1. Vector analysis in spetial relativity
A vector in an inertial frame can be written as
In another inertial frame which moving in the drirection with vecocity
and are connected by the Lorentz transformation
And the basis has the relationship
The Lorentz transformation has an inverse
2. The metric tensor
The scalar product of two vectors is defined as
or, it can be written as
are called ‘components of the metric tensor’.
3. Definition of tensors
A tensor of type is a mapping from vectors into the real numbers, which is linear in each of its arguments.
A tensor is a rule which give s the same real number independently of the reference frame in which the vectors’ components are calculated.
Notice that the tensor is the mapping itself, not the value of this mapping.
The components in a frame of a tensor of type are the values of the mapping when its arguments are the basis vectors of the frame .
For example, the components of the metric tensor is
4. The (0, 1) tensors: one-form
A tensor of the type is called a covector, a covariant vector, or a one-form. We can denote a one-form as . The set of all one-forms satisfies the axioms for a vector space, which is called the ‘dual vector space’ to distinguish it from the space of all vectors like . The components of are
Since , the above equation also defined the basis
Eq. 8 will give us
and the basis changes as
5. The (0, 2) tensors
The components of an arbitrary tensor is
The basis is the inner product of one-form basis
6. Metric as a mapping of vectors into one-form
When the metric tensor has been supplied with a vector, it can be treated as a one-form
since is symmetric,
its components are
which means if , then .
Matrix has an inverse, we denote it as , which has the same value of . Then, given we can find
7. (M, N) tensors
An tensor is a linear function of one-forms and vectors into the real numbers.
For instance, if is a tensor then it requires a one-form and a vector to give a real number . It has components
In a new frame,
The upper indices are called ‘contravariant’ since they transform contrary to the basis vectors and lower ones ‘covariant’.
Schutz, Bernard. A first course in general relativity. Cambridge university press, 2009.