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I often hear that Minkowski spacetime is non-euclidean. Euclidean geometry is characterized by Euclid's five postulates being true. Which of those postulates are untrue in Minkowski spacetime (if any), and what physical consequences do we observe from them?

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    $\begingroup$ The question is quite meaningless until you specify how you're going to interpret words like "point", "line" and "angle" in Minkowski spacetime. Is the postulate "every grib contains a grob" true in Minkowski spacetime? Yes or no, depending entirely on how you interpret the words "grib" and "grob". $\endgroup$ – WillO Oct 25 '18 at 22:16
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    $\begingroup$ @WillO - "The question is quite meaningless ..." - and yet Jerry Schirmer was able to provide an answer. Try to be nicer with your comments. $\endgroup$ – Peter4075 Oct 26 '18 at 11:33
  • $\begingroup$ @peter4075: jerry was able to provide an answer contingent on a particular interpretation for the word "angle". With a different (and equally reasonable) choice the answer would be different. $\endgroup$ – WillO Oct 26 '18 at 12:36
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    $\begingroup$ @WillO: but that's what a postulate is, a choice for what you mean by terms. $\endgroup$ – Jerry Schirmer Oct 26 '18 at 13:37
  • $\begingroup$ interesting. isn't minkowski spacetime already endowed with a metric (minkowski metric) and then the notion of distance, angle, etc ... naturally follow ? i'm honestly asking, not contradicting what either of you have said. $\endgroup$ – marjimbel Oct 26 '18 at 13:46
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The Pythagorean distance formula doesn't hold for arbitrary shapes, thanks to the negative sign in the metric. It's also pretty easy to say that boosts obey hyperbolic angle addition rules rather than circular ones. Since the postulate about the congruency of right angles is needed to prove the Pythagorean distance relation, and angle addition rules for timelike intervals are different than those for spacelike intervals, one would conclude that the "all right angles are congruent" postulate doesn't hold-- the "right angle" between two null directions is different than that between two spacelike directions.

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  • $\begingroup$ interesting. do you have a way to show intuitively that angle addition rules are different for timelike vs spacelike intervals, or a reference ? also, i guess minkowski space itself isn't hyperbolic. the "space of boosts" is hyperbolic in the sense that an event gets mapped to another event within a hyperboloid. if minkowski was hyprbolic, then it would have negative curvature, no? it is in fact flat, in the sense of zero riemman curvature tensor. $\endgroup$ – marjimbel Oct 26 '18 at 11:46
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    $\begingroup$ @marjimbel ... concerning the addition angle rules, look at the Yaglom reference in my answer. I think my answer is a first step in that direction. Eventually, one would use the cross-ratio to define the angle in that geometry. $\endgroup$ – robphy Oct 26 '18 at 12:43
  • $\begingroup$ cool. i will. just to summarize so far. we saying that both postulates 1 (existence of line segments) and 4 (congruent right angles) fail in Minkowski, but not the parallel postulate ? $\endgroup$ – marjimbel Oct 26 '18 at 13:50
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    $\begingroup$ Is the fourth postulate fundamentally about right-angles? Or is it really about something else (homogeneity and isotropy) which we are here characterizing with right-angles? $\endgroup$ – robphy Oct 26 '18 at 17:16
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    $\begingroup$ @robphy: Euclid phrases it in terms of right angles, but I'm sure it's equivalent to timelike into spacelike anisotropy here. $\endgroup$ – Jerry Schirmer Oct 26 '18 at 22:27
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Euclidean geometry is characterized by Euclid's five postulates being true

This is true, but what characterizes non-Euclidean geometry are deviations from the parallel postulate, specifically.

Minkowski space with the flat Minkowski metric has submanifolds that have non-Euclidean geometry (hyperbolic geometry). This means that Euclid's parallel postulate is violated: basically, if the parallel postulate does hold for a given geometry, then the sum of the angles of a triangle is $\pi$ radians since the separation between parallel lines is constant. The geometry is non-Euclidean when the sum of the angles of a triangle is greater than (spherical) or less than (hyperbolic) $\pi$ radians, since the separation between parallel lines increases or decreases, respectively.

EDIT: After thoughtful feedback in the comments, I tried to make it clear that the fifth postulate is violated on the submanifolds of a Lorentzian manifold, whereas the entire Minkowski spacetime is indeed flat (and affine meaning Euclid/Playfair parallelism holds on the Lorentz manifold).

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    $\begingroup$ the hyperbolic properties refer to the geometry of the "space of boosts", no ? if minkowski spacetime itself had a hyperbolic geometry then it would have negative curvature, which it doesn't since it is a "flat space" in the sense of zero riemman curvature tensor. what am i missing ? $\endgroup$ – marjimbel Oct 26 '18 at 3:58
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    $\begingroup$ So I think we agree that the kinematic space of a minkowski spacetime is hyperbolic. The thing is, "flatness" is a property of the metric chosen, and we use the flat minkowski metric in my answer. Apologies for not specifying that before. $\endgroup$ – N. Steinle Oct 26 '18 at 14:36
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    $\begingroup$ More to your question: Minkowski space is not Reimannian, but it has submanifolds that are which have hyperbolic geometry (negative curvature), while the whole Minkowski manifold is flat, in the sense you mean, when endowed with the flat Minkowski metric (also called the Lorentz metric). $\endgroup$ – N. Steinle Oct 26 '18 at 14:47
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    $\begingroup$ The hyperbolic aspect of Minkowski space involves the way angles are measured, using the arc of a unit hyperbola. In Euclidean geometry, angles are measured using the arc of a unit circle. In both cases, no aspect of the Parallel Postulate is involved here. Both (and also the Galilean spacetime) are so-called "affine geometries" where the Parallel Postulate holds. This is also discussed in the Yaglom reference in my answer. $\endgroup$ – robphy Oct 26 '18 at 16:25
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    $\begingroup$ @marjimbel indeed, I mean that the fifth postulate is violated within that submanifold, which determines the negative curvature. All three postulates are violated in Minkowski space: I totally agree that the violation of the first and third postulates result in the unique causal structure of 4D Minkowski spacetime, but I maintain that it is the violation of Euclid's parallel postulate that causes the kinematics to be hyperbolic (by definition of course). $\endgroup$ – N. Steinle Oct 26 '18 at 18:19
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Minkowski spacetime violates Euclid's First Postulate, when expressed in a form like the Playfair postulate. Physically, this violation tells us something about the Causal Structure in spacetime.

The first postulate can be expressed like the "dual" version of the fifth postulate:
"Given a point, and a line not through that point, there exists no point on that line which cannot be joined (by an “ordinary” line) to the given point."

So, for special relativity, I write it like:
"Given an event, and a worldline not experiencing that event, there exists infinitely many events on that worldline which are not “timelike-related” to the given event."

For Galilean relativity,
"Given an event, and a worldline not experiencing that event, there exists one event on that worldline which is not “timelike-related” to the given event."

(I think the Parallel Postulate (suitably expressed, e.g. Playfair) is okay for Galilean and Minkowski spacetimes. I think it's what allows us to draw parallel lines... like the ends of a meterstick on a spacetime diagram.)

update:
This viewpoint is what I physically interpreted from
I.M. Yaglom's "A simple non-Euclidean geometry and its physical basis : an elementary account of Galilean geometry and the Galilean principle of relativity"
https://archive.org/details/ASimpleNon-euclideanGeometryAndItsPhysicalBasis/page/n237 .
See Figures 188a-c on page 220, which I inserted below. robphy-interprets-FirstPostulate-Yaglom-p220

(From Yaglom, p220)
In accordance with the requirement of congruence of any two lines in the plane, it is natural to define the "Minkowskian plane" as the points of the (ordinary) plane and the lines of one kind—say, the first. In this connection we note that in Euclidean geometry every point on a line a can be joined by a line to a point A not on a (Fig. 188a); in Galilean geometry, each line a contains a unique point, "parallel" to A, which cannot be joined by an (ordinary) line to A (Fig. 188b); in Minkowskian geometry, there are infinitely many points on a line a of the first kind which cannot be joined to A by means of lines of the first kind (Fig. 188c). On the other hand, all three geometries share the property that through any point A not on a there passes a unique line not intersecting a (a unique line parallel to a).

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  • $\begingroup$ this is very interesting and i had never heard of an example of a geometry that deviates from euclid's in the first postulate instead of the parallel postulate. i think the parallel postulate being violated leads to "curved" spaces, such as hyperbolic. however, minkowski is "flat". i struggle to reconcile the fact that Minkowski is flat with the fact that it is a non-euclidean geometry, and that is what lies behind my initial question $\endgroup$ – marjimbel Oct 26 '18 at 4:03
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    $\begingroup$ I agree with your characterization. Minkowski spacetime is a flat (zero-curvature) space. It is a Cayley-Klein geometry. I will update my answer with a reference. $\endgroup$ – robphy Oct 26 '18 at 9:22
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    $\begingroup$ You're implicitly defining "lines" to be timelike geodesics. This is an interesting choice, and it might be justifiable, but in some ways it would be more natural to consider all geodesics (timelike, null, or spacelike) as the "lines" in spacetime. $\endgroup$ – Michael Seifert Oct 26 '18 at 16:39
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    $\begingroup$ Note that if you adopt @MichaelSeifert's choice, the axiom that gets violated is one Euclid considered so obvious that he didn't even state it explicitly, namely Hilbert's first axiom of congruence (roughly, if $\overline{AB}$ is a line segment, and $\gamma$ a ray with vertex $C$, there is some $D$ on $\gamma$ such that $\overline{AB}$ is congruent to $\overline{CD}$). $\endgroup$ – Micah Oct 26 '18 at 16:45
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    $\begingroup$ The discussion here illustrates the point I'm trying to make in my comments on the main question. Whether Euclid's first postulate is true in Minkowski spacetime depends entirely on what meanings you choose to assign to the words "line" and "point". This answer makes one possible assignment of meanings. There are alternatives, as pointed out by @MichaelSeifert . Until you specify the interpretation, the question by itself is meaningless. $\endgroup$ – WillO Oct 26 '18 at 20:31
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This is from Schutz's A First Course in General Relativity (second edition, page 117):enter image description here

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