Two dimensional Lorentzian vs Euclidean geometry

[pervect]

Reading a somewhat long and argumentative thread here inspired the following unrelated question in my mind:

Where does a 2 dimensional flat Lorentzian geometry depart from Euclidean geometry as axiomatized by Euclid? I.e. Euclid's axioms (in modern language) can be taken to be:

1. A straight line segment can be drawn joining any two points.

2. Any straight line segment can be extended indefinitely in a straight line.

3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

4. All right angles are congruent.

5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

We can interpret straight lines in Euclidean geometry as "geodesics" in the Lorentzian geometry. I believe the difficulties start to arise in axioms 3 and 4. A meaningful notion of a "circle" becomes difficult, perhaps we'd replace it with a hyperbola? The concept of rapidity between time-like geodesics is promissing, but doesn't seem to be a sufficiently general replacement for "angle".

 

[Orodruin]

I would say that all of the axioms inherently has an assumption on the existence of a metric. From 3 on, you start dealing with things which are not really the same as in Euclidean space if you replace the metric by a pseudo metric. As you say, circles become hyperbolas, I guess we can live with that. Number 4 should be downright false in Minkowski space - you can have several types of vector pairs with an inner product of zero. For example, you can take a time-like and a space-like vector or you can take two space-like vectors. These two sets cannot be transformed into each other by an isometry (i.e., Lorentz transformation).

 

[robphy]

It fails the first axiom.
...but satisfies the fifth axiom (in Playfair's formulation).
In fact, you can write the first in a form that resembles the projective dual of Playfair.

 

[Orodruin]

It fails the first axiom.

How? It is perfectly possible to have a geodesic connecting any two points in Minkowski space.

 

[robphy]

Have a look at page 18 of an old poster of mine:
http://www.aapt.org/doorway/Posters/SalgadoPoster/SalgadoPoster4.htm

 

[Orodruin]

Have a look at page 18 of an old poster of mine:
http://www.aapt.org/doorway/Posters/SalgadoPoster/SalgadoPoster4.htm

There is nothing saying that the line has to be time-like. Space-like geodesics are also geodesics.

 

[robphy]

There is nothing saying that the line has to be time-like. Space-like geodesics are also geodesics.

The idea comes from I.M. Yaglom's "A Simple Non-Euclidean Geometry and its Physical Basis".
There he distinguishes "lines of the first ["ordinary"], second, and third kind" (corresponding to timelike, spacelike, and lightlike).
It is interesting that the first and fifth axioms can be reformulated in a similar way (via the Playfair formulation) [related by duality].
So, when the first axiom is formulated as such (following Yaglom),
Minkowski fails that axiom since there are point-events that cannot be connected by timelike lines.

The whole scheme is part of what is called "Cayley-Klein Geometry" (what I call Spacetime Trigonometry).

 

[Orodruin]

So how do you treat the 4th axiom? By only allowing time-like geodesics, there will be no lines at right angles.

While it may be interesting to consider other options, I would consider the direct generalisation of straight lines to general (space-like, time-like, or null) geodesics to be the natural one.

Also to add to my first response, I did not consider the two-dimensional restriction. You cannot have space-like vectors which are orthogonal in two dimensions. However, you still have two different types of right angles, those between a time-like and a space-like geodesic and those between a null geodesic and itself. These form two distinct equivalence classes under Poincaré transformations.

 

[robphy]

So how do you treat the 4th axiom? By only allowing time-like geodesics, there will be no lines at right angles.

While it may be interesting to consider other options, I would consider the direct generalisation of straight lines to general (space-like, time-like, or null) geodesics to be the natural one.

Also to add to my first response, I did not consider the two-dimensional restriction. You cannot have space-like vectors which are orthogonal in two dimensions. However, you still have two different types of right angles, those between a time-like and a space-like geodesic and those between a null geodesic and itself. These form two distinct equivalence classes under Poincaré transformations.

I don't know how to handle the fourth axiom.
My goal in my poster (and in answering the OP)
was to identify a Euclidean axiom where Minkowski fails... thus, rendering it as non-euclidean.

The rest of the poster is exploring the Cayley-Klein geometry approach to those 9 two-dimensional geometries
and how it can be applied to physics... specifically, trying to formulate physical concepts in Galilean and Minkowskian relativity in a unified way.
Considering the rest of the euclidean axioms may be interesting... but that hasn't been my main focus.

In the Cayley-Klein approach, the emphasis is on projective geometry (not Euclidean geometry).

 

[robphy]

Possibly interesting reading on the "fourth postulate":
http://blogs.scientificamerican.com...the-deal-with-euclide28099s-fourth-postulate/
http://www.cut-the-knot.org/triangle/pythpar/Fifth.shtml
https://williewong.wordpress.com/2009/07/01/straightedge-and-compass-constructions/
... it's more about homogeneity, rather than about "right angles".

 

[pervect]

I hadn't contemplated breaking up lines into three types, but it does seem to make the concept of "angle" a lot more meaningful. Rapidity as the concept of angle between two timelike geodesics makes perfect sense, my concept of angle with null geodesics would be that they're at a right angle to everything (which is distinctly odd from a Euclidean perspective), and I I'm probably still confusing myself over what "angle" might be between space-like geodesics (especially if we restrict ourself to one spatial dimension), but I probably don't really need to figure that out at this point.

And the consequence of breaking up lines into the different types would be that the first axiom doesn't hold anymore.

Thanks for the references, I haven't read all of them at this point yet.

 

[PeterDonis]

my concept of angle with null geodesics would be that they're at a right angle to everything

Actually they're not. Null vectors are orthogonal to themselves, but not to null vectors in different directions. For example, in two dimensions, the null vector $(1, 1)$ is not orthogonal to the null vector $(1, -1)$. Nor are null vectors orthogonal to all non-null vectors; for example, $(1, 1)$ is not orthogonal to either the timelike vector $(1, 0)$ or the spacelike vector $(0, 1)$. (In three dimensions, null vectors in the t-x plane would be orthogonal to spacelike vectors in the y-z plane.)