So for example a while back I discussed the Riemann tensor Rabcd. This was a thing that ate 3 tangent vectors and spit out a tangent vector... which is why there are 3 subscripts and one superscript.
Namely, if we parallel translate a tangent vector u around the little parallelogram of size epsilon whose edges point in the directions of the tangent vectors v and w, it changes by a little bit. Namely, it changes by the tangent vector whose component in the a direction is
- epsilon2 Rabcd vb wc ud + terms on the order of epsilon3
Here we sum over b, c, and d. The thing "vb" is the component of the vector v in the b direction... in whatever the hell coordinate system we happen to be using. And remember, indices like a,b,c,d range from 0 to 3 if we are working in 4d spacetime.
Oz says: "OK, I bet in reality it's not quite as simple as this however. "
Well, that was a pretty precise definition a large class of tensors, but not of the most general kind.
Here it is again, more formally so you will feel the suffering normally associated with education:
A tensor of "rank (0,k)" at a point p of spacetime is a function that takes as input a list of k tangent vectors at the point p and returns as output a number. The output must depend linearly on each input.
A tensor of "rank (1,k)" at a point p of spacetime is a function that takes as input a list of k tangent vectors at the point p and returns as output a tangent vector at the point p. The output must depend linearly on each input.
I am avoiding defining tensors of rank (n,k) for other values of n, because there is actually a fair amount of physics I can do without dragging them in. Eventually I would need to explain them, and you'd see they weren't much worse.
Oz remarks: "I take it that "linear way" is a fundamental property of a Tensor."
Indeed!!!!!!!!!! It's a branch of Linear Algebra.
Oz comments: "In general I assume there are an infinity of possible tangent vector directions like v,w above defining some parallelogram of size epsilon (which some nasty person will presumably tend to zero). I presume this is operational for a well behaved space over which tangent vectors are definable."
Sure, there are infinitely many tangent vectors at a point. This is not so bad. For example, note that the output
Rabcd vb wc ud
(let's ignore that epsilon junk and the niggly minus sign) depends linearly on the inputs u, v, and w, so we don't need to know it for *infinitely* many choices of u, v, and w to figure out what it will be for all possible choices. Linearity keeps life simple.
Oz notes: "Somebody has sneakily brought in some co-ordinates whilst nobody was looking (a,b,c ....). Now telling me they are local just won't do and nor will telling me they are 'whatever co-ordinates you desire'. I have (in GR contexts) been bludgeoned into realising that I need to reconsider the concept of 'co-ordinates' altogether."
Good. There's nothing like a good bludgeoning now and then. Well, certainly they are local coordinates, because you would be hard pressed to flatten out a whole pumpkin and impose *global* coordinates on it, rendering it a mere plane. And certainly the coordinates above are indeed "whatever coordinates you desire". But you are starting to sound like a mathematician --- high praise in my book --- with your complaint about the unpleasant appearance of coordinates. So to reward you, I will explain how it works without coordinates. You'll see it's much simpler.
The Riemann tensor is a tensor of rank (1,3) at each point of spacetime. Thus it takes three tangent vectors, say u, v, and w as inputs, and outputs 1 tangent vector, say R(u,v,w). As usual, the output depends linearly on each input. The Riemann tensor is defined like this:
Take the vector w, and parallel transport it around a wee parallelogram whose two edges are the vectors epsilon u and epsilon v , where epsilon is a tiny number longing to approach zero. The vector w comes back a bit changed by its journey; it is now a new vector w'. We then have
w' - w = -epsilon2 R(u,v,w) + terms of order epsilon3
Note: I am now saying just what I said before, but without those yucky coordinates! If you insist on using coordinates, I will say "Go ahead! Pick any that you like! I don't care which!" The only effect will be to turn the above elegant equation into the grungier but sometimes more practical one:
w'a - wa = - epsilon2 Rabcd vb wc ud + terms of order epsilon3