A tensor is a physical value that transform in certain ways.

How to construct tensors

Linear combination of tensors of the same kind

Multiply tensors with different indices (direct product). This gives symmetric tensors;

Contraction. This reduces the total number of indices by 1.

Differentiation/Division. Similar to direct product.

Special tensors

(4-invariant)

Levi-Civita symbol:

(Wikipedia) ... Also, the specific term "symbol" emphasizes that it is not a tensor because of how it transforms between coordinate systems, however it can be interpreted as a tensor density.

Special topics

Current Densities and Electromagnetism

If we have a simple particle that follows .

Defining , This generalizes to

This does not manifestly look like a 4-vector. For: 1. the Dirac delta is (3) multiple, and 2. explicit use of . Now we can turn it in to a manifestly covariant form by add a delta term and a integral

which also means that is a scalar. Then

is indeed a 4-vector.

is conserved

Observe that ,
This can also be written as .

Electromagnetic tensor

From Maxwell's equation:

Source equations:

Non-source equations:

We can write this down using 4-vectors/tensors. First introduce electromagnetic tensor defined as follows:

The source equations can be written as

The non-source equation can be written as

Equation of motion for EM

Energy-Momentum Tensor

The conservation of charse arises from the fact that current density is a 4-vector. In order for momentum to conserve, we need a order-2 tensor to correspond to the field.

The tensor of a particle looks like this:

which is manifestly a Lorentz tensor.

Electromagnetic field itself also contains energy. This energy can be described in terms of energy-momentum tensor:

Then we can guess the relation between the tensors: