In Goldstein Classical Mechanics Chapter 7 (3rd edition, page 287), the authors classify vector fields as follows:
Name | Time Portion | Space Portion | (Magnitude$)^2$ | Type |
---|---|---|---|---|
Coordinate | $ct$ | $\mathbf{r}$ | $c^2t^2 - r^2$ | spacelike, null, or timelike |
Velocity | $\gamma c$ | $\gamma \mathbf{v}$ | $c^2$ | timelike |
Momentum | $\displaystyle \frac{E}{c}$ | $\mathbf{p}$ | $m^2c^2$ | timelike |
Force | $\displaystyle \frac{\gamma}{c} \frac{\mathrm{d}E}{\mathrm{d}t}$ | $\displaystyle \gamma \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t} = \gamma \mathbf{F}$ | $-\left(\mathbf{F}_{\mathrm{Newtonian}}\right)^2$ | spacelike |
Current density | $\gamma pc$ | $\gamma \mathbf{J}$ | $\rho^2 c^2$ | timelike |
where $\gamma = \displaystyle \frac{1}{\sqrt{1 - \beta^2}}$ and $\beta = v/c$. Other symbols have their usual meanings.
Now, I understand what it means to have a spacelike, timelike, or null separation between coordinates. But what does it mean when the authors use this term for a vector field? Apart from the mathematical notion of the norm of the vectors of these fields being of a certain sign everywhere what does it physically mean to say that the momentum field is always timelike and the force field is always spacelike?