## Ask Slashdot: Math Curriculum To Understand General Relativity? 358

Posted
by
timothy

from the braver-man-than-I-am dept.

from the braver-man-than-I-am dept.

First time accepted submitter sjwaste writes

*"Slashdot posts a fair number of physics stories. Many of us, myself included, don't have the background to understand them. So I'd like to ask the Slashdot math/physics community to construct a curriculum that gets me, an average college grad with two semesters of chemistry, one of calculus, and maybe 2-3 applied statistics courses, all the way to understanding the mathematics of general relativity. What would I need to learn, in what order, and what texts should I use? Before I get killed here, I know this isn't a weekend project, but it seems like it could be fun to do in my spare time for the next ... decade."*
## Easier way to learn it (Score:4, Informative)

Save yourself some trouble and get Relativity; The Special and the General Theory by Einstein himself. In his words "The work presumes a standard of education corresponding to that of a university matriculation examination..." however note those words

were written in 1916 and education standards are somewhat lower now. What used to be required for admission are often not

learned during university at all.

I know I have read it several times now and when I finish and sit and think a bit I'll almost 'get it' before retreating from the gates of madness. Think Cthulhu.

But I think it boils down to not only can we not exceed C we can't go slower either. Everything moves at C and the axis of that motion we perceive as time. And everything else we call reality is the contortions required to make that so under all circumstances.

## Re: (Score:2)

He asked about general relativity. IIRC, general requires much more math than special. Special relativity can be handled by linear algebra very well.

For instance:

http://www.math.rochester.edu/people/faculty/chaessig/students/Adams(S10).pdf

## Re:Easier way to learn it (Score:4, Interesting)

Madness indeed. I got quite deep into physics and calculus at university and hit a brick wall with multivariable calculus. I believe that you'll need the multivariable calculus skills in order to get any reasonable grip on general relativity. You'll also need a strong physics background: force, mass, acceleration, rotary motion, etc. Having read Einstein's book on special relativity, I'd definitely say start there. It's pretty clear and amazingly intuitive. The Feynman lectures on physics are probably the best physics textbook ever. I wonder too if you might find a class on it online -- maybe Harvard or MIT:

http://ocw.mit.edu/courses/physics/8-962-general-relativity-spring-2006/ [mit.edu]

## Re: (Score:2, Interesting)

General relativity? I'm doubtful. To even phrase it you need to know something about Reimannian manifolds (see http://en.wikipedia.org/wiki/Einstein_field_equations#Mathematical_form), which is way beyond something you'd see in standard calculus or even most undergrad math programs. Sure there are a lot of intuitive concepts that can be expressed without the math, but unless you understand the math, it's hard to see how things like frame-dragging are predicted by the theory.

## Re:Easier way to learn it (Score:4, Informative)

I don't need to understand math in order to understand that a baseball hit up at an angle will follow a parabolic trajectory to the earth. The same can hold for much of physics; it's possible to understand a few expected behaviors without needing to understand every little detail and every mathematical concept that backs it up.

http://en.wikipedia.org/wiki/Introduction_to_general_relativity [wikipedia.org]

That's a decent starter, without too much math. (IANAP... there are probably better introductions, that's just an obvious find.) In fact, learning about these things may get one interested enough to care about the math, and to learn the intricate details.

## Re:Easier way to learn it (Score:4, Interesting)

## Re:Easier way to learn it (Score:5, Insightful)

Unless you can work out on your own how to put numbers on it, your understanding is imperfect. Being able to run some numbers through an equation doesn't mean you understand it even as well as the guy who doesn't know any maths but knows where to stand to catch the ball.

## Re:Easier way to learn it (Score:4, Insightful)

No, not really. You can get a very superficial understanding of what is going on without any maths, but you are just fooling yourself if you think that understanding is anything remotely like what you understand once you've actually worked with the maths.

## Re: (Score:3, Interesting)

You can understand the outcomes without the math. You can NOT understand the "why" without the math. I'll leave it as an at home exercise whether those people you know actually understand general relativity, or just know the implications of it.

## Re: (Score:3)

I think more specifically, you can understand the outcomes if someone spoon feeds it to you bit by bit and answers your questions. If you want to "understand general relativity" the math is mandatory in my opinion.

## Re: (Score:2, Interesting)

No, the usual math completely obscures the "why". The "why" of GR can only be understood WITHOUT the gratuitously ugly tensor formulation.

## Re:Easier way to learn it (Score:5, Insightful)

Even Einstein himself never claimed to understand the why of GR. GR is all about the math. The vague analogies sometimes bandied about aren't science. They are flights of fancy and completely unproven and were only ever used to try to explain the math to people who didn't understand the long tensor calculus equations. The math itself

isthe science. There is no way around the equations. GR cannot be explained with natural language. Only with mathematics.## Re: (Score:3, Insightful)

You can not understand the

whywith math, math only helps you calculating it To. actually understand it you will have to step up one abstraction level to philosophy.## Re:Easier way to learn it (Score:5, Informative)

For the interested [gutenberg.org].

## Re: (Score:2)

## Re:Easier way to learn it (Score:5, Insightful)

+1 on this and all related posts: Relativity is about physics, about beautiful physics, and is not about math.

There are bits of relativity for which Einstein had to go math-shopping: He knew what the physics must look like, he needed to know if the mathematicians had any tools that matched what he wanted to express (they did, Lorentz transformations being one of the most important).

Note: I have a physics degree, which means I have studied more math than anything else. The math is important to express the physics precisely, important to get useful answers to specific questions. But the physics come first. (There's the old trope of the physics prof saying "set C to 1 so you can see the physics happening.)

Read about and try to reproduce Einstein's thought experiments. Start with the one about travelling at the speed of light, and what you would see as you approached C (hint: if you travel at C, photons can only reach you from in front, from along your axis of travel). Think about the "falling in an elevator" experiment. These get you a long way to the principle of equivalence, the principle of relativity, etc.

Only once you have some idea of the physics should you attempt to tackle the math - and by that time, you'll be starting to get a good idea of what the math might look like.

Do not attempt to learn the math first and thereby get to the physics. There lies madness.

## Re:Easier way to learn it (Score:4, Informative)

That's not really true. Dirac went looking to remove the square from E=mc^2 since it allowed for the possibility of negative matter and energy. Eventually he came up with a solution using matrices, which as it happened once again left the door wide open for negative matter and energy and ultimately lead to the prediction and subsequent discovery of antimatter. In this case the maths directly lead to a major advance in physics.

Without maths, how would physicists even theorise anything? All they would have is their intuition which is at best useless and at worst an active hindrance to the the discovery and understanding of major advances in physics of the 20th century and beyond.

## Re:Easier way to learn it (Score:4, Informative)

Where are my mod points when there's an AC comment to be rescued from obscurity.

Myvirtualiddoesn't offer bad advice but the AC's comment is also spot on. You only get so far without the math.I studied physics but never took classes on topology and differential manifolds and this severely hampers me in getting a good grip on GR although I am perfectly comfortable with Einstein's thought experiments.

So in a sense I agree with myvirtualid's stance: Don't start with the math but you will need it later and then the math may lead you to completely new insights as pointed out by the AC.

## Re:Easier way to learn it (Score:4, Informative)

To put a quantitative understanding to your qualitative understanding of the theory from various authors in the field it becomes real simple: Calculus I, II, III [Multi-variable], Linear Algebra, Probability and Statistics for Engineers [Math 171,172,273,220 and 360 respectively at Washington State University], Differential Equations [ODEs Math 315 at WSU], Vector Analysis [Tensor Calculus Math 375 at WSU] and Intermediate Differential Equations [ODE/PDE for Nonlinear Dynamics Math 415 at WSU]. Then add onto that a foundation in Classical Mechanics and Modern Physics for Engineering should suffice.

But I don't think you want to really know it more than just to understand in common language how to explain it in it's most vulgar sense.

## Re:Easier way to learn it (Score:5, Insightful)

But I think it boils down to not only can we not exceed C we can't go slower either. Everything moves at C and the axis of that motion we perceive as time. And everything else we call reality is the contortions required to make that so under all circumstances.

Sir –

I wouldn't quite describe it that way, from the perspective of the epiphany Einstein must have had. I don't think it's that complex, and in any case I think it's more beautiful than that. As a matter of interest, perhaps someone will find the following worth reading.

We have space, and it's where we live. This space is physical but can be represented by representations in our brains and on various media, which representations we call physics.

We make rules in physics to reflect what happens in our space, our reality. Some rules we can see, and they are generally intuitive. For example: Two points - places - are distinct when not the same position, and these points are indivisible (identity). Also, two lines added together make a third line, regardless of the order those lines are added in (commutativity). Three lines can be added in any order to equal the same distance (associativity). Two lines never meet (parallelism). This is the Euclidian space [wikipedia.org], and applying such to our universe is Newtonian physics (aka classical physics).

Suppose though that the physical world in which we live is not Euclidean, contrary to our observations and intuition. Suppose in this world parallel lines in our world meet at infinity. We can call this a Lobachevsky space [wikipedia.org] (also known as a hyperbolic geometry), and its principles formed the essential breakthrough in general relativity.

Once one accepts as axiomatic that we live in a Lobachevskian space, the acceleration of mass becomes governed (for reasons beyond the scope of this note) - otherwise we would violate other rules (e.g. identity). Hence the perception of time slows in lieu of infinite acceleration (imagine two trains travelling at the speed of light towards each other; to each other they would appear to be travelling only at the speed of light - not, as one might expect, twice the speed of light - because time relative to each other slows; contrast a stationary that expects both to pass at the speed of light in opposite directions). This effect is observed and compensated for in our Global Positioning System [ohio-state.edu].

All to say, by changing our perspective from representing our accepted physical world as a Euclidean geometry to something unintuitive, a Lobachevskian geometry, we arrive at the ability to represent and predict what happens in our physical world.

The consequences inherent to the axiomatic perspective of living in Lobachevskian space are commonly and collectively referred to as "general relativity", and they are non-trivial. The underlying premise that commenced that perspective is itself quite simple.

## Re: (Score:2)

> The faster an object moves through space the slower it moves through time

> (relative to the observer of course) and vice-versa.

We said exactly the same thing. I stated it from the viewpoint of the object while you stated it from the observer p.o.v. If I set out for Alpha Centuri at .5C (from Earth's viewpoint) you see me going .5C on a course for our closest neighbor. But on the ship, as soon the engines cut off, I see myself at rest (what we call moving at C on the T axis) with the sort of wierd

## Re: (Score:2)

Time dilation works largely because as we approach C our mass also increases. At .5% of C (not 50%) our mass will have effectively doubled and time will be noticeably slower than home, to keep accelerating at one G you need more energy to push the increased mass. As you approach 50% of C your mass will be thousands of times greater than at "rest" the energy required to continue to keep accelerating is unimaginable if it were possible to get within 95% of C your mass would be nearly infinite and it would r

## Re: (Score:2)

## Re: (Score:3)

That's why it's important to study, read and listen.

## Re:Easier way to learn it (Score:4, Informative)

## ok reading it several times is easy (Score:2)

Reading the book and "thinking" that you grok relativity is a much easier task.

I know plenty of people that think they have it down pat. However there are quite a few time dilation scenarios that will cause a paradox if you don't have the model dead right. The frames of reference are a bitch.

## have basic calculus and vectors? (Score:5, Informative)

## Re:have basic calculus and vectors? (Score:4, Informative)

## A lot of work (Score:3, Informative)

Linear Algebra, Differential Equations, Advanced Calculus, Partial Differential Equations, Electromagnetism, Waves, Introduction to Astronomy, Special Relativity, Differential Geometry

## Re: (Score:2)

Yes... there is a lot of math. That is almost all I remember from my attempt to learn it in PH236 at a small technical school in southern cal. Memories of Christoffel symbols, Riemann Curvature, and covariant derivatives dance in my head. I find learning the math pretty dry without some physics behind it. There are on-line class notes which might be helpful and studying with friends (if you can find someone that shares your illness).

http://www.amazon.com/Gravitation-Physics-Charles-W-Misner/dp/0716703440/ [amazon.com]

B

## Re: (Score:3)

Ohh... I did find class notes for 236. I assume that the prof will be happy you're learning too so here's a link:

http://www.pma.caltech.edu/~ph236/yr2010/index.html [caltech.edu] [caltech.edu]

by the way... learn to use google, and Amazon... Slashdot is full of old hosers, and now even Cmdr Taco has left!

## Re: (Score:2)

I tried MTW as an EE and didn't get very far. The first problem is that it's a graduate-level textbook, which means it wants you to do all the work yourself. The second is that it's a graduate-level *physics* textbook, which means it assumes you have advanced undergrad knowledge of E&M, mechanics, etc., not just a first-year physics course. Beautiful book, though -- worth owning as a work of art if nothing else. Wheeler also co-authored a book on special relativity which is targeted more at the undergra

## Differential Geometry is the key (Score:2)

Springer has an OK book on Differential Geometry, and then you want to move on to Gravitation, by Misner, Thorne, and Wheeler.

## Re: (Score:2)

## What do you really want to do ? (Score:3)

What do you really want to do ? (My guess is that you are not sure.)

If you want to be able to write down and solve Einstein equations for some case, you need vector and tensor algebra, geometry and calculus. Many people who work in GR never do this (for others, it's all they do). If you are interested in some more particular case (black holes or gravitational radiation, say), you need to understand Einstein's equations at some level, plus whatever approximations or simplifications are used in that area (transverse traceless gauge or post-Newtonian approximations, for example). Also, you should get to where you understand Lorentz transforms in your sleep. If you can't do and understand Lorentz transforms, the actual GR math will likely be beyond you.

What I would recommend is to buy Misner, Thorne and Wheeler [amazon.com], and read and follow "track 1." I would allocate 1 year for that.

## Re: (Score:2)

I can not speak for the original submitter, but I'd like to pose a "for example" for you: I've been trained as an electrical engineer. Most of the higher physics I've been taught have been the quantum mechanics stuff more relevant to semiconductors, microelectronics, etc. But I've always been interested in space/astronomy since I was a kid. During my Uni days, my course material would occasionally tease me with tid-bits like "....but relativistic effects become important in GPS applications where the sy

## Re: (Score:2)

Speaking of which, what always amaze me is that Maxwell's equations, written half a century before Einstein's special relativity, are actually fully compatible with it.

You need not modify them to work with special relativity like you would, for example, kinetic energy.

Special relativity is 'engrained' inside Maxwell's equations.

And what truely blew my mind is the revelation that the Magnetic field is just a relativistic effect of the Electrical field. See Pr. Feynman's Physics Course for a detailed explanat

## Re: (Score:2)

I agree with almost everything you said, except about using Gravitation to learn General Rel...

Do you have any other recommendations for rigorous explanations, that are more motivating, and less dense and obscure?

The book is (and the authors are) impressive, but I couldn't recommend it.

## Re: (Score:2)

## To understand or to fully understand? (Score:2)

To understand some of it, a little of differential forms, tensors, differential equations should be enough (i assume analysis and linear algebra to be present already) - maybe 2 or 3 months for the basics.

To understand it fully and make own calculations at the state of the art - the same subjects and all related math fields. Think about something like 1-2years if you have a talent for it.

## Road to Reality by Roger Penrose (Score:2)

The Road to Reality : A Complete Guide to the Laws of the Universe

by Roger Penrose

http://www.amazon.com/Road-Reality-Complete-Guide-Universe/dp/0679454438 [amazon.com]

Likely the most serious math book you will find in a retail, consumer bookstore. An excellent read and essential to truly understanding modern physics.

## Re: (Score:2)

## Not a matter of math (Score:3, Interesting)

The actual math needed to understand the basics of relativity[1] is actually quite simple. If you've had calculus, you have more than you need.

The hard part is wrapping your brain around the concepts and the fact that the rules you use to interact with the world around you are a subset of the rules of the universe.

A book I have recommended several times for people who want to start learning about physics is 'Asimov on Physics'. Dr. Asimov was a master of explaining difficult science in a way that laymen could understand.

[1] Going beyond the basic, or getting into odder corners of general relativity, is another matter.

## incorrect (Score:2)

## Re: (Score:2)

Well, exponentiation is also useless - you can just use multiple multiplications. Tensors save a lot of typing, and allow one to think about problems on higher level.

## General relativity is part of physics series (Score:2)

When I was an undergraduate engineering student, I learned relativity from my university's physics department as part of a lower-division series of classes. A typical series looks like this:

Now, as for the math classes, you would usually take many previous math classes (or concurrently) as part of the physics prerequisit

## Re: (Score:2)

I'm pretty surprised that General Relativity was part of a basic physics sequence... I think you mean Special Relativity, which is basically (linear) algebra and is a small departure from classical physics.... I too studied Special Rel in a freshman class at a small school in Pasadena... Then I sat in on Ph236 where I tried to grasp part of General Rel as taught by Kip Thorne (who helped write Gravitation, a book which demonstrates it's weighty topic)... Mostly I learned math, and my final understanding tod

## Re: (Score:2)

Ohh... I did find class notes for 236. I assume that the prof will be happy you're learning too so here's a link:

http://www.pma.caltech.edu/~ph236/yr2010/index.html [caltech.edu]

by the way... learn to use google, and Amazon... Slashdot is full of old hosers, and now even Cmdr Taco has left!

## Re: (Score:2)

oops this wasn't post I was following up on... and comes off as harsh, but looking at the class notes is still worth a try...

## all that is really needed (Score:2)

before you take anything, read "sphereland" to help open your mind.

repeat as necessary until you "get it"

then take vector calculus, field theory, and tensor analysis

(and of ourse, any pre-requisites)

you should now be well eqipped to understand both the

concepts and undrerlying math.

cheers

## special and GENERAL relativity (Score:2)

My understanding is that, while related, general relativity requires tensor analysis (aka vector calculus). Special relativity can be thought of as a 'correction' to Newton's laws of motion. General relativity is more kin to 'altering the topology of the universe' (lack of a better phrase).

prerequisites:

calc I and II

Math for special relativity:

-linear algebra (possibly modern algebra)

good pdf:

http://www.math.rochester.edu/people/faculty/chaessig/students/Adams(S10).pdf

Math for general relativity:

-vector/t

## Re: (Score:2)

Better to think of Newton's laws as an "approximation" to the laws of special relativity, rather than the other way around.

## Re: (Score:2)

I use the term "correction" in the mathematical sense; a correction is the exact opposite of an approximation.

## What are your goals? (Score:2)

## Re:What are your goals? (Score:4, Insightful)

If its just a hobby I don't understand why you would want to know the in-depth details since you probably wont be playing with equations most of the time.

On the contrary, if it's a hobby he's probably interested in reading and playing with the various speculative equations for warp drive and time travel - for example, the Alcubierre Drive [wikipedia.org], or Kip Thorne's wormholes [wikipedia.org]. Which has nothing to do with everyday physics, but everything to do with science fiction worldbuilding and geeky entertainment. Certainly that's what I would do if I understood enough of GR to get to the "test the equations" stage.

## Moths to a flame (Score:4, Informative)

Can't really understand it without the math, but over the decades innumerable "popular science" authors have attempted to write about general relativity for the "common man", with no math beyond maybe pythagoras.

Its kind of like having a verbal understanding of ohms law, without actually knowing how to divide. "So you increase the resistance and the current drops, assuming constant voltage, ok?". On a small scale its easier to understand the little bits, but its hard to grasp the entire thing.

One thing to look out for is relativity was "cool" some decades ago, so anything with a tenuous connection, will have GR on the cover and some pictorial representation of an elderly Einstein. Kaufman has a famous book for beginners "cosmic frontiers of general relativity" but note that only a few chapters talk about G.R., the rest is 40 year old black hole research. A better title would have been "black hole physics in the 70s, and related topics.". Its a perfectly good book, just not quite what you're asking for.

Another oddity is no one every provides a pix of Einstein when he did his famous work as a young man, only pictured as an elderly dude. Other scientists don't get that treatment; Feynman's "popular press photos" are all from his middle age when he was earning his 2nd Nobel, Tesla is usually portrayed as a steampunk vampire young goth man...

## Re: (Score:2)

Feynman's "popular press photos" are all from his middle age when he was earning his 2nd Nobel

I'm not aware that Feynman received the Nobel prize more than once. (Sorry to pick on unimportant details here, I'm just so concerned about the children, that they don't get miseducated.)

## check this online math text first... (Score:2)

Read this pdf online, chapter by chapter, and do the exercises. It should take weeks:

http://virtualmathmuseum.org/Surface/a/bk/curves_surfaces_palais.pdf [virtualmathmuseum.org]

If you understand the pdf well, you can probably then take on a graduate level general relativity text directly. If not, you should refresh your trigonometry and calculus first, I suppose.

## MTW - GRAVITATION (Score:3)

From the preface:

This is a textbook on gravitation physics (Einstein's "general relativity" or "geometrodynamics"). It supplies two tracks through the subject. The first track is focused on the key physical ideas. It assumes, as a mathematical prerequisite,

only vector analysis and simple partial-differential equations.It is a really fun book to read at the first track level; especially if you are not on the hook for the homework.

## Re: (Score:2)

The book Gravitation (black with an apple on the cover) by Misner, Thorne and Wheeler is the one you want. The book is thick (over 1200 pages) but it teaches you General Relativity and the math you need to understand it. They teach it to undergrads (juniors and seniors) at Caltech. It also help if you have a grasp of Differential Geometry. It should take about a year to learn.

byteherder

## "the math of GR" -- how much math is that? (Score:5, Informative)

You've made an admirable attempt to define your question clearly, but you didn't quite succeed. General relativity can be understood at a variety of mathematical levels, so saying you want to understand "the mathematics of general relativity" doesn't really pin it down.

The other issue is that you haven't defined your physics background. If you really want to understand GR, you need to be fairly sophisticated in physics.

The first thing I'd suggest is that you build a solid foundation of understanding in special relativity. The best intro to SR is Taylor and Wheeler, Spacetime Physics, and you already have the math background to understand that.

Physically, GR is a field theory. The first field theory was electromagnetism. E&M is a lot easier to understand than GR, because it takes place on a fixed background of flat spacetime, and it also connects directly to everyday experience. The more intuition and technical skill you can build up in the context of E&M, the better prepared you'll be for GR. For someone ambitious about going far in physics, the best intro to E&M is Purcell, Electricity and Magnetism. Purcell uses vector calculus, and he tries to teach you all the vector calc you need as he goes along. However, you will want some of the preparation provided by a second-semester calc course, and you will probably also have an easier time if you can also study from a separate book on vector calculus. Here [wisc.edu] is a free online calc book that I like, and here [mq.edu.au] is a free vector calc book you could use. When you're learning second-semester calc, I'd suggest you skip the integration tricks that form the bulk of such a course; they're largely irrelevant to your goal, and nowadays you can use Maxima or integrals.com for that kind of thing.

With that background, you're more than prepared to start studying GR at the level of Exploring Black Holes, by Taylor and Wheeler.

If you want to go on after that and understand GR at a higher mathematical level, you could try an upper-division undergrad book such as Hartle or my own free book [lightandmatter.com], and then maybe move on to a graduate-level texts. The mathematics used in graduate-level texts is typically introduced explicitly in the text itself; basically tensors and calculus on a manifold. You don't need any more math prerequisites than vector calculus before diving in. The classic graduate text is Misner, Thorne, and Wheeler. I would still recommend it wholeheartedly, except that it's now decades out of date. A more modern alternative is Carroll; there is a free online version [caltech.edu], plus a more complete and up to date print version. Other GR books worth owning are General Relativity by Wald and The Large-Scale Structure of Space-Time by Hawking and Ellis.

## Re: (Score:2)

or my own free book [lightandmatter.com]

Just to say thanks for making those textbooks freely available :).

Best Regards.

## Math prerequisites (Score:2)

## Read some Pop-phys books (Score:2)

Just read "black holes and time warps" by Kip Thorn.

## Same Question for Particle Physics (Score:2)

Can I ask the same question for particle physics -- specifically non-abelian gauge theories. I'd like to be able to under stand the Higgs mechanism and supersymmetry properly and how the particles emerge from the symmetries of the fields.

My pure maths background is quite strong, but I stopped doing applied somewhere in my second undergraduate year and have forgotten most of the more advanced bits of it. So I have a hazy memory of curvilinear coordinates, and an even hazier one of Hamiltonians and Lagrangian

## Re: (Score:2)

## Re: (Score:2)

Can I ask the same question for particle physics -- specifically non-abelian gauge theories. I'd like to be able to under stand the Higgs mechanism and supersymmetry properly and how the particles emerge from the symmetries of the fields.

My pure maths background is quite strong, but I stopped doing applied somewhere in my second undergraduate year and have forgotten most of the more advanced bits of it. So I have a hazy memory of curvilinear coordinates, and an even hazier one of Hamiltonians and Lagrangians. I can still more or less remember my SR course. On the positive side, I understand Lie groups and Lie algebras and their representation theory pretty well.

The problem with particle physics is that the math background required is often not taught in math departments. The fact that you have studied Lie groups, Lie algebras, and their representations is very good. You are luckier than most. Keep in mind, though, that in particle physics you often need to deal with infinite dimensional representations, whereas many math courses I've seen are limited to finite dimensional (matrix) representations of groups. Also keep in mind that one of the most basic symmetry

## Susskind's lectures (Score:2, Informative)

Leonard Susskind's Modern Physics lectures on the Stanford University's channel on youtube are excellent.

http://www.youtube.com/watch?v=hbmf0bB38h0

## Roughly speaking, learn maths. (Score:2)

## Gravity - by Hartle (Score:2, Informative)

Gravity, by Hartle. It's the textbook we used in the undergrad GR course, so geared towards those with some math, without being too difficult, abstract, or esoteric. If you know college calculus and vectors, I think it does a good job of explaining any of the other math you need along the way. And if you have any questions, a bit of web searching will fill in any holes.

## Easy (not so) GR (Score:2)

## Read Schutz (Score:3)

Many introductory general relativity books give you some of the math background you need. A very good one in that regard is Bernard Schutz: A First Course in General Relativity, Cambridge University Press, ISBN 0-521-27703-5. It begins with a very good introduction to special relativity, and then develops the math needed for basic GR. I would avoid Misner, Thorne, and Wheeler. The 2 track approach is confusing, and the math is thrown at you in bits and pieces as you need it, making it hard to see the big picture.

If you are interested in math courses to take, multi-variable calculus, then differential geometry are good choices. If there are separate courses on tensor calculus or tensor analysis, they are good, but that material is often just taught as part of differential geometry. For really advanced stuff, like cosmology, you might need some topology as well.

## I. Calculus. II. Differential Equations. III L... (Score:2)

First off, you should pick up an undergraduate text on "Modern Physics," which should include a really basic intro to both special and general relativity. Any text will do, but I own the one by Tipler/Llewellyn. This kind of text will be fairly light on the math, but will include some. This will also get you started with some really basic problems which should show that while you may not fully understand General Relativity (GR), you can do some really basic problems (e.g. gravitational redshift).

I. Calcu

## Some concrete book suggestions (Score:3, Informative)

The Geometry of Physics, Theodore Frankel; An excellent introduction to differential geometry and its application not just to GR but to other areas of physics as well. Highly recommended.

A First Course in General Relativity, Bernard Schutz; I found this book helpful in some specific areas -- notably understanding the notions of the stress-energy tensor.

Gravitation, Charles Misner, Kip Thorne, & John Wheeler; This is the classic text, and is comprehensive and comprehensible. I like Wheeler's way of thinking about physics, and it shows through here. There is the standard joke, that this is a text which not only discusses gravitation, but also attempts to demonstrate it by its high mass.

## I have a special relativity simulator (Score:2)

During the 80s I wrote an interactive three-dimensional special relativity simulator. It was a wire frame simulation and ran under DOS. I recently tried it on a Windows XP machine and it still works. (It did not work when I tried on a Mac under Parallels/XP, so it appears that one needs an actual Windows machine, not a virtual machine.) When I first ran it during the 80s I simulated a famous scene from the first 3D relativistic simulation done at MIT during the 50s and I got the same results: lamp posts tha

## My list (Score:2)

So I've both taken GR as an undergrad/grad student, and now taught it to both. My undergrad was in math, grad school physics. To understand modern GR (singularity theorems, black holes, cosmology, lensing effects etc) from a math background the subjects that really help are:

1) Special Relativity. This is an easier intro that really comes out of the end of electrodynamics courses (ie, why there's that pesky 'c' in Maxwell's equations that doesn't seem Gallilean invariant). There are outstanding lecture notes

## GR... (Score:2)

GR is suitable as a 4th year or graduate course in physics. The undergrad is a bit sketchy but manageable. So really whatever the math requirements at your school are for 3rd year or 3.5 years of an undergrad in physics and you'll be there. As with most problems in physics there's a few different ways to formulate them, so your instructor may choose the one most appropriate given the available prereqs (and depending on how much time they have they might teach a lot of the math you need in the class).

Typi

## Ray d'Inverno's Introducing Einstein's Relativity (Score:3)

Here's the Amazon link if you are interested (although your university library may have it, mine did which is where I discovered this gem): http://www.amazon.com/Introducing-Einsteins-Relativity-R-dInverno/dp/0198596863 [amazon.com]

## First Understand Special Relativity (Score:2)

"What is happening on Mars right now?"

If you know that this question is meaningless and why, then you are ready to study general relativity.

Otherwise take a course in Special Relativity or read and study "Spacetime Physics" by E F Taylor and J A Wheeler. Wheeler once told me that he believed that every figure should have as much information as 10 pages of text, and some figures in "Spacetime Physics" come near his goal.

IMHO most scientists who can perform the algebra and solve problems in Special Relativity

## Free your mind... (Score:2)

and realize there is no spoon.

Yeah I know, what a horrible opening, but it really applies.

Think about the utter simplicity and beauty of the equation of E=MC^2.

Read a "Brief History of Time" cover to cover about 10 times but don't try to dig into what he is saying, take it on face value, because he is explaining it, you just have let it sink in.

What will really bake your noodle is when you realize that

everythinghas infinite energy.## Schaum's outline on tensors (Score:2)

It was all a little beyond me; a friend once tried to explain to me the metric tensor [wikipedia.org] but I couldn't get it in my thick head

Steps to take: if that wikipedia article is gobbledygook, go read Schaum first (you probably don't need to understand the whole book but you need the tensor notation at l

## Tensor Calculus and Reimannian Geometry (Score:2)

## Answer from a Grad Student (Score:5, Informative)

What approach you use depends on how well you want to understand. I am going to assume that you want to understand the equations and how to manipulate them --- that when asked about the anomalous procession of Mars, you could sit down with a pencil and graphing calculator for an hour and tell them that GR accounts for ~40 arcseconds/century. To get there, you will need to cover a series of courses: Classical Mechanics, Linear Algebra, Special Relativity, Multivariable Calculus, and then General Relativity. If you also study Electromagnetism and Differential Equations, you will get a bit more out of it, but those subjects are not necessary.

Classical Mechanics (prereqs: none):You don't need anything beyond an AP physics level understanding of mechanics, but you do need that. MIT has all of the 8.01 (classical mechanics) lectures online. [mit.edu]Linear Algebra (prereqs: none):You need to understand what a vector is, what a matrix is, what a linear transformation is, and what traces and determinants are. You probably have this knowledge from stats. If not, trys Jacob [amazon.com] or any similar text.Multivariable Calculus (prereqs: Linear Algebra):A standard undergrad book is fine. You need to know how to transform variables and use multivariable differential operators. A standard course is online. [mit.edu]Special Relativity (prereqs: Classical Mechanics, Linear Algebra):Special Relativity is essential for understanding General Relativity. Of particular importance is the 4-vector notation and the Lorentz transformation. A. P. French [amazon.com] is one of the classic textbooks.General Relativity (prereqs: Special Relativity, Multivariable Calculus):The nice thing about introductory Physics texts is that they teach you all the differential geometry you need to understand. The unfortunate thing is they tend to be aimed at Physics graduate students. There are a few undergrad textbooks, but they are not as rigorous and not as worthwhile to read. The classic General Relativity textbook is Misner, Wheeler, Thorne, but MWT is better as a reference text than as a first course. Better textbooks would be Wald, General Relativity, and Carroll, Spacetime and Geometry [amazon.com]. Of the two, I would recommend the latter.You should keep in mind that the texts will be hard and the learning curve will be steep. The best way to understand the material is to do most of the problems in the undergraduate books or all the problems in the graduate texts, and ideally, have someone read over your problem sets. It will, however, be rewarding.

## A couple of texts that will help and a web page: (Score:3)

Several of the preceding responses have covered much of what you'll need.

If you've not had any exposure to tensor analysis, I'd recommend a gentle introduction called: A Brief on Tensor Analysis by James Simmonds.

If you're still needing a grounding in vector calculus Div, Grad, Curl and All That. is a good overview of it.

At least one has recommended Wald as a text. I'd recommend Gravitation by Misner, Thorne and Wheeler. Which one you prefer will become apparent pretty quickly.

And definitely, you will need a quite solid grounding in Special Relativity.

For doing the tensor manipulations with a computer program, GRtensorII for Maple was one I've used.

My instructor in it, Dan Finley at UNM has a page for the class he teaches on it at: http://panda.unm.edu/Courses/Finley/p570.html [unm.edu]

One warning, Dan is not one to "spare the rod" when it comes to the mathematics. (Which to me, is a good thing.)

It's a worthy goal, but one that will take a lot of determination, work and preparation. Unfortunately, I had to drop out of Finley's class due to my full time job boiling over (we lost two other employees, and I had to cover). It's been 15 years, but someday I still intend to get back to it.

## subject (Score:2)

You're going to need tensor calculus. Probably the best way to get a curriculum is to look at whether your school offers this, then look at the prerequisites for the class and work your way down. It will require a minimum of several semesters of calc (these would have been calc 1, 2 and 3 at my school), a theory or proofs course, probably abstract algebra/real analysis, linear algebra, differential equations (if it's offered as a separate course from calc 2 & 3), and a solid grounding in vectors.

## Lies (Score:2)

To fully appreciate special and general relativity, you should really take the normal courseload of physics and calc that work up to it.

Because, in the beginning you learn algebra and then you learn physics with it using standard equations like d=rt.

Then, you take your first or second calc class and take something like mechanics or dynamics and realize everything you learned was lie. Everything was a special case and physics is truly based on calculus.

Then, you take your third and fourth calc (vector calc

## my thoughts on this (Score:5, Informative)

On the physics side, I recommend looking at classical mechanics, special relativity, and the history of physics research (theory and experiment) during this critical time. I think it's important to know not just the results, but why they came around to that line of thinking. The history is also something you can do for entertainment or inspiration while you're building up the considerable list of prerequisites for the general theory.

The math side is very hard. As I see it, most of the math is under a vague title, "differential geometry". There are three main parts: differentiation and integration in multiple variables (generally, you're working in "3+1" variables for general relativity and dealing with partial differential equations in this space); manifold theory; and Riemannian geometry (which manifests in general relativity as the very similar Minkowski geometry). I mention partial differential equations above. They're nice to know, but not essential for the theory.

The first can be found in the end of college calculus books. Such treatments generally suffer from ignoring differential forms. I have a specific recommendation here. While you are going through that calculus book, also read "Differential Forms with Applications to the Physical Sciences" by Harvey Flanders. It is a smallish Dover book with a good treatment of differential forms (and their use in multi-variable differentiation, integration, and differential equations).

Manifold theory is one of the more interesting contributions of mathematics to the world. The idea is that you have an object, called a "manifold", that looks, locally like a fixed dimension Euclidean space at each point of the manifold. The dimension of the Euclidean space is in turn the dimension of the full manifold. For example, the surface of the Earth crudely looks like a plane with wrinkles (ignoring holes like arches and tunnels and whether you consider the top or bottom of oceans as "surface"). But it's sort of ball-shaped while a plane is infinite in extent.

On a plane, you can label the entire plane with a pair of coordinates so that each point of the plane has a unique coordinate and vice versa. Not so with the surface of Earth. However, you can map local pieces of the Earth's surface to a plane one-to-one and onto. That is typical behavior for a manifold.

The fundamental concept is that a manifold has local behavior and description provided by a particular set of "coordinate charts" which lead to global behavior and descriptions over the entire manifold. How that's done is hard to understand, but powerful in application. There are consistency conditions on that set of coordinate charts that allow for various structures (such as the subsequent "Reimannian metric") defined in terms of one coordinate chart to be converted via some change of variables algorithm to become in terms of another coordinate chart which happens to overlap with the first.

Finally, there's Riemannian geometry and its analogue, Minkowski geometry for general relativity. The idea here is that you have a manifold with an additional structure, a "metric" which defines a sort of inner product on the tangent vector fields of the manifold as well as a distance between points on the manifold. The Minkowski metric is no longer a true metric. One of the coordinates has become "time-like" resulted in a single dimension with negative length. You can't measure distance any more with the metric, but you still have the inner product property on the tangent vectors, which are now called phase vectors and can be used to describe velocity and momentum in a system with several space-like and one time-like coordinates.

And that's enough to describe general relativity, as a physical system operating on a manifold with a Minkowski metric which has three space-like coordinates and one time-like coordinate (dimension "

## Penrose - Road to Reality (Score:3)

Hey great thread! I can confidently state that I'm in lower percentile of the posters here regarding physics and math (I'm just above the random trolls and bellow everyone else). I found Penrose - The Road to Reality a great overview starting with math I already understood, educating me about some concepts I didn't get before and ending up with today's physics of which I understood, charitably ... uh ... 10 percent ... cough ... I already had a tourist knowledge of higher math but my actual arithmetic is a disgrace and I found Penrose kept me on the horse longer than other texts.

And I've been flamed for recommending this book for reasons I didn't understand in the past so YMMV.

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the one you draw, assuming one the |x> is one glyph means semi-direct product. http://en.wikipedia.org/wiki/Semidirect_product [wikipedia.org]

if you meant |${SOME_NAMES}> it is the bra-ket notation : http://en.wikipedia.org/wiki/Bra_vector [wikipedia.org]

for more help with the notations, wikipedia is your friend @ http://en.wikipedia.org/wiki/List_of_mathematical_symbols [wikipedia.org]

## Re:Add on question: Quantum Mechanics. (Score:4, Interesting)

I didn't follow Bra-Ket notation at all until I read up on the history of it. For me, it helped a lot to know Dirac invented it, and that it was needed because it applied to Hilbert spaces, and that Hilbert developed that concept a few years before Dirac got started, and that John von Neumann was the guy who actually named Hilbert's concept "Hilbert Spaces". Why did those things matter?

1. Hilbert was discussing infinities, and he was familiar with Cantor's work (and liked it) so he was using the modern definition of infinities (plural), where there are multiple trans-finites possible. His math was meant to cover all that, and the use of it for QM was a limited case. Some events can be described using a quite limited number of spatial dimensions and the results will be understandable with a little calculus or even trig if you just understand how to take the notation used and put it into actual equations. For example, there's a Hilbert for a three dimensional Euclidean space. Other (particularly in QM) events need many spatial dimensions to describe, sometimes even an infinite number.

2. The Ket part of the notation is about those vectors in a Hilbert space. You could represent that Euclidean space I mentioned with just a Ket notation, for example. Since Hilbert spaces can have either a finite number of dimensions or an infinite number, and can entail complex numbers, the Bra part becomes needed when the Hilbert space has complex numbers involved. The Bra and Ket together are a short way of writing a formula for a complex conjugate, and the whole can be expressed just as a complex number. These can be mathematically manipulated by partial differential equations. Any person with a fair knowledge of Linear Algebra can derive information from them, secure that the treatment is mathematically both complete and rigorous. That seems to be the real point of the notation, it gets results into a form where the rest of the process uses math that's regarded as rock solid.

3. Dirac invented other math for areas where the completeness condition of all Hilbert Spaces didn't apply. He called some of these "rigged Hilbert Spaces" . He proved people could use the Bra-Ket system and similar operations to describe those QM events, but the results won't technically be proven to be correct in an absolute mathematical sense. many working physicists do it anyway.

4. People tend to refer to Feynman for a good source to understand all this and not mention von Neumann as much, but it looks like von N. was historically quite involved in it. Maybe some of what he wrote on QM could clarify Bra-Ket notation better for you than the standard modern textbooks.

## Szabo and Ostlund, Schaum outline on tensors (Score:2)

But you need to have a background in high-school level linear algebra first, i.e. you need to know how to work with a simple n-dimensional orthogonal basis set of vectors, otherwise Szabo and Ostlund is probably too difficult.

I

## Re:A question borne of helplessness... (Score:4, Insightful)

You could have left off the first paragraph and provided an informative response. I was going to post something about MIT's online courseware, too. But you had to preface a useful bit of information with a put-down. Welcome to slashdot where innocent questions are met with derision and insults.

## Re: (Score:2, Flamebait)

Welcome to slashdot where innocent questions are met with derision and insults.

It was also a lazy question, one that a simple Google search for "general relativity" could have answered. I agree with the parent poster that if he can't be bothered to dig a little on his own, he's never going to take the time to study it anyways.

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Well I did Google the subject, and there were lots of pages describing the topic at a high level, including the mathematics and pointers to deeper treatments. Considering that you were on a ten-year plan, I don't think you needed the specific help you as claimed in your analogy.

Reading up on general sources and diving deeper as you saw fit doesn't require a post to Slashdot.

## Re: (Score:2)

The internet is a terrible, TERRIBLE, source for a proper scientific education free from bias.

Right. Because of humans. Luckily humans don't make books or any other sources of information. They just dwell on the internet, and there's absolutely no useful information there! That's why you can accept everything you hear or read as long as it didn't come from the internet.

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