Reference: UCI OpenCourseWare - GR 04 (Playlist)
Scalars, Vectors, and Tensors (Contd.)
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.
- 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.
Current Densities and Electromagnetism
If we have a simple particle that follows .
, 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.
This can also be written as
From Maxwell's equation:
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
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: