Cauchy Surface Research Articles

Analytic Hadamard States, Calderón Projectors and Wick Rotation Near Analytic Cauchy Surfaces

We consider the Klein–Gordon equation on analytic spacetimes with an analytic Cauchy surface. In this setting, we prove the existence of pure analytic Hadamard states. The proof is based on considering an elliptic operator obtained by Wick rotating the Klein–Gordon operator in a neighborhood of a Cauchy hypersurface. The Cauchy data of Hadamard two-point functions are constructed as Calderon projectors (suitably generalized if the hypersurface is non-compact) for the elliptic operator.

Rough initial data and the strength of the blue-shift instability on cosmological black holes with Λ > 0

We consider the wave equation on Reissner–Nordström–de Sitter and more generally Kerr–Newman–de Sitter black hole spacetimes with . The strength of the blue-shift instability associated to the Cauchy horizon of these spacetimes has been the subject of much discussion, since—in contrast to the asymptotically flat case—the competition with the decay associated to the region between the event and cosmological horizons is delicate, especially as the extremal limit is approached. Of particular interest is the question as to whether generic, admissible initial data posed on a Cauchy surface lead to solutions whose local (integrated) energy blows up at the Cauchy horizon, for this statement holds in the asymptotically flat case and would correspond precisely to the blow up required by Christodoulou’s formulation of strong cosmic censorship. Some recent heuristic work suggests that the answer is in general negative for solutions arising from sufficiently smooth data, i.e. there exists a certain range of black hole parameters such that for all such data, the arising solutions have finite local (integrated) energy at the Cauchy horizon. In this short note, we shall show in contrast that, by slightly relaxing the smoothness assumption on initial data, we are able to prove the analogue of the Christodoulou statement in the affirmative, i.e. we show that for generic data in our allowed class, the local energy blow-up statement indeed holds at the Cauchy horizon, for all subextremal black hole parameter ranges. We present two distinct proofs. The first is based on an explicit mode construction while the other is softer and uses only time translation invariance of appropriate scattering maps, in analogy with our previous (Dafermos and Shlapentokh-Rothman 2017 Commun. Math. Phys. 350 985–1016). Both proofs use statements concerning the non-triviality of transmission and reflexion, which are easy to infer by o.d.e. techniques and analyticity considerations. Our slightly enlarged class of initial data is still sufficiently regular to ensure both stability and decay properties in the region between the event and cosmological horizons as well as the boundedness and continuous extendibility beyond the Cauchy horizon. This suggests thus that it is finally this class—and not smoother data—which may provide the correct setting to formulate the genericity condition in strong cosmic censorship.

A singularity theorem for Einstein\u2013Klein\u2013Gordon theory

Hawking’s singularity theorem concerns matter obeying the strong energy condition (SEC), which means that all observers experience a nonnegative effective energy density (EED), thereby guaranteeing the timelike convergence property. However, there are models that do not satisfy the SEC and therefore lie outside the scope of Hawking’s hypotheses, an important example being the massive Klein–Gordon field. Here we derive lower bounds on local averages of the EED for solutions to the Klein–Gordon equation, allowing nonzero mass and nonminimal coupling to the scalar curvature. The averages are taken along timelike geodesics or over spacetime volumes, and our bounds are valid for a range of coupling constants including both minimal and conformal coupling. Using methods developed by Fewster and Galloway, these lower bounds are applied to prove a Hawking-type singularity theorem for solutions to the Einstein–Klein–Gordon theory, asserting that solutions with sufficient initial contraction at a compact Cauchy surface will be future timelike geodesically incomplete. These results remain true in the presence of additional matter obeying both the strong and weak energy conditions.

Renyi relative entropies and renormalization group flows

Quantum Renyi relative entropies provide a one-parameter family of distances between density matrices, which generalizes the relative entropy and the fidelity. We study these measures for renormalization group flows in quantum field theory. We derive explicit expressions in free field theory based on the real time approach. Using monotonicity properties, we obtain new inequalities that need to be satisfied by consistent renormalization group trajectories in field theory. These inequalities play the role of a second law of thermodynamics, in the context of renormalization group flows. Finally, we apply these results to a tractable Kondo model, where we evaluate the Renyi relative entropies explicitly. An outcome of this is that Anderson’s orthogonality catastrophe can be avoided by working on a Cauchy surface that approaches the light-cone.

The Poisson brackets of free null initial data for vacuum general relativity

A hypersurface composed of two null sheets, or ‘light fronts’, swept out by the two congruences of future null normal geodesics emerging from a spacelike 2-disk can serve as a Cauchy surface for a region of spacetime. Already in the 1960s free (unconstrained) initial data for vacuum general relativity were found for hypersurfaces of this type. Here the Poisson brackets of such free initial data are calculated from the Hilbert action. The brackets obtained can form the starting point for a constraint free canonical quantization of general relativity and may be relevant to holographic entropy bounds for vacuum gravity. Several of the results of the present work have been presented in abbreviated form in the letter (Reisenberger 2008 Phys. Rev. Lett. 101 211101).

Existence of CMC Cauchy surfaces from a spacetime curvature condition

In this note we present a result establishing the existence of a compact CMC Cauchy surface from a curvature condition related to the strong energy condition.

Causality and Legendrian linking for higher dimensional spacetimes

Let ( X m + 1 , g ) be an ( m + 1 ) -dimensional globally hyperbolic spacetime with Cauchy surface M m , and let M ˜ m be the universal cover of the Cauchy surface. Let N X be the contact manifold of all future directed unparameterized light rays in X that we identify with the spherical cotangent bundle S T ∗ M . Jointly with Stefan Nemirovski we showed that when M ˜ m is not a compact manifold, then two points x , y ∈ X are causally related if and only if the Legendrian spheres S x , S y of all light rays through x and y are linked in N X . In this short note we use the contact Bott–Samelson theorem of Frauenfelder, Labrousse and Schlenk to show that the same statement is true for all X for which the integral cohomology ring of a closed M ˜ is not the one of the CROSS (compact rank one symmetric space). If M admits a Riemann metric g ¯ , a point x and a number ℓ > 0 such that all unit speed geodesics starting from x return back to x in time ℓ , then ( M , g ¯ ) is called a Y ℓ x manifold. Jointly with Stefan Nemirovski we observed that causality in ( M × R , g ¯ ⊕ − t 2 ) is not equivalent to Legendrian linking. Every Y ℓ x -Riemann manifold has compact universal cover and its integral cohomology ring is the one of a CROSS. So we conjecture that Legendrian linking is equivalent to causality if and only if one can not put a Y ℓ x Riemann metric on a Cauchy surface M .

Quantum reference frames via transition amplitudes in timeless quantum gravity

We propose an algorithm of extracting Schr\"odinger theories under all viable physical time from the Einstein-Hilbert path integral, formulated as the timeless transition amplitudes $\hat{\mathbb{P}}:\mathbb{K} \to \mathbb{K}^*$ between the boundary states in a kinematic Hilbert space $\mathbb{K}$. Each of these Schr\"odinger theories refers to a certain set of quantum degrees of freedom in $\mathbb{K}$ as a background, with their given values specifying moments of the physical time. Restricted to these specified background values, the relevant elements of $\hat{\mathbb{P}}$ are transformed by the algorithm into the unitary propagator of a corresponding reduced phase space Schr\"odinger theory. The algorithm embodies the fundamental principle of quantum Cauchy surfaces, such that all the derived Schr\"odinger theories emerge from one timeless canonical theory defined by $\hat{\mathbb{P}}$ as a rigging map, via the relational Dirac observables referring to the corresponding backgrounds. We demonstrate its application to a FRW loop quantum cosmological model with a massless Klein-Gordon scalar field. Recovering the famous singularity-free quantum gravitational dynamics with the background of the scalar field, we also obtain in another reference frame a modified Klein-Gordon field quantum dynamics with the background of the spatial (quantum) geometry.

Embeddings of non-positively curved compact surfaces in flat Lorentzian manifolds

We prove that any metric of non-positive curvature in the sense of Alexandrov on a compact surface can be isometrically embedded as a convex spacelike Cauchy surface in a flat spacetime of dimension $$(2+1).$$ The proof follows from polyhedral approximation.

Gravitational closure of matter field equations

The requirement that both the matter and the geometry of a spacetime canonically evolve together, starting and ending on shared Cauchy surfaces and independently of the intermediate foliation, leaves one with little choice for diffeomorphism-invariant gravitational dynamics that can provide consistent evolution equations to the coefficients of a given system of matter field equations. Concretely, we show how starting from any linear local matter field equations whose principal polynomial satisfies three physicality conditions, one may calculate coefficient functions which then enter an otherwise immutable set of countably many linear homogeneous partial differential equations. Any solution of these so-called gravitational closure equations then provides a Lagrangian density for any type of tensorial geometry that features ultralocally in the initially specified matter Lagrangian density. Thus the given system of matter field equations is indeed closed by the so obtained gravitational equations. In contrast to previous work, we build the theory on a suitable associated bundle encoding the canonical configuration degrees of freedom, which allows to include necessary constraints on the geometry in practically tractable fashion. By virtue of the presented mechanism, one thus can practically calculate, rather than having to postulate, the gravitational theory that is required by specific matter field dynamics. For the special case of standard model matter one obtains general relativity.

Rigidity in vacuum under conformal symmetry

Moitvated in part by [3], in this note we obtain a rigidity result for globally hyperbolic vacuum spacetimes in arbitrary dimension that admit a timelike conformal Killing vector field. Specifically, we show that if M is a Ricci flat, timelike geodesically complete spacetime with compact Cauchy surfaces that admits a timelike conformal Killing field X, then M must split as a metric product, and X must be Killing. This gives a partial proof of the Bartnik splitting conjecture in the vacuum setting.

The wave equation in Friedmann-Robertson-Walker space-times and asymptotics of the intensity and distance relationship of a localised source

Variations in redshift measurements of Type 1a supernovae and intensity observations from large sky surveys are an indicator of a component of acceleration in the rate of expansion of space-time. A key factor in the measurements is the intensity-distance relation for Maxwell’s equations in Friedmann-Robertson-Walker (FRW) space-times. In view of future measurements of the decay of other fields on astronomical time and spatial scales, we determine the asymptotic behavior of the intensity-distance relationship for the solution of the wave equation in space-times with an FRW metric. This builds on previous work done on initial value problems for the wave equation in FRW space-time [Abbasi, B. and Craig, W., Proc. R. Soc. London, Ser. A 470, 20140361 (2014)]. In this paper, we focus on the precise intensity decay rates of the special cases for curvature k = 0 and k = −1, as well as giving a general derivation of the wave solution for −∞ < k < 0. We choose a Cauchy surface {(t, x) : t = t0 > 0} where t0 represents the time of an initial emission source, relative to the Big Bang singularity at t = 0. The initial data [g(x), h(x)] are assumed to be compactly supported; supp(g, h) ⊆ BR(0) and terms in the expression for the fundamental solution for the wave equation with the slowest decay rate are retained. The intensities calculated for coordinate time {t : t > 0} contain correction terms proportional to the ratio of t0 and the time differences ρ = t − t0. For the case of general curvature k, these expressions for the intensity reduce by scaling to the same form as for k = −1, from which we deduce the general formula. We note that for typical astronomical events such as Type 1a supernovae, the first order correction term for all curvatures −∞ < k < 0 is on the order of 10−4 smaller than the zeroth order term. These correction terms are small but may be significant in applications to alternative observations of cosmological space-time expansion rates.

Remarks on entanglement entropy in string theory

Entanglement entropy for spatial subregions is difficult to define in string theory because of the extended nature of strings. Here we propose a definition for Bosonic open strings using the framework of string field theory. The key difference (compared to ordinary quantum field theory) is that the subregion is chosen inside a Cauchy surface in the "space of open string configurations". We first present a simple calculation of this entanglement entropy in free light-cone string field theory, ignoring subtleties related to the factorization of the Hilbert space. We reproduce the answer expected from an effective field theory point of view, namely a sum over the one-loop entanglement entropies corresponding to all the particle-excitations of the string, and further show that the full string theory regulates the ultraviolet divergences in the entanglement entropy. We then revisit the question of factorization of the Hilbert space by analyzing the covariant phase-space associated with a subregion in Witten's covariant string field theory. We show that the pure gauge (i.e., BRST exact) modes in the string field become dynamical at the entanglement cut. Thus, a proper definition of the entropy must involve an extended Hilbert space, with new stringy edge modes localized at the entanglement cut.

Linearization instability for generic gravity in AdS spacetime

In general relativity, perturbation theory about a background solution fails if the background spacetime has a Killing symmetry and a compact spacelike Cauchy surface. This failure, dubbed as {\it linearization instability}, shows itself as non-integrability of the perturbative infinitesimal deformation to a finite deformation of the background. Namely, the linearized field equations have spurious solutions which cannot be obtained from the linearization of exact solutions. In practice, one can show the failure of the linear perturbation theory by showing that a certain quadratic (integral) constraint on the linearized solutions is not satisfied. For non-compact Cauchy surfaces, the situation is different and for example, Minkowski space having a non-compact Cauchy surface, is linearization stable. Here we study, the linearization instability in generic metric theories of gravity where Einstein's theory is modified with additional curvature terms. We show that, unlike the case of general relativity, for modified theories even in the non-compact Cauchy surface cases, there are some theories which show linearization instability about their anti-de Sitter backgrounds. Recent $D$ dimensional critical and three dimensional chiral gravity theories are two such examples. This observation sheds light on the paradoxical behavior of vanishing conserved charges (mass, angular momenta) for non-vacuum solutions, such as black holes, in these theories.

Brunn-Minkowski Theory and Cauchy's Surface Area Formula

We use Brunn-Minkowski theory and well-known integration tools to prove Cauchy's surface area formula, which states that the average area of a projection of a convex body is equal to its surface area up to a multiplicative constant in the dimension. Using the same technique, we give a straightforward extension to intrinsic moment vectors of a convex body.

Topology and Singularities in Cosmological Spacetimes Obeying the Null Energy Condition

We consider globally hyperbolic spacetimes with compact Cauchy surfaces in a setting compatible with the presence of a positive cosmological constant. More specifically, for 3+1 dimensional spacetimes which satisfy the null energy condition and contain a future expanding compact Cauchy surface, we establish a precise connection between the topology of the Cauchy surfaces and the occurrence of past singularities. In addition to (a refinement of) the Penrose singularity theorem, the proof makes use of some recent advances in the topology of 3-manifolds and of certain fundamental existence results for minimal surfaces.

Redshift and contact forms

It is shown that the redshift between two Cauchy surfaces in a globally hyperbolic spacetime equals the ratio of the associated contact forms on the space of light rays of that spacetime.

Reconstruction of plasma shape and eddy current profile based on modified cauchy condition surface method in merging spherical tokamak

The spherical tokamak merging start-up is one of the central solenoid-free start-up method. For controlling the merging actively, it is essential to identify the plasma shapes during the merging process. In the present work, the modified Cauchy condition surface method is applied to demonstrate the reconstruction of the plasma boundary shapes as well as the eddy current profiles at three different representative phases in the merging start-up process (a. initial phase, b. merging phase, c. after merging). Profiles of magnetic flux and plasma boundary have been reconstructed accurately using noiseless magnetic sensor signals and reconstructed with a fairly good accuracy using sensor signals with 3% noise. The eddy current profile have been also reconstructed with a fairly accuracy.

Upper bound for diameter of cosmological black holes and nonexistence of black strings

The diameter of the apparent horizon, defined by the distance between furthest points on the horizon, in spacetimes with a positive cosmological constant $\varLambda$ has been investigated. It is established that the diameter of the apparent horizon on the totally umbilic partial Cauchy surface cannot exceed $2\pi/\sqrt{3\varLambda}$. Then, it is argued that arbitrarily long black strings cannot be formed in our universe.

Distinguishability of black hole microstates

We use the Holevo information to estimate distinguishability of microstates of a black hole in anti-de Sitter space by measurements one can perform on a subregion of a Cauchy surface of the dual conformal field theory. We find that microstates are not distinguishable at all until the subregion reaches a certain size and that perfect distinguishability can be achieved before the subregion covers the entire Cauchy surface. We will compare our results with expectations from the entanglement wedge reconstruction, tensor network models, and the bit threads interpretation of the Ryu-Takayanagi formula.