Summary of 4-Vectors

We started with the position vector in Minkowski space.

$$x_\mu=(ct,x,y,z)$$

$$x_\mu x_\mu = -c^2t+x^2+y^2+z^2=s^2$$

An important dot product is that of the difference between two spacetime points. The dot product above gives the ``distance'' $s$ in Minkowski space from the origin.

The difference between spacetime points for a single particle is an important case. We use the dot product of this difference with itself.

$$(\Delta x_\mu)^2\equiv (x^{(2)}_\mu-x^{(1)}_\mu)(x^{(2)}_\mu-x^{(1)}_\mu)=\Delta s^2=-c^2\Delta\tau^2$$

The time difference in the particles rest frame $\tau$ is called the proper time and is demonstrated to be a scalar quantity in the above equation.

We define the velocity 4-vector with the equation.

$$v_\mu={dx_\mu\over d\tau}$$

$$v_\mu v_\mu=-c^2$$

We define the momentum 4-vector with.

$$p_\mu=m v_\mu$$

$$p_\mu p_\mu=-m^2c^2$$

We have shown that in the non-relativistic limit, the 4-momentum is consistent with.

$$p_\mu=({E\over c},\vec{p})$$

We accept this as being the components of the momentum 4-vector giving us the dot product of the momentum 4-vector with itself.

$$p_\mu p_\mu = -{E^2\over c^2}+p^2$$

$$E^2=(pc)^2+(mc^2)^2$$

The dot product of the momentum 4-vector and the position 4-vector

$$p_\mu x_\mu=-Et+\vec{p}\cdot\vec{x}$$

is related to the phase of waves. For example in quantum mechanics, a free particle with a definite momentum is represented by the plane wave.

$$\psi=e^{i(\vec{p}\cdot\vec{x}-Et)/\hbar}$$

We define the Force 4-vector.

$$F_\mu={\partial p_\mu\over\partial \tau}$$