High School Solving the Kerr metric in the program Maxima

[exmarine]

Does anyone know how to get Maxima to solve the Kerr metric? I enter the terms for that metric that I found on Wikipedia. It tries to print out the Einstein tensor (covariant, leinstein(true)) and the expressions are so long that it literally locks up my computer. And isn’t the Kerr metric a vacuum solution, with that tensor equal to zero?
I can get Maxima to correctly solve all the other metrics I know about, so I think I am running it correctly.

 

[Ibix]

Are you setting flags like ratchristof to true? If you don't then Maxima won't automatically simplify expressions for intermediate steps like the Christoffel symbols and the expressions can get complicated. That's usually when I find trouble with Maxima output - when it's decided that it's going to show a+b as a+c+b-c.

 

[exmarine]

I am using: RATFAC:TRUE;
Does that cover the intermediate steps too? I will check the help file.
Thanks.

 

[m4r35n357]

I am using: RATFAC:TRUE;
Does that cover the intermediate steps too? I will check the help file.
Thanks.

I found the whole factoring/simplification thing a bit random, but looking back I used ratsimp a lot, and trigsimp a little bit . . .

One other tip, if you can't get it to simplify exactly how you want, you can sometimes get it to return a zero for the difference against a user-specified simplification, if you see what I mean.

 

[Ibix]

You could also try mcs:ratsimp(mcs) after calling christof().

 

[exmarine]

Ok, nothing works. I went back to the MTW textbook and tried their version of the Kerr metric. I thought maybe the Wikipedia version might be wrong, or I might have messed up in specifying it to Maxima. But I got the same result, i.e., Maxima could not solve it, Einstein tensor (covariant) was not going to be zero, and the expression(s) were so long that it locked up my laptop – even after increasing the configuration settings.

So my final question is: How do you guys satisfy yourselves that the Kerr metric actually has all those properties claimed for it? By faith? (‼ I don’t believe anything I can’t reasonably prove for myself.) And how did the early guys find those properties, like Kerr himself, certainly without the assistance of a computer program?

PS. For what it is worth: Yes, I have been through the help file quite a bit. And there are 2 insight type articles on here that I have downloaded and studied. And I have been through Crowell’s textbook which has many examples of using Maxima. In fact his book is how I was introduced to using Maxima – THANKS! And yes, I have successfully used Maxima to solve all the other metrics I know about. If someone could write another insight article specifically on solving the Kerr metric, it would be greatly appreciated.

 

[pervect]

There's some output from a different program, GRTensor, at http://grtensor.phy.queensu.ca/NewDemo/kerr/kerr.html. I assume that the goal here is to compute the Ricci (or the Einstein tensor) from the Kerr metric and show that it is zero, to demonstrate that that metric is a vacuum solution of Einstein's field equations.

But that doesn't necessarily answer the question of why one should believe it. Believing it requires trusting the program, and (in this case) the publisher of the website who published the output of the program, but why should one trust either? With access to the program (which isn't free, unfortunately, like Maxima), one could avoid having to trust the website, but one would still need to trust the program itself.

This isn't that much different than trusting Maxima,and trusting GRTensor, I suppose. But why should one trust either? Using the program(s) for a bit, and comparing it's output to hand calculations and peer-reviewed published sources might help build trust. Is that "proof"? Why should one trust peer-reviewed publications, and how much should one really trust them?

I don't have a good general answer, but my observation is that technology seems to (for the most part) work. So I trust that bridges won't (usually) fall down if I drive over them, I trust that my car (for that matter) will usually run, and if I'm lost I might trust that my GPS is the best tool for getting to my destination. Though one can think of situations where any of these things might not be true - bridges fail, cars break down, GPS devices can malfunction, or even be manipulated by people with the proper military access.

For me personally, I mostly trust that technology works, and I believe that this is not a happy accident, but it's due to the fact that the people who created the technology knew what they were doing, and thus I trust them. I trust that if I get a steel tape to measure distances calibrated by the appropriate national standards institute, it will be accurate, and I trust the theoretical basis on how the standards were set, that these standards "make sense" and work well.

 

[Ibix]

I just tried it and it worked ok, although I had to simplify the Ricci tensor element by element. Here's the code I used:

/* Load ctensor and set up a Kerr-Newman (rotating charged black hole) spacetime */
load(ctensor);
ct_coordsys(kerr_newman);
/* ...and set the charge to zero to get Kerr */
lg:substitute(0,e,lg);
/* Generate and simplify the inverse metric */
cmetric(false);
ug:trigsimp(ug);
/* Generate Christoffel symbols and the Ricci tensor */
christof(mcs);
ricci(true);
/* The elements of the Ricci tensor don't seem to simplify automatically,*/
/* even if I set ctrgsimp:true. So do it manually. */
for i:1 thru 4 do
    for j:1 thru 4 do
        ric[i,j]:trigsimp(ric[i,j]);
/* Display a matrix of zeros */
cdisplay(ric);

 

[exmarine]

Thanks for the code! Will try it and hopefully learn.

Trust? Well I established trust in Maxima by comparing small steps at a time with my hand calcs, and that it gets the conventional accepted results...

 

[m4r35n357]

Thanks for the code! Will try it and hopefully learn.

Trust? Well I established trust in Maxima by comparing small steps at a time with my hand calcs, and that it gets the conventional accepted results...

If you still want more examples, I also have some old code that calculates from explicitly defined Boyer-Lindquist coordinates all the way down to scalar curvature and kretchmann scalar via the usual routes.

 

[exmarine]

Yes please, any examples appreciated! Maybe attached as text files makes it easier to grab.

The code above from (Ibix) works, but only if I suppress the printout of the ricci tensor before it is simplified: ricci(false). Apparently those un-simplified terms are what clogs up my computer.

Next I am interested in the geodesics (to understand more about frame dragging for example). Those expressions now look intractable. Is there some way to get Maxima to substitute (and then use!) theta equal to pi/2 for example?

Thanks!

 

[m4r35n357]

Yes please, any examples appreciated! Maybe attached as text files makes it easier to grab.

Attached (stupid filter didn't accept .wxm so just rename it). It's a few years old so I had to remind myself that it still worked.

You just need to run the top cell to enable ctensor, run the Boyer-Lindquist (new) cell (the non-new one generates the inverse metric by matrix inverse, the new one is entered explicitly), then run the "symbols to einstein" cell, and stand well back!.

 

[m4r35n357]

Is there some way to get Maxima to substitute (and then use!) theta equal to pi/2 for example?

I don't know if there is a better or more "official" way, but I use

xtheta: %pi / 2$

and delete the leading "x" as and when I want to apply it.

 

[Ibix]

Is there some way to get Maxima to substitute (and then use!) theta equal to pi/2 for example?

The way I've done it from time to time is:

theta:%pi/2;
substitute(theta,'theta,lg);

Note the apostrophe at the beginning of the second theta. That says "I mean theta as a literal", so substitute will look in lg for instances of $\theta$ and replace them with the value $\pi/2$ that was defined in the line above. If you do that before calling cmetric() then all of the stuff you derive will have $\theta=\pi/2$ built in.

@m4r35n357 and I have both written numerical Kerr geodesic calculators based off work by Kraniotis - https://arxiv.org/abs/gr-qc/0405095. You might like to look at how he derives things - it produces much more tractable equations than the epic ones you get from straightforward application of the geodesic equation.

And yes, my experience of Maxima is that it's LaTeX-type layout is what takes ages to render, not the actual algebra.

 

[exmarine]

You guys are great, helping me learn stuff here in my dotage. Your file looked like a batch file (*.mac instead of *.wxmx), so that is how I ran it. It bombed out around line 259(?) where you set theta to pi/2, and a subsequent depend statement did not appreciate that. So I x-ed theta back out and it ran, but eventually locked up my computer. I am starting to suspect my problems are with the Window interface part when the output is too voluminous. I’ll have to explore that more, but if I keep the printout small, ask for only one thing at a time, etc., it seems to work.

The substitute statement with the apostrophe did the trick of making Maxima actually use pi/2, at least for collapsing the trig functions. It would be nice if it could also recognize that the derivative of pi/2 is zero, but I can live with that.

The paper by Kraniotis (and Whitehouse) and his references were fascinating. (Maybe I should ask this question in a new thread.) But I wondered how their idea about determining the cosmological constant from the rotation curves in galaxies is faring now? As he says, his finding that said constant is negative is opposite that found from the supernovae data. His earlier paper was long before (~2000) some anticipated satellite launches (~2010), that have certainly happened by now. Does anyone know if newer observations support or contradict their idea?

Finally, the Kerr metric that Maxima gives matches the one given in MTW for the diagonal terms, but I can only get the off-diagonal term to match Maxima if “a” (the angular momentum per unit mass) is equal to 1. Any idea what that is all about?

Thanks.

 

[m4r35n357]

Your file looked like a batch file (*.mac instead of *.wxmx), so that is how I ran it.

Assuming you are talking about my file, it is not a batch file, it is a wxmaxima data file (named relativity.wxm on my system), so you heed to File/Open it in the GUI and run it from there as it is made up of lots of different cells. You then run just the cells I mentioned (SHIFT-ENTER with the cursor somewhere in the cell). HTH.