What does it mean when one says that a vector field is spacelike, timelike, or null separated?

Question

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?

Answer 1

The "canonical" meaning of a vector in general, in a geometric sense, is a displacement: if you have a point $P$ in a suitably homogeneous space, such as Euclidean three-dimensional space, then

$$P + \mathbf{v}$$

is a point displaced in the direction and through the distance encoded within $\mathbf{v}$. Likewise, if $Q$ and $P$ are two points, then

$$Q - P$$

is the vector that represents the displacement from $P$ to $Q$.

The same applies to space-time vectors: we can use them to represent displacements between space-time points, just as we can use them to represent displacements between purely spatial points. Thus, it is natural to suggest that if we can classify the separations between points into those categories, that we should also analogously be able to classify vectors producing those separations when added to one or the other point, into the same categories.

Hence, we can define that a vector is spacelike, timelike, or lightlike if it produces, respectively, a space-time point (or event) that is spacelike, timelike, or lightlike separated from another such point when added to that given point. That is, $\mathbf{v}$ is space/time/light-like if $P$ and $P + \mathbf{v}$ are space/time/light-like separated, respectively.

From this, you can prove that, in turn, in terms of the length of the vector,

$$||\mathbf{v}|| = \sqrt{v_t^2 - v_x^2 - v_y^2 - v_z^2}$$

that

• $\mathbf{v}$ is timelike if $||\mathbf{v}||$ is nonzero and real,
• $\mathbf{v}$ is lightlike if $||\mathbf{v}||$ is zero,
• $\mathbf{v}$ is spacelike if $||\mathbf{v}||$ is nonzero and imaginary.

And this likewise can, then, be extended to define analogous notions for other vectors that are not dimensionally suitable to represent displacements directly. And we can then also likewise analogously extend it further to vector fields: a vector field $\mathbf{F}$ is (s/t/l)-like if at every space-time point $P$, $\mathbf{F}(P)$ is (s/t/l)-like as we have just defined and generalized.

Finally, physically, the 4-momentum is never spacelike (not "always timelike" - the 4-momentum of photons is lightlike) because it, in effect, represents the displacement within space-time that a physical object is undergoing while it is moving. And a displacement in a space-like direction would represent a motion faster than light. And as far as we have discovered, nothing travels faster than light.

Answer 2

That a vector $V$ is timelike, lightlike/null, or spacelike indicates solely that $V^{\mu}V_{\mu}=V_{0}^{2}-V_{1}^{2}-V_{2}^{2}-V_{3}^{2}=V_{0}^{2}-{\bf V}^{2}$ is positive, zero, or negative. What that means in different specific case varies somewhat.

For the four-momentum $p^{\mu}$: Since the momentum components are the canonical conjugates to the position variables,* $p^{\mu}$ is the generator of spacetime translation. This is automatically true quantum mechanically, and it is also true classically, when the theory is interpreted with Poisson brackets implementing transformations or through Hamilton's Principle, with the trajectory of a particle being the configuration of Least Action. Qualitatively, that $p^{\mu}$ must be timelike (or lightlike for a massless particle) means that the spacetime trajectory followed by a particle must also be timelike (or lightlike**).

The situation for the four-velocity is the same, since the four-velocity is just defined by rescaling the four-momentum by $m^{-1}$. The current density is slightly more complicated, but if $J^{\mu}$ is viewed as generated by the motion of a (large) collection of pointlike charges, ${\bf J}=\sum_{n}q_{n}{\bf v}_{n}\delta^{3}[{\bf r}-{\bf r}_{n}(t)]$, then its timelike nature also descends from that the timelike nature of $p^{\mu}$.

The requirement that the four-force be spacelike is a consequence of the fact that a particle of mass $m$ must, both before and after it experiences a four-impulse $\Delta I^{\mu}=F^{\mu}\Delta t$ satisfy the same energy-momentum relation, $E^{2}-{\bf p}^{2}c^{2}=m^{2}c^{4}$. Because $|\partial E/\partial{\bf p}|=|{\bf v}|<c$, the energy must change less than the momentum under the impulse (modulo factors of $c$), meaning the four-impulse (and hence four-force) must have a larger $|{\bf F}|$ than $F_{0}$; in other words, the four-force is spacelike.

*This is not true formally, for the time components (since the time $t$ is not a dynamical variable), but the same qualitative statements still apply to the time components.

**Barring reflections, which change the momentum from one lightlike vector to another.

(I have edited the question to refer to "vectors," rather than "vector fields," since only the coordinate and current density are vector fields, rather than single vectors. A vector field would be a vector-valued function of position, whereas the momentum of a particle is just a function of time, having no meaning except at the instantaneous location of the particle.)