What, exactly, are invariants?

[vanhees71]

In grad school, learning more tensorial methods and the abstract-index notation from Wald's text helped.
But it wasn't until I learned about operational definitions of distance and time measurements via radar-methods on a spacetime diagram from Geroch, did things finally click for me. For me, I could now tie together the verbiage of introductory texts, the notations of vectors and tensors in coordinate form and in abstract-index form, and physics connected to observation using light signals and clocks (which is more relativistic in spirit than rods and clocks).

That's exactly my point, but remember the very heated discussion we had in these forums, when I dared to make the point that a reference frame is not simply an abstract "coordinate patch" in a pseudo-Riemannian manifold but something made of real things in the lab ;-)).

On my own in grad school, I stumbled upon Yaglom's "A simple non-Euclidean geometry and its physical basis", which introduced me to Klein and the Cayley-Klein geometries. I also tumbled upon Schouten's "Ricci Calculus" and "Tensor Analysis for Physicists", which introduced me to visualizing tensors.

Ideally, one really should try to be fluent in (and fluid in, in the sense of "being able to inter-connect") all of these methods.

My $0.03.

FACK.

 

[Freixas]

(on the diagrams anyway--I realize the proper lengths are invariant)

I don't seem to be able to edit my post. This parenthetical comment is incorrect. Proper lengths are invariant, but I don't think there is such as thing as "the proper length of a worldline". From the point of view of the observer whose worldline we draw, they are always at position 0 and the length of their journey is also 0. Sorry for the error.

 

[PeterDonis]

Proper lengths are invariant, but I don't think there is such as thing as "the proper length of a worldline".

The corresponding invariant along a worldline (i.e., a timelike curve) is proper time.

 

[PeterDonis]

Here are two Minkowski spacetime diagrams. Both show the same worldlines relative to different frames, but the lengths (on the diagrams anyway--I realize the proper lengths are invariant) and the angles are not obviously invariant.

You're thinking of it backwards.

Take an ordinary object, say a coin, and look at it from different angles. Its appearance will be different, but its physical properties will not change. The physical properties are invariants.

Your two spacetime diagrams are the analogue of looking at an ordinary object from two different angles. The appearances are different, but the physical properties do not change. The physical length along a given curve does not change even though its apparent length changes--just as the coin's physical size and shape does not change even though its apparent size and shape does. Similar remarks apply to angles.

 

[Freixas]

Your two spacetime diagrams are the analogue of looking at an ordinary object from two different angles. The appearances are different, but the physical properties do not change. The physical length along a given curve does not change even though its apparent length changes--just as the coin's physical size and shape does not change even though its apparent size and shape does. Similar remarks apply to angles.

My diagrams were in response to this exchange:

If I look at a spacetime diagram, I still have to be careful about what is an invariant and what is not.

Lengths along of curves and angles between curves where they intersect. In other words, geometric invariants.

If I look at a spacetime diagram, all I have is a projection with respect to a particular frame of reference. My point (and yours, apparently) is that looking at this projection doesn't give me any insights into what is and isn't an invariant.

The only way I know of to look at objects in spacetime geometrically is through one or more projections. Is there an alternate geometrical representation for spacetime curves in which the lengths and angles are invariant?

 

[PeterDonis]

My point (and yours, apparently) is that looking at this projection doesn't give me any insights into what is and isn't an invariant.

No, that isn't my point. My point is the opposite: you already know that the lengths along worldlines and the angles between them are invariants. You don't learn that by looking at the diagram. You learn that by understanding what relativity says as a physical theory and understanding what actual measurements you make to know lengths along worldlines and angles between them. You don't make those measurements just by looking at a diagram, any more than you measure the size and shape of a coin by just looking at it from some angle.

Is there an alternate geometrical representation for spacetime curves in which the lengths and angles are invariant?

You are chasing a phantom here. Basically, what you're looking for is a diagram of 4-dimensional locally Lorentzian geometry that represents all lengths and angles exactly. This is a fool's errand. You can't even have such a diagram for the surface of the Earth, which is a 2-dimensional locally Euclidean geometry and so is much closer to what we can visualize than 4-d spacetime is.

What you need to understand is that you can do physics without such an exact diagram. And then learn how.

 

[robphy]

Draw a circle with two radii and label the smaller angle between the radii,
then draw the tangent through the tip of one of the radii to form a triangle [after suitable extending the segments].
Can you identify invariants and non-invariants in this figure?

You could check by rotating the page and
seeing if you get the same results by using the same procedures.

You could also use suitable geometrical tools to measure various quantities.

I think that once this is understood, then you can ask the same kinds of questions for special relativity.
(You might even translate your "frame of reference" and projection notions into their
Euclidean analogues... to see re-interpret what you are asking yourself.)

 

[Freixas]

No, that isn't my point. My point is the opposite: you already know that the lengths along worldlines and the angles between them are invariants.

Sorry, there are a lot of posts here. I think @robphy was the one suggesting that I could use geometry; he clarifies his approach in #42.

This is a fool's errand.

Agreed. I didn't think this was possible, but mathematicians have a lot of tricks and I'm hardly the one to provide a theorem showing this is impossible.

 

[robphy]

The only way I know of to look at objects in spacetime geometrically is through one or more projections. Is there an alternate geometrical representation for spacetime curves in which the lengths and angles are invariant?

My light-clock diamonds approach is an attempt to represent the proper-time along piecewise inertial worldlines... and do so with a physical mechanism in the spirit of relativity.
Rapidity ("spacetime-angles" between timelike tangents) can be seen as
areas of sectors in unit-hyperbolas ("circles").

These are taking advantage of the fact that
the equality of areas on a plane in euclidean geometry
also holds in Minkowski geometry---this is an affine notion, not a metrical one.

This tries to follow the spirit of various "proof without words" demonstrations.

 

[Freixas]

Draw a circle with two radii and label the smaller angle between the radii,
then draw the tangent through the tip of one of the radii to form a triangle [after suitable extending the segments].
Can you identify invariants and non-invariants in this figure?

I have a problem with the question. The most basic definition of an invariant is "a function, quantity, or property which remains unchanged when a specified transformation is applied." Things might be invariant with respect to one transformation and not invariant with respect to another.

In S.R., I believe invariants are invariant with respect to changing the frame of reference. If my math is right (always suspect), S.R. invariants aren't invariant with respect to changing the simultaneity convention.

With the problem you gave, it's not clear what transformation(s) you are picturing. You mentioned rotation; the answer to your questions might be different if I applied a shearing transform. Given specific radii, the angles of the triangle are invariant with respect to rotation, translation, and scaling, but not with respect to shearing.

 

[robphy]

I have a problem with the question. The most basic definition of an invariant is "a function, quantity, or property which remains unchanged when a specified transformation is applied." Things might be invariant with respect to one transformation and not invariant with respect to another.

In S.R., I believe invariants are invariant with respect to changing the frame of reference. If my math is right (always suspect), S.R. invariants aren't invariant with respect to changing the simultaneity convention.

With the problem you gave, it's not clear what transformation(s) you are picturing. You mentioned rotation; the answer to your questions might be different if I applied a shearing transform. Given specific radii, the angles of the triangle are invariant with respect to rotation, translation, and scaling, but not with respect to shearing.

I'm thinking about a high-school geometry problem.
We draw the figure and we study it,
without laying down a grid, or rotating it into some standard position, or
sequentially rotating it into standard positions to analyze certain segments.
We use Euclidean geometry, the Pythagorean theorem, trigonometric definitions, etc...
(Maybe we have gotten used to rotating the diagram to see that its orientation doesn't matter.)

For such a problem, we won't be shearing.
(For Galilean spacetime diagrams, I might be shearing... but that's a "rotation", a Galilean transformation.
One can learn to read such diagrams like a diagram for Euclidean geometry,
without having to transform to a particular reference frame to analyze.)

---

Concerning change the simultaneity convention... i don't know.
I haven't really thought about it.
My focus hasn't ventured into variants of special relativity,
but rather on how Minkowski space is one of a family Cayley-Klein geometries.

Presumably, the choice of simultaneity convention will imply that
either the hyperbola as "circle" is not representative of the metric and/or
the tangent line to that hyperbola is not used to define orthogonality,
which mimics the scheme that applies Euclidean geometry and Galilean spacetime geometry.

I would consider looking into how to formulate such simultaneity conventions
in terms of invariant structures in special relativity. What gets perturbed?

 

[Dale]

the angles are not obviously invariant. I'm not sure how looking at a diagram clarifies what is and isn't an invariant.

This is where it is worth writing things in a manifestly covariant formulation. The length is invariant because it can be written as $$\int_P \sqrt{g_{\mu\nu} dx^\mu dx^\nu}$$ and an angle is invariant because it can be written $$\cos(\theta)=\frac{g_{\mu\nu} x^\mu y^\nu}{\sqrt{g_{\mu\nu} x^\mu x^\nu} \sqrt{g_{\mu\nu} y^\mu y^\nu}}$$

Once you write a quantity in a manifestly covariant formulation then you immediately know that it is invariant under any coordinate transformation.

 

[robphy]

Of course, @Dale 's calculation shows the invariance, when you are given $x^\mu$ and $y^\mu$.

If however, the $y^\mu$ is tied to an arbitrary choice of axis.
Then the angle is a scalar tied to choice of axis.
This angle, however, is not an invariant, independent of the choice of axis.

In your diagram, I think you were referring to the angle made in the diagram of your various frames.
So, such an angle isn't independent of the choice of axis, as you said.

 

[vanhees71]

Note that, of course, angles make only sense in Euclidean metrics, not in Lorentzian pseudo-metrics.

 

[Dale]

Note that, of course, angles make only sense in Euclidean metrics, not in Lorentzian pseudo-metrics.

I disagree. Angles between spacelike vectors in a Lorentzian manifold are exactly the same as angles between vectors in Euclidean manifolds. Angles between timelike vectors in a Lorentzian manifold are related to relative velocities and there is no reason that they should not be considered to make sense. Null vectors clearly don't work, and I am not sure about combinations of timelike and spacelike vectors, but to broadly say that angles don't make sense in Lorentzian manifolds goes too far, IMO.

 

[DrGreg]

I agree with Dale. If anyone doesn't like using the word "angle" between timelike vectors, you can call it "rapidity" instead, but it's essentially the same concept geometrically.

 

[robphy]

I disagree. Angles between spacelike vectors in a Lorentzian manifold are exactly the same as angles between vectors in Euclidean manifolds.

One needs to be careful here.
When a set of a spacelike vectors is orthogonal to a given timelike vector (so these spacelike vectors are parallel to a hyperplane of simultaneity, and are thus "spatial vectors" to that timelike observer),
then "Angles between these spacelike vectors in a Lorentzian manifold are exactly the same as angles between vectors in Euclidean manifolds."

However, on a (1+1)-spacetime diagram.
The t- and t'-axes are timelike vectors, and x- and x'- axes are spacelike vectors.
Note: the "angle" between the x- and x'-axes is not a Euclidean angle...
it's numerically equal to the rapidity between the timelike t- and t'-axes.

 

[DrGreg]

One needs to be careful here.
When a set of a spacelike vectors is orthogonal to a given timelike vector (so these spacelike vectors are parallel to a hyperplane of simultaneity, and are thus "spatial vectors" to that timelike observer),
then "Angles between these spacelike vectors in a Lorentzian manifold are exactly the same as angles between vectors in Euclidean manifolds."

However, on a (1+1)-spacetime diagram.
The t- and t'-axes are timelike vectors, and x- and x'- axes are spacelike vectors.
Note: the "angle" between the x- and x'-axes is not a Euclidean angle...
it's numerically equal to the rapidity between the timelike t- and t'-axes.

This is something I hadn't considered before, but after thinking about this, I think we deal with this as follows:

Given two spacelike 4-vectors $\textbf{K}$ and $\textbf{L}$ evaluate$$
\lambda = \frac{ g( \textbf{K}, \textbf{L} ) } { \sqrt{| g( \textbf{K}, \textbf{K} ) |} \, \sqrt{| g( \textbf{L}, \textbf{L} ) |}} \, .
$$If $|\lambda| \leq 1$, there is an associated "Euclidean" angle $\cos^{-1} \lambda$.

If $|\lambda| \gt 1$, there is an associated rapidity $\cosh^{-1} |\lambda|$, in the sense indicated in the second half of robphy's post.

That seems to make sense, or have I overlooked something?

---

Footnote added Sun 18 Jul: The "Euclidean" angle could be $\cos^{-1} (-\lambda)$, depending on which sign convention you use for the metric.

 

[Dale]

One needs to be careful here.
When a set of a spacelike vectors is orthogonal to a given timelike vector (so these spacelike vectors are parallel to a hyperplane of simultaneity, and are thus "spatial vectors" to that timelike observer),
then "Angles between these spacelike vectors in a Lorentzian manifold are exactly the same as angles between vectors in Euclidean manifolds."

However, on a (1+1)-spacetime diagram.
The t- and t'-axes are timelike vectors, and x- and x'- axes are spacelike vectors.
Note: the "angle" between the x- and x'-axes is not a Euclidean angle...
it's numerically equal to the rapidity between the timelike t- and t'-axes.

Agreed, and I have no problem with appropriate caveats and highlighting things that make spacetime angles different. I just think that geometrically they are “angles” in the same sense that geometrically spacetime intervals are “lengths” and it is worth bringing in that sort of reasoning and intuition (with appropriate caveats etc.)

 

[robphy]

Agreed, and I have no problem with appropriate caveats and highlighting things that make spacetime angles different. I just think that geometrically they are “angles” in the same sense that geometrically spacetime intervals are “lengths” and it is worth bringing in that sort of reasoning and intuition (with appropriate caveats etc.)

There is definitely an intuition to be imported.
One just needs some care
and one has to allow a weakening strict Euclidean thinking to embrace these analogous variations.
(I guess one could pseudo- prefix a bunch of terms.)

From a Cayley-Klein geometry ( https://en.wikipedia.org/wiki/Cayley–Klein_metric ) viewpoint,
angle and distance are "dual" concepts...
one is a measure of separation between lines that meet at a point, and
the other is a measure of separation between points that are joined by a line.
When expressed as a cross-ratio ( https://en.wikipedia.org/wiki/Cross-ratio ),
their similarities are clearer.

(On a sphere, angle and distance are more similar.
This is so-called double-elliptic in the Cayley-Klein geometries.
Anti-deSitter space is doubly-hyperbolic.
Galilean is doubly-parabolic.
Hyperbolic space has a hyperbolic-measure for distances but an elliptic/circular measure for angles. )

At the same time, one should be aware of some special properties that happen to arise in (say) the Euclidean case, that don't hold for the more general case. So, one should focus on the general case to develop the subject in a unified way... then make note of special situations that may arise.

 

[Hornbein]

In defense of Albert, I find his paper best at convincing the reader that special relativity corresponds to reality. It's very down to Earth. Once reader is convinced of this they can move on to formalisms that are easier to work with.

 

[robphy]

In defense of Albert, I find his paper best at convincing the reader that special relativity corresponds to reality. It's very down to Earth. Once reader is convinced of this they can move on to formalisms that are easier to work with.

In your sentence, instead of "best", I would use words like: insightful, breakthrough, groundbreaking, revolutionary, pioneering, first .

If I were to introduce relativity to an absolute beginner with the goal to understand the essence of relativity,
I would not start with Einstein.
I would probably start with Bondi... Relativistic kinematics that is distilled and more to the point.

If I were to introduce relativity to a more mathematically-mature reader
with the goal to suggest that relativity forces us
to rethink kinematics, dynamics, and better-understand electrodynamics,
I might suggest Einstein (among others).I wouldn't say, to an absolute beginner,
that Einstein's paper (by itself) would convince the reader that special relativity corresponds to reality.

I would say that
other people said that Einstein's paper suggests that special relativity corresponds to reality
and that I would try to read Einstein's paper because of that,
and that I would likely seek other formalisms that are easier to work with
(for me) to better understand Einstein's insights and the subsequent results from Einstein's insights.I would probably say similar things about the groundbreaking works by Newton, Maxwell, Schrodinger, etc...
For these great works,
I would seek other formalisms that are easier to work with
(for me) to better understand their insights and the subsequent results from their insights.

 

[vanhees71]

I disagree. Angles between spacelike vectors in a Lorentzian manifold are exactly the same as angles between vectors in Euclidean manifolds. Angles between timelike vectors in a Lorentzian manifold are related to relative velocities and there is no reason that they should not be considered to make sense. Null vectors clearly don't work, and I am not sure about combinations of timelike and spacelike vectors, but to broadly say that angles don't make sense in Lorentzian manifolds goes too far, IMO.

I can understand what you mean by angles between space-like vectors, i.e., you can define them as in Euclidean space as
$$\cos \theta = -\frac{a \cdot b}{\sqrt{-a \cdot a} \sqrt{-b \cdot b}},$$
where $-$ signs are due to my west-coast choice of the signature (+---).

What do you mean by an "angle" between time-like vectors? May be rapidities as in coordinates for Bjorken flow?

 

[DrGreg]

I can understand what you mean by angles between space-like vectors, i.e., you can define them as in Euclidean space as
$$\cos \theta = -\frac{a \cdot b}{\sqrt{-a \cdot a} \sqrt{-b \cdot b}},$$
where $-$ signs are due to my west-coast choice of the signature (+---).

What do you mean by an "angle" between time-like vectors? May be rapidities as in coordinates for Bjorken flow?

See my post #53 (including my quote from robphy's post #52).

When, in my notation, $\textbf{K}$ and $\textbf{L}$ (your $a$ and $b$) are both timelike, then $|\lambda| \geq 1$ and so $\cosh^{-1} |\lambda|$ is the rapidity between the 4-velocities that are parallel to $\textbf{K}$ and $\textbf{L}$.

 

[vanhees71]

Ok, but that's then an extension of the usual definition of an angle in Euclidean geometry. A rapidity is not an angle!

 

[robphy]

Some old posts on "[pseudo-]angles" in special relativity:

• https://www.physicsforums.com/threa...-like-and-time-like-vectors.74115/post-554189
• https://www.physicsforums.com/threads/angles-in-minkowski-space.991133/post-6364136

I mentioned that the Gauss-Bonnet theorem might be the way to nail down definitions of
angles [with signs, if needed] between 4-vectors (not-necessarily of the same type).
Here are some papers that pursue this approach:

• A relativistic version of the Gauss-Bonnet formula
  Garry Helzer
  J. Differential Geom. 9(4): 507-512 (1974). DOI: 10.4310/jdg/1214432546
  https://projecteuclid.org/journals/...ss-Bonnet-formula/10.4310/jdg/1214432546.full
• The Gauss–Bonnet theorem for Cayley–Klein geometries of dimension two
  Alan S. McRae
  New York J. Math. 12 (2006) 143–155.
  https://nyjm.albany.edu/j/2006/Vol12.htm
  https://nyjm.albany.edu/j/2006/12-8p.pdf

 

[Dale]

Ok, but that's then an extension of the usual definition of an angle in Euclidean geometry. A rapidity is not an angle!

Rapidity is an “angle” in the same sense that the spacetime interval is a “length”. I have seen you post enough about relativity that I know you understand both this concept and also the typical misuse of terminology involved.

Since we want to encourage a geometric understanding of relativity it is unavoidable to use geometric words. The words are appropriate because they represent a reasonable Lorentzian generalization of the corresponding Euclidean concept. The fact that they are generalizations means that care must be taken and appropriate caveats/warnings should be given. It does not mean that the generalizations do not make sense as you claimed

 

[vanhees71]

Well, yes, I'm always trying to avoid this abuse of Euclidean language, which is very misleading.

 

[Dale]

Well, yes, I'm always trying to avoid this abuse of Euclidean language, which is very misleading.

There are recent cases where the misleading terminology has indeed led to long discussions with some novices here, where they seem to be unable to understand the ways that the generalization differs from the Euclidean concept. So I do see that risk too.

However, I also see the benefit in leveraging student’s existing geometrical knowledge. It is a risk-benefit trade off. Some students will be harmed by the risks and some will be helped by the benefits.

 

[robphy]

Well, yes, I'm always trying to avoid this abuse of Euclidean language, which is very misleading.

• When most people say “geometry”, the drawings done in the style of Euclid come to mind.
• But, some of us think of "geometry" in the sense of Felix Klein (as you say in #33, "providing a symmetry group"), possibly without a diagram in sight.

So, we have allowed the word "geometry" to be generalized beyond the more common Euclidean interpretation. (The alternative would be to use a prefix [Klein-, generalized- , pseudo- ] or a new word altogether.

Some of us think the notions of "length", "angle", "metric", "dot product", etc..
can also be generalized beyond the more common Euclidean/Riemannian interpretation.
Of course, a definition needs to be provided... and care must be taken.
But I think we do a pretty-good job in that
we don't need to always use a prefix [generalized- , pseudo- ] or a new word altogether
(except to make contact with already established terms: e.g. rapidity).

(Maybe not the best analogy but...
Maybe it's like some "re-boot" or "re-imagined" version of classic TV shows and movies.
One introduces a new viewpoint by relying on the classic version [likely, easier to sell the idea],
rather than introduce a new title altogether.)

From another point of view,
appropriately generalizing the meaning of words
is akin to a "unification" of previously disparate concepts.

• E.g. "gravity" in the sense of "mgh" near the Earth surface
  and "gravity" in the universal inverse-square law.
• I think hyperbolic geometry ( by Gauss, Lobachevsky and Bolyai)
  and Minkowski's spacetime geometry idea are examples... generalizing the geometry of Euclidean plane,
  all developed consistently and all as examples of Cayley-Klein geometries.

 

[Sagittarius A-Star]

Well, yes, I'm always trying to avoid this abuse of Euclidean language, which is very misleading.

I think it becomes is clear, if the expressions "circular angle" and "hyperbolic angle" are used. But they still contain both the word "angle".

In other words, this means just as how the circular angle can be defined as the arclength of an arc on the unit circle subtended by the same angle using the Euclidean defined metric, the hyperbolic angle is the arclength of the arc on the "unit" hyperbola subtended by the hyperbolic angle using the Minkowski defined metric.

Source:
https://en.wikipedia.org/wiki/Hyperbolic_angle#Relation_To_The_Minkowski_Line_Element

 

[Freixas]

When most people say “geometry”, the drawings done in the style of Euclid come to mind.

Novice here.

Yes, that's what I assume. At some point, it sunk in that when you guys said "angle", you were talking about relative velocity.

But, some of us think of "geometry" in the sense of Felix Klein (as you say in #33, "providing a symmetry group"), possibly without a diagram in sight.

When someone says "geometry", I think "diagram". When someone uses geometric terms in an advanced way, it leads to my confusion at the bottom of my post #32.

I hate to bring up my classification scheme, but relative velocity looks to me like a type III invariant: the velocity of any object with respect to a given frame of reference is a relative velocity. My approach may be clunky because I don't have the advanced geometrical knowledge of the people in this group, but it let's me limp along.

 

[Freixas]

By the way, the talk of relative velocity leads me to another question I've had. Are there any conditions in which we talk about the relative velocity of two observers when they are not colocated?

If we have two frames of reference (even comoving frames), I picture them as a sort of "field" in which every point of one frame has the same relative velocity with respect to the the colocated point of the other frame. More precisely, I might say that if I have two colocated objects, each at rest with one of the two frames, their relative velocity will be the same. But, given the same set of objects, can I say anything about their relative velocities if they are not colocated?

In novice problems, we are given examples where two object are both moving inertially. Their relative velocity is generally presented as an invariant independent of their location.

If you have a problem in which one observer is accelerating, there is even a formula that tells me the relative velocity of that observer to one in the initial rest frame of the accelerating observer. I give the formula $t$ (with respect to the rest observer) and it gives me $v$, the relative velocity of the two observers.

If we have two accelerating observers, we could pick the comoving frame of one and determine the relative velocity. We have a problem, of course; if we reverse whose comoving frame we use, we don't get the same lines of simultaneity, so the relative velocity could be different.

In all three cases, actually, the lines of simultaneity for the two objects are never the same unless they are at rest with respect to each other. But because at least one object is moving inertially, we get the same relative speed regardless of how we angle each object's line of simultaneity.

When I talk about the relative velocities of two non-colocated objects, am I always dealing with what @PeterDonis calls "an artifact of the choice of coordinates" or am I ever dealing with invariants? It seems to me that the answer should all be one way or the other.

 

[Dale]

Are there any conditions in which we talk about the relative velocity of two observers when they are not colocated?

As long as spacetime is flat, that is fine. If spacetime is curved then there is no unique way to compare the relative velocity of spatially separated worldlines.

 

[robphy]

I hate to bring up my classification scheme, but relative velocity looks to me like a type III invariant: the velocity of any object with respect to a given frame of reference is a relative velocity. My approach may be clunky because I don't have the advanced geometrical knowledge of the people in this group, but it let's me limp along.

As I suggested earlier,
it might be best to start with Euclidean geometry because the notions of invariance are already there and are likely more familiar.
Then, proceed to special relativity which adds additional likely non-intuitive notions to the situation.

Draw any diagram ( with circles, lines, points, angles,etc…) in the Euclidean style… lay down a grid (Cartesian style).

Describe features of that diagram with your classification scheme.
(A concrete labeled diagram might help.)
Once you do that, it will be easier to consider the special relativity version of the situation.

(This is a strength of the geometric analogy. Many times issues raised about special relativity are also featured in Euclidean geometry and Galilean relativity. However, it seems that special relativity is the one of three that has to defend itself.)