in the presence of rotation, an additional rescaling is required for the angular momentum density components in the directions orthogonal to the boost. As a result of the Lorentz contraction, the thickness of the body in the direction of the boost shrinks. The body takes the form of a pancake, and its gravitational field is localized in the null plane. We discuss light and particle scattering in this gravitational field, and calculate the scattering parameters associated with the gravitational memory effect. We also show that by taking the inverse of the Penrose transform, one can use the obtained scattering map to study the gravitational lensing effect in the rest frame of a massive spinning object.

Penrose limit of MNa solution and spin chains in three-dimensional field theories

In three dimensions, a theory with spontaneous breaking of supersymmetry was described holographically by the Maldacena-Nastase (MNa) solution. We revisit the issue of its Penrose limit, and compare the resulting pp wave and its string eigenstates to a sector of the theory on 5-branes reduced on S3, and a spin chain-like system. We obtain many of the features of the pp wave analysis, but a complete matching still eludes us. We suggest possible explanations for the resulting partial mismatch, and compare against other spin chains for 3-dimensional field theories with gravity duals.

Toroidal tidal effects in microstate geometries

Tidal effects in capped geometries computed in previous literature display no dynamics along internal (toroidal) directions. However, the dual CFT picture suggests otherwise. To resolve this tension, we consider a set of infalling null geodesics in a family of black hole microstate geometries with a smooth cap at the bottom of a long BTZ-like throat. Using the Penrose limit, we show that a string following one of these geodesics feels tidal stresses along all spatial directions, including internal toroidal directions. We find that the tidal effects along the internal directions are of the same order of magnitude as those along other, non-internal, directions. Furthermore, these tidal effects oscillate as a function of the distance from the cap — as a string falls down the throat it alternately experiences compression and stretching. We explain some physical properties of this oscillation and comment on the dual CFT interpretation.

On type 0 string theory in solvable RR backgrounds

Motivated by a possibility of solving non-supersymmetric type 0 string theory in AdS5× S5 background using integrability, we revisit the construction of type 0 string spectrum in some solvable examples of backgrounds with RR fluxes that are common to type IIB and type 0B theories. The presence of RR fluxes requires the use of a Green-Schwarz description for type 0 string theory. Like in flat space, the spectrum of type 0 theory can be derived from the type II theory spectrum by a (−1)F orbifolding, i.e. combining the untwisted sector where GS fermions are periodic with the twisted sector where GS fermions are antiperiodic (and projecting out all spacetime fermionic states). This construction of the type 0 spectrum may also be implemented using Melvin background that allows to continuously interpolate between the type II and type 0 theories. As an illustration, we discuss the type 0B spectrum in the pp-wave background which is the Penrose limit of AdS5× S5 with RR 5-form flux and also in the pp-wave background which is the Penrose limit of AdS3× S3× T4 supported by mixed RR and NSNS 3-form fluxes. We show that increasing the strength of the RR flux increases the value of the effective normal ordering constant (which determines the mass of the type 0 tachyon in flat space) and thus effectively decreases the momentum-space domain of instability of the ground state. We also comment on the semiclassical sector of states of type 0B theory in AdS5× S5.

Penrose limits in non-Abelian T dual of Klebanov-Tseytlin background

In this paper we consider the Klebanov-Tseytlin background and its non-Abelian $T$-dual geometry along a suitably chosen $SU(2)$ subgroup of isometries. We analyze the Penrose limits along various null geodesics of both geometries. We observe that the Klebanov-Tseytlin geometry does not admit any pp-wave solutions. However, the $T$-dual background gives rise to a pp-wave solution upon taking the Penrose limit along some appropriate null geodesic. We comment on the possible gauge theory dual for our pp-wave background.

String excitation by initial singularity of inflation

We discuss excitation of string oscillation modes by an initial singularity of inflation. The initial singularity of inflation is known to occur with a finite Hubble parameter, which is generally lower than the string scale, and hence it is not clear that stringy effects become significant around it. With the help of Penrose limit, we find that infinitely heavy oscillation modes get excited when a singularity is strong in the sense of Krolak’s classification. We demonstrate that the initial singularities of Starobinsky and hill top inflation, assuming the slow roll inflation to the past infinity, are strong. Hence stringy corrections are inevitable in the very early stage of these inflation models. We also find that the initial singularity of the hill top inflation could be weak for non-slow roll case.

Penrose limit for holographic duals of deformations

We explore bosonic string sigma models on warped BTZ × S 3 both in the plane wave as well as beyond plane wave limit. Using the light cone gauge, we obtain the corresponding Hamiltonian and therefore the spectrum associated with the pp wave strings. Our analysis reveals a constant shift in the spectrum arising as a result of TsT transformations along the isometries of the target space manifold. Imposing the energy positivity constraint on the CFT2 spectrum, we estimate an upper bound on the shift. We also estimate corrections as we go beyond the plane wave limit. Finally, we perform calculations using conformal gauge which reveals identical spectrum for the pp wave strings and thereby shows the equivalence between the two approaches.