The accretion disks around black holes consist of infalling matter boosted almost to the speed of light making collisions with opposite counterpart. This is the rough picture occurring near black holes or other strongly gravitating centers that produce observed phenomena such as astrophysical jets. A toy model that can be considered imitating such a process is colliding null sources in general relativity. We present such a simple model projected into the plane of null coordinates that takes into account only neutral sources. We show that even at such a simplified model, uncharged and non-rotating, it is possible to obtain jet-like ejections albeit they lie below the horizon. In the present study the spacetime consists of either one of i) a cloud of strings, ii) a global monopole, iii) a particular model of bumblebee gravity, all described by a similar class of stress-energy tensor. There are gravitational waves accompanying the null sources and naturally collision of gravitational waves is also taken into account. After the collision, the spacetime contains both null and non-null sources, followed by trailing gravitational radiations. Locally the interaction region of the colliding null-sources and gravitational waves is isometric to the static background spacetime.
Colliding Einstein-Maxwell-Scalar fields need not necessarily doomed to become in a spacelike singularity. Examples are given in which null singularities emerge as intermediate stages between a spacelike singularity and a regular horizon.
We present two recently obtained solutions of the Einstein equations with spherical symmetry and one additional Killing vector, describing colliding null dust streams.
New exact vacuum solutions with various singularities in the plane-symmetric spacetime are shown, and they are applied to the analysis of inhomogeneous cosmological models and colliding gravitational waves. One of the singularities can be true null singularities, whose existence was locally clarified by Ori. These solutions may be interesting from the viewpoint of the variety of cosmological singularities and the instability problem of Cauchy horizons inside black holes.
The Einstein equations are integrated in the presence of two (incoming and outgoing) streams of null dust, under the assumptions of spherical symmetry and staticity. The solution is also written in double null and radiation coordinates and it is reinterpreted as an anisotropic fluid. Interior matching with a static fluid and exterior matching with the Vaidya solution along null hypersurfaces is discussed. The connection with two-dimensional dilaton gravity is established.
We present a characteristic algorithm for computing the perturbations of a Schwarzschild spacetime by means of solving the Teukolsky equations. Our methods and results are expected to have direct bearing on the study of binary black holes presently underway using a fully {\em nonlinear} characteristic code \cite{Gomez98a}.
It was believed that when gravitational, electromagnetic and scalar waves interact, a spacelike curvature singularity or Cauchy horizon develops because of mutual focusing. We show with an exact solution that the collision of Einstein-Maxwell-Scalar fields, in contrast to previous studies, predicts singularities on null surfaces and that this is a transition phase between spacelike singularities and regular horizons. Divergences of tidal forces in the null singularities is shown to be weaker relative to the spacelike ones. Using the local isometry between colliding plane waves and black holes,…
The known exact solutions of Einstein's vacuum field equations modeling the gravitational fields of pure gravitational radiation involve wave fronts which are either planar or roughly spherical. We describe a scheme designed to check explicitly whether or not the wave fronts collide. From the spacetime point of view the scheme determines whether or not the null hypersurface histories of the wave fronts intersect and, in particular, allows easy identification of the cases in which the null hypersurfaces do not intersect.
Single pulse of null dust and colliding null dusts both transform a regular horizon into a space-like singularity in the space of colliding waves. The local isometry between such space-times and black holes extrapolates these results to the realm of black holes. However, inclusion of particular scalar fields instead of null dusts creates null singularities rather than space-like ones on the inner horizons of black holes.
In a recent work on the quantization of a massless scalar field in a particular colliding plane wave space-time, we computed the vacuum expectation value of the stress-energy tensor on the physical state which corresponds to the Minkowski vacuum before the collision of the waves. We did such a calculation in a region close to both the Killing-Cauchy horizon and the folding singularities that such a space-time contains. In the present paper, we give a suitable approximation procedure to compute this expectation value, in the conformal coupling case, throughout the causal past of the center of t…
Following a previous work on the quantization of a massless scalar field in a spacetime representing the head on collision of two plane waves which fucus into a Killing-Cauchy horizon, we compute the renormalized expectation value of the stress-energy tensor of the quantum field near that horizon in the physical state which corresponds to the Minkowski vacuum before the collision of the waves. It is found that for minimally coupled and conformally coupled scalar fields the respective stress-energy tensors are unbounded in the horizon. The specific form of the divergences suggests that when the…
The stress-energy tensor of a matter shell whose history coincides with a null hypersurface in the Einstein-Cartan gravity is revisited. It is demonstrated that with a proper choice for the torsion discontinuity taken to be orthogonal to the null hypersurface and consistent with the antisymmetric property of torsion tensor, the modified expression for the asymmetric stress-energy tensor is automatically tangent to the hypersurface. The main differences with respect to a previous work are addressed
Bell-Szekeres (BS) solution for colliding electromagnetic waves in Einstein-Maxwell (EM) theory describes also colliding waves in nonlinear electrodynamics (NED) with an emergent cosmological constant. Our NED model covers the first leading orders to the well-known Heisenberg-Euler (HE) type in a particular gauge of pure magnetic field. Prior to the problem of collision we obtain dyonic solution for the considered NED theory in a conformally flat spacetime which has both electric and magnetic fields with constant invariants. Our sole finding is that null currents inevitably arise in the proces…
We present a space--time model of the collision of two homogeneous, plane impulsive gravitational waves (each having a delta function profile) propagating in a vacuum before collision and for which the post collision space--time has constant curvature. The profiles of the incoming waves are $k\,δ(u)$ and $l\,δ(v)$ where $k, l$ are real constants and $u=0, v=0$ are intersecting null hypersurfaces. The cosmological constant $Λ$ in the post collision region of the space--time is given by $Λ=-6\,k\,l$.
In a recent work an approximation procedure was introduced to calculate the vacuum expectation value of the stress-energy tensor for a conformal massless scalar field in the classical background determined by a particular colliding plane wave space-time. This approximation procedure consists in appropriately modifying the space-time geometry throughout the causal past of the collision center. This modification in the geometry allows to simplify the boundary conditions involved in the calculation of the Hadamard function for the quantum state which represents the vacuum in the flat region befor…
We use null hypersurface techniques in a new approach to calculate the retarded waveform from a binary black hole merger in the close approximation. The process of removing ingoing radiation from the system leads to two notable features in the shape of the close approximation waveform for a head-on collision of black holes: (i) an initial quasinormal ringup and (ii) weak sensitivity to the parameter controlling the collision velocity. Feature (ii) is unexpected and has the potential importance of enabling the design of an efficient template for extracting the gravitational wave signal from the…
A method is presented which allows for the numerical computation of the stress-energy tensor for a quantized massless minimally coupled scalar field in the region outside the event horizon of a 4D Schwarzschild black hole that forms from the collapse of a null shell. This method involves taking the difference between the stress-energy tensor for the in state in the collapsing null shell spacetime and that for the Unruh state in Schwarzschild spacetime. The construction of the modes for the in vacuum state and the Unruh state is discussed. Applying the method, the renormalized stress-energy ten…
The general form of the surface stress tensor of an infinitesimally thin shell located on a rotating null horizon is derived, when different interior and exterior geometries are joined there. Although the induced metric on the surface must be the same approached from either side, the first derivatives of the metric need not be. Such discontinuities lead to a Dirac $δ$-distribution in the Einstein tensor localized on the horizon. For a general stationary axisymmetric geometry the surface stress tensor can be expressed in terms of two geometric invariants that characterize the surface, namely th…
We study the quantization of a scalar field on a classical background given by the Szekeres Class of solutions, which represent the collision of two gravitational plane waves with constant polarization. These solutions consist of two approaching gravitational plane waves moving in a flat background and an interaction region which always contains a curvature singularity. Following a suitable approximate procedure, introduced in a previous paper, we propose a way to compute the vacuum expectation value of the stress-energy tensor throughout the causal past region of the collision center in the q…
Recently, Larry Ford and Tom Roman have discovered that in a flat cylindrical space, although the stress-energy tensor itself fails to satisfy the averaged null energy condition (ANEC) along the (non-achronal) null geodesics, when the ``Casimir-vacuum" contribution is subtracted from the stress-energy the resulting tensor does satisfy the ANEC inequality. Ford and Roman name this class of constraints on the quantum stress-energy tensor ``difference inequalities." Here I give a proof of the difference inequality for a minimally coupled massless scalar field in an arbitrary two-dimensional space…
Solutions to the mode equation for a quantized massless scalar field outside a black hole that forms from the collapse of a null shell: Late-time behaviors and computation of the stress-energy tensor
The physical situation of the collision and subsequent interaction of plane gravitational waves in a Minkowski background gives rise to a well-posed characteristic initial value problem in which initial data are specified on the two null characteristics that define the wavefronts. In this paper, we analyse how the Abel transform method can be used in practice to solve this problem for the linear case in which the polarization of the two gravitational waves is constant and aligned. We show how the method works for some known solutions, where problems arise in other cases, and how the problem ca…
For the quantised, massless, minimally coupled real scalar field in four-dimensional Minkowski space, we show (by an explicit construction) that weighted averages of the null-contracted stress-energy tensor along null geodesics are unbounded from below on the class of Hadamard states. Thus there are no quantum inequalities along null geodesics in four-dimensional Minkowski spacetime. This is in contrast to the case for two-dimensional flat spacetime, where such inequalities do exist. We discuss in detail the properties of the quantum states used in our analysis, and also show that the renormal…
Using the Brown-York prescription for the definition of quasilocal gravitational energy-momentum tensor on a boundary and also complete canonical structure on a null boundary which has been found recently \cite{Aghapour:2018icu}, we propose a similar stress tensor on the null boundary. Then we exploit this stress tensor to compute the quasi-local energy and angular momentum for some well-known gravitational solutions. We have found that in addition to reference spacetime method for regularizing total energy, in the case of null boundary we can add a possible counterterm so avoiding embedding d…
We consider a situation in which two metrics are joined at a null hypersurface. It often occurs that the union of the two metrics gives rise to a Ricci tensor that contains a term proportional to a Dirac delta-function supported on the hypersurface. This singularity is associated with a thin distribution of matter on the hypersurface, and following Barrabes and Israel, we seek to determine its stress-energy tensor in terms of the geometric properties of the null hypersurface. While our treatment here does not deviate strongly from their previous work, it offers a simplification of the computat…
The Brown-York stress tensor provides a means for defining quasilocal gravitational charges in subregions bounded by a timelike hypersurface. We consider the generalization of this stress tensor to null hypersurfaces. Such a stress tensor can be derived from the on-shell subregion action of general relativity associated with a Dirichlet variational principle, which fixes an induced Carroll structure on the null boundary. The formula for the mixed-index tensor $T{}^i{}_j$ takes a remarkably simple form that is manifestly independent of the choice of auxiliary null vector at the null surface, an…
When two non-interacting plane impulsive gravitational waves undergo a head-on collision, the vacuum interaction region between the waves after the collision contains backscattered gravitational radiation from both waves. The two systems of backscattered waves have each got a family of rays (null geodesics) associated with them. We demonstrate that if it is assumed that a parameter exists along each of these families of rays such that the modulus of the complex shear of each is equal then Einstein's vacuum field equations, with the appropriate boundary conditions, can be integrated systematica…
A method is given to compute the stress-energy tensor for a massless minimally coupled scalar field in a spacetime where a black hole forms from the collapse of a spherically symmetric null shell in four dimensions. Part of the method involves matching the modes for the in vacuum state to a complete set of modes in Schwarzschild spacetime. The other part involves subtracting from the unrenormalized expression for the stress-energy tensor when the field is in the in vacuum state, the corresponding expression when the field is in the Unruh state and adding to this the renormalized stress-energy …
We consider the retarded solution to the scalar, electromagnetic, and linearized gravitational field equations in Minkowski spacetime, with source given by a particle moving on a null geodesic. In the scalar case and in the Lorenz gauge in the electromagnetic and gravitational cases, the retarded integral over the infinite past of the source does not converge as a distribution, so we cut off the null source suitably at a finite time $t_0$ and then consider two different limits: (i) the limit as the observation point goes to null infinity at fixed $t_0$, from which the ``$1/r$'' part of the fie…
The Tolman-Bondi and Vaidya solutions are two solutions to Einstein equations which describe dust particles and null fluid, respectively. We show that it is possible to match the two solutions in one single spacetime, the Tolman-Bondi--Vaidya spacetime. The new spacetime is divided by a null surface with Tolman-Bondi dust on one side and Vaidya fluid on the other side. The differentiability of the spacetime is discussed. By constructing a specific solution, we show that the metric across the null surface can be at least $C^1$ and the stress-energy tensor is continuous.
The Averaged Null Energy Condition (ANEC) states that the integral along a complete null geodesic of the projection of the stress-energy tensor onto the tangent vector to the geodesic cannot be negative. Exotic spacetimes, such as those allow wormholes or the construction of time machines are possible in general relativity only if ANEC is violated along achronal geodesics. Starting from a conjecture that flat-space quantum inequalities apply with small corrections in spacetimes with small curvature, we prove that ANEC is obeyed by a minimally-coupled, free quantum scalar field on any achronal …
We consider the effective stress-energy tensors for the foreground and background sectors in ghost-free bimetric gravity. By considering the symmetries of the theory, we show that the foreground and background null energy conditions (NECs) are strongly anti-correlated. In particular, the NECs can only be simultaneously fulfilled when they saturate, corresponding to foreground and background cosmological constants. In all other situations, either the foreground or the background is subject to a NEC-violating contribution to the total stress-energy.
Exact solutions describing the spherical collapse of null fluids can contain regions which violate the energy conditions. Physically the violations occur when the infalling matter continues to move inwards even when non-gravitational repulsive forces become stronger than gravity. In 1991 Ori proposed a resolution for these violations: spacetime surgery should be used to replace the energy condition violating region with an outgoing solution. The matter bounces. We revisit and implement this proposal for the more general Husain null fluids including a careful study of potential discontinuities …
To gain more insights about what could be the source of the Alcubierre warp drive, a technique designed to find an orthogonal basis from the metric expression is used. It is deduced that exotic fluids can produce the same stress-energy tensor as the Alcubierre warp drive. For different configurations of warp acceleration and velocity, constraints on density, pressure, total positive mass, total negative mass, absolute value of the equation of state as well as null-energy condition violation are provided. The Alcubierre warp drive Einstein tensor expression and the optimization of fluid propert…
We investigate a stress-energy tensor for a CFT at strong coupling inside a small five-dimensional rotating Myers-Perry black hole with equal angular momenta by using the holographic method. As a gravitational dual, we perturbatively construct a black droplet solution by applying the "derivative expansion" method, generalizing the work of Haddad (arXiv:1207.2305), and analytically compute the holographic stress-energy tensor for our solution. We find that the stress-energy tensor is finite at both the future and past outer (event) horizons, and that the energy density is negative just outside …
An example is described in which an asymptotically flat static vacuum Weyl space-time experiences a sudden change across a null hypersurface in the multipole moments of its isolated axially symmetric source. A light-like shell and an impulsive gravitational wave are identified, both having the null hypersurface as history. The stress-energy in the shell is dominated (at large distance from the source) by the jump in the monopole moment (the mass) of the source with the jump in the dipole moment mainly responsible for the stress being anisotropic. The gravitational wave owes its existence prrin…
The Barrabès-Israel theory of light-like shells in General Relativity is used to show explicitly that in general a light-like shell is accompanied by an impulsive gravitational wave. The gravitational wave is identified by its Petrov Type N contribution to a Dirac delta-function term in the Weyl conformal curvature tensor (with the delta-function singular on the null hypersurface history of the wave and shell). An example is described in which an asymptotically flat static vacuum Weyl space-time experiences a sudden change across a null hypersurface in the multipole moments of its isolated axi…
We demonstrate that the anisotropic stress-energy supporting the Kiselev black hole can be mimicked by being split into a perfect fluid component plus either an electromagnetic component or a scalar field component, thereby quantifying the precise extent to which the Kiselev black hole fails to represent a perfect fluid spacetime. The perfect fluid component carries either an electric or a scalar charge, which then generates anisotropic electromagnetic or scalar fields. This in turn generates anisotropic contributions to the stress-energy. These in turn induce forces which partially (in additi…
This paper develops a theory of thin shells within the context of the Einstein-Cartan theory by extending the known formalism of general relativity. In order to perform such an extension, we require the general non symmetric stress-energy tensor to be conserved leading, as Cartan pointed out himself, to a strong constraint relating curvature and torsion of spacetime. When we restrict ourselves to the class of space-times satisfying this constraint, we are able to properly describe thin shells and derive the general expression of surface stress-energy tensor both in its four-dimensional and in …
The averaged null energy condition has known violations for quantum fields in curved space, even if one considers only achronal geodesics. Many such examples involve rapid variation in the stress-energy tensor in the vicinity of the geodesic under consideration, giving rise to the possibility that averaging in additional dimensions would yield a principle universally obeyed by quantum fields. However, after discussing various procedures for additional averaging, including integrating over all dimensions of the manifold, we give a class of examples that violate any such averaged condition.
The characteristic initial value problem has been implemented as a robust computational algorithm (the PITT NULL CODE), with direct application to binary black holes. The event horizon can be analyzed by characteristic techniques as a stand-alone object using an analytic conformal model which gives new insight into the intrinsic geometry of binary black holes. When applied to a non-axisymmetric horizon, the model reveals substantially new features. Colliding black holes generically go through a toroidal phase before they become spherical. The conformal structure of the horizon supplies part of…
We report here on a new method for calculating the renormalized stress-energy tensor (RSET) in black-hole (BH) spacetimes, which should also be applicable to dynamical BHs and to spinning BHs. This new method only requires the spacetime to admit a single symmetry. So far we developed three variants of the method, aimed for stationary, spherically symmetric, or axially symmetric BHs. We used this method to calculate the RSET of a minimally-coupled massless scalar field in Schwarzschild and Reissner-Nordstrom backgrounds, for several quantum states. We present here the results for the RSET in th…
It has been known for several decades that Einstein's field equations, when projected onto a null surface, exhibits a structure very similar to non-relativistic Navier-Stokes equation. I show that this result arises quite naturally when gravitational dynamics is viewed as an emergent phenomenon. Extremising the spacetime entropy density associated with the null surfaces leads to a set of equations which, when viewed in the local inertial frame, becomes identical to the Navier-Stokes (NS) equation. This is in contrast with the usual description of Damour-Navier-Stokes (DNS) equation in a genera…
Quantum inequalities are constraints on how negative the weighted average of the renormalized stress-energy tensor of a quantum field can be. A null-projected quantum inequality can be used to prove the averaged null energy condition (ANEC), which would then rule out exotic phenomena such as wormholes and time machines. In this work we derive such an inequality for a massless minimally coupled scalar field, working to first order of the Riemann tensor and its derivatives. We then use this inequality to prove ANEC on achronal geodesics in a curved background that obeys the null convergence cond…
The Averaged Null Energy Condition (ANEC) states that the integral along a complete null geodesic of the projection of the stress-energy tensor onto the tangent vector to the geodesic cannot be negative. ANEC can be used to rule out spacetimes with exotic phenomena, such as closed timelike curves, superluminal travel and wormholes. We prove that ANEC is obeyed by a minimally-coupled, free quantum scalar field on any achronal null geodesic (not two points can be connected with a timelike curve) surrounded by a tubular neighborhood whose curvature is produced by a classical source. To prove ANEC…
In this paper, we describe an extremely efficient method for computing the renormalized stress-energy tensor of a quantum scalar field in spherically-symmetric black hole spacetimes. The method applies to a scalar field with arbitrary field parameters. We demonstrate the utility of the method by computing the renormalized stress-energy tensor for a scalar field in the Schwarzschild black hole spacetime, applying our results to discuss the null energy condition and the semi-classical backreaction.
We study wormhole solutions in the framework of f (R,T) gravity where R is the scalar curvature, and T is the trace of the stress-energy tensor of the matter. We have obtained the shape function of the wormhole by specifying an equation of state for the matter field and imposing the flaring out condition at the throat. We show that in this modified gravity scenario, the matter threading the wormhole may satisfy the energy conditions, so it is the effective stress-energy that is responsible for violation of the null energy condition.
We consider the properties of stress-energy tensors compatible with a Null Big Bang, i.e., cosmological evolution starting from a Killing horizon rather than a singularity. For Kantowski-Sachs cosmologies, it is shown that if matter satisfies the Null Energy Condition (NEC), then (i) regular cosmological evolution can only start from a Killing horizon, (ii) matter is absent at the horizon, and (iii) matter can only appear in the cosmological region due to interaction with vacuum. The latter is understood phenomenologically as a fluid whose stress tensor is insensitive to boosts in a particular…
We use the subleading soft-graviton theorem to construct an operator $T_{zz}$ whose insertion in the four-dimensional tree-level quantum gravity $\mathcal{S}$-matrix obeys the Virasoro-Ward identities of the energy momentum tensor of a two-dimensional conformal field theory (CFT$_2$). The celestial sphere at Minkowskian null infinity plays the role of the Euclidean sphere of the CFT$_2$, with the Lorentz group acting as the unbroken $SL(2,\mathbb{C})$ subgroup.
Modes emerging out of a collapsing black hole are red-shifted to such an extent that Hawking radiation at future null infinity consists of modes that have energies beyond the Planck scale at past null infinity. This indicates that physics at the Planck scale may modify the spectrum of Hawking radiation and the associated stress-energy tensor of the quantum field. Recently, it has been shown that, the T-duality symmetry of string fluctuations along compact extra dimensions leads to a modification of the standard propagator of point particles in quantum field theory. At low energies (when compar…
