PDA

View Full Version : What is a Tensor?



Himanshu Raj
2007-Jan-16, 05:31 AM
I have studied about general relativity, not all, but the non-mathematical part of it. I have come across physical quantities called tensors. I just know that a tensor is a physical quantity that has different magnitude in different direction. Nothing more. I have read in few place that mathematics behind general relativity is very beautiful- which describesour universe. But I don't understand a bit of any word that's written the text books of GR (null geodesic, tensors, co-variant & contravariant vector transformation, levi-cetiva symbol, covariant diffrentiation, etc). I think I should know the basic mathematics behind it before proceeding on to GR. Please tell me what basic mathematics should I know to understand General Relativity.

tusenfem
2007-Jan-16, 11:21 AM
Well, first pick up a mathematics book in which matrix calculus is explained, e.g. the wonderful book by Marsden & Tromba "Vector Calculus". If you understand vectors and matrices, then you can continue to try and understand GR.

Argos
2007-Jan-16, 01:04 PM
Good for a start (http://www.physlink.com/Education/AskExperts/ae168.cfm). More (http://en.wikipedia.org/wiki/Tensor) and more (http://mathworld.wolfram.com/Tensor.html).

Tensor
2007-Jan-16, 02:00 PM
Heheheheheheheh ;) :D

publius
2007-Jan-16, 06:51 PM
Well, first pick up a mathematics book in which matrix calculus is explained, e.g. the wonderful book by Marsden & Tromba "Vector Calculus". If you understand vectors and matrices, then you can continue to try and understand GR.

Well, I'll be doggone -- that's one of my favorites as well. I picked that book up in the university bookstore one time -- it's a darn good text on vector calculus. There's wonderful stuff in there that will help you with GR -- the notion of tangent vectors and parameterizing curves according to arc length.

-Richard

Tensor
2007-Jan-16, 07:16 PM
Well, I'll be doggone -- that's one of my favorites as well. I picked that book up in the university bookstore one time -- it's a darn good text on vector calculus. There's wonderful stuff in there that will help you with GR -- the notion of tangent vectors and parameterizing curves according to arc length.

-Richard

I wish I would have found that before I picked up a couple of others. It was very clearly presented.

DyerWolf
2007-Jan-16, 08:43 PM
How approachable is this 'wonderful book by Marsden & Tromba "Vector Calculus"'?

I was math averse in my youth. Despite the slow ossification of matter between my ears during these latter years, I recently discovered the usefulness of mathematics (if only as a way to get off Nereid's 'hitlist').

Can a guy who last studied math 20 years ago find a foothold?

sirius0
2007-Jan-16, 11:15 PM
I think there comes a point where anyone with abillity or potential (some would say that even these are internal choices). Has to teach themselves maths. Either this occurs as an internal dialouge whilst the lecturer is ranting :) or alone with a book etc. The turning point for me in my degree was when I realised that I could only learn in private. A saying I kept saying to myself was that "Confidence is not a pre-requisite" the maths will work regardless as to how I feel about it! I went from a near fail in year 11 (never did year 12) to finishing a batchelor in physics as a mature age student 20 years later with an average credit (couple of HDs in there too!) in maths. I still struggle a little but do get there. Because I am working as an engineer now I am getting my brain stimulated by those physicists that post here and reading their links etc.
I think maths is acheivable for you as long as you are cheeky enough to start. Confidence is not a pre-requisite but it will probably be the reward.

Cougar
2007-Jan-17, 02:50 AM
Good for a start (http://www.physlink.com/Education/AskExperts/ae168.cfm). More (http://en.wikipedia.org/wiki/Tensor) and more (http://mathworld.wolfram.com/Tensor.html).
That first link says....


"As might be suspected, tensors can be defined to all orders. Next above a vector are tensors of order 2, which are often referred to as matrices... a two dimensional array... An example of a second order tensor is the so-called inertia matrix (or tensor) of an object. For three dimensional objects, it is a 3 x 3 = 9 element array that characterizes the behavior of a rotating body....

There are yet more complex phenomena that require tensors of even higher order. For example, in Einstein's General Theory of Relativity, the curvature of space-time, which gives rise to gravity, is described by the so-called Riemann curvature tensor, which is a tensor of order four. Since it is defined in space-time, which is four dimensional, the Riemann curvature tensor can be represented as a four dimensional array (because the order of the tensor is four), with four components (because space-time is four dimensional) along each edge. That is, in this case, the Riemann curvature tensor has 4 x 4 x 4 x 4 = 256 components! [Fortunately, it turns out that only 20 of these components are mathematically independent of each other, vastly simplifying the solution of Einstein's equations]."

So if the Riemann curvature tensor is no mere matrix but only "can be represented as a four dimensional array," how does one represent such an array on a two dimensional piece of paper or chalkboard?

publius
2007-Jan-17, 03:14 AM
So if the Riemann curvature tensor is no mere matrix but only "can be represented as a four dimensional array," how does one represent such an array on a two dimensional piece of paper or chalkboard?


That's one of my favorite jokes, actually: GR is so complicated, it takes 4dimensions to just visualize the mathematical structures themselves!

But that's just a joke. You can program a 4D array in a computer as easy as pie can't you? You just use four subscripts and do operations with them by "walking" over 4 indices, rather than two or three.

You don't need to be able to visualize the structure, although it helps. You have to draw on your experience with lower order objects you can visualize to see how do to the "index gymnastics" of all that mess. Other operations raise and lower the rank of the tensor(s) in the expression. For example, the divergence of a vector gives you a scalar. That can be seen as a rank-1 to rank-0 operation. And you can generalize that and define a general divergence operator so that the divergence of a rank-2 tensor is a vector. And then consider the gradient operator. That takes a rank-0 object and gives a rank-1 vector object. Well, you can define things so a gradient of a vector is a rank-2 tensor.

There are rules and a notation based on those rules that allows you to to write quite terse looking expressions, but those expression involve the index gymnastics. Learning those gymnastics is one of the big things -- it tells you what to do with the components to build higher and lower rank objects.

And finally, while tensors can be though of as arrays like this, they are more than that. For example, while you can represent a rank-2 tensor as a square matrix, not all 2D matrices are a proper rank-2 tensor. There are important rules about what a real tensor is.

-Richard