General Relativity/What is a tensor?

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A tensor is a powerful abstract entity. And while its abstractness makes it a somewhat difficult thing to describe, we can begin to get a feel for what a tensor is through a non-abstract example.

Suppose you are sailing. The wind is coming from a certain direction and can be described as a vector, a directional quantity. Now there are many ways to represent this vector. For example, you can represent it as a speed from a certain direction (${\vec {w}}$). Alternatively, you can break it up into components and describe the vector as a combination of a certain amount of wind from the east and another amount from the south (${\vec {w}}'$). But despite the different ways of describing this vector, there is still this underlying abstract thing — the wind speed from a certain direction.

Now there is another important vector — the force that the wind produces when it hits the sail. If the direction of the force were always the same as the direction of the wind, then we could represent the relationship with a scalar, which just multiplies the wind vector by a constant factor to get the force vector.

However, life isn't that easy because the force is not always in the direction of the wind. In fact it usually isn't. So we can't represent the relationship between the wind and the force by a simple scalar. However, there is one important useful fact that we can use — the relationship between the wind speed and the force on the sail is (approximately at low speeds) linear. That is, if you double the speed of the wind, you double the force. The function that computes the force from the wind is a linear operator. The fact that this operator is linear lets us represent it in terms of a matrix, relative to a given basis:

$T={\begin{pmatrix}x&y\\z&v\end{pmatrix}}$

Note that just like you can change the way you represent the speed of the wind(${\vec {w}}$ and ${\vec {w}}'$), and the force it produces (${\vec {F}}$ and ${\vec {F}}'$), you can change the way you represent the operator that connects the two. In fact, whenever you change the representations of the wind and the forces, you will have to change the matrix in order to talk about the same situation. However, just like a vector is an abstract thing that can represent the wind or the force that the wind produces when it hits the sail, there is another kind of abstract thing that you can use to represent the relation between these things. In symbols

${\vec {F}}=T\cdot {\vec {w}}$ ${\vec {F}}'=T'\cdot {\vec {w}}'$

That thing in the middle, that T, is an example of a tensor. It is defined by the way in which ${\vec {F}}$ relates to ${\vec {w}}$. Tensors are really abstract things, but we now can begin to see the power of this abstractness. For one thing you can do algebra with tensors. So, say instead of one sail T, you have two sails T and U. We can represent the total force

${\vec {F}}=T\cdot {\vec {w}}+U\cdot {\vec {w}}$

Now because T and U are matrices and are linear we can combine them to form a new tensor V.

${\vec {F}}=(T+U)\cdot {\vec {w}}$ ${\vec {F}}=V\cdot {\vec {w}}\ \mathrm {where} \ V=T+U\,$

We can also multiply two tensors together. We have the force that the wind produces on the ship. Now the force that the ship produces on the water, when wind is acting on ship with force $F_{wind}$ can be represent also as a tensor (the same way we did with wind, sail and ship):

${\vec {F}}_{ocean}=W\cdot {\vec {F}}_{wind}$ ${\vec {F}}_{ocean}=W\cdot V\cdot {\vec {w}}$ ${\vec {F}}_{ocean}=X\cdot {\vec {w}}$ where tensor $X=W\cdot V=W\cdot (T+U)\,$ characterizes ship and both sails.

So what is a tensor?

Now that we have seen an example of a tensor, we can be more explicit about what a tensor is. A tensor is a linear function. In the case that we described, a tensor takes a vector and turns it into another vector. We can be more general and talk about tensors that turn a vector into a scalar or a scalar into a vector.

At this point, you should get some sense as to why tensors are important in general relativity. General relativity is all about matter changing the way that distances work. How do you find a distance? Well you take a vector and put it into a function. If the distance is short enough the function is (approximately) linear, and you can describe it as a tensor.

Special tensors

One special tensor is called the Kronecker delta tensor. It is just the identity matrix.

An identity is like 0 for normal addition (adding 0 to a number keeps it the same) and 1 for multiplication (multiplying a number by 1 gives the number itself).

The identity matrix is often represented by δⁱ_j, where

$\delta _{j}^{i}=\left\{{\begin{matrix}1&{\mbox{if}}&i=j\\0&{\mbox{if}}&i\neq j\end{matrix}}\right.$

Actually, the fact that $\delta _{j}^{i}$ is actually a tensor (rather than just a symbol that has two indices) is due to the fact that a mixed tensor that has this as its components in any one coordinate frame, will have the same components in any frame! Technically, we call such entities numerically invariant tensors.

Applications of tensors

Tensors are used in many places.

Apart from general relativity, tensors are used extensively in Continuum Mechanics. Tensors can be used to specify the stress at any point in the continuum. Such tensors are called stress tensors. The stress at any small surface (a tiny region around a point) in the continuum can be obtained by the matrix multiplication of this stress tensor and the unit vector (column vector) that is normal to the surface.

General relativity in one page

There are two things that can be described with tensors.

One is the curvature. A tensor describes how much things are bent. Using this tensor you can do things like calculate distances and angles, and figure out the shortest path through an area of space.

The second is stress-energy which is roughly how much energy and momentum exists in a particular location and what direction it is flowing in.

The basic equation of general relativity relates these two tensors. That's the basic idea. Everything else involves just getting used to doing the math.

Exercises

(Note: What I'd really like to do here is to have people add their answers to an answers page where they can see other people's answers. So feel free to add your answers HERE)

1) Describe an example of a tensor.

2) Suppose you have a situation in which the response of the sail to the wind is non-linear. How can you describe this in terms of tensors?

3) Can pressure be expressed as tensor?