Hi, Carl,
CarlB said:
Okay, the program of writing Newton and Schwarzschild in Cartesian (see post #11 for references to literature) coordinates without trig has been completed and the results test out nicely... Chris, I really appreciate your help on this problem, but I don't have time to bring you up to speed when you can do it for yourself with little effort. If you are willing to swear that you've carefully read each post in the rest of the thread, and at least looked at the articles that were linked to it, then I will answer whatever questions you have left, and we won't clutter up the thread going over the same old same old. Actually, after looking through the thread it appears that I really should rewrite the secret reasons for working on this sort of thing.
Well, I don't think you are giving me enough credit for "reasonable effort" (I can see that for your part, you don't think I am giving you sufficient credit for knowledge and careful work, and you may be right!), but it seems that you have now recognized some of the expository deficiencies of your previous posts, and I applaud your attempt to rectify the lacunae you noticed.
By the way, if you said previously that you have a background in geometric algebra, I must have either missed or forgotten that comment. As it happens, many years ago I knew a fair amount about Cayley-Dickson algebras (see for example my posts on noneuclidean trigonometries, which will probably be familiar to you but not to many others here), and I do know the papers by Doran, so I think I can say with some confidence that it was not nearly so apparent as you thought that this was part of the background for your computations. (By the way, there is an important point about the Doran chart for the Kerr vacuum which we can discuss if you like, which illustrates my point about unanticipated subtleties which can easily trip up newbies.)
Also, if you had seen as many inaccurate plots drawn by enthusiastic computer jocks for fun, or as many accurate but terribly misinterpreted plots, as I have done, you would probably appreciate why it is in your own best interests to quickly convince sceptics that your plots do not fall into these categories!
But even if you don't appreciate why I am skeptical by default, please not that following up on my suggestions should be
fun and if you have indeed made no errors of math or interpretation should greatly strengthen the presention of your plots.
OK, I hope this clears the air!
Let me try to continue by annotating Carl's post, saying some things which I thing should help others to follow along.
CarlB said:
Painleve coordinates:
ds^2 = -dt^2 + (dr-(2/r)^{0.5} dt)^2 + r^2 d\phi^2.
The Painleve chart is one of the most important charts on the Schwarzschild vacuum solution, and I've been a big fan of it ever since I independently "discovered" it some time around 1984. (Much later I learned it was introduced by Painleve in 1921.)
Like the Eddington chart, it comes in ingoing and outgoing flavors and the ingoing (outgoing) Painleve chart covers the same region as the ingoing (outgoing) Eddington chart. We are interested in ingoing Painleve chart here because Carl is (I take it) planning to study the motion of freely and radially infalling observers who pass from the "right exterior" into the "future interior" region. The Painleve chart is in fact ideally adapted for this purpose. To obtain it from the exterior Schwarzschild chart
ds^2 = -(1-2m/r) \, dt^2 + \frac{dr^2}{1-2m/r} + r^2 \,<br />
\left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right),
-\infty < t < \infty, \; 2 m < r < \infty, \; <br />
0 < \theta < \pi, \; -\pi < \phi <\pi
put
T = t + \int \frac{\sqrt{2m/r}}{1-2m/r} \, dr<br />
= t + \sqrt{8 m r} - 4m \operatorname{arctanh} \left( \sqrt{\frac{r}{2 m}}\right)
which gives
ds^2 = -(1-2m/r) \, dT^2 + 2 \, \sqrt{2m/r} \, dT \, dr + dr^2 + r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right),
-\infty < T < \infty, \; 2m < r < \infty, \; 0 < \theta < \pi, \; -\pi < \phi <\pi
I'm too lazy to draw pictures, and I've told this story before, but let me try to give some indication of how I discovered this transformation before I knew any calculus (!). Looking at the pictures in MTW, my idea was to "pull down" the constant time slices by defining a new time coordinate such that the new constant time slices would be everywhere orthogonal to the world lines of the Lemaitre observers (whom I learned about from the little textbook by Dirac), in order to combine the best features of the ingoing Lemaitre and ingoing Eddington charts. You can actually do this if you know only the appropriate trignometry and trust a table of integrals :-/ and that's what I did.
Now we immediately notice two things. First, the domain can be extended from the right exterior into the "future interior" by increasing the range of the radial coordinate 0 < r < \infty (we'll need to interpret this carefully inside the horizon, but it still deserves the name of "Schwarzschild radial coordinate"). Second, the constant Painleve coordinate time slices T=T_0 have the metric of euclidean three-space! This was my first noteworthy "discovery" in gtr, so you can see why I am so fond of the Painleve chart!
With a bit of insight (or practice!) we can read off an "obvious" coframe field:
<br />
\sigma^0 = -dT, \;<br />
\sigma^1 = \sqrt{\frac{2m}{r}} \, dT + dr, \;<br />
\sigma^2 = r \, d\theta, \;<br />
\sigma^3 = r \, \sin(\theta) \, d\phi<br />
whose dual frame field consists of the unit vector fields (first timelike, last three spacelike)
<br />
\vec{e}_0 = \partial_T - \sqrt{\frac{2m}{r}} \, \partial_r, \;<br />
\vec{e}_1 = \partial_r, \;<br />
\vec{e}_2 = \frac{1}{r} \, \partial_\theta, \;<br />
\vec{e}_3 = \frac{1}{r \, \sin(\theta)} \, \partial_\phi<br />
This frame field is suitable for studying the physical experience of observers who fall in freely and radially "from rest at r=\infty". That is, the world lines of this class of observers are precisely the integral curves of the timelike unit vector field from our frame field, and the three spacelike vector fields then give the "spatial frame vectors". IOW, our frame field attaches the "local Lorentz frame" of these observers to each event. The point of introducing this frame here is to point out that in the Schwarzschild vacuum geometry, the timelike vectors \vec{e}_0 are everywhere orthogonal to the hyperslices T=T_0. IOW, in a sense, for these particular observers, we can consider their "space at a time" to be
flat, but of course the spacetime itself is not flat. (A pedantic aside: the spatial frame vectors have vanishing Fermi derivatives, after projection into the orthogonal hyperplane elements, so this frame is in fact
inertial nonspinning, which is the closest we can come in a curved Lorentzian manifold to a global Lorentz frame as in the standard formalism of str.
We can simultaneously employ alternative frame fields to study other families of observers, and this is in fact the best way to clear up the kind of confusion so often evident in discussions of alleged "paradoxes". I have done that in great detail elsewhere and it's off topic here, so moving on:
By the spherical symmetry, we can suppress one angular coordinate by restricting attention to the equatorial plane \theta=\pi/2. Also, the mass parameter can be set to one by rescaling coordinates, so let's do that too. This gives the line element Carl wrote down, after renaming T \mapsto t.
Going one step further, we can temporarily suppress \phi also and draw the local light cones at various events using the frame vectors
\vec{e}_0 = \partial_T - \sqrt{\frac{2m}{r}} \, \partial_r, \; \vec{e}_1 = \partial_r
Switching to a more computer graphics friendly notation,
\vec{A}(T,r) = (1, -\sqrt{\frac{2m}{r}} ), \; \vec{B}(T,r) = (0,1)
Fixing event P = (T_0,r_0), and pretending we are working in Minkowski spacetime (really, we are working in the tangent space at our event),
draw a segment to the event P+\varepsilon \, (\vec{A}+\var{B}),
then from that event to the event P+\varepsilon \, (\vec{A}-\var{B}),
then back to P. That's the top half of our local light cone. Draw the frame vectors to the same scale (i.e. scalar multiplied by \varepsilon) to see how our "local Lorentz frame" relates to the shape of our "local light cones". Now draw some more of these gadgets, to scale. Draw some in the exterior, some in the interior, and some on the horizon itself. Notice how the light cones progressively shear inward as the Schwarzschild radius decreases. This is really a way of visualizing how our local light cones, which are can be considered as a feature of the local "conformal structure" although perceptive readers will note that I am treating them as a "metrical structure", relate to the Killing vector field \partial_t, which is timelike on the right exterior (hence our solution is "static" there), but null on the horizon, and spacelike in the future interior.
Next, if we want to study general geodesics, it makes sense to write down the geodesic equations. The easiest way to do this is to read off the geodesic Lagrangian and then compute the Euler-Lagrange equations; putting the result in "monic" form we obtain the geodesic equations in their standard form, from which we can read off the Christoffel symbols, if we so desire. Alternatively, we can use the methods of Cartan (see MTW). I'll spare you the gory mess: suffice it to say that we get something quite a bit messier than what we obtain for the Schwarzschild exterior chart (see the discussion of effective potentials in MTW for a convenient method of analyzing the geodesics in the interior, in terms of the Schwarzschild exterior chart).
Now, anyone who did the "coordinate tutorial" I posted long ago to sci.physics.relativity knows very well that adopting the "right" coordinate chart can
greatly simplify the analysis of the geodesics! As the active reader has now seen, while the Painleve chart has many virtues, it does not succeed in simplifying the geodesic equations if that is our goal; indeed, it makes them appear somewhat more complicated.
Another reason for exploring multiple charts is that Lie's symmetry methods can be much easier to crank through, which can enable us to discover "especially symmetrical" geodesics. In particular, if we didn't already know about the circular photon orbits at r=3m, we would discover them by a systematic study of this kind. Actually, Lie's methods are probably even more relevant for the problem of solving the wave equation, Klein-Gordon equation, Dirac equation, etc., on curved spacetimes, where changing charts can again have drastically beneficial effects, and where again this can be detected (even sometimes predicted) using a Lie theoretic symmetry analysis of the equation to be solved. This game is important for physics for various reasons; one which should sound familiar to many readers here is the study of perturbations, e.g. we might want to know how radiation of various kinds is "scattered" by a Schwarzschild object.
At this point, I'd highly recommend that anyone who hasn't much experience comparing say the wave equation in various charts should pause and collect a bunch of familiar charts for the Schwarzschild vacuum (paying careful attention to noting the coordinate ranges), such as Schwarzschild, static and spatially isotropic, Lemaitre, Eddington, Painleve, Kruskal-Szekeres, and figure out the transformations between them. Next review how to compute the wave operator using the elegant methods of Cartan (see MTW, or my long ago expository post to sci.math.relativity called "The Joy of Forms"), and compute for your charts. Check your results by applying the transformations you found earlier.
OK, so now let's confront the question: how can we find charts in which the geodesic equations, or the wave equation, or whatever, look "as simple as possible"? I don't think there's an algorithmic answer to that, but there are some basic things to try.
One approach is to try to "rationalize" our coordinates. For example, the usual polar spherical line element
<br />
ds^2 = dr^2 + r^2 \; \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right)<br />
contains trig functions. We can rationalize this by setting \eta=\cos(\theta), so that the line element becomes
<br />
ds^2 = dr^2 + r^2 \; \left( \frac{d\eta^2}{\eta^2} + (1-\eta^2) \, d\phi^2 \right)<br />
which contains only rational expressions in the coordinates.
A particularly dramatic example of this arises in a much more general context which includes the solution under discussion: in studying the Ernst vacuums, a major breakthrough was the recognition that the "rational prolate spheroidal chart" results in a dramatic simplication of the Ernst equation (that is, the "master equation" in the set of equations to which we can reduce the vacuum EFE in studying the Weyl canonical chart for a stationary axisymmetric spacetime). If you're curious, the rational prolate spheroidal chart for E^3 has the line element
\frac{ds^2}{A^2} = (\chi^2-\xi^2) \, \left( \frac{d\chi^2}{\chi^2-1} + \frac{d\xi^2}{1-\xi^2} \right) + (\chi^2-1) \, (1-\xi^2) \, d\phi^2,
1 < \chi < \infty, \; -1 < \xi < 1, \; -\pi < \phi < \pi
where A > 0 is a scale factor. The transformation from the standard cylindrical chart is
z = A \, \chi \, \xi, \; r = A \, \sqrt{\chi^2-1} \, \sqrt{1-\xi^2}
Using the elegant exterior calculus methods of Cartan, you can verify with surprisingly little effort that the Laplace operator becomes, for an axisymmetric function,
\Delta = \frac{1}{A^2 \, (\chi^2-\xi^2)} \left( \partial_\chi (\chi^2-1) \partial_\chi + \partial_\xi (1-\xi^2) \partial_\xi \right)
Quite remarkably, this is symmetric under interchanging \chi, \xi, which Chandrasekhar took good advantage of in his work on the Ernst vacuums! Even better, you can readily attack this by the elementary method of separation of variables, obtaining a useful series expansion for asymptotically vanishing harmonic functions in terms of Legrendre polynomials/functions. This looks completely different from the analogous expression we'd obtain by analyzing the same equation in a cylindrical or polar spherical chart! (This is far more impressive for the Ernst equation, where as I said it turns out that the rational prolate spheroidal Weyl canonical chart on a stationary axisymmetric spacetime achieves several substantial simplifications of several interesting cases of the EFE for this situation, including vacuum, electrovacuum, minimally coupled massless scalar fields, dusts, and so on.)
Another approach is to try harmonic coordinates. These are charts in which the coordinates, considered as monotonic functions on the domain of our chart, are harmonic functions (in the "Laplace-Beltrami sense": they satisfy the curved spacetime wave equation).
Still another is try geometric algebra, which is apparently what Carl is working on, ultimately in the context of trying to simplify some equation other than the geodesic equation. Geometric algebra is a kind of elaboration of the formalisms of vector calculus, exterior calculus, and quaternionic calculus. So far it has been studied mainly in the UK, especially Cambridge, and I think Carl will agree that, like Hamilton's pursuit of his program of rewriting nineteenth century physics in terms of quaternions, geometric algebra is either ignored or considered by many observers to be of dubious value, given the intricacy of the notation. However, its enthusiastic proponents (and back in the day, I knew someone studying this stuff at Cambridge pretty well) can point to a few notable successes, including the Doran chart for the Kerr vacuum, which generalizes the Painleve chart (aha!) to the Kerr vacuum, which was obtained using the methods of geometric algebra. As a simpler example, it often happens when using frame fields that one wishes to "despin" a spinning frame by rotating by just the right angle about just the right axis of rotation (said angle and axis defined at each event) to kill the projected Fermi derivatives, and this kind of operation would indeed probalby be more conveniently carried out in geometric algebra, except in special cases. For example, the "obvious" coframe read off the Doran chart is in fact dual to a spinning frame, i.e. its timelike unit vector is indeed what we want, but the spacelike unit vectors need to be "despun" to obtain a nonspinning inertial frame analogous to the Lemaitre frame discussed above. However, this Doran-Lemaitre frame can in this case be obtained easily by elementary methods, once one has the Doran chart in hand, and I have little doubt that the Doran chart itself can be obtained without recourse to geometric algebra.
CarlB said:
First step is to convert them to "Cartesian" form.
Looking back, I see that Carl originally referred to "Cartesian coordinates" for the Schwarzschild vacuum solution, which in the context of this forum, or elementary gtr generally, is terribly confusing because those in the know will fear that someone is trying to treat a curved spacetime using coordinates which can exist only in a flat spacetime! However, eventually it turned out that Carl just meant that he is applying a transformation given by a formula familiar from vector calculus:
<br />
x = r \, \sin(\theta) \, \cos(\phi), \;<br />
y = r \, \sin(\theta) \, \sin(\phi), \;<br />
z = r \, \cos(\theta)<br />
Although of course we are working in part of the Schwarzschild vacuum, not Euclidean three-space, this makes perfect sense,
so long as we are careful to restrict the ranges of the coordinates to ensure that we obtain a well defined diffeomorphism on the overlap of the domains (in this case, they are almost the same--- as an exercise, what is the "bad locus" which must be excised from the domain of the original Painleve chart but which we can reasonably hope to include in the variant Carl is switching to?)
CarlB said:
Following post #2, we find:
\begin{array}{rcl}<br />
ds^2 &=& -dt^2 + ((xdx+ydy)r^{-1}-\sqrt{2}\;r^{-0.5} dt)^2 + (y^2dx^2 + x^2dy^2 -2xy\;dx\;dy)r^{-2},\\<br />
&=&-dt^2 + dx^2 + dy^2 + 2r^{-1}dt^2 -2\sqrt{2}\;r^{-1.5}(x\;dx\;dt+y\;dy\;dt),\\<br />
I = \left(\frac{ds}{dt}\right)^2 &=&\frac{2}{r}-1 + \dot{x}^2 + \dot{y}^2-2\sqrt{2}r^{-1.5}(x\dot{x}+y\dot{y})<br />
\end{array}
The idea is to apply the Euler Lagrange equations to the integral over t for s:
s = \int \sqrt{\left(\frac{ds}{dt}\right)^2}\;dt = \int \sqrt{I}\;dt
to obtain the equations of motion in a Cartesian coordinate system.
In the standard "geodesic Lagrangian", the dots refer to differentiation with respect to an affine parameter, which for timelike geodesics we can take to be an arc length paramter. Here, however, Carl's dots refer to differentiation with respect to the Painleve time coordinate, which I am denoting by T. But recall that in the Painleve chart, the Painleve time coordinate happens to give the arc length parameterization of the Lemaitre congruence. The expressions Carl quotes (attributing them to pervect)
\frac{d}{d t} \left[ \frac{1}{2 \sqrt{I}} \left( \frac{\partial I}{\partial \dot{x}} \right) \right] = \frac{1}{2 \sqrt{I}} \frac{\partial I}{\partial x}
do agree with the Euler-Lagrange equations obtained via elementary variational calculus from the standard geodesic Lagrangian, except for what would be an inessential scale factor
if I didn't depend upon time. Pervect?
