What is the real Physical Meaning of tensor?

Question

I have read something about tensor calculus from Arfken and Corvet but all of them are more some mathematical algebra for tensors. what happens in reality and nature when we use tensors in our equations?

Answer 1

Physical quantities are (mostly) calculated and observed within a coordinate system, and depend on it. So if I say that some object has velocity $\vec{v}%$, it means that I measured this vector after I set the origin and orientation of the coordinate system. Now let's say that someone else has decided to define a different coordinate system, then he might observe a different velocity of the same object $\vec{v}'$. However, there should be a way that him and I can communicate our observed quantities and agree that both are the same, when adjusting for coordinate system. Tensor are all the quantities that behave "properly" when adjusting for coordinate systems, and much of tensor calculus deals with how to translate different obervables between different coordinate systems.

So for example a vector such as $\vec{v}$ is a rank-$1$ tensor, and all rank-$1$ tensors have the same rules as how to take them from one coordinate system to another. So $\vec{r}$ and $\vec{v}$ would both translate by the same set of equations between my coordinate system and the other's scientist. A scalar is a rank-$0$ tensor, which means that it doesn't change. Similarly, there are rules for how rank-$2$ tensors etc. will transfer.

Now, as you can imagine, a lot of the definitions depend on what coordinate transformations we allow and are dealing with. So if we only discuss rotations in real space, for example, then the kinetic energy $E=m\vec{v}^2/2$ is a scalar. However if we allow boosts (that is - translating to different inertial frames), then this is no longer a proper quantity, and one must construct different vectors (this time $4$-vectors) that will lead to different scalars (for example the relativistic energy $E^2 = m^2c^4+\vec{p}^2c^2$).

But basically, we would like to describe all our quantities by proper tensors, so they will not be bound to a specific arbitrary coordinate system, but could be universal.

Answer 2

Some quantities like temperature, pressure, etc. are well described by a real function of coordinates, called scalars.

But many other quantities do not fall into this category: Velocity, acceleration, relative position, electric and magnetic fields, stress/strain have not just magnitude, but also direction. These, are well described by a function of coordinates that return 3 or 4-dimensional tuples, called vectors, or more pompously, tensors of rank 1.

In the case of stress and electromagnetic fields, it make even more sense to describe them in terms of algebraic combinations of vectors that result in oriented surfaces, which are tensors of rank 2. The rules to combine vectors algebraically to obtain higher dimensional oriented objects are called geometric algebras, and generalize well to any number of dimensions

When we talk in terms of coordinate components, tensorial quantities like the velocity vector can be written as tuples $v_x, v_y, v_z$, but we need to remember that if we change the basis or orientation of our coordinate system we are not really changing the underlying physics, only our description. Hence the map between the components of our tensorial quantity must transform between coordinate systems according to a recipe that depends only on the properties of space, and ignore everything else on the problem

The same thing happens with tensors of rank higher than 1: We can write down tuples of tuples $\sigma_{xx}, \sigma_{xy}, [...] , \sigma_{zz}$ that represent a stress tensor inside a material, on some coordinates, and if the tuples describe a physical tensor, then they will transform between coordinate systems according to the appropiate geometric recipe