# Are there more distinctive names of “null curves” with certain additional properties (absence of “chord curves”)?

In this answer (to the question "In general relativity, are light-like curves light-like geodesics?", PSE/q/76170) a particular example of a curve is discussed whose "tangent is everywhere null" and which is therefore called a "null curve". I'll restate the example curve explicitly as

$$\nu : \mathbb R \rightarrow \cal M,$$

together with a coordinate function

$$\mathbf r : \cal M \rightarrow \mathbb R^{1,2}; \qquad \mathbf r := \{~t,~x,~y~\}$$

such that

$$\mathbf r \circ \nu[~\lambda~] := \{~t_{\nu}[~\lambda~],~x_{\nu}[~\lambda~],~y_{\nu}[~\lambda~]~\} = \{~\lambda,~\text{Cos}[~\lambda~],~\text{Sin}[~\lambda~]~\}.$$

The calculation of the corresponding "tangent" value proceeds (roughly) via

\begin{align} \left( \frac{d}{d\lambda}[~t_{\nu}[~\lambda~]~] \right)^2 - \left( \frac{d}{d\lambda}[~x_{\nu}[~\lambda~]~] \right)^2 - \left( \frac{d}{d\lambda}[~y_{\nu}[~\lambda~]~] \right)^2 & = & \\ 1 - \left( -\text{Sin}[~\lambda~] \right)^2 - \left( \text{Cos}[~\lambda~] \right)^2 & = 0\end{align} for all values $\lambda$ in the domain of "null curve" $\nu$.

Now, interestingly, for any two distinct values $\lambda := a$ and $\lambda := b$ from the domain of "null curve" $\nu$, there exist curves (which in the following are suggestively called "chord curves") $$\kappa_{ab} : [~a, ~b~] \subset \mathbb R \rightarrow \cal M,$$ such that $$\kappa_{ab}[~a~] = \nu[~a~], \qquad \kappa_{ab}[~b~] = \nu[~b~]$$ and: the "tangent of $\kappa_{ab}$" is everywhere positive.

As one concrete case consider $$\mathbf r \circ \kappa[~\lambda~] := \{~t_{\kappa}[~\lambda~],~x_{\kappa}[~\lambda~],~y_{\kappa}[~\lambda~]~\} =$$ $$\{~\small{\lambda,~\text{Cos}[~a~] + \left( \frac{\lambda - a}{b - a} \right) \left( \text{Cos}[~b~] - \text{Cos}[~a~] \right),~\text{Sin}[~a~] + \left( \frac{\lambda - a}{b - a} \right) \left( \text{Sin}[~b~] - \text{Sin}[~a~] \right)~}\},$$

with the corresponding "tangent" value

$$\left( \frac{d}{d\lambda}[~t_{\kappa}[~\lambda~]~] \right)^2 - \left( \frac{d}{d\lambda}[~x_{\kappa}[~\lambda~]~] \right)^2 - \left( \frac{d}{d\lambda}[~y_{\kappa}[~\lambda~]~] \right)^2 =$$ $$\small{1 - \left( \frac{\text{Cos}[~b~] - \text{Cos}[~a~]}{b - a} \right)^2 - \left( \frac{\text{Sin}[~b~] - \text{Sin}[~a~]}{b - a} \right)^2 = 1 - 4 \left( \frac{\text{Sin}[~(b - a)/2~]}{b - a} \right)^2 \gt 0.}$$

(As a sidenote: similarly, for given distinct values $\lambda := a$ and $\lambda := b$ from the domain of "null curve" $\nu$, one may ask for "chord curves" whose "tangent" value should be everywhere negative; but I have not been able to construct a corresponding concrete example. In the following, it is not necessary to distinguish "positive chord curves" from "negative" ones, if cases of the latter exist at all. Relevant is only that the "tangent" value of a "chord curve" exists everywhere and is not "null" anywhere.)

My question is:
Is there some specific name (i.e. more distinctive than "just another null curve") referring to "null curves" which do not have any "chord curves" at all? (Are such special cases of "null curves" perhaps called "null geodesics"?)

• Ī once participated in a related discussion on one forum, now at this link. Maybe you will extract something useful from it. – Incnis Mrsi Oct 20 '14 at 9:34

As in your post, let $\gamma : [a,b] \rightarrow \mathcal{M}$ be a smooth curve onto a Lorentzian manifold.

You might be interested in this math.SE post, indicating that all points in more than two Lorentzian dimensions can be joined by a spacelike curve, so space-like chord curves are utterly uninteresting, as they can be constructed from taking a small tube around $\gamma$ from $\gamma(a)$ to $\gamma(b)$ and letting the spacelike path wind around that tube.

The existence of time-like chord curves implies that $\gamma$ is not a maximum of the length functional, and that therefore $\gamma$ is not a length-maximizing geodesic. I am not yet sure if it implies that $\gamma$ is no geodesic at all, since geodesics on Lorentzian manifolds are weird. However, if time-like chord curves exist between every $\gamma(t)$ and $\gamma'(t)$ for every $t,t'\in [a,b]$, then $\gamma$ is no geodesic, since Lorentzian time-like geodesics are locally length-maximizing.

Thus, being a null geodesic indeed implies that, locally, there are no time-like chord curves.

• ACuriousMind: "You might be interested in this math.SE [MSE/q/502752] post, indicating that all points in more than two Lorentzian dimensions can be joined by a spacelike curve [...] constructed from taking a small tube around $\gamma$ [...] and letting the spacelike path wind around that tube." -- Ah, of course! (+1. That's explained so well, I put off looking at MSE/q/502752 for now ;) I'd even give +3 for your kind use of that "seat-of-my-pants" terminology "chord curve". (Did I indeed guess the 'technical term'? Hawing-Ellis is expensive! ...) – user12262 Jul 23 '14 at 5:10
• ACuriousMind: "space-like chord curves are uninteresting [...] Thus, being a null geodesic indeed implies that, locally, there are no time-like chord curves." -- 1:Conversely, my question still is: Is there a null curve without any time-like chord curves (whether "local", or otherwise) not called a "null geodesic"? (Also, I wonder how the term "light-like curve" relates to either of these ...) 2:Could you please make explicit (and surely you've got the means to try) how to determine for a null curve with (at least) one time-like chord curve whether that's "local", or not? – user12262 Jul 23 '14 at 12:06
• @user12262: For $2.$, I am afraid I, at least, cannot. We only know that there are neighbourhoods for every point on $\mathcal{M}$ on which the geodesics are length-maximizing (that's what locally means), not how big they are (for a sphere (admittedly Riemannian), the maximal neighborhood would be the hemispheres, but I've no idea how to proceed for general manifolds). For, $1.$ ("light-like" and "null" are synonymous) but I am again unable (at least for now) to find either a rigorous argument showing that time-like chords imply no geodesics or construct a counterexample. – ACuriousMind Jul 23 '14 at 12:38
• ACuriousMind: "For 1."light-like" and "null" are synonymous" -- Really!? ... I looked up what I suppose(d) to be the source of all that "-likeness" terminology: H. Minkowski, "Raum und Zeit", (1909). There appears (in translation) "space-like", and "time-like", but, curiously, no mentioning of "light-like". Instead: of certain "cones" having to do with "sending light" or "receiving light"; i.e. surely what's called "light cone". So I'll try to be more careful ... – user12262 Jul 23 '14 at 18:57
• @user12262: The old masters are not always the place to go - their terminology is often quite different from what we use today, and often, their approaches are a weird mix between "intuition" and rigorous computation. In the field of "Lorentzian geometry" as we know it today, we call (for a metric with signature $(-+++)$), vectors $v$ with $g(v,v) < 0$ time-like, $g(v,v) = 0$ light-like (more physical) or null (more mathy) and $g(v,v) > 0$ space-like. The terminology extends then to curves if the tangent vector of the curves is in one of these three classes at every point of the curve. – ACuriousMind Jul 23 '14 at 18:57