In particular with p 0 the metric reduces to the usual eddingtonfinkelstein coordinates 1. The metric ds2 is singular only at r note that r is not the radius of anything. Gullstrand painlev e gp coordinates were discovered by allvar gullstrand and paul painlev e in 19211922. The ingoing coordinates are such that the time coordinate follows the proper time of a freefalling observer who starts from far away at zero velocity, and the spatial slices are flat. In einsteins theory of general relativity, the schwarzschild metric also known as the schwarzschild vacuum or schwarzschild solution is the solution to the einstein field equations that describes the gravitational field outside a spherical mass, on the assumption that the electric charge of the mass, angular momentum of the mass, and universal cosmological constant are all zero. This makes gp coordinates a natural choice for a model of the gravitational force by an exchanged particle which we will call the graviton. Pdf regular coordinate systems for schwarzschild and other. Gullstrand painlev e gp coordinates were discovered by allvar gullstranda 1 and paul painlev e 2 in 19211922. The gp observer uses another, identical ribbon make it blue and lays it down between r2 and r1. For an outward moving photon, we get dr q 2gm r 1 dt.
Physics stack exchange is a question and answer site for active researchers, academics and students of physics. Painlevegullstrand pg coordinate system, the metric is not diagonal, but asymptotically flat. The force of gravity in schwarzschild and gullstrand. Generalized painlevegullstrand metrics request pdf. Show that these coordinates cover both regions i and ii of the maximally extended schwarzschild solution, and that the metric of the constant time surfaces tp const is the. Geodesics in generalized painlevegullstrand coordinates and. Gravitational collapse in painlev egullstrand coordinates. Different metric solutions and coordinates describing a black hole will be studied in relation to these transformations of parity, time, and energy mass. The blue line is an example of one of the v constant lines. Classical limit for dirac fermions with modified action in. The riemann tensor is computed from the metric of the space. Again, the motion along the red curve is up and to the left.
Since other coordinates like the ingoing eddingtonfinkelstein coordinates are far superior. The painlevegullstrand extension a black hole fallacy the term. But avoid asking for help, clarification, or responding to other answers. In these coordinates, the horizon is the black hole horizon nothing can.
Gullstrand metric tensor has an off diagonal element so that it is regular at the schwarzschild radius and has a. The kerr metric can then be interpreted as describing space flowing on. The kerr metric can then be interpreted as describing space flowing on a curved riemannian 3manifold. We construct a coordinate system for the kerr solution, based on the zero angular momentum observers dropped from infinity, which generalizes the painlevegullstrand coordinate system for the schwarzschild solution. Among the most popular coordinate systems that are regular at the horizon are the kruskalszekeres and eddingtonfinkelstein coordinates. Generalized painleve gullstrand metrics chunyu lin. Jan 27, 2011 the spreadsheet i have produced to get a better feel for the results being predicted by both the schwarzschild metric and the gullstrandpainleve version simply uses numeric integration to get an approximation. Generalized painlevegullstrand metrics sciencedirect.
Likewise, the outgoing eddingtonfinkelstein coordinates are obtained by replacing t with the null coordinate. Painlevegullstrand synchronizations in spherical symmetry. The stationary limit arises as the set of points on this. For spherically symmetric spacetimes, we show that a painlevegullstrand synchronization only exists in the region where. General relativity, black hole, metric extensions, ricci tensor, escape velocity.
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. It seems like the schwarzschild metric only has historical relevance. Mar 08, 2009 painlevegullstrand coordinates for the kerr solution natario, jose 20090308 00. The metric g gp looks like a generalised gullstrandpainleve metric but with an hyperbolic space part k1. The argument can also be run in the reverse direction. The painlevegullstrand extension a black hole fallacy. A new hypothetical solution to gravitational singularities. One such system, introduced by painleve and gullstrand in the 1920s, is especially simple and pedagogically powerful. Q is the charge of the bh, mp is the plank mass, while m is the bh mass.
The painleve gullstrand extension a black hole fallacy the term. I provide herein the proof that the painleve gullstrand. First and foremost, the gullstrand painleve coordinates are not an independent solution of einsteins field equation, but rather an adjustment of the schwarzschild solution to a different coordinate reference, such that the apparent coordinate singularity at rrs is avoided. The continuation of the schwarzschild metric across the event horizon is a wellunderstood problem discussed in most textbooks on general relativity. Schwarzschild metric from wikipedia, the free encyclopedia in einsteins theory of general relativity, the schwarzschild solution or the schwarzschild vacuum describes the gravitational field outside a spherical, uncharged, nonrotating mass such as a non.
Painlevegullstrand coordinates in the present context, although they are becoming more widely used22, 23. Gravitational collapse in painlevegullstrand coordinates 1. Photon paths inside the event horizon 3 the particle starts off at rgm 2 and its world line travels up and to the left until it hits r0. For spherically symmetric spacetimes, we show that a painleve gullstrand synchronization only exists in the region where. Suppose spacetime is filled with observers, all freefalling radially inward with the same energy per unit mass e. Painleve gullstrand metric the acoustic geometry can be written in the canonical form ds2. Painlevegullstrand coordinates for the kerr solution core. An admirable exception is the text exploring black. Plotted are the light cones at various values of r. The reader can reinsert mby making the reverse substitution. That is, for a spherical body of radius r the solution is valid for. The force of gravity in schwarzschild and gullstrandpainleve. And is some sense for retard eddingtonfinkelstein time for an observer at future. Meanwhile, for the matter solution in the case of the gravitational collapse, the metric is assumed to be of the form ds2.
General relativity and geometry 230 9 lie derivative, symmetries and killing vectors 231 9. Painlevegullstrand global rain coordinates a general. Outline radial distance in schwarzschild spacetime. Pdf painlevegullstrand synchronizations in spherical symmetry. Thus we start from the charged black hole metric in the pg coordinates ds2 dt2 dr vrdt2. They call the singularity at the schwarzschild radius a coordinate singularity. Painlevegullstrand metric the acoustic geometry can be written in the canonical form ds2. The force of gravity in schwarzschild and gullstrandpainlev. For a sphere, the metric is given by the functions ga11 2, gg 12 210, and ga22 22sin.
It seems like the condition on the coordinates that all lightcones must point in the same coordinate axis leads to the problems of the metric at the horizon. Painlevegullstrand coordinates for the kerr solution. It is based on a black hole with a mass of 10,000 suns and steps the radius r by 0. Spherical spacelike geometries in static spherically. Since r is not the radius there is no a priori reason to suppose that it must. Pdf a painlevegullstrand synchronization is a slicing of the. Our first objective in this paper is to popularize another set of coordinates, the painlevegullstrand. Gravitational collapse in painlev\egullstrand coordinates. What is our universe now claude bernard university lyon 1.
This is a plot of the light cones of the vr coordinates where the v axis is a straight line slanted up to the left. The ingoing eddingtonfinkelstein coordinates are obtained by replacing the coordinate t with the new coordinate. Distance of two shells in gullstrandpainleve coordinates. Note the presence of the cross term, and note that it is the only term that is affected by the sign of a i. Thanks for contributing an answer to physics stack exchange. Painlevegullstrand coordinates for the kerr solution arxiv. Jun 10, 2017 in the original painlevegullstrand coordinates for the schwarzschild metric, the new time coordinate is defined in terms of the original time and radial coordinates, as follows. A number of methods have been employed by cosmologists to effect what they call an extension of their schwarzschild solution, to remove the singularity at their schwarzschild radius rs 2g mc 2. Download limit exceeded you have exceeded your daily download allowance. All descriptions are a conglomeration of local measurements. Surprisingly, the gullstrandpainleve metric is widely neglected in texts on general relativity. In the metric components above, the mass of the black hole is m, its angular momentum per unit mass is a, and its electric charge is q. A particle orbiting in the schwarzschild metric can have a stable circular orbit with r 3r s.
Interpreting gullstrandpainleve coordinates physics forums. Homework 1 solution, physics 776, spring 2005 painlevegullstrand. From the specified metric, we compute, first, the connection, which is not itself a tensor. Generalized painlevegullstrand metrics chopin soo dept. The painlevegullstrand extension a black hole fallacy vixra. Painleve 1922, gullstrand 1922, kanai, siino, hosoya 2011. The schwarzschild metric is a solution of einsteins field equations in empty space, meaning that it is valid only outside the gravitating body. Pdf regular coordinate systems for schwarzschild and. The stationary limit arises as the set of points on this manifold where.
Role of the gullstrandpainleve metric in acoustic black holes. Ii, we can express the schwarzschild metric in the painlev. Unlike the schwarzschild form, the painlevegullstrand metric tensor has an off diagonal element so that it is regular at the schwarzschild. Gullstrand painleve coordinates are a particular set of coordinates for the schwarzschild metric a solution to the einstein field equations which describes a black hole. For convenience, we will do this both with the schwarzschild and gp coordinates. Geodesics in generalized painlevegullstrand coordinates and tunneling process from a schwarzschild black hole article pdf available in international journal of theoretical physics 549.
967 1152 1480 312 908 536 108 1489 597 979 648 679 61 544 778 1185 724 714 949 142 10 922 252 1303 999 933 1444 1455 482 1489 312 1105 1155 876 546 938 49 491