Summary of 4-Vectors

We started with the position vector in Minkowski space.

\bgroup\color{black}$ \displaystyle x_\mu=(ct,x,y,z)$\egroup
\bgroup\color{black}$ \displaystyle x_\mu x_\mu = -c^2t+x^2+y^2+z^2=s^2$\egroup
An important dot product is that of the difference between two spacetime points. The dot product above gives the ``distance'' \bgroup\color{black}$ s$\egroup 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.

\bgroup\color{black}$ \displaystyle (\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$\egroup
The time difference in the particles rest frame \bgroup\color{black}$ \tau$\egroup 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.

\bgroup\color{black}$ \displaystyle v_\mu={dx_\mu\over d\tau}$\egroup
\bgroup\color{black}$ \displaystyle v_\mu v_\mu=-c^2$\egroup

We define the momentum 4-vector with.

\bgroup\color{black}$ \displaystyle p_\mu=m v_\mu$\egroup
\bgroup\color{black}$ \displaystyle p_\mu p_\mu=-m^2c^2$\egroup
We have shown that in the non-relativistic limit, the 4-momentum is consistent with.
\bgroup\color{black}$ \displaystyle p_\mu=({E\over c},\vec{p})$\egroup
We accept this as being the components of the momentum 4-vector giving us the dot product of the momentum 4-vector with itself.
\bgroup\color{black}$ \displaystyle p_\mu p_\mu = -{E^2\over c^2}+p^2$\egroup
\bgroup\color{black}$ \displaystyle E^2=(pc)^2+(mc^2)^2$\egroup

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

\bgroup\color{black}$\displaystyle p_\mu x_\mu=-Et+\vec{p}\cdot\vec{x} $\egroup

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.

\bgroup\color{black}$\displaystyle \psi=e^{i(\vec{p}\cdot\vec{x}-Et)/\hbar} $\egroup

We define the Force 4-vector.

\bgroup\color{black}$ \displaystyle F_\mu={\partial p_\mu\over\partial \tau}$\egroup

Jim Branson 2012-10-21