In the previous post, we took a tour through the literature and found that when general relativity is translated into the elegant mathematical language of the geometric algebra, the natural coordinates for a black hole turn out to be Painleve or Gullstrand-Painleve coordinates instead of the more common Schwarzschild coordinates. Our next post will derive the equations of motion for orbits in this coordinate system, but before we get into the difficult mathematics, we should take a quick look at the Painleve coordinates.
First of all, most of my readers will know that in general relativity, the choice of coordinates is quite arbitrary. Both Schwarzschild and Painleve coordinates describe the same object, the gravity field of a gravitating object which is spherically symmetric (and therefore non rotating), i.e. they are all descriptions of the black hole. In this sense they correspond to the same solution to Einstein’s field equations, which is sometimes called “Schwarzschild’s Solution”, or the “Schwarzschild Metric”. This is a little confusing, “Schwarzschild” was the person who found the Schwarzschild metric and he found it using the Schwarzschild coordinates, so his name is used twice here.
I guess I should put a pretty picture from the gravity simulation that resulted from all this here so it will show above the fold. This is a set of “knife edge” orbits, that is, orbits that quite nearly fall into the black hole but do not. Due to the time spent near the black hole, whose event horizon is marked in gray, the test masses get huge precession:
We will be discussing the less pretty, but more mathematical subject of Painleve and Schwarzschild coordinates in this post.
For those of you who understand general relativity well enough to read the physics literature, let me put a couple useful references about Painleve coordiantes here.
Regular coordinate systems for Schwarzschild and other spherical spacetimes [Karl Martel, Eric Poisson, 2000]
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 to popularize these coordinates. Our other purpose is to provide generalizations to the Painleve-Gullstrand coordinates, first within the specific context of Schwarzschild spacetime, and then in the context of more general spherical spacetimes.
Those who are just a little less educated in general relativity, but aren’t willing to sit through a boring pedagogical lecture by yours truly might nevertheless enjoy this paper on The River Model of Black Holes, whose subject is a generalization of Painleve coordinates to rotating black holes, and how it can be used to give an intuitive understanding for what black holes are about. It is also a useful reference paper for the coordinate systems you end up with when you apply geometric algebra to rotating and charged black holes:
This paper presents an under-appreciated way to conceptualize stationary black holes, which we call the river model. The river model is mathematically sound, yet simple enough that the basic picture can be understood by non-experts. %that can by understood by non-experts. In the river model, space itself flows like a river through a flat background, while objects move through the river according to the rules of special relativity. In a spherical black hole, the river of space falls into the black hole at the Newtonian escape velocity, hitting the speed of light at the horizon. Inside the horizon, the river flows inward faster than light, carrying everything with it. We show that the river model works also for rotating (Kerr-Newman) black holes, though with a surprising twist. As in the spherical case, the river of space can be regarded as moving through a flat background. However, the river does not spiral inward, as one might have anticipated, but rather falls inward with no azimuthal swirl at all. Instead, the river has at each point not only a velocity but also a rotation, or twist. That is, the river has a Lorentz structure, characterized by six numbers (velocity and rotation), not just three (velocity). As an object moves through the river, it changes its velocity and rotation in response to tidal changes in the velocity and twist of the river along its path. An explicit expression is given for the river field, a six-component bivector field that encodes the velocity and twist of the river at each point, and that encapsulates all the properties of a stationary rotating black hole.
There are many, much more arbitrary, coordinates that work for the Schwarzschild metric than the two systems we will discuss. In relativity, coordinates describe events. To describe the relationship between two coordinates, we define how to translate an event described in one coordinate system into the other coordinate system. For the case of the Schwarzschild and Painleve coordinates, the translation between them is very simple. Of the four coordinates, usually taken in spherical coordinates to be radius r, time t, and angles , only the time t is different between the two coordinate systems. To move back and forth between the coordinate systems, one modifies the time t by adding an amount that depends on the radius (and not on the angle or on the time). The “t” we’re talking about here is “coordinate time”, that is, it is not proper time, but instead, in the theory of general relativity, it is just a label that is used to distinguish diffferent events.
The orbits of Painleve and Schwarzschild coordinates are therefore also closely related. If we begin with the “same” initial condition (being very careful to “match” the initial radial velocities) and then ignore the time dependency, a particle will draw out the same orbit in these two coordinate systems. The radius and angles are the same, they differ only in the definition of the time that applies to the event right?
Well this is not quite right. If you drop a particle into a black hole, so that it goes past the point of no return and goes into the “event horizon”, Schwarzschild coordinates will not allow the particle to actually cross the event horizon. Painleve coordinates allow the particle to go all the way to the singularity at the center in finite coordinate time. If we model the SAME test particle with the SAME initial conditions in the two corodinate systems, the two coordinate systems will show the same orbital path, but with the particle arriving at different times.
An Example Orbit
As an illustration of the relationship between the two coordinate systems, let’s look at the physical example of a black hole and shoot a test mass into it. We will use units of seconds, and put M = G = 1. This puts the event horizon at a distance of 1 light-second from the singularity. Conveniently, these are the coordinates that are used in my gravity simulation and since it draws orbits in both Schwarzschild and Painleve coordinates, we can use it to draw nice pictures showing the difference between the two coordinates.
We’ll begin with the particle at r=4 and t=0, and with an initial orbital velocity of 0.4c directed perpendicular to the radial direction. Such initial conditions describe a particle that will fall into the black hole. Let us look at the sequence of events that describe the orbit from time t=0 to time t=17.3 seconds:
In the above illustration, the black hole’s event horizon is marked in gray. The test particle begins at the right moving up the page. There are two orbits shown, a Painleve orbit (in blue) and a Schwarzschild orbit (in green). Since the two orbits end up covering the same radius and angular coordinates, the two oribts coincide. But the Painleve orbit, at the time given, has moved farther along than the Schwarzschild orbit. In fact, the Schwarzschild orbit will end at the event horizon and will never penetrate it.
The Advantages of Schwarzschild Coordinates
The event horizon is only a coordinate singularity of the Schwarzschild coordinates, it’s not like the singularity at the center of the black hole which is an essential singularity. Given that Painleve coordinates avoid this singularity, it’s at least a little surprising that Schwarzschild coordinates are used at all; and that’s before considering the fact that Painleve coordinates are natural to use in the context of the Dirac equation.
Let’s compare the two coordiante systems in terms of how they define the Schwarzschild metric. In Schwarzschild coordinates, the metric looks like:
while Painleve coordinates look like:
Since the transformation between the two coordinate systems only changes t and does not involve the angular coordinates, the angular contribution to the metric, is the same. Schwarzschild coordinates are distinguished from the Painleve coordinates in that Schwarzschild coordinates have no cross term. This is more natural for special relativity and this is why they’ve been used so much more over the years.
The Arrow of Time
Painleve coordinates have a square root, and in the above equation I’ve left left a in front of that square root. Mathematically, either sign gives a solution to Einstein’s field equations. The + sign gives a black hole, the – sign gives a “white hole” . A white hole is the time reversal of a black hole. In this sense, Painleve coordinates define an arrow of time at a microscopic level in general relativity. This arrow of time is not generally appreciated, see the Wikipedia article for example, which omits it.
The arrow of time shows up in the cross product term . In Schwarzschild coordinates there are no such cross product terms and the resulting coordinates are symmetric with regard to time reversal. Thus Painleve coordinates treat the future differently than they do the past; they define an arrow of time.
Under the usual interpretation of general relativity, the choice of coordinate is arbitrary and consequently the fact that Painleve coordinates defien an arrow of time is not enough to actually choose an arrow of time. It is only when general relativity is rewritten in the language of geometric algebra (i.e. Dirac spinor) that one ends up with a theory that has an arrow of time. This is a significant theoretical advantage for the geometric algebra version of general relativity over the usual tensor formulation.
Rotating Black Holes
The calculations that we will describe in the next post will be for a non rotating black hole, that is, Painleve coordinates. It is important to note that the more general case, that of a rotating and charged black hole, is covered by a generalization of Painleve coordinates. The author hasn’t converted that coordinate system into the form of a Newtonian force due to the inadequacy of his mathematical tools (uh, I use the free tool MAXIMA rather than Mathematica which is probably superior but quite expensive).
The solution to Einstein’s field equations for the charged, rotating case is known as the Kerr-Newman metric. It can also be fit into a geometric algebra form. From The River Model of Black Holes, equation (30) the generalization of Painleve coordinates to the Kerr-Newman metric is:
where and are vectors that depend on the charge and mass. They are time-like and space-like, respectively. See the River Model paper for the details.
The form of the “Kerr Newman” metric given above is particularly simple and elegant. This is another advantage to Painleve coordinates. The generalization of the Schwarzschild metric to the charged, rotating, case is sufficiently complicated that I don’t feel like typing up the LaTeX for it here. If you want to see the nasty thing, visit the Wikipedia article on Kerr-Newman.
In this and the previous post we saw that the elegant geometric algebra implies Painleve coordinates for a charged, rotating, black hole that are considerably simpler and more elegant than the usual solution given in most general relativity text books, a solution that gives an arrow of time. In our next post we will return to the mathematical problem of redefining the orbits of Schwarzschild metric as a gravitational acceleration using Painleve coordinates. Maybe we will eventually get to work on doing the same thing for the charged rotating black holes, but don’t hold your breath.