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It has been stated here that:

we can say, without introducing a coordinate system, that the interval associated with two events is timelike, lightlike, or spacelike
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This assertion appears at variance with

Therefore I'd like to know:
How can be determined whether the interval associated with two events (which are given or characterized and distinguished by naming, for either event, the distinct individual participants which had been coincident at that event) is for instance "timelike", without using any coordinates and coordinate system?

Is it correct that the interval associated with two events (given as described above) is "timelike" if and only if there exists at least one participant who took part in both of these events?

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How about this for a more "physical" definition: two points in space-time are time-like separated if and only if a massive particle starting at one could, if subjected to appropriate finite forces, reach the other.

Replace "massive" with "massless" to get the definition of light-like separation. If neither is possible the the points are space-like separated.

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  • $\begingroup$ Nathaniel: Yes -- that's certainly the kind answer I had expected as motivation/justification for the quoted statement. My/The conclusion is: representing the notions "timelike", "lightlike", "spacelike" on the basis of coordinates/manifolds, such as in Wikipedia is too narrow. Instead, these notions are definable (or understood outright) without reference to coordinates/manifolds. "massive"; "massless" -- I wouldn't presume such notions outright, but define them as sketched in physics.stackexchange.com/a/37714 $\endgroup$ – user12262 Feb 28 '14 at 17:49
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Let $\lambda, \mu, \nu$ be functions on the reals to points (events) in spacetime. Let these be "straight" curves, in the sense that $\lambda', \mu', \nu'$ each all have the same direction for all values of their parameters. For example, $\lambda(u) = \lambda_0 + lu$ is a simple case, as $\lambda'(u) = l$. The vector $l$ is the vector along the direction of the straight curve.

Given these three straight curves, we can define events $A$, $B$, $C$ that are the intersections of $\lambda, \mu$; $\lambda, \nu$; and $\mu, \nu$. $A$ is point defined by the parameters $u, v$ such that $\lambda(u) = \mu(v)$. If some material objects are following these worldlines, then the objects meet at these intersections, at these events.

The rest of the process of determining whether the interval between, for example, $A$ and $B$ is timelike, spacelike, or lightlike is then similar to Julio Parra's answer: say we want to determine whether the interval between $A$ and $B$ is spacelike, timelike, or lightlike. The common worldline between these two events is $\mu$. We would then integrate the derivative to find the length of the spacetime interval between these two events along the worldline $\mu$:

$$s^2 = \int_{v_A}^{v_B} \eta[\mu'(v), \mu'(v)] \, dv$$

And the sign of $s^2$ then determines whether the interval is spacelike, timelike, or null (it depends on the metric).

I think the key point is the identification of some connecting straight worldline between two events--some "observer" or "participant" who takes part in both events of interest. EDIT: no other connecting worldline will give the same answer, but the worldline that extremizes the answer is considered canonical for defining this quantity. In the flat spacetime case, that's the straight path.

In general, we can talk about whether vectors are spacelike, timelike, or lightlike without considering two particular events. These are properties of directions in spacetime, not necessarily pairs of events.

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  • $\begingroup$ Muphrid: "Let $\lambda, \mu, \nu$ be functions on the reals to points (events)" -- Oh no!, no coordinates, please! But let's go on: "in spacetime. [...] The vector $l$ is [...]" -- This seems to presume that the set of events under consideration ("spacetime") were given as a en.wikipedia.org/wiki/Vector_space or at least with (all) intervals already known even quantitatively. That's hardly "events being given (only) by naming participants", as required in my question. (contd.) $\endgroup$ – user12262 Feb 28 '14 at 18:04
  • $\begingroup$ Muphrid: "I think the key point is the identification of some connecting straight worldline between two events--some "observer" or "participant" who takes part in both events of interest." -- There you go! (+1 on this item only.) Btw.: Why did you require "straight worldlines", instead only (more generally) "smooth" ones, or (even) "equal sign for all intervals between pairs of elements of the connecting arc"? "[...] properties of directions in spacetime, not necessarily pairs of events." -- In my question I took this detour via en.wikipedia.org/wiki/Causal_structure#Curves $\endgroup$ – user12262 Feb 28 '14 at 18:13
  • $\begingroup$ @user12262 Regarding the straight curves $\lambda, \mu, \nu$: I have not chosen any coordinate system at that time. Yes, there is a presumption that we're in a Minkowskian vector space. That is the foundation of special relativity and allows us to talk about points and such without reference to any particular coordinate system. Are you particularly looking for an answer that make no reference to Minkowski vector space? $\endgroup$ – Muphrid Feb 28 '14 at 18:20
  • $\begingroup$ @user12262 Regarding your second point, I used straight curves here for simplicity because then $\mu'$ is constant and $s^2 = \eta(\mu', \mu') (\Delta v)^2$. But as I said, "Any other connecting worldline should give the same answer; this is merely the simplest approach." $\endgroup$ – Muphrid Feb 28 '14 at 18:22
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    $\begingroup$ @user12262: It sounds like you want someone to say, for example, a massive observer could follow a worldline connecting two timelike events, while a photon could follow a worldline connecting two lightlike events, and if neither is the case, then the interval between those events is spacelike. That's all well and good, but then you're just exchanging stuff you don't understand in math for stuff you don't understand about physics: e.g. how do we know which worldlines massive observers can follow? $\endgroup$ – Muphrid Feb 28 '14 at 19:34
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One has not necessarily to know all this quantitative interval stuff to distinguish time-likeness. Suppose an observer lives on a body without (significant) atmosphere, such as our Moon. All events observed by means of light (EM radiation) belong to observer’s light cone and hence are separated by a null interval from the observer. All events observed indirectly (either by reflected or scattered light, or by their material traces) belong to observer’s past cone and hence are separated by a time-like interval from the observer. Events separated by a space-like will not be seen in any way, as well as the entire future cone (with both null and time-like intervals). But the observer can influence events of the future cone, whereas events with space-like intervals are inaccessible in any direction, from observer’s present moment.

Suppose, we pinged our lunar resident-observer (event P), he replied (event Q), and we received the reply (event R). Intervals P–Q and Q–R are null. Interval from anything on Earth before P and Q is time-like, with Q at the future end. Interval from Q and anything on Earth after R will also be time-like, with Q at the past end. Intervals from Q and our history between P and R are space-like: neither can our resident-observer at Q know anything about P⋯R (P excluded), nor can it influence anything in P⋯R (R excluded).

Ī answered the question with respect to the physical spacetime. If the question is about mathematical formalism to deal with spacetime without coordinates, then examples from the previous paragraph can be interpreted as a partial order relation.

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  • $\begingroup$ Incnis Mrsi: "[...] reflected or scattered light, or by their material traces" -- So, applying this terminology to what I asked specificly at the end of my question text: Do you agree that two events ($\mathcal{P, Q}$) are related by a timelike interval 1: if there exists at least one "material trace" which took part in both of these events; Or 2: if there exists at least one "material trace" who to part in distinct events $\mathcal{A, B, Q}$ (in that order) where $\mathcal{A, P}$ as well as $\mathcal{B, P}$ are lightlike; [... to be continued] $\endgroup$ – user12262 Oct 28 '14 at 21:27
  • $\begingroup$ [... continued] Or 3: if there exists at least one "material trace" who to part in distinct events $\mathcal{A, B, P}$ (in that order) where $\mathcal{A, Q}$ as well as $\mathcal{B, Q}$ are lightlike? $\endgroup$ – user12262 Oct 28 '14 at 21:27
  • $\begingroup$ Strictly speaking, two endpoints of a time-like interval are not necessarily causally related in fact, but with respect to @user12262’s points: 1 – if we speak about massive bodies, then yes; 2 – light-likeness of A–P and B–P (future endpoints are P) doesn’t imply anything about P–Q; 3 – certainly no if future endpoints are Q. By the way, situations 2 and 3 are not possible in Special Relativity. $\endgroup$ – Incnis Mrsi Oct 29 '14 at 12:02
  • $\begingroup$ Incnis Mrsi:"Strictly speaking, two endpoints of a time-like interval are not necessarily causally related" -- Alright; causal relations, as considered in geometry are not quite the same as causation, e.g. due to some "material trace propagating". "[...] situations 2 and 3 are not possible in Special Relativity." -- But they certainly are (read the descriptions carefully again ...) $\endgroup$ – user12262 Nov 3 '14 at 21:04
  • $\begingroup$ @user12262: in Minkowski’s world, if interval A–B is time-like, then:     • future light cones of A and B never intersect;     • past light cones of A and B do not intersect.  It is, generally, not true in a curved spacetime. $\endgroup$ – Incnis Mrsi Nov 6 '14 at 7:11
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For example:

enter image description here

From "Gravitation and Spacetime" via Google Books


Added to address a comment below:

My question was intended to be addressed without referring to any coordinates; including no coordinates (!) such as "clock times T, T1, T2" or somesuch.

But these clock times aren't coordinates. The reading of a single clock is the proper time along the worldine of the clock, i.e., it is an invariant and coordinate independent measure of time. From the Wikipedia article "Proper Time":

In relativity, proper time is the elapsed time between two events as measured by a clock that passes through both events.

Moreover, in the 2nd paragraph of the text above we have:

The radar-ranging procedure permits us to bypass the coordinate system, so we can measure the spacetime interval without the use of coordinates.

Consider a clock at rest with respect to you. The 3 orthogonal spacelike directions are evident as is the (Lorentzian) orthogonal timelike direction. The ticks of the clock gives a proper time parameter along the timelike worldline of the clock.

We can define the coordinate time, in the frame in which the clock is at rest, as the proper time given by the clock. And, we can define space coordinates as described above.

In your comment, you put the cart before the horse. The coordinates were defined by the direct measurement of the interval, not the other way around.

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  • $\begingroup$ My question was intended to be addressed without referring to any coordinates; including no coordinates (!) such as "clock times $T$, $T_1$, $T_2$" or somesuch. (But of course my question referred to the quoted statement by you, Alfred Centauri. So -- did you not mean your statement as strictly as I meant my question? ...) However, reading in your answer: "[...] it emits a light signal [...] and it receives the echo", apparently concerning some particular participant (one-and-the-same "it") -- does this not suffice to determine "timelike"-ness, without mentioning any coordinates? $\endgroup$ – user12262 Feb 28 '14 at 5:11
  • $\begingroup$ @user12262, it is either clear or it isn't. You've been, I believe, deliberately obtuse in the past and I believe that is the case now. I have no further interest in this as I don't see any possible profit in it. $\endgroup$ – Alfred Centauri Feb 28 '14 at 12:19
  • $\begingroup$ For the benefit of other readers, I have updated my answer to address the OP's comment above. $\endgroup$ – Alfred Centauri Feb 28 '14 at 13:44
  • $\begingroup$ Alfred Centauri -- You should learn: (1) That a clock is not necessarily a good clock (cmp. MTW Fig. 1.9). (2) That, and How it is measured whether for some real-valued (or integer-valued) coordinate parametrization $T$ of given indications of a participant (e.g. in terms of "number of ticks") the duration (a.k.a. "prope time") of that participant from any one "tick" to "the next tick" was constant, or not (cmp. MTW Box 16.4). (3) How to determine “timelike”-ness without using a coordinate system (from answers of other users given here). Do you concede any of these points? $\endgroup$ – user12262 Feb 28 '14 at 18:46

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