'Gravitoelectromagnetometrics' software

[tehno]

Not the happiest title I know .
Anyway,here is what I want to ask experts in General relativity ,if there are such people hanging around here:
There are (commercially available) softwares dealing with problems in classical electrodinamics.All of them rely,in one way or another, on application of 4 Maxwell equations.
Are there any known and good sofwares developed for simulating problems in General relativity ?There are those 10 Einstein equations,but then I doubt anybody will need them in everydays life ,except a very few physicist working in the field of GR.

 

[ranger]

I can't answer your question, but could you point me to the software packages that deal with classical electrodynamics?

 

[tehno]

Your question is appropriate to ask in Electrical Engineering subforum.Not here.

 

[cesiumfrog]

You might like to google GRworkbench; there should be a major release in a year (two at most).

 

[ranger]

Your question is appropriate to ask in Electrical Engineering subforum.Not here.

I asked becuase your initial post indicates that you know of such software packages.

 

[Stingray]

Simulating Einstein's equation without (significant) approximation is extremely difficult. It has only very recently been known how to keep a simulation running for more than a few time steps. Getting working simulations and extracting useful information from them is very much at the forefront of current research. We're a long way from "canned" programs.

There are probably publically-available programs out there which simulate some very simple aspects of GR; perhaps the motion of test bodies around a static black hole and things of that nature. I'm not aware of any, though.

 

[Chris Hillman]

I want to ask experts in General relativity ,if there are such people hanging around here

If you mean "individuals who have studied widely used textbooks" (that's the "Wikipedia definition" of an "expert"), then yes, several; if you mean "authors who have written a widely used gtr textbook" (that's closer to the "academic definition" of an "expert"), then no, not as far as I know.

Are there any known and good sofwares developed for simulating problems in General relativity ?

Depending upon what you have in mind, perhaps yes. You said "simulating problems"--- did you mean "solving problems"? Or perhaps even just "making computations"?

GRTensorII is very easy to use and very well suited for finding exact solutions and studying their properties. It is not very well suited for making "index gymnastics" type computations, but some other packages attempt to provide tools for these computations. There are further specialized packages for making yet other kinds of computations. See http://www.math.ucr.edu/home/baez/RelWWW/software.html for some links.

There are those 10 Einstein equations,but then I doubt anybody will need them in everydays life ,except a very few physicist working in the field of GR.

That's a very odd comment considering the question you just asked.

Simulating Einstein's equation without (significant) approximation is extremely difficult. It has only very recently been known how to keep a simulation running for more than a few time steps. Getting working simulations and extracting useful information from them is very much at the forefront of current research. We're a long way from "canned" programs.

There are probably publically-available programs out there which simulate some very simple aspects of GR; perhaps the motion of test bodies around a static black hole and things of that nature. I'm not aware of any, though.

Just wanted to clarify that Stingray obviously interpreted your question in terms of numerical relativity. I'd just add that this is really too demanding for amateurs; on the other hand, any sufficiently advanced student can play with toy models using something like GRTensorII. (But Stingray, if you really meant to suggest that exact solutions do not abound or that there are not powerful methods other than numerical simulations, I can give you many citations proving that this is not at all true.)

Incidently, it might be worth pointing out that a particularly bizarre crank who somehow seems to have acquired a small but almost cultish following has apparently asked his adherents to go out and find software which will enable him to solve his so-called "field equation". The sad thing is that this individual appears utterly unable to recognize that his own repeated statements clearly show just how badly he has misunderstood the beautiful mathematics he claims to employ. A good analogy would be someone who claims to be not only a rabid baseball fan but even "the next Hank Aaron", yet who keeps talking about the "batting the puck into the net" and "John McEnroe galloping around the ballrink"--- I mean, sometimes it's hard to keep a straight face!

My point is that harsh experience impels me to add this caveat: it is important to recognize that you cannot successfully use software to do mathematical physics unless you have enough background to know what you are doing, and enough insight/judgement to correctly interpret your results. A good rule of thumb is that if you don't know how, in principle, to work a certain computation by hand, you probably shouldn't try to do it by computer! But I trust this caveat is superfluous in your case, tehno.

 

[Stingray]

But Stingray, if you really meant to suggest that exact solutions do not abound or that there are not powerful methods other than numerical simulations, I can give you many citations proving that this is not at all true.

I certainly didn't mean that! I had in mind a relativistic generalization of the various Newtonian "solar system" simulators that are floating around. These basically allow to you to watch an arbitrary configuration of point masses evolve in time.

 

[Chris Hillman]

Good-oh, plus a caveat about Newtonian solar system simulations

OK, glad I simply misunderstood! I'd add the caveat that I haven't verified in detail those Newtonian simulators either; a good spot check might be try to compare with known results such as maximality for a historic solar exclipse. But for example kstars gives impressive results on this spot check, plus or minus a century from now, but it certainly would not be reliable over millennia, much less millions of years--- if for no other reason than that it does not take gtr into account, which eventually will cause serious trouble for the inner planets, some asteroids, etc.. And I wouldn't be suprised if some Newtonian solar system simulators out there are not even accurate over one year!

 

[Stingray]

There are certainly many types of solar system simulators out there. I remember making one myself back when I first learned Newton's laws. My algorithm was probably complete junk. It was fun to watch though.

I think modern research-level simulators use symplectic techniques (for studying solar system stability and such). They stick in lots of other things by hand, including GR effects for Mercury. The claimed accuracy is up to a billion years as I recall.

 

[Chris Hillman]

I think modern research-level simulators use symplectic techniques (for studying solar system stability and such). They stick in lots of other things by hand, including GR effects for Mercury. The claimed accuracy is up to a billion years as I recall.

Right, although I recall a few hundred million years for the Digital Orrery. Some citations I already mentioned in another thread:

James Applegate, M. Douglas, Y. Gursel, Gerald Jay Sussman, Jack Wisdom, The outer solar system for 200 million years, Astronomical Journal, 92, pp 176-194, July 1986, reprinted in Lecture Notes in Physics #267 -- Use of Supercomputers in Stellar Dynamics, Springer Verlag, 1986.

James Applegate, M. Douglas, Y. Gursel, P Hunter, C. Seitz, Gerald Jay Sussman, A digital orrery, in IEEE Transactions on Computers, C-34, No. 9, pp. 822-831, September 1985, reprinted in Lecture Notes in Physics #267, Springer Verlag, 1986.

The conclusion is that the solar system is not after all stable:

Gerald Jay Sussman and Jack Wisdom, Chaotic evolution of the solar system, Science, 257, 3 July 1992.

Gerald Jay Sussman and Jack Wisdom, Numerical evidence that the motion of Pluto is chaotic, Science, 241, 22 July 1988.

 

[coalquay404]

I agree that GRtensorII is a capable piece of software, particularly if you're interested in investigating the properties of various exact solutions. However, it's not very extensible, and I believe that it's mature software to the extent that it's no longer actively being developed. (I am of course open to correction on this.)

For what it's worth, Kasper Peeters at the AEI is working on a new CAS called http://www.aei.mpg.de/~peekas/cadabra/ which seems to show a great deal of potential. It's designed primarily with field theories in mind (i.e., field theories on a fixed background), but should nonetheless be useful if you're interested in such things or, for example, studying QFTs on curved spacetimes. Unfortunately, it's very new and runs only on Linux and OS X. Binaries exist, but it takes a bit of work to get them up and running.

 

[Chris Hillman]

I agree that GRtensorII is a capable piece of software, particularly if you're interested in investigating the properties of various exact solutions. However, it's not very extensible,

I don't know what you mean by "extensible", but it might be worth pointing out that the GRTensorII website doesn't try very hard to show what it is good for. Come to think of it, the way that I use this package is perhaps not terribly close to what the authors suggest. For example, I never use their built in command to create a frame definition, I directly edit the *.mpl file using vim, and I rarely need to type anything into Maple in order to run GRTensorII commands.

About extensibility: again, I don't know what you mean by "not very extensible", but it might be worth stating that the grdef command is very convenient for defining arbitrary tensorial quantities and then computing them routinely for zillions of families of Lorentzian spacetimes. Also, don't underestimate the importance of the ease with which gtr handles frames; this is really one of its strongest points for studying specific families of spacetimes. And don't forget that despite some awkwardness in passing information with other Maple packages, the very fact that it is a Maple package means that it is easily integrated with all the other fine stuff Maple can do (I already gave a slick example for solving the Killing equations using the powerful casesplit command). There is a built-in tensorial package for Maple, but this is far less convenient. Overall, GRTensorII is by far the best package all around I've come across, for messing around with gtr.

For what it's worth, Kasper Peeters at the AEI is working on a new CAS called http://www.aei.mpg.de/~peekas/cadabra/ which seems to show a great deal of potential. It's designed primarily with field theories in mind (i.e., field theories on a fixed background), but should nonetheless be useful if you're interested in such things or, for example, studying QFTs on curved spacetimes. Unfortunately, it's very new and runs only on Linux and OS X. Binaries exist, but it takes a bit of work to get them up and running.

Cool, but QFT in a fixed spacetime is something completely different, although I entirely approve of things which run only on Linux or OS X because a scientist really has no business using an unstable OS.

I haven't been keeping up with Mathematica packages, but surprisingly enough it seems that there is still no real competitor for GRTensorII coming from that camp. (Ricci, developed by Jack Lee with one of my former graduate student colleagues at UW, is really geared toward Riemannian geometry, not Lorentzian geometry, despite the origins in studying Einstein manifolds, which in mathematics usually means a Riemannian manifold such that the Ricci tensor is a constant scalar multiple of the metric tensor.

 

[cesiumfrog]

For what it's worth, Kasper Peeters at the AEI is working on a new CAS called http://www.aei.mpg.de/~peekas/cadabra/ which seems to show a great deal of potential. It's designed primarily with field theories in mind (i.e., field theories on a fixed background), but should nonetheless be useful if you're interested in such things or, for example, studying QFTs on curved spacetimes. Unfortunately, it's very new and runs only on Linux and OS X. Binaries exist, but it takes a bit of work to get them up and running.

How familiar are you with this project?

Forgetting the quantum specialisations, I'm wondering how it compares to other systems (eg. mathematica, maple, axiom or maxima). Is it able to take a metric (or calculate one, eg. Schwarzschild) and produce covariant derivatives, Rieman tensor components, etc? How scriptable is it? Can it numerically solve ODEs? Plot graphs?

 

[Chris Hillman]

Hi, cesiumfrog,

I'm wondering how it compares to other systems (eg. mathematica, maple, axiom or maxima). Is it able to take a metric (or calculate one, eg. Schwarzschild) and produce covariant derivatives, Rieman tensor components, etc? How scriptable is it? Can it numerically solve ODEs? Plot graphs?

I just skimmed the manual, and it looks like it is NOT a CAS in the sense of Maple or Mathematica, but a framework for a performing certain QFT computations if you are willing to write (in C++) the necessary routines yourself. Almost the first thing they say is that by design they avoided writing a language for programming in Cadabra itself. In sharp contrasts, everything I would consider a CAS, including Maple, Mathematica, and more specialized packages such as Macaulay2, Singular, GAP and so on, do provide a programming environment. (Macaulay2 is particularly fun because of its strongly typed language which conforms perfectly to mathematical usage.)

According to the manual, Cadabra can compute
$$g_{ab}, \, g^{ab}, \, C_{abcd}, \, R_{abcd}$$
and it can perform some symmetrizations and some other algebraic manipulations. But it appears to be far, far less flexible than GRTensorII, and certainly far less ready for daily use by students.

At a glance, it looks like Cadabra cannot solve differential equations. In fact, at a glance it is not clear that it can really differentiate, unless you write your own routines to say evaluate covariant derivatives of tensorial quantities.

OTH, it is free, so perhaps one could pick up some of the slack with Maxima; the fact that Cadabra is written in C++ does at least mean (I think) that it can be run as a stand alone program, so if you can figure out how to pass data to Maxima you might be able to get some of the functionality most of us would demand in a proper CAS.

GRTensorII is also free, but since it is a Maple package you do need to obtain Maple, which is certainly not free (except for academics at universities with a generous site license). OTH, despite some glitches, for the most part, GRTensorII meshes very well with all the nifty stuff Maple can do (I've played with Maxima and hope that this project continues to develop, but right now Maxima is pretty pathetic compared to Maple or Mathematica and I hardly dare hope for the situation to change drastically any time soon).

 

[Stingray]

Right, although I recall a few hundred million years for the Digital Orrery. Some citations I already mentioned in another thread:

Thanks for the references. They sound interesting.

By the way, have you had much experience with MathTensor? I've never actually used it (or seen anyone use it), but it seems like it would be useful. Unless I'm missing something important, GRTensor is severely limited by having to work with specific metrics and component values.

 

[coalquay404]

Well, by "extensible" I was actually referring to the awkward manner in which output from GRTensor has to be passed to other Maple packages. This is of course possible, but I don't believe it is something which was an explicit design goal of GRTensor. I agree with the comment about Mathematica, by the way. Any time I need to use a CAS I go straight to Mathematica - I think Maple's become horribly bloated over time - so it's surprising that nothing new has been produced for it.

How familiar are you with this project?

Forgetting the quantum specialisations, I'm wondering how it compares to other systems (eg. mathematica, maple, axiom or maxima). Is it able to take a metric (or calculate one, eg. Schwarzschild) and produce covariant derivatives, Rieman tensor components, etc? How scriptable is it? Can it numerically solve ODEs? Plot graphs?

I started playing with it only a couple of weeks ago so I don't know if I'd describe myself as being familiar with it. From what I've seen so far though, I am pretty impressed. As I said, it's main goal is to be useful in "field theories" so if you're working with, for example, a theory in which you need to deal with a twelve-form and fermionic degrees of freedom, it seems to save an awful lot of pen-and-paper anguish.

Covariant derivatives and the like can be handled using the Derivative property. For example, you can define an object called "nabla" and assign it the property of being a derivative thus:

\nabla{#}::Derivative

Then if you have some one-form $A_{a}$ you can output the derivative of $A_a$ by typing

\nabla{A_{a}};

(Notice the similarity between this notation and that of TeX?) Since Derivative is a class which doesn't necessarily commute, you can then define covariant derivatives in a straightforward way.

As I said, I'm only learing this at the moment, but I think it's got potential.

EDIT: Having read Chris' post it should probably be emphasised that Cadabra is not to be considered a CAS on a par with something like Maple. It's definitely not something that I would point an undergrad towards since better alternatives exist for the problems which he or she would be likely to come up against.

 

[Chris Hillman]

Well, by "extensible" I was actually referring to the awkward manner in which output from GRTensor has to be passed to other Maple packages. This is of course possible, but I don't believe it is something which was an explicit design goal of GRTensor.

Agreed, but I pointed out that this "awkwardness" is the exception, not the rule.

I agree with the comment about Mathematica, by the way. Any time I need to use a CAS I go straight to Mathematica - I think Maple's become horribly bloated over time - so it's surprising that nothing new has been produced for it.

Good grief, you completely misread what I said. Please reread it more carefully!

I started playing with it only a couple of weeks ago so I don't know if I'd describe myself as being familiar with it. From what I've seen so far though, I am pretty impressed. As I said, it's main goal is to be useful in "field theories" so if you're working with, for example, a theory in which you need to deal with a twelve-form and fermionic degrees of freedom, it seems to save an awful lot of pen-and-paper anguish.

Right, the main goals of Cadabra appear to be completely different from GRTensorII; they are not really comparable at all.

Covariant derivatives and the like can be handled using the Derivative property. For example, you can define an object called "nabla" and assign it the property of being a derivative thus:

\nabla{#}::Derivative

Then if you have some one-form $A_{a}$ you can output the derivative of $A_a$ by typing

\nabla{A_{a}};

(Notice the similarity between this notation and that of TeX?) Since Derivative is a class which doesn't necessarily commute, you can then define covariant derivatives in a straightforward way.

So, can you enter say the Schwarzschild metric tensor in a convenient way, compute the covariant derivative of the Riemann tensor, and extract components with respect to some frame field? (Looks like they call a frame a vierbein.) I did notice the tex thing (they make much of that in the manual) and agree that is an attractive feature. But when you say "define covariant derivatives", I think you mean that the user is expected to see how to literally tell Cadabra the definition of a covariant derivative in gory detail, which might be a bit daunting for students.

As I said, I'm only learing this at the moment, but I think it's got potential.

Agreed, it does sound like this might eventually become a useful tool.

EDIT: Having read Chris' post it should probably be emphasised that Cadabra is not to be considered a CAS on a par with something like Maple. It's definitely not something that I would point an undergrad towards since better alternatives exist for the problems which he or she would be likely to come up against.

I do think you should have mentioned that from the get-go, particularly since it was not clear (at least not to me) that the OP is even a currently enrolled undergraduate student.

 

[Chris Hillman]

MathTensor

By the way, have you had much experience with MathTensor?

No, apart from Maple, I use hardly any commercial software.

Unless I'm missing something important, GRTensor is severely limited by having to work with specific metrics and component values.

No, you are correct--- this is what I was referring to in my post when I said it is unsuited to index gymnastics.

OTH, if you have some metric in mind, typically a metric with some undetermined functions, but not one which is completely general, then GRTensorII is perfectly suited to study this family of spacetimes, for example by determining which ones are vacuum solutions, or which have the right eigenstructure to be candidate perfect fluid solutions (after this, you still need to check positivity and energy conditions and ideally some other things too). For this reason, it is very effective at finding exact solutions using the Ansatz method. Because it supports frames/coframes, it is ideally suited to computing physically meaningful quantities ("physical components").

There is a need, I think, for a package which can perform index gymnastics in a similarly convenient manner. (Actually, there are some, written by various researchers, but these are buggy, have limited availability, often are not very portable, and suffer from other flaws.)

 

[coalquay404]

Good grief, you completely misread what I said. Please reread it more carefully!

I don't think I did misread what you say, although I should perhaps have quoted the relevant text. My comment was in reference to this:

I haven't been keeping up with Mathematica packages, but surprisingly enough it seems that there is still no real competitor for GRTensorII coming from that camp.

What I was trying to get across was that (a) I'm surprised we haven't yet seen any really first-rate packages for Mathematica which make tensorial calculations both easy and powerful and (b) apart from my need to make use of GRTensor I tend to avoid Maple like the plague.

Right, the main goals of Cadabra appear to be completely different from GRTensorII; they are not really comparable at all.

Simply because the stated goals are not the same as those of GRTensor doesn't mean that the packages themselves are not comparable. For what it's worth, I think a user should be able to use Cadabra in much the same way as GRTensor were one so inclined.

So, can you enter say the Schwarzschild metric tensor in a convenient way, compute the covariant derivative of the Riemann tensor, and extract components with respect to some frame field? (Looks like they call a frame a vierbein.) I did notice the tex thing (they make much of that in the manual) and agree that is an attractive feature. But when you say "define covariant derivatives", I think you mean that the user is expected to see how to literally tell Cadabra the definition of a covariant derivative in gory detail, which might be a bit daunting for students.

I suppose it depends on your interpretation of "convenience". ;-) Again, I'm not suggesting that Cadabra is necessarily a better tool for what the OP has in mind than, say, GRTensor. What I am saying is that it has the potential to be both more powerful and, perhaps, more useful. Regardless, if I've alerted some people to it's existence I'll take that as a plus.

I do think you should have mentioned that from the get-go, particularly since it was not clear (at least not to me) that the OP is even a currently enrolled undergraduate student.

I'm not certain of the OP's status either since he didn't bother to tell us but as I said, others who read this thread might be looking for something along the same lines so I don't think it's really any harm to alert people to some of the choices available.

 

[Chris Hillman]

Cadabra vs. GRTensorII for gtr comps (NOS)

Hi, coalquay404,

In an earlier post in this thread, you wrote, referring to my own post,
"I agree with the comment about Mathematica, by the way. Any time I need to use a CAS I go straight to Mathematica - I think Maple's become horribly bloated over time - so it's surprising that nothing new has been produced for it." I took this to mean that you thought I had unfavorably compared Maple with Mathematica; I did not, and in fact I use Maple, not Mathematica. And I don't agree with your statement that "nothing new has been produced for [Maple]". For the record.

What I was trying to get across was that (a) I'm surprised we haven't yet seen any really first-rate packages for Mathematica which make tensorial calculations both easy and powerful and (b) apart from my need to make use of GRTensor I tend to avoid Maple like the plague.

Fair enough. For the record, wherever possible, I urge serious researchers to use both Mathematica and Maple and attempt to constantly check one against the other for anything important. Maple even offers some conversion code to facillate importing results from a Maple worksheet into a Mathematica worksheet. Both of these packages are very capable, and they both have passionate fans. I feel they each have significant advantages, and also significant disadvantages.

Simply because the stated goals are not the same as those of GRTensor doesn't mean that the packages themselves are not comparable. For what it's worth, I think a user should be able to use Cadabra in much the same way as GRTensor were one so inclined.

Well, I guess the only way to find out would be for me to email some typical Maple worksheets recording some of my own playdates with GRTensorII and have you email me back postscript or pdf versions showing how you reproduced that session with Cadabra. It would actually be very cool if you were right! You did mention the issue of continued development, and that is always a concern; the worst nightmare of Maple or Mathematica users is that those companies might go bankrupt, potentially threatening decades of data. As you may know, GAP passed an important milestone recently when this enormous project was handed over peaceful to a second generation of committed open source developers.

I suppose it depends on your interpretation of "convenience". ;-) Again, I'm not suggesting that Cadabra is necessarily a better tool for what the OP has in mind than, say, GRTensor. What I am saying is that it has the potential to be both more powerful and, perhaps, more useful. Regardless, if I've alerted some people to it's existence I'll take that as a plus...I don't think it's really any harm to alert people to some of the choices available.

I'd like to know more if I have underestimated the power of Cadabra. But you should really show me some specifics if you want me to take your claims seriously, since on casual inspection I didn't see much obvious evidence for them. A good place to start would be to clarify what kinds of computations you think could be conveniently performed in Cadabra. (E.g. general index gymnastics? Computations of specific components of some tensor with respect to a specific coframe field? Newman-Penrose formalism? Linear perturbations? What about general computations such as solving differential equations, Groebner basis computations, integration, graphing?) If these form a distinct set from those which I claim can be conveniently performed in GRTensorII, perhaps the truth is that these two packages potentially complement one another well. It would also be helpful if you could clarify how much work is required to make the "definitions" mentioned in the manual.

Gloss: NOS="not otherwise specified".

 

[pervect]

What I was trying to get across was that (a) I'm surprised we haven't yet seen any really first-rate packages for Mathematica which make tensorial calculations both easy and powerful and (b) apart from my need to make use of GRTensor I tend to avoid Maple like the plague.

Could you fill us in on what you don't like about Maple or prefer about Mathematica?

I haven't had a chance to evaluate Mathematica, but the one thing I really don't like about Maple is is symbolic integration. The problem is that I give Maple functions which are real-valued in some domain of interest, and often get symbolic integrals that do not evaluate to real numbers in that domain. If there's a way of telling Maple what the domain of interest is, I haven't found it - assume doesn't seem to do the job.

Other than that, I have no complaints about maple.

As far as passing objects to Maple from Grtensor, I find that either grarray or grcomponent does a good job.

 

[tehno]

Depending upon what you have in mind, perhaps yes. You said "simulating problems"--- did you mean "solving problems"? Or perhaps even just "making computations"?

GRTensorII is very easy to use and very well suited for finding exact solutions and studying their properties. It is not very well suited for making "index gymnastics" type computations, but some other packages attempt to provide tools for these computations. There are further specialized packages for making yet other kinds of computations. See http://www.math.ucr.edu/home/baez/RelWWW/software.html for some links.

Thanks for your answer Chris Hillman.At least ,there is something _^^_outthere.
Yes ,I meant computations (simulations) .
Defining the appropriate "geometry" and boundary conditions in the model and go with flow of field equations.
I used to work with and on simulating packages for problems dealing with (classical) electromagnetism .So ,I wondered if some (considerable) effort was put in developing specialized software for GR.
I'm no expert in GR .

A good rule of thumb is that if you don't know how, in principle, to work a certain computation by hand, you probably shouldn't try to do it by computer! But I trust this caveat is superfluous in your case, tehno.

Sarcasm ?

 

[coalquay404]

Apologies for the lateness of my reply - things are quite busy here at the moment.

Hi, coalquay404,

In an earlier post in this thread, you wrote, referring to my own post,
"I agree with the comment about Mathematica, by the way. Any time I need to use a CAS I go straight to Mathematica - I think Maple's become horribly bloated over time - so it's surprising that nothing new has been produced for it." I took this to mean that you thought I had unfavorably compared Maple with Mathematica; I did not, and in fact I use Maple, not Mathematica. And I don't agree with your statement that "nothing new has been produced for [Maple]". For the record.

"For the record", that's not what I said at all. I said that I was surprised that nothing new has been produced for Mathematica which could rival GRTensor. The words I used were neither ambiguous nor misleading, and I'm quite unsure as to how you were confused by them.

Fair enough. For the record, wherever possible, I urge serious researchers to use both Mathematica and Maple and attempt to constantly check one against the other for anything important. Maple even offers some conversion code to facillate importing results from a Maple worksheet into a Mathematica worksheet. Both of these packages are very capable, and they both have passionate fans. I feel they each have significant advantages, and also significant disadvantages.

Perhaps this is more a comment on the way in which I work than anything else, but I would never rely on Maple or Mathematica for "anything important". Given the nature of what I work on, I'd never trust a CAS for anything which I might later want to write up. I view them as handy tools with which to play around with various ideas, but I've never (and probably will never) use them to produce any work that I'm willing to stand behind.

Well, I guess the only way to find out would be for me to email some typical Maple worksheets recording some of my own playdates with GRTensorII and have you email me back postscript or pdf versions showing how you reproduced that session with Cadabra. It would actually be very cool if you were right! You did mention the issue of continued development, and that is always a concern; the worst nightmare of Maple or Mathematica users is that those companies might go bankrupt, potentially threatening decades of data. As you may know, GAP passed an important milestone recently when this enormous project was handed over peaceful to a second generation of committed open source developers.

Unfortunately, this is a pretty busy time of year around these parts what with the undergraduates getting ready for the end of term and so on. As a result, I don't have the time nor, in fact, the inclination, to begin a detailed comparison of Cadabra and GRTensor. If you're interested in doing so, Cadabra is well documented and it should be straightforward to come up with a strategy for comparing the two.

I'd like to know more if I have underestimated the power of Cadabra. But you should really show me some specifics if you want me to take your claims seriously, since on casual inspection I didn't see much obvious evidence for them. A good place to start would be to clarify what kinds of computations you think could be conveniently performed in Cadabra. (E.g. general index gymnastics? Computations of specific components of some tensor with respect to a specific coframe field? Newman-Penrose formalism? Linear perturbations? What about general computations such as solving differential equations, Groebner basis computations, integration, graphing?) If these form a distinct set from those which I claim can be conveniently performed in GRTensorII, perhaps the truth is that these two packages potentially complement one another well. It would also be helpful if you could clarify how much work is required to make the "definitions" mentioned in the manual.

Gloss: NOS="not otherwise specified".

I'm not sure why I feel the need to point this out, but I am not acting as some type of evangelist for Cadabra. My reason for mentioning it was that I thought it may have been germane to the OP's question and that I think it's an interesting piece of software.