PC1672 Advanced dynamics

1.4 Space-time four-vectors

The location of an event in space-time can be represented by the 4-vector ${\sf x}=(x_1,x_2,x_3,x_4)$ where ${\bf x}=(x_1,x_2,x_3)$ is its position and $x_4={\rm i}ct$ (notation). This has a meaning which is the same for all observers, although the four components used to represent it take different values in different inertial frames. For example, under a boost to a frame moving with velocity in the -direction introduced before, the components transform to

		$\displaystyle x'_1 = \gamma(V)\left(x_1 + {\rm i}{V\over c}x_4\right)$
		$\displaystyle x'_2 = x_2$
		$\displaystyle x'_3 = x_3$
		$\displaystyle x'_4 = \gamma(V)\left(- {\rm i}{V\over c}x_1+x_4 \right)$

where

$\begin{displaymath}\gamma(V)={1\over\sqrt{1-V^2/c^2}}\end{displaymath}$

Just as the components of all vectors transform in the same way under some rotation of the axes, the components of all four vectors transform in the same way under a particular Lorentz transformation.

All 4-scalars take the same value in all inertial frames. An example is

$\begin{displaymath}{\sf x}\cdot{\sf x}=\sum_{\mu=1}^4x_\mu^2=s^2\end{displaymath}$

the square of the invariant distance from the origin of space time.

Principle of ovariance: If we express physical laws in terms of 4-scalars and 4-vectors (and in some cases 4-tensors) then these laws will be invariant under Lorentz transformations: they will have the same forms in all inertial frames.

An important example of a 4-vector is the 4-velocity of a particle. This is defined as the ratio of the change in the particle's space-time position to the change in its proper time

$\begin{displaymath}{\sf U}={{\rm d}{\sf x}\over {\rm d}\tau}\end{displaymath}$

In a frame where the particle has velocity ${\bf u}=(u_1,u_2,u_3)$ its 4-velocity is

$\begin{displaymath}{\sf U}=\gamma(u)(u_1,u_2,u_3,{\rm i}c)\end{displaymath}$

Another useful 4-vector is

$\begin{displaymath}{\sf k}=(k_1,k_2,k_3,{\rm i}\omega/c)\end{displaymath}$

where ${\bf k}=(k_1,k_2,k_3)$ is the wave vector of a wave and $\omega$ is its frequency. This allow us to write the amplitude of the wave in the invariant form

$\begin{displaymath}\psi({\bf x},t)=A\cos({\sf k}\cdot{\sf x}+\phi)\end{displaymath}$

To transform a 4-vector under, for example, the boost in the -direction, we simply substitute the components of that 4-vector for the components of ${\sf x}$ in the expressions above. This can be applied to the 4-velocity to obtain the relativistic formula for addition of velocities, and to the wave vector to get the relativistic Doppler shift.

Notation: Here sans-serif type will be used to denote 4-vectors, while bold roman type will used for 3-vectors and ordinary type for scalar quantities such as the magnitudes of vectors. See Examples 1 for more details.

Textbook references

K&K 14.3
M 10.4-5
M&T 14.9
Feynman lectures, volume I, 17

Mike Birse
17th May 2000