Basically, I need to learn Riemannian geometry (the metric tensor), how to derive the field equations and Schwarzschild's solution; and how to "obtain non-zero curvature components". So does anybody know if A First Course in General Relativity (Schutz) covers these topics well? Also I'm just now taking my first calculus course, (which doesn't cover partial differential equations, or vector calculus). So will I have the prerequites required?

Thanks!

__________________
The first principle is that you must not fool yourself - and you are the easiest person to fool.
~~~ Richard Feynman ~~~
It is imperative in science to doubt.
~~~ Richard Feynman ~~~
Common sense is not so common
~~~ Voltaire ~~~

I don't know, but I am also interested in finding out what kind of math background one needs before studying this subject.

But in case you don't know, Standford has a huge series of physics lectures by Leonard Susskind online. There's over 20 hours of lecture on General Relativity alone:

http://www.youtube.com/view_play_lis...8BDEEBA6BDC78D

And here are all the Standford videos:

http://www.youtube.com/profile?user=...view=playlists


Rob

Patience, young man, patience! Rome was not built in a day. What you're asking is sort of like a first year medical student wanting to do brain surgery next week, or a freshman engineering student who wants to build a supersonic airplaine next week by hand.

I'm not trying to discourage, mind you, just being realistic. Patience! You'll need a few years of the standard math and physics courses to get the groundwork. Differential geometry and tensor calculus is fairly advanced, and you need a good understanding of partial differential equations, vector calculus, linear algebra, and all that good stuff. The physics base is important as well to give you that sort of intuitive feel for what all the math is trying to model.

-Richard

So do you need partial differential equations, vector calculus, linear algebra, tensor calculus, differential geometry, etc in order to fully understand the concepts or the specific book, or both? I know that in order to get a complete understanding of the math I'll need to take a whole lot more courses, however I was hoping I the book would fill me with the bare minimum needed. Basically, I'm trying to get enough knowledge/understanding of the Schwarzschild solution and "non-zero curvature components" in order to help out with a black hole computational research project with a professor. Of course I would relearn the material in university in order to get a complete understanding. The professor stressed to me that it would be a ton of work, but he also said it was doable - I'm just trying to find a book that doesn't kill me with the math, but still provides me with the knowledge. So is this book to tall a order?

__________________
The first principle is that you must not fool yourself - and you are the easiest person to fool.
~~~ Richard Feynman ~~~
It is imperative in science to doubt.
~~~ Richard Feynman ~~~
Common sense is not so common
~~~ Voltaire ~~~

Taylor and Wheeler's Exploring Black Holes: Introduction to General Relativity will give you a tour of the Schwarzschild metric using pretty straightforward algebra and a tickle of calculus. It'll let you calculate orbits, light deflection, gravitational time dilation, and many other things, but all by taking Schwarzschild more or less as a given.
Your list of requirements digs considerably deeper, though. Does that come from your professor?

Grant Hutchison

Well he said that for a start I needed to learn: geodesic motion and gravity as curvature, the Schwarzschild solution, obtaining the non-zero curvature components of Schwarzschild geometry. So I assumed Riemannian geometry (metric tensor), and the field equations would be necessary, however I'm now unsure of what learning those concepts actual entails

__________________
The first principle is that you must not fool yourself - and you are the easiest person to fool.
~~~ Richard Feynman ~~~
It is imperative in science to doubt.
~~~ Richard Feynman ~~~
Common sense is not so common
~~~ Voltaire ~~~

I don't know the book by Schutz.

You are quite a long way from having the prerequisites for differential geometry.

You will need to understand multi-variable calculus well, at a level that is usually reflective of a junior-senior level class that includes at least the inverse function theorem and implicit function theorem and an introduction to differential forms. You also need at least passing acquaintance with elementary point-set topology and algebraic structures such as rings.

A good start might be something along the lines of Mike Spivak's book Calculus on Manifolds.

Then you can try a book on differential geometry that includes a treatment of Riemannian Geometry. Perhaps something like Barrett O'Neill's Semi-Riemannian Geometry with Applications to Relativity or Gravitation by Misner, Thorne and Wheeler.

That course will give you a solid foundation in Riemannian geometry. You should be warned that it is a rather demanding and rigorous path, and you may find that there are a lot of gaps that you will have to fill in along the way. Differential geometry is a fairly complex subject.

There are less demanding books on GR, and a good one seems to be Essential Relativity, Special, General and Cosmological by Wolfgang Rindler.

In any case the usual background required to study the subject of differential geometry is quite a bit greater than just introductory calculus. It falls under the heading of serious mathematics, and if you are now taking introductory calculus of one variable, you have not yet encountered what I am calling serious mathematics.

BTW a young fellow by the name of Albert Einstein had trouble with mathematics at this level of complexity, and he sought help from a mathematician friend. So don't be overly concerned if you find the going a bit tough at the start.

Would a course like this: http://ocw.mit.edu/OcwWeb/Mathematic...Home/index.htm provide me with enough of a foundation from which I could self-study sections of the other courses? Obviously, I'm not going to be able to take linear algebra, vector calculus, differential equations, partial differential equations, and differential geometry before university, but would it be possible to get by (i.e. know enough about; geodesic motion and gravity as curvature, the Schwarzschild solution, and obtaining the non-zero curvature components of Schwarzschild geometry) by self-studying small sections of those courses?

__________________
The first principle is that you must not fool yourself - and you are the easiest person to fool.
~~~ Richard Feynman ~~~
It is imperative in science to doubt.
~~~ Richard Feynman ~~~
Common sense is not so common
~~~ Voltaire ~~~

Like everyone else says, be patient. The thing is that you still don't have the background for that course. Multi-variable calculus is a second year course, and it sounds like you're starting your first year of calculus. I know you want to run, but you really need to learn to walk first.

If you're determined to do self study, start with with single-variable calculus. Yes, it's the course you're taking now, but you need to get through that material before you're ready for multivariable calculus.

Based on what I see with regard to the text and general level, that course is a pre-requisite for the pre-requisites to study introductory differential geometry. The calculus class that you said you are taking or about to take is a pre-requisite for this one.

The class appears to be a good class, but basically third-semester calculus with some basic vector analysis. That will put you in a position to understand a more advanced class that handles introductory differential forms and calculus in dimensions above 3.

The sorts of questions that you are asking are far more advanced than what you may think that they are. It takes quite a bit of work to even define what curvature means in higher dimensions. It took Riemann to figure out the right definitions. He was rather bright.

Publius gave you good advice. Don't rush things too much. You can't really understand special relativity until you understand Newtonian mechanics well. You can't understand Newtonian mechanics well until you understand calculus, in dimension at least 3 very well. You can't understand general relativity until you understand special relativity reasonably well. And then to understand general relativity you need to either under stand Riemannian geometry or be able to learn it at the same time as you are learning general relativity. This is decidedly non-trivial mathematics and non-trivial physics.

You said " Obviously, I'm not going to be able to take linear algebra, vector calculus, differential equations, partial differential equations, and differential geometry before university." But the hard fact is that without all of those things one cannot even begin to define what a geodesic is. Riemannian geometry is not a simple subject.

You need to be aware that there are two meanings to the term "Riemannian geometry" depending on the author. In many cases "Riemannian geometry" implies the existence of a positive-definite metric. That case is reasonably intuitive, since you get a "distance" function that behaves the way that you expect distance to work, and the underlying manifold is a topological metric space in the usual sense based on the distance that comes from the Riemannian metric. In other cases "Riemannian geometry" merely implies the existence of a non-degenerate quadratic form. This is sometimes called semi-Riemannian or pseudo-Riemannian geometry. In that case the "distance" that comes with the metric can be negative and two distinct points can be at zero "distance" from one another. The underlying manifold is not a topological metric space with this notion of "distance". General relativity works with a pseudo-Riemannian geometry. That increases the level of mathematical abstraction involved substantially.

If you jump into this too quickly the result is likely to be extreme frustration. That would be too bad as it could well kill a very healthy interest. You need to walk before you run. To learn a subject like physics or mathematics you need to thoroughly understand the basics before you tackle the hard stuff.

What you might profitably do is study the subject at a somewhat less mathematically demanding level. You might find a class that is suitable. I think in fact that KenG is teaching something along those lines right now and he might be able to give you some good advice on sources.

My feeling is you will need to learn GR in the reverse direction from what you'll get if you start with rigorous mathematical texts. Instead, I'd say you should start with the specific list of skills that your professor gave you, and try to see how those skills relate to the project you are doing. This is a bit like a medical student learning only about the foot, instead of taking a full anatomy course, because you know you are only going to be dealing with feet.

Of course, if you want to be a podiatrist, you will need to understand the full human anatomy, because a human is a connected object. But the objectives you laid out will simply not permit you to learn GR "the right way", so I feel you will have to learn it arse-backwards. You'll have to learn all about feet, and gradually construct a vague sense that there is a human being attached to that foot.

For example, there is a huge difference between understanding the mathematical entities well enough to do proofs, or even understand proofs, versus being able to manipulate them to do calculations. It is not completely unheard of (believe me) for a student working with a professor to spend a year or more immersed in solutions to a particular equation, and still not have the least idea how that equation is derived, or in some cases, even understand what it means. That has to change eventually, because eventually you'll be fielding questions on your work and people will ask things like why did you use that equation instead of some other one (and the other one will have some real jargony sounding name). But to get you started on your project, you can escape with a much rougher idea of what the heck you are doing, and I believe that is inescapable.

Here is a suggestion.

Don't alter the classes that you are currently planning to take. They sound about right.

Find yourself a copy of Mike Spivak's little paperback Calculus on Manifolds. See if you can read and understand it.

If you find that relatively easy going, then you are quite exceptional and you might be able to jump from that directly to a text on differential geometry.

If you find that it is too hard at the moment, do not be surprised or discouraged. Just wait a bit, continue to study math and physics and you will eventually get to the point where you are ready to study the more advanced stuff.

In either case hang on to Spivak's book. It will serve you well when you get to the point where you are ready for that material.

I don't really have any sources to recommend, but I think a course that focuses on some of the general logic of relativity, rather than the mathematical relationships, would help you "end run" the more complete approach that is simply not an option for you. Recall that Einstein once said "Since the mathematicians have invaded relativity, I hardly understand it myself any more." This does not mean that the rigorous formulation of relativity is a bad thing, it means that it is not the first thing you need to understand about relativity. It can come later, once you get the basic gist of what kinds of things you are allowed to think about in relativity, and how those things connect to the equations you'll be using.

In terms of what those new things that you need to think about are, a good start is understanding what relativity, any relativity, is. First of all, realize that relativity has nothing to tell us about local phenomena. We have to have other theories to tell us what the observers "on the scene" will see in regard to any system (where "on the scene" means sharing the same inertial motion as the system in question. By inertial I just mean the system has no external forces on it other than gravity, or else you are only talking about part of the system. There are ways to handle forces and fields, it gets a lot harder.) What relativity does is to take what that observer sees, and translate it to what all other observers would see if they received information from that system, moving along some path other than the one the system itself followed. If you are solving geodesics, the "other path" will involve light signals coming from the system, so for you GR will be the study of how light gets from an observer who moved with the system and saw something that had nothing to do with relativity (probably nothing very interesting at all, a stationary particle perhaps), to an observer you care more about, like Earth. GR will be the study of what a particle that is stationary for an observer with extensive (i.e., comoving) experience of that particle, will look like under light propagation to some other less hypothetical observer.

As such, the only physics involved will be the propagation of light through a gravitating environment. But a funny thing will "happen on the way to the forum"-- when you are done calculating how light will propagate between those two observers, you will find that some "permanent" differences will get etched into the fabric of reality (where by "reality" I mean the quantities that are invariant to the observers, all observers will reckon them the same-- specifically, any observation is such an invariant, but the fact that that particular observer observed it must be included in the invariant, all observers must agree that is what that particular person will see).

For example, by "etched" into the reality, I mean that if the observer who took a different path than the system (the system is probably just a particle, which of course is perceived as stationary by the on-the-scene observer, though the source of the gravitiy would be moving, whereas the observer of interest will probably be stationary with respect to the source of gravity) ends up sharing two points on the path of the system, their total proper time (the time on their own clocks) elapsed between those crossings will not be the same. You can calculate that difference by tracking the light paths through the gravitating environment (and in the absence of gravity, you can have the same effect for a noninertial observer, an observer with a real external force on them), and you can interpret that calculation as tracing out a "geodesic".

Perhaps the last thing to get at this stage is that if we interpret a geodesic as a "real trajectory" of something, we should realize that the "trajectory" is just through our own coordinate system, not through real spacetime, because there's no such thing as the latter. Spacetime is a kind of mathematical milieu through which calculations pass at some stage, a bookkeeping system related to a coordinate choice. It might help motivate the mathematical calisthenics if you recognize that.

Spratley,


That question "How to derive the Einstein Field Equation" prompted me to chuckle, and was why I (and everyone else) responded with "be patient", etc. To see why, maybe this will help, a little Wiki article on the Einstein-Hilbert action (Hilbert was one of Einstein's friends from whom he required a little help as Dr. Rocket mentioned):

http://en.wikipedia.org/wiki/Einstein-Hilbert_action

Does that look like anything other than gobbledy-gook? You may ask what is "action", and what is "varying" it? And what's a Lagrangian? All that requires a few years of physics courses to appreciate. Once you have learned that, you'll realize that the "Principle of Least(Stationary, better) Action" is one of the most elegant principles of physics. It just gives you the warm fuzzies.

Your enthusiasm is admirable, but it's going to take patience and discipline to get where you want to be. Once you're there, you might well become one of the high priests of GR & Gravitation. But that's not going to happen overnight.

-Richard

Looking at that Einstein-Hilbert action page reminds of something important, which it mentions I think.

The EFE in the traditional approach is derived by varying the metric to get the stationary action. The connection is assumed to be the Christoffel or Levi-Civita one (ask Doc Rocket what all that means. ).

However, that's not the only way to do it. See the Palatini action and the Palatini variation. That varies the connection to get the field equations. As I understand it, that yields the good ol' EFE just like the simpler way, and shows the Levi-Civita connection is not just an assumption. Again, ask DocRock what all that means.

Apparently, varying the connection is the more general way to do it, and allows you to deal with much more abstract "stuff". The loop quantum gravity guys are enamored with this approach I think.

Anyway, what all that means in the big picture, I think, is this. All this stuff about spacetime being a geometry with curvature and all that is a just model, just a story we have concocted to get the right answers. The connection, which is basically about Ken's translating what a local observer on the scene sees to a distant observer is all that is important. There need to be any "reality" in between.

Whatever story, whatever mathematical structure you come up with in between to explain that connection doesn't really matter and shouldn't be thought of as "real".

-Richard

Actually the guy that I had in mind was Marcel Grossman. It was Grossman to whom Einstein turned with the request to find a mathematical system that might describe gravitation, and it was Grossman who turned Einstein on to Riemannian geometry and helped him with the mathematics of tensors.

Einstein did correspond with Hilbert, and they became good friends. But during the period when Einsein was trying to formulate the field equatios he and Hilbert were both studying that problem and were competing. The chronology is a bit cloudy, but there is a good case in the minds of some (Kip Thorne is apparently one) that Hilbert formulated the correct invariant field equations before Einstein. In any case they both published within days of one another. There in no disagreement however on who should get the credit for general relativity. Even Hilbert went out of his way to make certain that Einstein was given credit as the sole author of the theory.

Hilber was a FAR more powerful mathematician than Einstein. He was in fact, probably the most influential and powerful mathematician in the world at that time. He met with Einstein and they discussed Einstein's ideas with regard to his theory of gravitation. That is probably what started the competitive race to find the covariant field equations. Hilbert probably did find the right tensor equations first. But he gave credit for the theory to Einstein. He is quoted as saying "Every boy in the streets of Gottingen understands more about four-dimensional geometry than Einstein. Yet, in spite of that, Einstein did the work and not the mathematicians." After that race for the equations and the priority squabble in the press, they became great friends.

I'd certainly put David Hilbert up there in the top tier with the other greats, that's for darn sure. He's one of those names, like Euler, that comes up everywhere.

I was somewhat aware of the "competition" you mentioned between Hilbert and Uncle Al. I think that's the way they call it the Einstein-Hilbert Action, to credit that both came up with up independently. Of course, the EFE bears Einstein's name exclusively, and I agree it should. I think at first, Einstein was trying to set the Ricci curvature tensor directly equal to the stress-energy tensor, and that didn't work. The second term in what is now called the Einstein tensor involving the Ricci scalar and the metric comes from the correct Einstein-Hilbert Action.

There is some article I wish I could remember that discusses how all that and why the second term has to be there.

And back to Hilbert, something that chills me to the bone was a comment Hilbert made near the end. The Nazis had completely purged Jews from Gottingen (and that included Weyl and Noether). The infamous Nazi Education Minister was at some function with Hilbert and asked him how was Mathematics doing now that they have purged the undesirables out. Hilbert replied, "There is no Mathematics at Gottingen."

-Richard

In regard to the challenges of making "understanding how to derive the EFE" a goal, does anyone on this forum know how to derive the Einstein field equations? I don't mean can the follow the derivation, I mean can they understand/explain it. I certainly couldn't. You could Google John Baez and general relativity to get something that approaches insightful descriptions, but I think it is not only unrealistic to want to understand that derivation given the prerequisites, I think it's also unrealistic to expect patience to be the cure! The point is, being able to derive it really doesn't say much, it's kind of a mathematical exercise starting from an action that really has no intuitive meaning. I don't think the act of working with the equation is informed much at all by the process of deriving it. You'd do better just practicing with applications of the resulting equations.

Incidentally, if you care I can give you my take on why an action is a good way to go for deriving these equations. In Newtonian mechanics, you can use an action, which is a path-integral approach, or you can use Newton's law, which is a differential-equation approach. The philosophical difference is that in a path-integral approach, the question you ask is what path will be taken between two given endpoints. Basically, this means you think of the location of the particle at two points in time to be the given, and you ask what velocity do you need to give the particle to bridge those constraints, and what path does it take. It's like a "skeet shoot" approach, where you are calculating how you have to aim your gun to hit a target. In the process, you can derive a differential equation that connects each position and velocity with the next position and velocity, not for its own sake, but because it is an allowable piece of one of these path-integrated solutions.

Of course the other alternative is the Newton's-law approach that we tend to teach first, which treats the differential equation (A = F/m) as the fundamental driver of the dynamics, not a resultant of some allowed time-integrated path. Which one seems more fundamental depends on the question you ask-- if you know the position and velocity, it seems more direct to ask what's the acceleration (A = F/m), but if you know the endpoints of some path the particle needs to take, it seems more natural to ask what path bridging them gives a stationary action, and then simply note in passing that all such paths must locally obey A = F/m.

What then is the advantage of the action approach, if it only seems like a more natural approach in certain situations? The key advantage is that path-integrated approaches need no ontology of space and time, we never need to think of space and time as "real" to apply the concept of action. That's because philosophically, the contraint involved with the action is the endpoints of the path, expressed in any coordinates you like, and the path that connects them, in any coordinates you like, must be the one with stationary action. The coordinates are vastly more flexible than in the A=F/m approach, where A is supposed to be a "real" acceleration. The flexibility of the coordinates, when using an action approach, allows you to tailor your coordinates with total impunity to whatever bizarre constraints the system might be forced to obey, constraints that you do not wish to analyze in terms of "real" forces. The principle of stationary action will then automatically take those coordinates, which seemlessly integrate those bizarre constraints (pulleys, levers, strings, whatever), and convert them into a generalized form of A=F/m. You don't need to take A = F/m seriously this way, it depends on the coordinates.

So this is our hint as to why action is useful in GR. The main point of GR, it seems to me, is to recognize that any process can be described in terms of any arbitrary spacetime coordinates, and they all have to give the same reality once the coordinate translations are specified. So there can't be a "preferred" set of coordinates where A = F/m, and other coordinates have to first be translated before that equation can apply. Instead, laws of physics have to be an automatic algorithm for deciding if a hypothetical process is physically possible using any coordinate system to describe it. Since action is path-integrated, whether or not it is stationary does not care about coordinates, any coordinates that would serve to specify the path would also serve to determine if that path is physically possible or not. (To those who know what this means, all I've said is tantamount to the much more concise statement "action is a scalar.")

I don't quite understand the varying of the metric to get the stationary action -- yet. But I am working on it. Don't hold your breath.

But the general idea with a connection is that it an assignment of a covariant derivative to vector fields. This provides a means of parallel transport of tangent vectors, and it is the connection that tells you about curvature and torsion. Torsion and curvature are actually defined in terms of the connection. The connection determines geodesics -- geodesics are (parameterized) curves for which the tangent vectors are parallel along the curve.

If you happen to have a (pseudo) Riemannian metric defined on a manifold, then there is a unique connection that is torsion free and that under parallel transport preserves inner products (defined by the metric tensor) in the fibers of the tangent bundle. So, under the assumption that the torsion tensor is zero, if you know the metric you also know the connection. That connection is called the Levi-Civita connection, and what you do when you calculate Christoffel symbols is find that connection.

It is this relationship between the metric and the connection that makes Riemannian geometry what it is. That is the importance and the simplification accorded to the lack of torsion from a mathematical perspective. If you don't impose the zero torsion condition then things get more complicated and you get into a more general theory of connections. I think that this is jumping-off point for Einstein-Cartan theory which uses a gadget that is called a Cartan connection after Elie Cartan who developed the theory. I will have to do a lot more reading and studying to understand this stuff. It is pretty complicated. I'm still trying to figure out GR in rigorous mathematical style and this other version is much more difficult.

Elie Cartan developed the foundations for most of modern differential geometry -- differential forms, theory of Lie groups, and a lot more. Basically the successor to Riemann. He was the right guy at the right time to apply very sophisticated techniques in differential geometry and formulate a generalization of GR that does not make the zero torsion assumption.

Okay, so I guess the next question is: how much can I learn about geodesic motion, gravity as curvature, the Schwarzschild solution, and obtaining the non-zero curvature components of Schwarzschild geometry without a ton of advanced math? I've read all the "popular" books, and a couple "intermediate" ones (such as Gravity From the Ground Up), so I guess I need to look for something that uses math, yet doesn't kill me with differential geometry, etc. I still have two years of high school left, so I could probably learn a lot of the math required for a complete understanding before university, but I really need to be done learning what ever I'm going to by the end of the spring so I can apply it on the research over the summer.

So do you think Schutz's A First Course in General Relativity is going to be too hard (with the maths)?

__________________
The first principle is that you must not fool yourself - and you are the easiest person to fool.
~~~ Richard Feynman ~~~
It is imperative in science to doubt.
~~~ Richard Feynman ~~~
Common sense is not so common
~~~ Voltaire ~~~

It might be easier to start not by defining a set of skills to lear, but rather by defining a set of tasks that need to be done. Knowing what tasks you are expected to perform as your part in the research project will tell you what specific skills you will need to get the job done. If you are going to write software, for instance, it is always possible to solve an equation numerically without having the foggiest notion what you are doing mathematically. It's not the same kind of learning of course, but it does get the job done. So do you know what you are specifically going to do in this project? You may not really need to learn everything at once, or you may be able to go light on some topics and concentrate on others, to perform specific tasks.

You will quickly find out if you can handle the math simply by looking at Sean Carroll's book Spacetime and Geometry: An Introduction to General Relativity (Pearson/Addison-Wesley, 2004; the book grew out of Carroll's online Lecture Notes in General Relativity). Carroll has no "introduction to the math" chapters, so for learning the math end of the business, try Explorations in Mathematical Physics: The Concepts Behind an Elegant Language (Don Koks; Springer, 2006). His book includes chapters on tensors, differential geometry, and general relativity (including the Schwarzschild metric). Highly recommended for anyone with an interest in mathematical physics.

__________________
The point of philosophy is to start with something so simple as not to seem worth stating, and to end with something so paradoxical that no one will believe it. -- Bertrand Russell

Explorations in Mathematical Physics looks like a interesting book; from reading some reviews it seems like it covers more topics, but less in-depth which is exactly what I need! My only question is do I have the background to handle the book? A few reviews said you need to have taken linear algebra and differential equations. As for tasks vs skills I see what you're saying, but my professor said that I couldn't go anything until I understood concepts and could obtain those things. After talking to you guys I know that I will not be able to fully understand much of the math or concepts fully, so I need to focus on learning the bare minimum. Is there a way to acquire those skills in say the next 9 months?

I really appreciate you guys taking the time to help a novice like me!

__________________
The first principle is that you must not fool yourself - and you are the easiest person to fool.
~~~ Richard Feynman ~~~
It is imperative in science to doubt.
~~~ Richard Feynman ~~~
Common sense is not so common
~~~ Voltaire ~~~

Have you asked the professor the questions you're asking us? That is, have you asked him about good books to read to help you prepare for your project next summer? I would expect him to have some good recommendations.

The only way to find out if you can handle the material is for you to try.

If want some advice, I strongly recommend that you learn a good bit of linear algebra (not just matrix manipulation) and real calculus of several variables before you try. To do otherwise is rather like trying to read Moliere in the orginal without bothering to learn French.

But go ahead and try. If you find that you can't handle the subject, then all that is lost is some time while you go back and learn the basics. That will provide you more progress than you will obtain by simply wondering if you can do it.

Tobin Dax - My professor recommended Schutz's book for learning the physics - however he's left it entirely up to me as to how to learn the math that Schutz's book requires.

Dr. Rocket - Well I'm pretty sure I will not be able to understand much of anything right now (I haven't started single variable calculus yet), I'm just trying to get an idea of what I'll need to learn (after I after finish single variable calculus) before I can handle Schutz's book.

I'm definitely going to learn vector calculus, and then I'll try to learn linear algebra, differential equations, differential geometry, etc as I need it

__________________
The first principle is that you must not fool yourself - and you are the easiest person to fool.
~~~ Richard Feynman ~~~
It is imperative in science to doubt.
~~~ Richard Feynman ~~~
Common sense is not so common
~~~ Voltaire ~~~

I took a look at some of the pages in Schutz's book through Amazon.com

There is no way you are going to follow his treatment of tensors without knowing basic linear algebra and mult-variable calculus. The good news is that linear algebra is pretty easy and you already know that you need calculus of several variables.

I am curious about this research project. What sort of project involves high school students in a numerical calculation regarding black holes ? How did you become involved in this ? Some sort of advanced physics program for high school students ? Is there a team involved, with perhaps some post docs ?

Yes, that is what I was saying as well. I'm very concerned that this task will be impossible unless you structure it in terms of clearly defined goals that sound like "find the solutions of equation X to answer some particular question" rather than "learn multivariate calculus this week and differential geometry next week." If you try to do the latter, there won't be enough of a context to motivate you, you'll forget on Wednesday 95% of what you learned on Tuesday. Look at the skills you either have now or are just a few months away from (like computer programming, or differential equation solving, etc.), and figure out which ones are going to occupy you day to day on this project.

The OP also asked about deriving Schwarzschild. Wiki has a page on that:

http://en.wikipedia.org/wiki/Derivin...child_solution

That's a down and dirty way to do it, and one that certainly stands on the shoulders of giants. You know how that goes. You work and work and pull your hair out trying to solve something and success finally comes. And then you polish it up, get the deadweight out and come up with some simple and slick. But never let the simplicity and slickness of the final product detract from how hard getting there was.

IIRC, Hilbert also had a hand in polishing Schwarzschild's original derivation, and the form we all use today is really Hilbert's.

-Richard

The project is mostly computational in nature - it has to do with studying the orbits of neutrinos around black holes. My professor said he had worked on the project earlier with a few advanced undergrads (who he told me were accused of being grad students because the project was so advanced). So I'm not sure what kind of structure remains. Basically, I read a couple of things about the Intel talent search and it seem like an awesome idea - doing advanced and interesting research while in high school. So I emailed a few professor that did research with areas I was interested to see what kind of opportunities existed. Anyways, a professor agreed to let me help out with their project (I didn't want to just be a "lab rat") and gave me some books to study. I just talked to him and he told me that I Schutz covers all the math beyond calculus that you need, so I'm going to take a crank at that this winter and see how it goes!

I just wanted to think all you guys again for helping me out

__________________
The first principle is that you must not fool yourself - and you are the easiest person to fool.
~~~ Richard Feynman ~~~
It is imperative in science to doubt.
~~~ Richard Feynman ~~~
Common sense is not so common
~~~ Voltaire ~~~

That's what I was thinking. Baez gives you something like this:

But what does Einstein's equation mean? Well, it turns out that that a = b = 0 component of this equation can be translated into plain English as follows:

Take any small ball of initially comoving test particles in free fall. As time passes, the rate at which the ball begins to shrink in volume is proportional: to the energy density at the center of the ball, plus the flow of x-momentum in the x direction there, plus the flow of y-momentum in the y direction, plus the flow of z-momentum in the z direction.

Even better, if this holds in every coordinate system, Einstein's equation holds. So, all of general relativity can be recovered from this one sentence if one really thinks hard. In practice, though, it's best to master tensors and learn how to calculate with them to extract the predictions of general relativity.

Doesn't sound too tough.

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Everyone is entitled to his own opinion, but not his own facts.