Talk:General Relativity/What is a tensor?

cool :) i like this page. Boud 12:19, 12 January 2006 (UTC)

I also like this page. I'm trying to teach myself relativity using an actual textbook. I'd like to help make the wikitextbook on relativity accessable. Itzamna 27 Jan 2009

helpful. i am reading penrose 'the road to reality' and his explanation of tensors is very abstract (although a fascinating book overall). i have searched the web for an easier intro to tensors. this page is a good start as it uses real examples. further explanation especially around the super and subscript notation used with tensors would be good. (ib 6 Feb 09)

T as a matrix
Hi, Could you specify please, what is denoted as x, y, z and v in the first formula? (The matrix of T)

Coordinate free paradigm
One can play around with the words. We talk about "abstract vectors" to remove ourselves from the specific representations of the vector, because we are so used to think of vectors as n-tuples of numbers, members of $$\mathbb{R}^n$$. That was the old way. So the new way is the "abstract" one. But in another perspective, the new way is actually more concrete. Young students raised in the new way of thinking may not at all know what you mean when you say that the vector exists as something abstract.

The coordinate-free way of thinking about vectors and linear operators just pretends there is a vector space V that is not at all the same as $$\mathbb{R}^n$$. In this thinking, any member of V is a physical entity with a physical direction and a magnitude with physical dimension (length, force, etc). The n-tuple of components is just an element of $$\mathbb{R}^n$$, not an element of V. This physical nature of v is why it does not seem particularly abstract.

Also the equations are equations of physical entities, not component equations. We have F = Tw where F is a member of the (tangent) space of forces, $$\mathcal{F}$$, w is a member of the space of wind speeds, $$\mathcal{W}$$, and T is a member of $$\mathcal{L}(\mathcal{W};\mathcal{F})$$, the space of linear operators $$\mathcal{W}\to\mathcal{F}$$.

I suggest that the text emphasizes the closeness to the physics, not the "abstractness". The "abstract" vectors and the "abstract" tensors or multilinear operators are (closer to) the real things that exist in the physical world independent of the observer that may choose a coordinate system and a representation.

It appears that some of the difficulty derives from a lack of a proper language to express things clearly. We talk about the "vector (x,y,z)" when we actually mean the vector $$x\vec{i}+y\vec{j}+z\vec{k}$$ where $$\vec{i}$$ etc, are physical basis vectors. Now, (x,y,z) is more succinct, and $$(x^{\mu})$$ even more so. But in introductory material we could use some way of making explicit that we mean a vector in the tangent space (or whichever space), not in $$\mathbb{R}^n$$. Perhaps we could introduce a name $$\sigma$$ of the coordinate system and write e.g. $$(x, y, z)_\sigma$$, to make it that element of V whose components are x,y,z relative to the coordinate basis. Do the same with the matrices that represent the linear operators. When you get to the index gymnastics, say that you are dropping the chart/basis annotation where unambiguous, to avoid clutter.

Cacadril (talk) 00:26, 31 March 2010 (UTC)