# Ask Slashdot: Math Curriculum To Understand General Relativity?

#### timothy posted more than 2 years ago | from the braver-man-than-I-am dept.

358
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."*

## gee I dunno what does you uni offer (-1, Troll)

## Osgeld (1900440) | more than 2 years ago | (#37235370)

cant be much if they cant even provide you with a counselor

## Re:gee I dunno what does you uni offer (0)

## Anonymous Coward | more than 2 years ago | (#37235390)

Who said he was still at uni?

## Re:gee I dunno what does you uni offer (0)

## Anonymous Coward | more than 2 years ago | (#37235446)

If you haven't got anything useful to say, keep your fucking mouth shut.

## Re:gee I dunno what does you uni offer (0)

## Anonymous Coward | more than 2 years ago | (#37235816)

## And the world will be just. (0)

## Anonymous Coward | more than 2 years ago | (#37235464)

Troll doesn't read / understand summary.

Troll writes trollish thing about summary.

Troll gets modded Troll.

And the world will be just.

## Easier way to learn it (3, Informative)

## jmorris42 (1458) | more than 2 years ago | (#37235378)

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:Easier way to learn it (1)

## ThorGod (456163) | more than 2 years ago | (#37235442)

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 (3, Interesting)

## sneakyimp (1161443) | more than 2 years ago | (#37235458)

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:Easier way to learn it (0)

## Lumpy (12016) | more than 2 years ago | (#37235554)

Incorrect. I know a couple of people that understand relativity but can not do the math. It's all about wrapping your head around the concepts. you can easily understand it if you use dumbed down math. you just cant be accurate.

Just like how you can calculate the area or circumference of a circle using 3 instead of Pi and still get a close answer.

## Re:Easier way to learn it (2, Interesting)

## Anonymous Coward | more than 2 years ago | (#37235748)

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 (3, Informative)

## bondsbw (888959) | more than 2 years ago | (#37235854)

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 (3, Interesting)

## bhagwad (1426855) | more than 2 years ago | (#37235930)

## Re:Easier way to learn it (3, Interesting)

## Swarley (1795754) | more than 2 years ago | (#37235856)

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:Easier way to learn it (2)

## sneakyimp (1161443) | more than 2 years ago | (#37235958)

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:Easier way to learn it (1)

## Anonymous Coward | more than 2 years ago | (#37235882)

The probably only think they understand it and can babble on about some popsci. Real understanding needs real math.

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

## Anonymous Coward | more than 2 years ago | (#37235482)

For the interested [gutenberg.org].

## Re:Easier way to learn it (1)

## Anonymous Coward | more than 2 years ago | (#37235484)

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.

No, it boils down to all matter moving at the same speed

in space-time. The faster an object moves through space the slower it moves through time (relative to the observer of course) and vice-versa. To get a clearer picture on the concepts of relativity such as this you might want to try something likeWhy Does E=mcby Brian Cox & Jeff Forshaw. Although that still won't help the OP as it's meant as a gentle introduction and therefore deliberately dodges the mathematical underpinnings when possible.## Re:Easier way to learn it (1)

## jmorris42 (1458) | more than 2 years ago | (#37235600)

> 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 stuff around me that astronomers typically only see in the very far off universe.

And it is thinking about that sort of thing, and the implications that follow from it, that leads to the gates of madness. Our ape brains aren't built for that sort of four dimensional thinking and without a lot of training we don't handle it well.

## Re:Easier way to learn it (1)

## Paracelcus (151056) | more than 2 years ago | (#37235924)

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 require the energy output of a quasar to power your ship!

Think of light speed as the event horizon of a black hole, mass = speed or the reverse, it really doesn't matter, time is dragged down by mass, not speed.

## Re:Easier way to learn it (1)

## ModernGeek (601932) | more than 2 years ago | (#37236036)

## Re:Easier way to learn it (0)

## Anonymous Coward | more than 2 years ago | (#37235504)

Dude. Just stop. That is not what Special Relativity says at all. I seriously question whether you've read that book "several times" as you claim, or if you just don't understand what you've read.

## ok reading it several times is easy (1)

## tempest69 (572798) | more than 2 years ago | (#37236004)

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.

## Re:Easier way to learn it (0)

## Anonymous Coward | more than 2 years ago | (#37235522)

I don't think this is a good idea, and your experience of almost getting it despite having read the book multiple times isn't a shining endorsement.

Modern textbooks are much more suitable for students, because they build upon decades of teaching experience. Just like you wouldn't read Newton's Principia to learn calculus, one shouldn't read a book by Einstein to learn GR.

## Re:Easier way to learn it (2)

## AstroMatt (1594081) | more than 2 years ago | (#37235756)

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

## myvirtualid (851756) | more than 2 years ago | (#37235864)

+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 (4, Informative)

## Anonymous Coward | more than 2 years ago | (#37235946)

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.

## Bing is your friend. (0)

## Anonymous Coward | more than 2 years ago | (#37235382)

Mathematics of Relativity [bing.com]

As you can see, the first two hits are to Wiki with a very nice synopsis of the math subjects required.

## Add on question: Quantum Mechanics. (0)

## Anonymous Coward | more than 2 years ago | (#37235388)

Like the OP I do some pop science reading now and then, like the /. articles.

One thing I usually don't get are the QM articles. Can someone point out a good online resource for those things? Like, for instance, what the hell those |x> bra-ket things are suppossed to mean. Wikipedia is generally only good for those kind of things if you know the stuff already and just need a quick reference.

## Re:Add on question: Quantum Mechanics. (2)

## yevelse (686791) | more than 2 years ago | (#37235444)

## Re:Add on question: Quantum Mechanics. (1)

## tulcod (1056476) | more than 2 years ago | (#37235848)

## Re:Add on question: Quantum Mechanics. (2)

## JonySuede (1908576) | more than 2 years ago | (#37235466)

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. (3, Interesting)

## Artifakt (700173) | more than 2 years ago | (#37235846)

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.

## A question borne of helplessness... (-1, Troll)

## lavalyn (649886) | more than 2 years ago | (#37235426)

You're actually asking readers to "construct you a curriculum," without even starting to discuss what you've found so far. That reeks of laziness and apathy. More important than actually going through the material is the motivation to get through it. You seem to be of the mind that you'll "get around to it." That's not motivation.

Still, that's not a very helpful reply, so I'll give you a hint: MIT OpenCourseWare. Or go to any university website, look through their "Physics" program, check the degree prerequisites, and start grabbing the textbooks for those courses. That'll be a comprehensive curriculum on its own.

## Re:A question borne of helplessness... (0)

## Anonymous Coward | more than 2 years ago | (#37235476)

Or just use the book Einstein wrote...

Then each of the math problems he has becomes an exercise to learn the math around it... He starts off with fairly simple math and works his way up more advanced stuff.

## Re:A question borne of helplessness... (3, Insightful)

## yog (19073) | more than 2 years ago | (#37235680)

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:A question borne of helplessness... (-1)

## Anonymous Coward | more than 2 years ago | (#37235814)

If those are your really school credits, and you can't understand "General Relativity" was punk-ass schools did you attend? Did you do any studies at all? Hmmmm... this seems to be another "Troll" to me. No one, & I mean no one at all with that much real schooling would ever written a request so alarmingly stupid!

## Re:A question borne of helplessness... (1, Flamebait)

## Raenex (947668) | more than 2 years ago | (#37236024)

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.

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

## rubycodez (864176) | more than 2 years ago | (#37235430)

## Re:have basic calculus and vectors? (3, Informative)

## rubycodez (864176) | more than 2 years ago | (#37235480)

## A lot of work (2, Informative)

## Xerxes314 (585536) | more than 2 years ago | (#37235432)

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

## Re:A lot of work (0)

## Anonymous Coward | more than 2 years ago | (#37235682)

Add to that you need to know what Covariant and Contravariant basis vectors are before you can even read the notation.

Or more simply you need to get a PhD in physics.

## It just takes patience (0)

## Anonymous Coward | more than 2 years ago | (#37235694)

I would recommend getting Schaum's Outline on Tensor Calculus and working through it. Calculus is a must... and linear algebra is useful. However, don't let the matrix math infest your brain too deeply, because that will make learning tensors harder.

Make sure you understand special relativity before you start... knowing a couple of different way to derive the Lorentz Transformation is a must. Learning how to derive Poisson's field equation is also important. It's also helpful to read about Mach's Principle.

Misner, Thorne, and Wheeler is a great book on General Relativity. I recommend working through it slowly.

It's a lot of work to learn GR on your own, but it you do so, you can gain a much deeper understanding that most people get from a class. The theory is deeply beautiful and profound.

## Re:It just takes patience (1)

## toonces33 (841696) | more than 2 years ago | (#37236022)

I found MTW to be rather schizophrenic when I used it - probably because there were 3 different authors trying to write a single book, and there seemed to be differences in style as you go from one chapter to the next.

The first time I went through the subject I found it difficult to comprehend some of the concepts. It was later that I was taking solid-state physics where we were doing a lot of work in K-space that it became clearer what they meant by MTW.

Understanding how tensors work really does help a lot, but if general relativity is the first exposure to the subject, it might be a little harder. A more common everyday example would be stress and strain tensors that are used to describe how objects are deformed under pressure. Again, my studies of solid state physics helped me here in that I ended up dealing with non-uniform solids.

## Re:A lot of work (0)

## Anonymous Coward | more than 2 years ago | (#37235790)

Not really. You can understand general relativity with a calculus course. In fact, it was very easily derived in my lowly calc3 course.

## see khan (-1)

## Anonymous Coward | more than 2 years ago | (#37235448)

khan is pretty amazing... not sure if he goes to the extent of what you are looking for, but it would give you a good start.

www.khanacademy.org

## What do you really want to do ? (2)

## mbone (558574) | more than 2 years ago | (#37235454)

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.

## Study geometry first (1)

## yevelse (686791) | more than 2 years ago | (#37235468)

## To understand or to fully understand? (1)

## drolli (522659) | more than 2 years ago | (#37235490)

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 (1)

## Salis (52373) | more than 2 years ago | (#37235492)

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:Road to Reality by Roger Penrose (1)

## tulcod (1056476) | more than 2 years ago | (#37235970)

## Not a matter of math (2, Interesting)

## cheebie (459397) | more than 2 years ago | (#37235514)

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.

## Re:Not a matter of math (0)

## Anonymous Coward | more than 2 years ago | (#37235840)

The actual math needed to understand the basics of relativity is actually quite simple.

Please don't offer advice on topics that you clearly know nothing about. Basic Caculus is sufficient for Special Relativity, but you'll notice that his question was about General Relativity, which is a totally different matter.

Basic Calculus is TOTALLY INADEQUATE for the General Theory. To even understand the language that GR is formulated in requires much, much beyond a semester of calc. When I took GR as an undergrad, I'd had multivariable calc (absolutely required), linear algebra, ordinary differential eq, and partial differential eq. Learning Differential Geometry on its own is kind of a thankless task as it's really only used (in any practical sense) for GR, so most GR textbooks teach you what you need to know. In fact, most GR texts spend the first half teaching you differential geometry.

Understanding tensors and contra- and co-variant vectors is pretty essential too. You can learn this as you go, as needed, but I would recommend having some command of it before you start. An advanced E&M or field-theory course should get you comfortable with the concepts you'll need.

## incorrect (1)

## peter303 (12292) | more than 2 years ago | (#37235872)

## I am a physics major (0)

## Anonymous Coward | more than 2 years ago | (#37235516)

I am a physics major, about to get my BA at the end of this semester. I'd say it possible to understand and use every major concept in physics if you understand every thing up to vector calculus and throw in some linear algebra and diff eq (under stand 2nd order should be adequate). Obviously the more math you know the better, but up to this level should be enough to understand most of the material. Just make sure you chose the right physics text books that will hold your hand through the first few chapters and you'll be fine. Honestly I learned most of my advance math skills from my physics text books.

## A good place to start (1)

## drmitch (1065012) | more than 2 years ago | (#37235520)

## Re:A good place to start (1)

## Anonymous Coward | more than 2 years ago | (#37235870)

I thought that quote was said about Quantum Mechanics rather than General Relativity.

## Physics Textbook (1)

## theideaexplorer (2008150) | more than 2 years ago | (#37235536)

## General relativity is part of physics series (1)

## jmcbain (1233044) | more than 2 years ago | (#37235542)

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 prerequisites. These classes would include three in calculus, linear algebra, differential equations, and vector analysis. I believe this is fairly typical for U.S. college engineering programs.

## Re:General relativity is part of physics series (1)

## kurthr (30155) | more than 2 years ago | (#37235886)

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 today is very limited.

Perhaps Special Relativity is what the poster means too, but it doesn't seem like it based on his concern, and it's not what he said. As others have mentioned General Relativity is a much bigger an more difficult topic involving Tensors and Differential Geometry.

Look at the two topics in Wikipedia:

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

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

Basically, Special Rel deals with the special case of inertial reference frames (eg those that are not accelerating or rotating). It explains Doppler RADAR, and is basically completely accepted by the scientific community. Special Relativistic Quantum mechanics (Dirac's Equation) is part of the Standard Model and necessary for some quantum chemistry and Fine Structure of the atom.

Complexly, General Rel deals with the more general case of all reference frames (eg including gravitation, acceleration, and rotation). It explains gravitational lensing and a portion of Mercury's orbital precession, but is still not completely accepted, because it's not known how to combine its concepts with Quantum Mechanics. String Theory is the most popular attempt... (also not really accepted),

I consider Quantum Electrodynamics QED and Quantum Chromdynamics QCD to be much more commonly understood and comprehensible. They are still hard (like math), but not absurd.

## all that is really needed (1)

## goffster (1104287) | more than 2 years ago | (#37235560)

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

## Biggest tip I can offer (-1)

## Anonymous Coward | more than 2 years ago | (#37235564)

Don't ask Slashdot. Seriously, as fun as this place can be, it isn't exactly chock full of reasoned arguments, rational debate, and a pool of recognized experts. Slashdot is full of mob mentality, dogma, and people who claim to have degrees in whatever subject gives them the false credibility they need to cheaply win an argument.

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

Visit your local University, Community College, or Library. Talk to people who have valid credentials, even a simple reference librarian is a better, more valid, source of information on topics like this. Talk to them about what you want to know, why you want to know it, and let them help you construct a plan, find the right resources, and learn what you need.

## Re:Biggest tip I can offer (1)

## cheekyjohnson (1873388) | more than 2 years ago | (#37235792)

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.

## special and GENERAL relativity (1)

## ThorGod (456163) | more than 2 years ago | (#37235572)

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/tensor calculus (class after calc III)

-(optional) complex analysis (adding the point at infinity gives you a rough idea of how topologies can be manipulated/changed. The business of finding poles and using the location of poles in integral domains might help to form some intuition, I'm not sure.)

As pointed out elsewhere, go straight to the source, as well. You'll want to study more than just Einstein's papers, possibly.

## Re:special and GENERAL relativity (1)

## krlynch (158571) | more than 2 years ago | (#37235730)

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

## Re:special and GENERAL relativity (1)

## ThorGod (456163) | more than 2 years ago | (#37235898)

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

## Susskind's Lectures (1)

## Anonymous Coward | more than 2 years ago | (#37235574)

http://www.youtube.com/playlist?list=PL6C8BDEEBA6BDC78D

Leonard Susskind has a series of free lectures on GR on youtube. They're quite excellent, and they don't assume much beyond basic multivariate calculus (partial derivatives)

## What are your goals? (1)

## Antisyzygy (1495469) | more than 2 years ago | (#37235576)

## Moths to a flame (3, Informative)

## vlm (69642) | more than 2 years ago | (#37235582)

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...

## check this online math text first... (1)

## wherrera (235520) | more than 2 years ago | (#37235612)

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.

## "Einstein's Universe" by Nigel Calder (0)

## Anonymous Coward | more than 2 years ago | (#37235614)

If you haven't already, you could start with "einstein's Universe" by Nigel Calder. It's a great introduction to relativity without the heavy math. Then learn the math as needed to explore specific parts in depth.

## Two words: Math and Physics (0)

## Anonymous Coward | more than 2 years ago | (#37235618)

First and foremost, you need a full introductory calculus course and calculus based physics course. There are many options- as far as books are concerned, I'm a fan of Stein's Calculus book and Halliday and Resnick's Physics book but you should also check out MIT's OpenCourseWare as well as Carnegie Mellon's open learning initiative. Once comfortable with the basics, you should get a book (or find a website) on linear algebra, a book on tensors, and a higher level geometry book that includes non-euclidian geometry. At this point, you could move on to a book on general relativity but you might consider getting a book on electromagnetism- it will give you a background on special relativity (David Griffiths book is great). As for general relativity, Robert Wald's text is a good intro, as is Sean Carroll's or James Hartle's. Carroll also has lectures online about gravitational waves (http://elmer.tapir.caltech.edu/ph237/CourseOutlineA.html). You might want to also check out Kip Thorne's Applications of Classical Physics (http://www.pma.caltech.edu/Courses/ph136/yr2008/ - not just GR but many interesting physics subjects). If you get this far and are still interested, look at Misner/Thorne/Wheeler's Gravitation (don't start with this book).

You might also consider getting the Schaum's outlines for the subjects in addition to/ instead of textbooks- many good examples and explanations.

## Online Textbook (0)

## Anonymous Coward | more than 2 years ago | (#37235640)

http://www.lightandmatter.com/lm/

This is a decent online textbook that covers basics physics concepts.

## You don't need it to read Slashdot (1)

## kitserve (1607129) | more than 2 years ago | (#37235674)

I have a degree in theoretical physics, from the UK's top science university, and in my final year I did a course on General Relativity, for which I scored 70% (i.e. a 1st). I then went on to do a PhD in maths (or math for the non-Brits).

Despite the above, I don't fully understand the maths of general relativity. It is really, *really* hard! Likewise for advanced particle physics and quantum mechanics. I get the principles (I think), but unless you're an Einstein type genius, the maths is essentially about learning the rules and applying them. It is not intuitive, and unless you're prepared to write down the equations and work through them for each situation you come across, the maths is going to remain completely opaque.

That said, I still enjoy reading about these subjects on Slashdot and elsewhere. I think it's much more a question of finding good explanations of what the maths means than feeling obliged to work through it yourself.

If you're really keen, I suggest starting with special relativity. The maths is much simpler, but it still requires working through to make sense of the more complex relativistic situations, e.g. questions of simultaneity and so on. If you can manage that and are still keen, come back to general relativity at that point!

## MTW - GRAVITATION (2)

## drerwk (695572) | more than 2 years ago | (#37235684)

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:MTW - GRAVITATION (1)

## byteherder (722785) | more than 2 years ago | (#37235926)

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? (4, Informative)

## bcrowell (177657) | more than 2 years ago | (#37235686)

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:"the math of GR" -- how much math is that? (1)

## bcrowell (177657) | more than 2 years ago | (#37235980)

I forgot, if you want try any of the books that go beyond the level of Exploring Black Holes, you're going to need to learn some linear algebra. There happens to be an excellent free book on this topic: http://joshua.smcvt.edu/linalg.html/ [smcvt.edu]

## Use the effing google (0)

## Anonymous Coward | more than 2 years ago | (#37235708)

Nobel Prize Gerard 't Hooft has already done that for you: HOW to BECOME a GOOD THEORETICAL PHYSICIST [science.uu.nl].

## How to become a GOOD theoretical physicist (0)

## Anonymous Coward | more than 2 years ago | (#37235732)

Gerard 't Hooft, who won the nobel price in physics by his theory of the holographic priniciple,

has written a nice list of subjects to master to become a theoretical physicist.

http://www.staff.science.uu.nl/~hooft101/theorist.html

## Math prerequisites (1)

## Americium (1343605) | more than 2 years ago | (#37235758)

## Read some Pop-phys books (1)

## modmans2ndcoming (929661) | more than 2 years ago | (#37235768)

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

## Don't take a broad approach (0)

## Anonymous Coward | more than 2 years ago | (#37235774)

"Many of us, myself included, don't have the background to understand them."

OK, step one: As a college grad you should have learned the difference between an object and a reflexive pronoun. So, let's make that "Many of us, me included, don't have the background to understand them." Or, perhaps "Many of us, including myself, don't have the background to understand them."

On to things mathematical. Don't try to gain a general understanding of differential geometry. It'll be years before you get where you want to be.

Step 2: Learn the basics of tensor notation in a Euclidean setting. Learn wha a metic is and how to measure infinitesimal distances using the local metic. This is trivial in a Euclidean space.

Step 3. Learn how to do this on a sphere. It's one of the easier nontrivial cases. Practice calculating geodesics and Christoffel symbols on the sphere.

Step 4. Learn how to put the GR mass-energy density into the metric tensor. Do this for a simple nontrivial case of a single massive object in an otherwise flat 2+1 dimensional space-time.

Without committing massive amounts of time to become a genuine expert, this is probably about as well as you will come to understand GR. It will certainly put you in a position to appreciate the popular articles on the subject.

Now, many interesting things in the world of physics are not actually GR, but relativistic QM for which you'll want to instead just study the special theory and probably the Dirac formulation of QM.

## Same Question for Particle Physics (1)

## stevelinton (4044) | more than 2 years ago | (#37235776)

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.

## Re:Same Question for Particle Physics (1)

## Landak (798221) | more than 2 years ago | (#37235844)

## Re:Same Question for Particle Physics (1)

## BitterOak (537666) | more than 2 years ago | (#37235950)

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 groups in physics, namely the Poincaré group, is neither semi-simple nor compact, and many math courses limit their study to compact or semi-simple Lie groups. In short, real physics is hard. Your background sounds very strong though, and you should be able to tackle a book like Weinberg's series The Quantum Theory of Fields, which is probably the best set of books on the subject that I know of.

Incidentally, another good sign is the fact that recent LHC experiments seem to be ruling out supersymmetry [slashdot.org]. I say that's a good thing, as SUSY makes all the math (literally!) twice as hard.

## Susskind's lectures (2, Informative)

## Anonymous Coward | more than 2 years ago | (#37235794)

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. (1)

## Landak (798221) | more than 2 years ago | (#37235802)

## You need to understand differential geometry (0)

## Anonymous Coward | more than 2 years ago | (#37235810)

To work with GR mathematically, you have to understand differential geometry. You have to be able to work with tensors, like the metric tensor.

## Gravity - by Hartle (2, Informative)

## Anonymous Coward | more than 2 years ago | (#37235812)

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.

## My path (0)

## Anonymous Coward | more than 2 years ago | (#37235818)

At each step I have digested the basics of that subject before moving on. And later reiterated and expanded my knowledge on them on a need to know basis. It has taken seven years and still counting.

Real analysis -> Euclidean vector analysis -> linear algebra -> functional analysis -> manifolds and differential forms -> tensors -> Riemannian geometry

## Easy (not so) GR (1)

## woboyle (1044168) | more than 2 years ago | (#37235820)

## Why learn the math? (0)

## Anonymous Coward | more than 2 years ago | (#37235850)

One can learn quite a bit about General Relativity without breaking out any math. In fact, GR is taught in introductory physics using introductory level math to great effect. It is a sexy topic and draws a broad audience. And like with many of Physics' big theories, a student can learn a tremendous amount about the cause and effect of forces and behaviors without necessarily learning the math behind them. It is very much like a philosophy.

Fundamental to the theory of GR is that time is constrained by the speed of light. That in itself is not obvious and has implications that trickle down to (among other things) solid-state physics. Understanding how those implications are manifested in the real world is fascinating. You can read layman books that give you a pretty broad understanding of GR (and other big concepts).

If on the other hand, your intent is to get in the business of postulating and predicting outcomes, then you do need to understand the math behind the concepts. But beware that the math may not bring you closer to understanding the concepts. Additionally, it is a rare topic in physics that can't be explained to the layman in words they understand. The best test of a student's understanding of physics is to have them explain to another (non-science) student the principles of the theory. Without that, it's all to easy for a student to learn the math without really grasping the reasoning behind it.

## Khan Academy ? (1)

## matt007 (80854) | more than 2 years ago | (#37235862)

It might not go up to relativity, but should get you most of the way there.

www.khanacademy.org

## Re:Khan Academy ? (0)

## Anonymous Coward | more than 2 years ago | (#37236042)

This website is awesome. Thanks for sharing!

## Read Schutz (2)

## BitterOak (537666) | more than 2 years ago | (#37235894)

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... (2)

## allwheat (1235474) | more than 2 years ago | (#37235904)

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. Calculus. Sounds like you already know some.

II. Differential Equations

A. Ordinary

B. Partial

III. Linear Algebra (Some texts teach ordinary differential equations and linear algebra together)

IV. Math Methods for Physicists (Arfken and Weber). Use this more for reference than for learning. Any math you need beyond the above set will be fairly specialized, so you can study by topic.

V. The best intro to relativity is in David J. Griffiths "Intro to Electrodynamics", a widely used textbooks for undergraduate physics majors. This only covers special relativity, but it's probably a really good place to start. For the graduate level, refer to Jackson's "Classical Electrodynamics," or possibly an easier equivalent.

VI. Another text by Griffiths is "Introduction to Elementary Particles", which includes some really useful stuff on relativity at the undergraduate level but for physics majors.

VII. (admission: I haven't studied General Relativity because I'm in another area of physics (CM), but I've harbored a secret desire to study it and maybe someday will steel away and do it.) A really common book is "Spacetime and Geometry: An Introduction to General Relativity" by Sean Carroll. I've flipped through this and it looks extremely well written, so when I do go ahead with my study, this is probably the book I'll select. Another good one is "A First Course in General Relativity" by Bernard Schutz. These are both graduate level texts, and I can't imagine there being an undergraduate level text.

This may take a long time and will be occasionally difficult, but it is certainly doable. Good luck.

## Stanford.edu lenard susskind helped me get a grasp (0)

## Anonymous Coward | more than 2 years ago | (#37235906)

Try out the Youtube lectured of Stanford university.

Leonard Susskind has made a bunch of lectures starting easy & building up to the maximum my head can handle without exploding;

http://www.youtube.com/watch?v=25haxRuZQUk&feature=list_related&playnext=1&list=SPA2FDCCBC7956448F is the course where my head said pop in lecture nr. 9 or so..;)

Thes are full-length classes (2 hours each, 12 vids in total in this course alone; & there are about 10 of them so 10*2 * 10 = 200 hours+ of vids explaining in a very good way (even i can understand the buildup going on there).

I've been doing this for the last year or 2; & it is hard; but very interesting.

(& it starts off 'easy' & builds on previous lectures).

Really; this is probably the best vids i've seen about our universe, string theory, relativity, & whatnot..

Why the guy has not yet gotten a Nobel Prize is beyond me; but probably will be given in the next 4-5 years or so :)

Good luck; & remember to take a brake after each vid; & watch them again after a week or so just so you start to grasp what you dont know in this world :)

## Pushing nerd buttons (0)

## Anonymous Coward | more than 2 years ago | (#37235932)

http://www.noob.us/humor/the-office-dwight-faces-nerd-torture-of-the-highest-form/

## courses (0)

## Anonymous Coward | more than 2 years ago | (#37236000)

I did my PhD in GR,

you need:

- calculus in several variables

- linear algebra

- some topology

- ideally something on partial differential equations

- differential geometry

- classical mechanics

then, take a deep breath... and go read General Relativity by Wald.

## Some concrete book suggestions (3, Informative)

## phage434 (824439) | more than 2 years ago | (#37236034)

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.