The group of rigid motions is considered to guide the search for a natural system of space-time coordinates in General Relativity. This search leads us to a natural extension of the space-times that support Painlevé--Gullstrand synchronization. As an interesting example, here we describe a system of rigid coordinates for the cross mode of gravitational linear plane waves.
Following on from two recent papers, here we examine the relationship between Newtonian gravitation and general relativity in more depth. This allows us to define a scalar potential which is just the proper time of the vector potential when the latter is interpreted as the geodesic velocity field. The results are closely related to spacetimes that admit Painlevé--Gullstrand synchronization.
We conduct numerical simulations in Painlevé-Gullstrand coordinates of a variety of K-essence-type scalar fields under spherically symmetric gravitational collapse. We write down generic conditions on the K-essence lagrangian that can be used to determine whether superluminality and Cauchy breakdown are possible. Consistent with these conditions, for specific choices of K-essence-type fields we verify the presence of superluminality during collapse, while for other type we do not. We also demonstrate that certain choices of K-essence scalar fields present issues under gravitational collapse in…
The Painleve-Gullstrand coordinates provide a convenient framework for presenting the Schwarzschild geometry because of their flat constant-time hypersurfaces, and the fact that they are free of coordinate singularities outside r=0. Generalizations of Painleve-Gullstrand coordinates suitable for the Kerr geometry have been presented by Doran and Natario. These coordinate systems feature a time coordinate identical to the proper time of zero-angular-momentum observers that are dropped from infinity. Here, the methods of Doran and Natario are extended to the five-dimensional rotating black hole …
A Painlevé-Gullstrand synchronization is a slicing of the space-time by a family of flat spacelike 3-surfaces. For spherically symmetric space-times, we show that a Painlevé-Gullstrand synchronization only exists in the region where $(dr)^2 \leq 1$, $r$ being the curvature radius of the isometry group orbits ($2$-spheres). This condition says that the Misner-Sharp gravitational energy of these 2-spheres is not negative and has an intrinsic meaning in terms of the norm of the mean extrinsic curvature vector. It also provides an algebraic inequality involving the Weyl curvature scalar and the Ri…
We construct an exact solution for the spherical gravitational collapse in a single coordinate patch. To describe the dynamics of collapse, we use a generalized form of the Painlevé-Gullstrand coordinates in the Schwarzschild spacetime. The time coordinate of the form is the proper time of a free-falling observer so that we can describe the collapsing star not only outside but also inside the event horizon in a single coordinate patch. We show the both solutions corresponding to the gravitational collapse from infinity and from a finite radius.
Painlevé-Gullstrand coordinates, a very useful tool in spherical horizon thermodynamics, fail in anti-de Sitter space and in the inner region of Reissner-Nordström. We predict this breakdown to occur in any region containing negative Misner-Sharp-Hernandez quasilocal mass because of repulsive gravity stopping the motion of PG observers in radial free fall with zero initial velocity. PG coordinates break down also in the static Einstein universe for completely different reasons. The more general Martel-Poisson family of charts, which normally has PG coordinates as a limit, is reported for stati…
We discuss the tension between the possible existence of Painleve-Gullstrand coordinate systems versus the explicit geometrical features of the Kerr spacetime; a subject of interest to Professor Thanu Padmanabhan in the weeks immediately preceding his unexpected death. We shall carefully distinguish strong and weak Painleve-Gullstrand coordinate systems, and conformal variants thereof, cataloguing what we know can and cannot be done -- sometimes we can make explicit global statements, sometimes we must resort to implicit local statements. For the Kerr spacetime the best that seems to be achiev…
In a previous paper we have considered the Regge-Wheeler equation for fields of spin $s=0$, $1$ or $2$ on the Schwarzschild spacetime in coordinates that are regular at the horizon. In particular, we have constructed in Eddington-Finkelstein coordinates exact solutions in terms of series that are regular at the horizon and converge on the entire open domain from the central singularity to infinity. Here we extend this earlier work in two different directions. Firstly, we consider in Eddington-Finkelstein coordinates a massive scalar field that can serve as a dark matter candidate. Secondly, we…
A nonstatic Schwarzschild black hole interior solution in Painleve-Gullstrand coordinates is proposed in this paper, by means of a coordinate transformation that changes the spatial coordinate but the timelike one is preserved. The timelike and null geodesic equations are obtained exactly, taking advantage that the spatial z-coordinate is cyclic.
We construct a coordinate system for the Kerr solution, based on the zero angular momentum observers dropped from infinity, which generalizes the Painleve-Gullstrand coordinate system for the Schwarzschild solution. The Kerr metric can then be interpreted as describing space flowing on a (curved) Riemannian 3-manifold. The stationary limit arises as the set of points on this manifold where the speed of the flow equals the speed of light, and the horizons as the set of points where the radial speed equals the speed of light. A deeper analysis of what is meant by the flow of space reveals that t…
We perform a numerical study of black hole formation from the spherically symmetric collapse of a massless scalar field. The calculations are done in Painlevé-Gullstrand (PG) coordinates that extend across apparent horizons and allow the numerical evolution to proceed until the onset of singularity formation. We generate spacetime maps of the collapse and illustrate the evolution of apparent horizons and trapping surfaces for various initial data. We also study the critical behaviour and find the expected Choptuik scaling with universal values for the critical exponent and echoing period consi…
We investigate a foliation of Schwarzschild spacetime determined by observers freely falling in the radial direction. This is described using a generalisation of Gullstrand-Painlevé coordinates which allows for any possible radial velocity. This foliation provides a contrast with the usual static foliation implied by Schwarzschild coordinates. The $3$-dimensional spaces are distinct for the static and falling observers, so the embedding diagrams, spatial measurement, simultaneity, and time at infinity are also distinct, though the $4$-dimensional spacetime is unchanged. Our motivation is conce…
Making use of the higher dimensional global embedding Minkowski spacetime (GEMS), we embed (3+1)-dimensional Schwarzschild and Reissner-Nordström (RN) black holes written by the Painlevé-Gullstrand (PG) spacetimes, which have off-diagonal components in metrics, into (5+1)- and (5+2)-dimensional flat ones, respectively. As a result, we have shown the equivalence of the GEMS embeddings of the spacetimes with the diagonal and off-diagonal terms in metrics. Moreover, with the aid of their geodesic equations satisfying various boundary conditions in the flat embedded spacetimes, we directly obtain …
It is well known that static spherically symmetric spacetimes can admit foliations by flat spacelike hypersurfaces, which are best described in terms of the Painlevè--Gullstrand coordinates. The uniqueness and existence of such foliations were addressed earlier. In this paper, we prove, purely geometrically, that any possible foliation of a static spherically symmetric spacetime by an arbitrary codimension-one spherical spacelike geometry, up to time translation and rotation, is unique, and we find the algebraic condition under which it exists. This leads us to what can be considered as the mo…
The properties of an anisotropic fluid outside a star or a black hole embedded in an expanding universe are investigated. One finds that, in Painleve-Gullstrand coordinates, the heat flux of the cosmological fluid vanishes, in spite of the nonzero value of a non-diagonal component of the stress tensor. The pressures and energy density of the fluid are regular at $r = 2m$, divergent at $r = 0$ and change sign at certain values of their arguments.
An obstruction to the implementation of spatially flat Painleve-Gullstrand(PG) slicings is demonstrated, and explicitly discussed for Reissner-Nordstrom and Schwarzschild-anti-deSitter spacetimes. Generalizations of PG slicings which are not spatially flat but which remain regular at the horizons are introduced. These metrics can be obtained from standard spherically symmetric metrics by physical Lorentz boosts. With these generalized PG metrics, problematic contributions to the imaginary part of the action in the Parikh-Wilczek derivation of Hawking radiation due to the obstruction can be avo…
We investigate Choptuik scaling in the spherically symmetric collapse of a massless scalar field in higher dimensions using Painleve-Gullstrand (P-G) coordinates. Our analysis confirms the presence in higher dimensions of the cusps in the periodic scaling relationship recently observed in four dimensional collapse. In addition, we address the issue of the asymptotic behaviour of the critical exponent as the number of spacetime dimensions gets large. Our results are consistent with earlier work suggesting that the critical exponent monotonically approaches 1/2 in this limit.
Painleve-Gullstrand metric of the black hole allows to discuss the fermion zero modes inside the hole. The statistical mechanics of the fermionic microstates can be responsible for the black hole thermodynamics. Fermion zero modes also lead to quantization of the horizon area.
The principle of equivalence is used to examine covariant descriptions of quantum phenomena within the global exterior of geometries described using Painlevé- Gullstrand coordinates, which are everywhere non-singular away from their center. The differences between the descriptions of observers stationary within the curvilinear geometry and those in a locally-flat freely-falling system that is momentarily at rest are found to depend only on local proper accelerations, becoming vanishingly small for increasingly distant observers. Away from the horizon, the local detections of outgoing massless …
Cosmology is most typically analyzed using standard co-moving coordinates, in which the galaxies are (on average, up to presumably small peculiar velocities) "at rest", while "space" is expanding. But this is merely a specific coordinate choice; and it is important to realise that for certain purposes other, (sometimes radically different) coordinate choices might also prove useful and informative, but without changing the underlying physics. Specifically, herein we shall consider the k=0 spatially flat FLRW cosmology but in Painleve-Gullstrand coordinates -- these coordinates are very explici…
We consider quasiparticle propagation in constant-speed-of-sound (iso-tachic) and almost incompressible (iso-pycnal) hydrodynamic flows, using the technical machinery of general relativity to investigate the ``effective space-time geometry'' that is probed by the quasiparticles. This effective geometry, described for the quasiparticles of condensed matter systems by the Painleve-Gullstrand metric, generally exhibits curvature (in the sense of Riemann), and many features of quasiparticle propagation can be re-phrased in terms of null geodesics, Killing vectors, and Jacobi fields. As particular …
Herein we explore the non-equatorial constant-$r$ ("quasi-circular") geodesics (both timelike and null) in the Painleve-Gullstrand variant of the Lense-Thirring spacetime recently introduced by the current authors. Even though the spacetime is not spherically symmetric, shells of constant-$r$ geodesics still exist. Whereas the radial motion is (by construction) utterly trivial, determining the allowed locations of these constant-$r$ geodesics is decidedly non-trivial, and the stability analysis is equally tricky. Regarding the angular motion, these constant-$r$ orbits will be seen to exhibit b…
It is demonstrated that with using Painlev{é}-Gullstrand coordinates in their quasi-Cartesian variant, the Hamiltonian functional for relativistic perfect fluid hydrodynamics near a non-rotating black hole differs from the corresponding flat-spacetime Hamiltonian just by a simple term. Moreover, the internal region of the black hole is then described uniformly together with the external region, because in Painlev{é}-Gullstrand coordinates there is no singularity at the event horizon. An exact solution is presented which describes stationary accretion of an ultra-hard matter ($\varepsilon\propt…
The Painleve-Gullstrand coordinates are extended to describe the black hole in the cosmological environment: the Schwarzschild-de-Sitter black hole, which has two horizons. The extension is made using the Arnowitt-Deser-Misner formalism. In this extension, which describes the metric in the whole range of radial coordinates $0<r < \infty$, there is the point $r=r_0$ at which the shift function (velocity) changes sign. At this point the observer is at rest, while the observers at $r<r_0$ are free falling to the black hole and the observers at $r>r_0$ are free falling towards the cosmological hor…
Generalized Painleve-Gullstrand metrics are explicitly constructed for the Kerr-Newman family of charged rotating black holes. These descriptions are free of all coordinate singularities; moreover, unlike the Doran and other proposed metrics, an extra tunable function is introduced to ensure all variables in the metrics remain real for all values of the mass M, charge Q, angular momentum aM, and cosmological constant Λ> - 3/(a^2). To describe fermions in Kerr-Newman spacetimes, the stronger requirement of non-singular vierbein one-forms at the horizon(s) is imposed and coordinate singularities…
We consider the quantum vacuum of fermionic field in the presence of a black-hole background as a possible candidate for the stabilized black hole. The stable vacuum state (as well as thermal equilibrium states with arbitrary temperature) can exist if we use the Painlevé-Gullstrand description of the black hole, and the superluminal dispersion of the particle spectrum at high energy, which is introduced in the free-falling frame. Such choice is inspired by the analogy between the quantum vacuum and the ground state of quantum liquid, in which the event horizon for the low-energy fermionic quas…
We find a specific coordinate system that goes from the Painlevé-Gullstrand partial extension to the Kruskal-Szekeres maximal extension and thus exhibit the maximal extension of the Schwarzschild metric in a unified picture. We do this by adopting two time coordinates, one being the proper time of a congruence of outgoing timelike geodesics, the other being the proper time of a congruence of ingoing timelike geodesics, both parameterized by the same energy per unit mass $E$. $E$ is in the range $1\leq E<\infty$ with the limit $E=\infty$ yielding the Kruskal-Szekeres maximal extension. So, thro…
The continuation of the Schwarzschild metric across the event horizon is almost always (in textbooks) carried out using the Kruskal-Szekeres coordinates, in terms of which the areal radius r is defined only implicitly. We argue that from a pedagogical point of view, using these coordinates comes with several drawbacks, and we advocate the use of simpler, but equally effective, coordinate systems. One such system, introduced by Painleve and Gullstrand in the 1920's, is especially simple and pedagogically powerful; it is, however, still poorly known today. One of our purposes here is therefore t…
In the paper we analyze the quantum-mechanical equivalence of the metrics of a centrally symmetric uncharged gravitational field. We consider the Schwarzschild metrics in the spherical, isotropic and harmonic coordinates, and the Eddington-Finkelstein, Painleve-Gullstrand, Lemaitre-Finkelstein, Kruskal metrics. The scope of the analysis includes domains of the wave functions of Dirac's equation, hermiticity of Hamiltonians, and the possibility of existence of stationary bound states of spin-half particles. The constraint on the domain of the wave functions of the Hamiltonian in a Schwarzschild…
An explicit fluid flow simulation of electromagnetic wave propagation in the gravitational field of a Schwarzschild black hole is given. The fluid has a constant refractive index and a spherically symmetric inward directed flow. The resulting form of the metric leads to a new coordinate system in which the Schwarzschild vacuum is written in Gordon's form. It is shown that a closely related coordinate system interpolates between the Kerr-Schild and Painleve-Gullstrand coordinates.
In terms of the Painlev{é}-Gullstrand-Lema{\^ı}tre coordinates a rather general scenario for the gravitational collapse of an object and the subsequent formation of a horizon is described by a manifestly $C^\infty$-metric. For a 1+1 dimensional model of the collapse the leading contributions to the Bogoliubov coefficients are calculated explicitely and the Hawking temperature is recovered. But depending on the particular dynamics of the collapse the final state represents either evaporation or anti-evaporation. The generalization of the calculation to 3+1 dimensions is outlined and possible im…
A simple observation about the action for geodesics in a stationary spacetime with separable geodesic equations leads to a natural class of slicings of that spacetime whose orthogonal geodesic trajectories represent freely falling observers. The time coordinate function can then be taken to be the observer proper time, leading to a unit lapse function. This explains some of the properties of the original Painlevé-Gullstrand coordinates on the Schwarzschild spacetime and their generalization to the Kerr-Newman family of spacetimes, reproducible also locally for the Gödel spacetime. For the stat…
Formulating a perfect fluid filled spherically symmetric metric utilizing the 3+1 formalism for general relativity, we show that the metric coefficients are completely determined by the mass-energy distribution, and its time rate of change on an initial spacelike hypersurface. Rather than specifying Schwarzschild coordinates for the exterior of the collapsing region, we let the interior dictate the form of the solution in the exterior, and thus both regions are found to be written in one coordinate patch. This not only alleviates the need for complicated matching schemes at the interface, but …
A strict derivation of the Schwarzschild metric, based solely on Newton's law of free fall and the equivalence principle, is presented. In the light of it, regarding Schwarzschild's coordinates as representing the point of view of a distant observer resting relative to the source of a centrally symmetric gravitational field, proves illegitimate. Such point of view is better represented by the Painleve-Gullstrand system of coordinates, which agrees with Schwarzschild's system with respect to its spatial coordinates and time scale, but disagrees with respect to the relation of simultaneity. A du…
We explore the connections between various coordinate systems associated with observers moving inwardly along radial geodesics in the Schwarzschild geometry. Painlevé-Gullstrand (PG) time is adapted to freely falling observers dropped from rest from infinity; Lake-Martel-Poisson (LMP) time coordinates are adapted to observers who start at infinity with non-zero initial inward velocity; Gautreau-Hoffmann (GH) time coordinates are adapted to observers dropped from rest from a finite distance from the black hole horizon. We construct from these an LMP family and a proper-time family of time coord…
Most general relativity textbooks devote considerable space to the simplest example of a black hole containing a singularity, the Schwarzschild geometry. However only a few discuss the dynamical process of gravitational collapse, by which black holes and singularities form. We present here two types of analytic models for this process, which we believe are the simplest available; the first involves collapsing spherical shells of light, analyzed mainly in Eddington-Finkelstein coordinates; the second involves collapsing spheres filled with a perfect fluid, analyzed mainly in Painleve-Gullstrand…
To see the back reaction of the Hawking radiation in a dynamical spacetime of the spherical gravitational collapse, we explicitly calculate the weak value of the energy-momentum tensor of the massless scalar field. The background geometry of a collapsing dust sphere is specified by using the Painleve-Gullstrand coordinates, in which the time coordinate coincides with the proper time of a free-falling observer and the metric tensor is regular at the event horizon. The result is that in the remote future the weak value diverges at the event horizon. We argue that since the semi-classical approxi…
The propagation of (massless) scalar, electromagnetic and gravitational waves on fixed Schwarzschild background spacetime is described by the general time-dependent Regge-Wheeler equation. We transform this wave equation to usual Schwarzschild, Eddington-Finkelstein, Painleve-Gullstrand and Kruskal-Szekeres coordinates. In the first three cases, but not in the last one, it is possible to separate a harmonic time-dependence. Then the resulting radial equations belong to the class of confluent Heun equations, i.e., we can identify one irregular and two regular singularities. Using the generalize…
We develop an effective framework for the $\barμ$ scheme of holonomy corrections motivated by loop quantum gravity for vacuum spherically symmetric space-times. This is done by imposing the areal gauge in the classical theory, and then expressing the remaining components of the Ashtekar-Barbero connection in the Hamiltonian constraint in terms of holonomies of physical length $\ell_{\rm Pl}$. The stationary solutions to the effective Hamiltonian constraint can be found exactly, and we give the explicit form of the effective metric in Painlevé-Gullstrand coordinates. This solution has the corre…
There is widespread disagreement about how the general covariance of a theory affects its quantization. Without a complete quantum theory of gravity, one can examine quantum consequences of coordinate choices only in highly idealized `toy' models. In this work, we extend our previous analysis of a self-gravitating shell model [1], and demonstrate that coordinate freedom can be retained in a reduced phase space description of the system. We first consider a family of coordinate systems discussed by Martel and Poisson [2], which have time coordinates that coincide with the proper times of ingoin…
