Tensors and General Relativity

Question

I've recently begun self studying general relativity, using mostly the material found in Robert Wald's "General Relativity", and almost right out of the gate one encounters the notion of a tensor. In Wald's book a $(k,l)$-tensor is simply defined as a multilinear map

$$T : \underset{k-times}{\underbrace{V^*\times \cdots \times V^*}} \times \underset{l-times}{\underbrace{V\times \cdots \times V}} \rightarrow \mathbb{R}$$

where $V$ is a vector space over the reals and $V^*$ is it's corresponding dual space.

My problem with tensors is not so much what they are but rather it stems from a more intuitive standpoint, namely, why use tensors in the first place? Although I suppose it's likely that if I had a better understanding of what they are I would see why they are useful.

Just to provide some context, my first brush with tensors was during an analysis course on manifolds and the discussion of general tensors of the above form ultimately lead to dealing specifically with differential forms, since then this is the only other time i've encountered them. In any case, both in that class and my recent investigation into GR it seems to be that tensors are the natural tool for analyzing manifolds from a differential viewpoint, and my question is: why is that?

To me they seem to be exactly what they are defined to be; multilinear maps, and I just don't see why they are the weapon of choice, per se, for dealing with these sort of ''small (differential-type) changes''.

If you have encountered anything that would aid in building intuition as to why tensors are useful in this context, I would love to hear/read it and it would be greatly appreciated.

Answer 1

There are several properties of tensor that we see reflected in physical quantities in the real world:

Coordinate system independence: a tensor's components transform to precisely counteract how vector and covector components transform under a change of coordinate system. This suggests the tensor is an object unto itself, not tied to a particular coordinate system, and such a notion jives with what we might expect from the real world: that physical, measurable quantities do not depend on the coordinate system chosen.

Generalization of vectors and matrices: Fully antisymmetric tensors correspond to "blades": planes and volumes and so on that can be added or multiplied by scalars in the same way you would do with vectors, and with similar geometric significance. Blades are a very important kind of tensor, and they are among the easiest tensors to visualize.

For matrices, consider that a general matrix might be seen as a linear map of a vector and a covector that produces a number: if $\omega$ is a covector and $v$ a vector with $M$ a matrix, then $\omega(M(v))$ is that scalar, and we could identify it with a tensor $m$ such that $m(v, \omega)= \omega(M(v))$. General tensors simply can take more possible arguments. They could even be said to take blades as arguments instead of plain vectors (the Riemann tensor field, for instance, can be said to take a 2-blade--an oriented plane--as one argument).

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In the context of differential geometry, you will deal with a mix of blades and other more general tensors. For instance, every oriented manifold of dimension $n$ has an $n$-blade that describes its orientation: a curve has a tangent vector, a surface has a tangent plane, and so on. This $n$-blade may vary in "direction" at different points. Knowing how it varies gives great insight into how the manifold is curved with respect to its embedding (if it's embedded).

Certain linear map of blades have proven useful for talking about properties of a manifold. Again, the Riemann tensor field is the gold standard here, and you would be best served by trying to understand why and how this tensor field describes curvature, how the commutator of covariant derivatives tells us that, and so on. Here, I would understate the notion that tensors as a whole are powerful tools--they're not, really--but specific tensors are, and what they have in common is that they give us answers that are ultimately coordinate system independent.