Reading for general relativity ~

I just want to get some reading on general relativity, who can help me?

Any respectable college (or even good HS) physics book will give you an excellent primer on GR. Did you learn any calculus yet? If you have, it will be easier. The hard part will be when you get to tensors. I would recommend Schaum's Outlines on Tensors (there are several). The reason why I bring it up is that GR is all written (and therefore can best be understood) w/ tensor notation.

Tensors are spatially dependent "interfunctions", since at least 2 different basis sets (usually fields) are interacting w/ one another (usually 2). So, for example, if you're sitting w/ a specific function in one basis set (some field) perched on top of (1,1,1) in an x1y1z1 plane and you start moving, let's say toward (-2, -1, 4), your function may not change but the tensor associated w/ your change in position may change, because the fields interact w/ one another differently at different points in the reference frame. Think of it as if another (for simplicity sake) cartesian plane X2,Y2,Z2 overlapped or existing in the same space as the first cartesian plane (x1,y1,z1).

To make matters worse, the reference frames may not map 1:1. Like in the stuff I do, they definitely don't (Electronic Electric Field Gradient versus an applied magnetic field inside a big magnet, Bzero).

-ss

Calculus? I just know a little~ Not good at it ~ I don't konw how "deep" I should go into calculus~ can you recommaned any reading to me either?
In addition I know nothing about Tensor~ can you help me?
Thx

To learn a bit more calculus I have found that "Calculus" by Stewart is a great book that can start you off from 'pre calculus' all the way to vector calculus.
And I would go with Solidspins suggestion for the book on tensors because I know nothing about tensors at this point in time.

Tensors are pretty deep into working with calculus. So start reading.

Schaum's isn't the best book to work with on Tensors, as it leaves some points required to work with GR indepth to the reader's responsibility. IOW it doesn't discuss them.

I'd say the best book dealing with GR (for those without a heavy math background) is this one. [amazon.com]

I haven't read it in a little while, but the deepest I remember it going into as for maths was Differential Geometry, which you should have covered already if you've done calculus in school already. If not, please shoot your math professor in the foot for cheating you on your education.

I don't know where you went to school, but in my 4 university level calculus courses, and 2 DE (ordinary and partial) courses we never covered differential geometry......unless we did under a different name, in which case "SNEAKY BASTARDS!!!" There is a 4th year math course specifically on differential geometry, and also a course specifically on tensor analysis. Plus the mathematical physics department at my school has one on GR, but I doubt that course is going to be very mathematically intense, since differential geometry isn't something the students have been taught, unless they are sneaky bastards!

Well, Differential Geometry is often also called Algebraic Geometry. It's basically Algebra formulated to deal with geometric plains of curvature. Which is exactly what GR is. Eistein pretty much created the field as we know it today, before him it was only used to describe how a curve on an object is shaped mathematically.

Differential Geometry as it is taught for GR is just a bit more preverse, and complex than that. It deals with curved planes, and layered approaches of geometry, where as there would be layers upon layers, within certain proportions, depending on application.

Most people now a days don't use Differential Geometry, because of Tensor Calculus, and the easier to use vector calculus. I prefer the geometry though, as it's easier to visualize.


Then again there's the metrics, which are basically calculus driven examples, that do the same as DG. I still say DG should be the first thing everyone learns, otherwise they miss the whole 60+ years of GR development, because that's how long it took them to get to the point where we can understand most of the math behind it. Einstein was brilliant, even if I don't agree with him. That math work was well beyond anyone else's work at the time.

"Differential Geometry is often also called Algebraic Geometry"?

No, that's not right. These are quite different subjects, although they certainly interact in lovely and useful ways, and they do share many concepts.

Algebraic geometry is concerned with things like solutions of systems of -polynomial- equations. One of the key ideas is the Nullstellensatz of Hilbert, which says that ideals in certain rings of polynomials correspond exactly to solution sets of such systems, which are called algebraic varieties. Another example of a topic in algebraic geometry would be studying configurations of lines in projective spaces.

Differential geometry is part of the study of smooth manifolds, a kind of geometric space which is much more general than an algebraic variety (but much less general than a topological manifold). In differential geometry, we study rates of things like functions change (kind of like a generalized vector calculus), and this leads to notions like bundling together algebraic things like vectors, covectors, tensors (multilinear operators), or differential forms to form "bundles". Then, a smooth section through a vector bundle or tensor bundle over a smooth manifold defines a vector field or tensor field on that manifold, and so forth.

Riemannian geometry (and Lorentzian geometry, the kind used in gtr) is part of differential geometry, not part of algebraic geometry, although there are many connections. Indeed, Roger Penrose, who has made many contributions to general relativity, was trained in algebraic geometry!

There are some fairly good articles about many of these topics at Wikipedia.

As to the original question, anyone interested in learning about gtr will be glad to know that there are a dozen excellent textbooks, some more suitable than others for self study. Some reading suggestions can be found at

http://math.ucr.edu/home/baez/RelWWW/reading.html

See also

http://math.ucr.edu/home/baez/relativity.html

for various other resources which might be helpful.

My questions are related to Differential Forms and Calculus of Forms vs Vectors and Vector Calculus (would like to get an intuitive understanding to contrast the two, not just definitions)

So far I managed to learn that mixed tensors include both types of objects (forms and vectors), and that forms "live" in the "dual" vector space of the vector space where regular vectors "live". The two types of objects can be transformed into each other using a metric tensor.

I also learnt there are operations on vectors (vector calculus) and on forms (exterior derivatives, wedge ,etc), and that the Physical Laws can be "expressed" using any of these two families of object; BUT they are not just a different notation from each other:

There are things that can be done with forms, that can't be done with vectors, and THAT makes the language of forms "better" (easier ?, simpler ?) to express the physical laws.

Can anyone comment on WHY that is the case, simple examples if possible, and if this is "modern" calculus why we all keep using the "old" vector calculus ?

Still confused, please help !

Try the book Gravity from the ground up, written by Bernard Schutz and published by Cambridge University Press.
It does not use college-level maths, so it might be a good start.

(I have not had the chance of reading it, yet.)

Well, the basic difference, at least as they are taught in university courses, is that vector calculus is developed without the concept of the manifold. In other words, the way vector calculus is taught assumes that space is Euclidean. Often this assumption is so deeply buried that it is not even mentioned. If the space is Euclidean and the coordinate system is Cartesian, then there is no difference between vectors and covectors, and simply transposing a vector produces a covector. There is no metric mentioned anywhere, because in Cartesian coordinates on Euclidean space, the metric turns out to be the identity, so it can be safely left out of calculations. This works as long as you stay exclusively within orthonormal coordinate systems. It will also work for cylindrical or spherical coordinate systems, as long as the local basis vectors are everywhere orthonormal.

On the other hand, if you use non-orthonormal basis vectors, there is a difference between vectors and covectors, and you must use a metric to transform one into the other. They also have opposite transformation properties, with vectors being contravariant and covectors covariant. When you go to any kind of curved space (a manifold) you have no choice but to use a metric, and it becomes impossible to pretend that vectors and covectors are the same thing any longer.

Where differential forms come into the picture is that you can derive local sets of basis vectors at each point from the coordinate system. It turns out that the partial derivative operator (differentiating in each coordinate separately) has all the properties we would expect a vector to have, and so you can simply DEFINE the basis vectors at each point to be the partial derivative operators. This makes vectors actually be operators acting on functions, where the functions are defined as functions of the coordinates on the manifold. The covectors then turn out to be exactly the differentials corresponding to the derivative operators, so that for example dx is a covector and d/dx is a vector.

About the wedge product: it is the same thing as an antisymmetric tensor product. If you take the ordinary tensor product A(*)B and make it antisymmetric by swapping terms and subtracting, you get A ^ B = A(*)B - B(*)A. The reason for this is so that integration will work properly. Wedge is a generalization of the cross product, which has the interpretation that its magnitude is the area of the parallelogram spanned by A and B. If A=B, then the area will be zero, and swapping A and B reverses the orientation of the area, so a good generalization to more dimensions would be to make the wedge product reverse sign whenever any two vectors are switched, which automatically makes the interpretation as n-dimensional volume work correctly. This is exactly what the wedge product does. There is nothing stopping you from taking the ordinary tensor product of differentials, but the result will be a tensor that does not have a physical interpretation as volume, and hence it won't make sense to use it in an integral. Notice that in orthonormal coordinate systems, forms such as dx ^ dy ^ dz reduce to just ordinary multiplication, dx dy dz, as seen in the integrals of a vector calculus course. So multiple integrals ARE using differential forms, you just don't have to think about it that way because of the Cartesian coordinate system.

I guess that the summary of my answer is that you do need the full apparatus of differential forms and vectors when working with curved space (as in General Relativity), but in the case of Euclidean space with Cartesian coordinates, the need for it, as well as the apparatus itself, just evaporates, and you are left with vector calculus. So you should use vector calculus when you don't actually need forms, because it is simpler and does the same job. If you want maximum generality, you would calculate with forms, because then your calculations would be valid on curved spaces as well as on Euclidean space.

Hope this helps!

--Stuart Anderson

In using where c = 8pi G = 1, the General Relativivity is G_ab = T_ab, but Baez has taken the trouble of spelling out this in English. The following is a collection of links on GR which roughly proceed from easiest to hardest.

http://www.astro.ucla.edu/~wright/relatvty.htm
http://math.ucr.edu/home/baez/einstein/einstein.html
http://www.physicsguy.com/ftl/html/FTL_part3.html
http://math.ucr.edu/home/baez/RelWWW/wrong.html
http://math.ucr.edu/home/baez/gr/outline2.html
http://math.ucr.edu/home/baez/gr/outline1.html
http://math.ucr.edu/home/baez/gr/oz1.html
Misner, Thorne, and Wheeler's Gravitation (one of the heavier books I own)
http://math.ucr.edu/home/baez/RelWWW/grad.html


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

Stuart mr_homm Anderson has given an excellent coverage of Ian's question.

How is the Universe expanding ?

I have read about the Universe expanding forever since the big bang, by analogy like bread rising due to the yeast creating bubbles inside...

"What" exactly is expanding ?

Is it just the matter moving into vacuum, "pre-existing" space or (this is what I find confusing) the "space" itself being created like the bread, in addition to the matter in the space ?

If the "space" (vacuum space ?) is being created and expanding, where is it expanding into ?

Related is the question of the "size" of the Universe (so many billion light years "in width" ?) . What is the meaning of this if the Universe does not have a boundary ?

Please explain what these measurements are made in reference to what ?

thanks.