Tensor Analysis in Special Relativity

1. Vector analysis in spetial relativity

A vector \vec{A} in an inertial frame \mathscr{O} can be written as

(1)   \begin{equation*} \vec{A}=A^{\alpha}\vec{e}_{\alpha} \end{equation*}

In another inertial frame \bar{\mathscr{O}} which moving in the x drirection with vecocity \mathbf{v}

(2)   \begin{equation*} \vec{A}=A^{\bar{\alpha}}\vec{e}_{\bar{\alpha}} \end{equation*}


A^{\bar{\alpha}} and A^{\alpha} are connected by the Lorentz transformation

(3)   \begin{equation*} A^{\bar{\alpha}} =\Lambda\indices{^{\bar{\alpha}}_{\beta}} A^{\beta} \end{equation*}

where

(4)   \begin{equation*} \Lambda = \begin{bmatrix} \gamma & -v\gamma & 0 & 0 \\ -v\gamma & \gamma & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{bmatrix} \end{equation*}

And the basis has the relationship

(5)   \begin{equation*} \vec{e}_{\alpha}=\Lambda\indices{^{\bar{\beta}}_{\alpha}} \vec{e}_{\bar{\beta}} \end{equation*}

The Lorentz transformation has an inverse

(6)   \begin{equation*} \Lambda\indices{^{\nu}_{\bar{\beta}}}(-\mathbf{v}) \Lambda\indices{^{\bar{\beta}}_{\alpha}}(\mathbf{v}) = \delta\indices{^{\nu}_{\alpha}} \end{equation*}

which will give us

(7)   \begin{equation*} A^{\nu} = \Lambda\indices{^{\nu}_{\bar{\beta}}}(-\mathbf{v}) A^{\bar{\beta}}  \end{equation*}

(8)   \begin{equation*} \vec{e}_{\bar{\mu}}  =\Lambda\indices{^{\alpha}_{\bar{\mu}}}(-\mathbf{v}) \vec{e}_\alpha{} \end{equation*}

2. The metric tensor

The scalar product of two vectors is defined as

(9)   \begin{equation*} \vec{A}\cdot\vec{B} =-A^0B^0+A^1B^1+A^2B^2+A^3B^3 \end{equation*}

or, it can be written as

(10)   \begin{equation*} \vec{A}\cdot\vec{B} =(A^{\alpha}\vec{e}_{\alpha})\cdot (B^{\beta}\vec{e}_{\beta}) =A^{\alpha}B^{\beta}(\vec{e}_{\alpha}\cdot\vec{e}_{\beta}) \equiv A^{\alpha}B^{\beta} \eta_{\alpha\beta} \end{equation*}

where

(11)   \begin{equation*} [\eta_{\alpha\beta}] = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 &1 \end{bmatrix} \end{equation*}

\eta_{\alpha\beta} are called ‘components of the metric tensor’.

3. Definition of tensors

A tensor of type (0, N) is a mapping from N vectors into the real numbers, which is linear in each of its N 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 \mathscr{O} of a tensor of type (0, N) are the values of the mapping when its arguments are the basis vectors \{\vec{e}_{\alpha}\} of the frame \mathscr{O}.

For example, the components of the metric tensor is

(12)   \begin{equation*} g(\vec{e}_{\alpha}, \vec{e}_{\beta})=\vec{e}_{\alpha}\cdot \vec{e}_{\beta}=\eta_{\alpha\beta} \end{equation*}

4. The (0, 1) tensors: one-form

A tensor of the type (0, 1) is called a covector, a covariant vector, or a one-form. We can denote a one-form as \tilde{p}. 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 \vec{A}. The components of \tilde{p} are

(13)   \begin{equation*} p_{\alpha} \equiv \tilde{p}(\vec{e}_{\alpha}) \end{equation*}

Since \tilde{p}=p_{\alpha}\tilde{\omega}^{\alpha}, the above equation also defined the basis \tilde{\omega}^{\alpha}

(14)   \begin{equation*} \tilde{\omega}^{\alpha}(\vec{e}_{\beta}) =\delta\indices{^{\alpha}_{\beta}} \end{equation*}

Eq. 8 will give us

(15)   \begin{equation*} p_{\bar{\beta}} =\Lambda\indices{^{\alpha}_{\bar{\beta}}}(-\mathbf{v})p_{\alpha} \end{equation*}

and the basis changes as

(16)   \begin{equation*} \tilde{\omega}^{\bar{\alpha}} =\Lambda\indices{^{\bar{\alpha}}_{\beta}}(\mathbf{v})\tilde{\omega}^{\beta} \end{equation*}

5. The (0, 2) tensors

The components of an arbitrary (0, 2) tensor \mathbf{f} is

(17)   \begin{equation*} f_{\alpha\beta}\equiv\mathbf{f}(\vec{e}_{\alpha}, \vec{e}_{\beta}) \end{equation*}

The basis is the inner product of one-form basis

(18)   \begin{equation*} \mathbf{f}=f_{\alpha\beta}\tilde{\omega}^{\alpha}\otimes\tilde{\omega}^{\beta} \end{equation*}

6. Metric as a mapping of vectors into one-form

When the metric tensor \mathbf{g} has been supplied with a vector, it can be treated as a one-form

(19)   \begin{equation*} \mathbf{g}(\vec{V}, ) \equiv \tilde{V}( ) \end{equation*}

since \mathbf{g} is symmetric,

(20)   \begin{equation*} \mathbf{g}( , \vec{V}) \equiv \tilde{V}( ) \end{equation*}

its components are

(21)   \begin{equation*} V_{\alpha}=\eta_{\alpha\beta} V^{\beta} \end{equation*}

which means if \vec{V} \to (a, b, c, d), then \tilde{V}\to(-a, b, c, d).

Matrix [\eta_{\alpha\beta}] has an inverse, we denote it as [\eta^{\alpha\beta}], which has the same value of [\eta_{\alpha\beta}]. Then, given \{V_{\beta}\} we can find \{V^{\alpha}\}

(22)   \begin{equation*} V^{\alpha}=\eta^{\alpha\beta} V_{\beta} \end{equation*}

7. (M, N) tensors

An (M, N) tensor is a linear function of M one-forms and N vectors into the real numbers.

For instance, if \mathbf{R} is a (1, 1) tensor then it requires a one-form \tilde{p} and a vector \vec{A} to give a real number \mathbf{R}(\tilde{p}; \vec{A}). It has components

(23)   \begin{equation*} \mathbf{R}(\tilde{\omega}^{\alpha}; \vec{e}_{\beta}) \equiv R\indices{^{\alpha}_{\beta}} \end{equation*}

In a new frame,

(24)   \begin{equation*} R\indices{^{\bar{\alpha}}_{\bar{\beta}}} =\Lambda\indices{^{\bar{\alpha}}_{\mu}}\Lambda\indices{^{\nu}_{\bar{\beta}}}R\indices{^{\mu}_{\nu}} \end{equation*}

The upper indices are called ‘contravariant’ since they transform contrary to the basis vectors and lower ones ‘covariant’.

 

Refference

Schutz, Bernard. A first course in general relativity. Cambridge university press, 2009.

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