For professionals - differential geometry

[TheDestroyer]

Hello guys,

I keep hearing that Euclidean parallel postulate was broken through differential geometry, can someone please explain how that happens, and in what sense?

I do understand differential geometry notations and tensors, so explanation with them is OK.

Thank you :)

 

[Jim]

No math needed, just eyeballs.
Trace the initially parallel line segments of two adjacent lines of longitude
on the surface of a globe, until the meet at the N. pole. Its there, that they
cross, clearly violating Euclid's straight line postulate.
Why: the Earth is not flat.
QED

 

[TheDestroyer]

Is it that simple? I heard that it has to do with differential geometry! and some complicated stuff!

are you talking about geodesics?

 

[Monocles]

It wasn't "broken" - mathematicians just thought, ok, so what happens when we assume that the parallel postulate is not true? From that, you get a form of non-Euclidean geometry. Euclidean geometry (i.e. using the parallel postulate) is still 100% valid.

 

[TheDestroyer]

What non-Euclidean spaces are you pointing to? is it 4 dimensions, or polar cordinates (as exmaples)? what type I mean violates it?

Thank you

 

[DrGreg]

The 2D surface of the Earth does not have a Euclidean geometry. Of course, you can consider it as a surface within Euclidean 3D space, but if you ignore the 3rd dimension and consider it purely as a two-dimensional space, it's not Euclidean. You say you understand differential geometry notation, so you should know what I mean by

$$ds^2 = r^2 (d\theta^2 + \cos^2 \theta \, d\phi^2)$$​

where $\theta$ is latitude and $\phi$ is longitude. There is no change of variables that would convert that into

$$ds^2 = dx^2 + dy^2$$​

which makes it non-Euclidean. But the answer Jim gave you in post #2 is a perfectly good answer.

 

[Klockan3]

I heard that it has to do with differential geometry!

Differential geometry is the study of curved spaces, one such example is the surface of a globe.

 

[TheDestroyer]

So we say that the Euclidean parallel postulates breaks if and only if there is curvature? that's what you're pointing to DrGreg?

 

[VKint]

One way of quantifying exactly how much parallel geodesics "deviate" from one another is through the aptly-named equation of geodesic deviation, which arises by considering a one-parameter family of geodesics $$\gamma_s (\lambda)$$ with affine parameter $$\lambda$$ (such a family is called a congruence). The congruence forms a smooth 2-submanifold with coordinate functions $$x^{\mu} (s,\lambda)$$; we can then consider the tangent vectors $$\displaystyle T^{\mu} = \frac{\partial x^{\mu}}{\partial \lambda}$$ and the "deviation vectors" $$\displaystyle S^{\mu} = \frac{\partial x^{\mu}}{\partial s}$$. These deviation vectors can be visualized as pointing from one geodesic toward the others. We then have, after some standard identities are applied,
$$A^{\mu} \equiv \frac{D^2}{d\lambda^2} S^{\mu} = R \indices{^{\mu}_{\nu \rho \sigma}} T^{\nu} T^{\rho} S^{\sigma} \textrm{,}$$
where $$A^{\mu}$$ is the relative "acceleration" between neighboring geodesics, $$\displaystyle \frac{D^2}{d\lambda^2}$$ denotes a second covariant derivative in the direction of $$T^{\mu}$$, and $$R \indices{^{\mu}_{\nu \rho \sigma}}$$ is the Riemann curvature tensor. (In general relativity, $$A^{\mu}$$ is interpreted as representing gravitational tidal forces on observers moving along geodesics in the congruence.)

Another way of quantifying non-Euclidean-ness in arbitrary surfaces is through the Gauss-Bonnet theorem, which states that the angle $$\alpha$$ between a vector parallel-transported around a closed curve $$\gamma$$ and the tangent vector to the curve changes by the total amount
$$\displaystyle \Delta \alpha = \frac{1}{2} \int R dA \textrm{,}$$
where the integral is carried out over the region enclosed by $$\gamma$$, and $$R$$ is the Ricci scalar (or twice the Gauss curvature, in the case of a surface). There are higher-dimensional analogues of this, but I'm not familiar with them. Perhaps a more illuminating statement of a special case of this amazing result is that for a "triangle" with geodesic sides in an arbitrary surface, the sum of the interior angles of the "triangle" is $$\pi$$ plus the integral of the Gauss curvature over the region inside the triangle. Thus, on negatively curved surfaces, triangles' angles add to less than $$180^{\circ}$$, while on positively-curved surfaces, they add to more than $$180^{\circ}$$. (The theorem that the interior angles of a triangle add to $$180^{\circ}$$ in Euclidean space is, I believe, equivalent to the parallel postulate.)

 

[TheDestroyer]

Thank you pal :), really thank you. I've somehow gotten the first grip on the answer :), especially the angle part. This is what I wanted to hear :D.

 

[Jim]

VKint has beautifully mathematicized what I have intuitized, and he is
correct about the tidal forces from GR.
I did not appreciate that the Gauss-Bonnet theorem would allow one
to calculate the angle thru which a vector deviates after parallel transport
around a closed curve. Neat !

 

[wofsy]

Originally, the parallel postulate was questioned on purely logical grounds after two thousand years of failed attempts at a proof. People began to wonder whether the parallel postulate was in fact instrinsic to geometry as had been believed since the time of Euclid and questioned whether it could actually be derived from simple axioms about the nature of lines and planes. These questions arose in the 18'th century. Gauss and Lobachevsky were among these early thinkers and they both found plane geometries that obeyed the axioms of lines and planes but where the parallel postulate was false. In these geometries the usual straight line was replaced by what appeared to be curved lines to the outside observer.

People continued to wonder then what the idea of straightness really was - in a quasi-philosophical way of thinking - if it was not the Euclidean idea. Gauss came up with the idea of paths of least constraint, an idea that already existed in early Physics through the Calculus of Variations. I do not know if geodesics were first defined as paths of least constraint on a surface but it certainly was one of the earliest ideas - way before the modern idea of a Riemannian connection.

Gauss realized that a path of least constraint has the following property that it shares in common with a Euclidean straight line: a particle moving at constant speed along the line has no acceleration. If you think about it, in practical terms that is in fact how you would tell if a path is straight or not.

 

[HallsofIvy]

So we say that the Euclidean parallel postulates breaks if and only if there is curvature? that's what you're pointing to DrGreg?

Basically, yes. If a space has 0 Gaussian curvature at every point, then it is identical to Euclidean space. Essentially, Gaussian curvature measures the deviation from Euclidean space at each point.

 

[wofsy]

Basically, yes. If a space has 0 Gaussian curvature at every point, then it is identical to Euclidean space. Essentially, Gaussian curvature measures the deviation from Euclidean space at each point.

This is only true locally. There are plenty of spaces with zero Gaussian curvature which are not globally identical to Euclidean space. For instance the torus can be give a metric of zero Gaussian curvature.

 

[wofsy]

Hello guys,

I keep hearing that Euclidean parallel postulate was broken through differential geometry, can someone please explain how that happens, and in what sense?

I do understand differential geometry notations and tensors, so explanation with them is OK.

Thank you :)

I am responding a second time because I believe there is a misconception in this thread about how differential geometry replaced Euclidean geometry historically.

Originally,people thought that space is intrinsically Euclidean, that Euclidean geometry is embedded in the very idea of space itself. Geometry was not thought of in terms of calculus or curvature but in terms of axioms that people believed uniquely defined the structure of space.These axioms were simple and described the way lines separate planes and how lines intersect with each other. One axiom stated that a line separates a plane into two half planes. Another says that two lines can intersect in at most one point. Another says that two points determine a unique line.

People felt that the Parallel Postulate which says that in a plane, there is a unique parallel to a given line passing through any point, should not be a postulate but was actually a necessary intrinsic feature of space. This meant that it should be provable from the other axioms. People tried to find this proof for two thousand years and failed repeatedly. The thing that got them was uniqueness. They could not show that there was only one parallel through a point rather than many. They were able to show that there is at least one parallel and I will describe how they did this.

18'th century mathematicians knew that if two lines are both perpendicular to a third line then they must be parallel. Why? Because if they were not then by symmetry they would have to intersect in both half planes and thus would intersect in two points, a violation of the axioms for lines. So they knew that parallels always exist. What they could not prove was that these parallels were unique. This made them question what straightness really meant and led them to accept the possibility that uniqueness wasn't true.

Gauss finally discovered an example of a plane geometry in which all of the axioms of space were still true but in which the parallel postulate failed. In his geometry there were infinitely many parallels through a point. This ended the search for a proof and wiped out two thousand years of belief in the Euclidean universe.

Gauss was able to arrive at this new axiomatic geometry because people were already questioning what the idea of straight really was and this allowed him to find a model of the axioms where lines were actually curves. But he did not use differential geometry nor the idea of Gauss curvature in these thoughts. He just realized that the axioms could be modeled using curves in a plane as lines and still hold, except for the parallel postulate itself.

After Gauss's new geometry was discovered, people thought that there were two possible intrinsic geometries for space, Euclidean and Gaussian, and Gauss went out and measured large triangles on the Earth to try to find out which one of the two was true for the universe. If the universe was Euclidean, the sum of the angles of a triangle would be 180 degrees, if it was Gaussian the sum would be less than 180 degrees. This was an Earth shattering event in the history of thought. People suddenly saw that the geometry of space must be determined empirically, that there is no way of knowing it a priori!

Gauss did not use differential geometry or the notion of Gaussian curvature in these early thoughts. It was only later that he realized that the deviation from 180 degrees in a geodesic triangle could be determined from the Gauss curvature and that knowing this, his new geometry could be modeled as a surface of constant negative curvature.

After these key initial discoveries mathematicians still continued to wonder what the idea of straightness really was. How the idea of a geodesic was adopted I am not sure but the notion of a path of least constraint, as Gauss called it, was already known from the calculus of variations, and differential geometry was already a well developed subject largely through the researches of Monge and his school in France. I am sure that the connection of differential geometry to non-Euclidean geometries was not made until Gauss discovered that Gaussian curvature is an intrinsic property of a surface. The whole idea in the search for the true geometry of space was to find its intrinsic properties. Until Gauss curvature was known to be intrinsic I can not see how it would have helped research on non-Euclidean geometry.

By the way, the sphere is not a model for a non-Euclidean plane geometry and was not considered as a candidate in early research. Lines on a sphere intersect in two points or conversely two points sometimes determine more than one line. That violates the axioms of space.

One might try to pass to projective space by modding out all of the opposite poles and letting the lines be the projections of great circles onto the quotient. Then two lines will intersect in exactly one point and there will be no parallel lines. Sadly, another axiom is then violated, the axiom that says that a line separates a plane into two half planes. It is hopeless. This is why the sphere did not play a role in the fall of the Parallel Postulate.

 

[quasar987]

Great post wofsy!

 

[wofsy]

Great post wofsy!

thanks.

That quote from David Hilbert - can you tell me about it?

 

[quasar987]

I'm afraid not. I've had it for as long as I can remember and I don't remember in what context it was said.

 

[LAncienne]

Hi, folks, a really good discussion! For a fun, accessible read on this topic, take a look at the second chapter of Roger Penrose's book, The Road to Reality.
Cheers

 

[LAncienne]

Regarding the Hilbert quote, just from context, I suspect that it was when both Einstein and Hilbert were separately putting the final touches on GR (remember the Einstein-Hilbert action, maybe?) It also may have been when someone pawned a piece of analysis to Emily Noether(sp?) (mathematician, natch!), who then came up with her profound result.

 

[Jim]

Hey Guys,
I think we've beaten this to death...time to move on