what a tensor is and what applications it has please.
In: Physics
A tensor is a number, or series of numbers, used to describe a physical property. They come in different degrees, called ranks, depending how much information they give.
For example,
Speed is scalar, or a 0th rank tensor. It’s a single number and implies a ‘magnitude’. For example a object’s speed is 5 miles per hour. No sense of direction, just the size of the property.
Velocity is a vector, or a 1st rank tensor, it’s a single number PLUS information on a direction. An object is falling 5 miles per hour *downwards*.
Velocity can also being explained as a magnitude (5 miles hour) but with 3 levels of direction, for example 5 degrees below the horizon (vertical), 46 degrees left (horizontal), and 87 degrees backwards (front/back). this is a 2nd rank tensor.
You can get into progressively more complicated tensors when you get to complicated physics and math, but just means more and more information about the thing you’re describing.
The world of tensors gets pretty important in higher level math and physics, if you were in a college level engineering program you’d being using tensors and learning complex math techniques to manipulate them, add/subtract them, etc.
So let’s say you want to talk about velocity of an object. What you would normally do is to draw an arrow in the direction of movement, in which the length is the speed, right? Basically, you represent it by a vector.
This arrow is essentially a (mathematical) tensor. It’s a geometric object.
But in physics, people care about observables, which are numbers. If you select a coordinate system, you can assign 3 coordinates to this arrow. At lower level of mathematics, this 3 coordinates numbers are somehow considered “the same” as the arrow, but it’s not. The 3 numbers represent an arrow. This distinction doesn’t really matter in Newtonian physics, because Newton believe in a single absolute coordinate system.
But in most other physical theories, there are no one true coordinate system. Instead, a physical theories have laws that are true under many coordinate system (which coordinate system depends on the theory). This is when the distinction matter. If you pick a different coordinate system, that same arrow will be assigned to different 3 numbers.
If you both believe that velocity is a real physical property that can be represented by a vector, and also believe that observables are the only way we can interact with the physical world, then the next conceptual step is obvious. Instead of an arrow, we represent velocity by a tuple of 3 numbers for every acceptable coordinate system, but also require that these tuples of 3 numbers are related in such a way that it could have came from an arrow. For example, if you have the tuple (1,0,0) for one coordinate system, but then for a different coordinate system in which the x and y axis got swapped, the new tuple should be (0,1,0). In other word, these tuple of 3 numbers are related by a coordinate transformation. This is an example of a (physics) tensor. They are what a mathematical tensor would be except that you remove the geometric object and replace it by all of its possible coordinate representation.
**BUT** that’s NOT the end of the story.
When physics talk about tensor, they usually mean tensor *field*. A tensor field assign a tensor to every position in space (or spacetime), in such a way that the tensor varies smoothly between points.
Once again, a mathematical tensor (field) would assign an arrow to every point, and can be represented diagrammatically by a bunch of arrow. And a physics tensor assign a function to every possible local system of coordinate – a function that assign to every tuple of space coordinate another tuple of vector coordinate – such that these set of functions transform in the right way between different coordinate system, such that these functions could have came from the coordinate representation of a field of arrows. The transformation rules are a bit more complicated, as different coordinate system don’t just assign different coordinates to vectors, but also to point in space.
Usually, the reason why physicist mention that something is a “tensor”, is because they need to emphasize the fact that it’s not some other geometric objects that are not tensor, and there are many of them when we’re in the context of a field, often obtained by taking derivatives. For example, the Ricci curvature tensor is a tensor, a fact that can be a surprise to people, since it is obtained by taking derivatives, so it seems like it might not be a tensor.
Now, all of the examples I mentioned so far are just (1,0)-tensor. Vector fields are (1,0)-tensor. Scalar fields are technically (0,0)-tensor, but you won’t often see that fact mentioned, for the simple reason that the transformation rules are obvious. What is especially interesting are higher rank tensor, which does not have easy visualization like arrow.
A (0,1)-tensor are covector, and can be thought to correspond to your choice of measurement of a vector in a specific direction. The choice of measuring velocity in the forward direction using unit of meter correspond to a covector. This allow vector to pair up with covector to produce a number: the covector measure the vector. This pairing allow us to think of a (1,0)-tensor as an object that take in 1 covector and produce a number, and a (0,1)-tensor as an object that take in 1 vector and produce a number.
This generalize upward and give higher rank tensor. A (m,n)-tensor is something that take in m covectors and n vectors to produce a number. Once again, the mathematical version is different from the physics version: mathematical tensor/tensor field is an abstract object that can be assigned coordinates, while physics tensor/tensor field are a bunch of number/functions that assign tuples to tuples and satisfy transformation rules.
One interesting example of a (2,0)-tensor is angular momentum, because low level physics usually introduce angular momentum as a vector instead. But once you need to deal with multiple coordinate system, cross product became your enemy, and it turned out that a lot of things defined using cross product need to be replaced by a different object, a 2nd rank tensor.
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