Consider a Lorentz transformation in which the new frame (primed coordinates) moves with velocity $v$ along the $z$ axis of the original frame (unprimed coordinates). For such a Lorentz "boost", show that
$$
\begin{aligned}
c t^{\prime} & =\cosh \theta c t-\sinh \theta z, \\
z^{\prime} & =-\sinh \theta c t+\cosh \theta z,
\end{aligned}
$$
with $x$ and $y$ unchanged; here, $\tanh \theta=v / c$. As $\cos i \theta=\cosh \theta$ and $\sin i \theta$ $=i \sinh \theta$, we see that the Lorentz transformation may be regarded as a rotation through an imaginary angle $i \theta$ in the $i c t-z$ plane.
By definition, any set of four quantities which transform like $(c t, \mathbf{x})$ under Lorentz transformations is called a four-vector. We use the notation
$$
(c t, \mathbf{x}) \equiv\left(x^0, x^1, x^2, x^3\right) \equiv x^\mu .
$$
According to the theory of special relativity, the total energy $E$ and the momentum $\mathbf{p}$ of an isolated system transform as the components of a four-vector
$$
\left(\frac{E}{c}, \mathbf{p}\right) \equiv\left(p^0, p^1, p^2, p^3\right)=p^\mu
$$
with the basic invariant $\left(E^2 / c^2\right)-\mathbf{p}^2$. The simplest system is a free particle, for which
$$
\frac{E^2}{c^2}-\mathbf{p}^2=m^2 c^2
$$
where $m$ is the rest mass of the particle. From now on, we revert back to the use of natural units with $c \equiv 1$ (see Section 1.4).
Just as in three-dimensional space, we may introduce the scalar product of two four-vectors $A^\mu \equiv\left(A^0, \mathbf{A}\right)$ and $B^\mu \equiv\left(B^0, \mathbf{B}\right)$
$$
A \cdot B \equiv A^0 B^0-\mathbf{A} \cdot \mathbf{B},
$$
which is left invariant under Lorentz transformations. Due to the minus sign, it is convenient to introduce a new type of four-vector, $A_\mu \equiv\left(A^0,-\mathrm{A}\right)$, so that the scalar product is
$$
A \cdot B=A_\mu B^\mu=A^\mu B_\mu=g_{\mu \nu} A^\mu B^\nu=g^{\mu \nu} A_\mu B_\nu .
$$
Here, we have introduced the (metric) tensor $g_{\mu \nu}$, which is defined by
$$
g_{00}=1, \quad g_{11}=g_{22}=g_{33}=-1, \quad \text { other components }=0
$$
(and similarly for $g^{\mu \nu}$ ). A summation over repeated indices is implied in (3.11). Upper (lower) index vectors are called contravariant (covariant) vectors. The rule for forming Lorentz invariants is to make the upper indices balance the lower indices. If an equation is Lorentz covariant, we must ensure that all unrepeated indices (upper and lower separately) balance on either side of the equation, and that all repeated indices appear once as an upper and once as a lower index.