About the geometry of general relativity

[Philippe]

We all know that the geometrical foundations of general relativity follow riemann's geometry and not the euclidean one.

About 3 months ago a lebanese mathmaticien claimed that he proved the 5th postulate of the euclidean geometry (about the parallel theorem) which is still with no proof... anyway he published a book (in french) in which he says that this proof implies that all other 2 geometries are wrong (including riemann's) and that it will affect general relativity..
you can check some stuff about him just search his name ( Rachid Matta) in google..
i don't know a lot about geometry stuff but do you think that his work is importent or that he is just some fool who wants to prove that einstein's work is wrong??

 

[Hurkyl]

i don't know a lot about geometry stuff but do you think that his work is importent or that he is just some fool who wants to prove that einstein's work is wrong??

Any proof of Euclid's fifth postulate is a proof that Euclidean geometry is logically inconsistent.

If this guy is using this alledged proof to argue that Euclidean geometry is the "right" way to do things, then he's way off.

 

[selfAdjoint]

His proof would have to derive the fifth postulate of Euclid from the other four.

I am sure his proof is wrong; I'll bet it relies on some un-Euclidean assumptions about lines and points.

Because spherical geometry with great circles as lines and their intersections as points satisfies the first four postulates of Euclid but not the fifth. And those definitions agree with what Euclid says lines and points do.

 

[JesseM]

Any proof of Euclid's fifth postulate is a proof that Euclidean geometry is logically inconsistent.

Why is that? I know it's already been shown that adding either the fifth postulate or its negation to the other four postulates will produce a consistent system, but that's not quite the same as what you said...

 

[clj4]

Can't be right. By assuming that the 5-th postulate to be false, Riemann and Lobachevsky derived two different, fully selfconsistent geometries. In one of them there is no parallel possible thru the external point, in the other geometry, there is an infinity of such parallels.
IF Rachid Matta could have proven the 5-th postulate from the other 4, the consequence would be an invalidation of the other two since the 5-th postulate had become a theorem of the Euclid geometry. But we already know that in the geometry on a sphere there is no one parallel that can be drawn thru ANY external point (all the "meridians" intersect at the poles)
Matta published his book in 2004, no Earth shattering consequences since then...Most probably his derivation is false.

 

[HallsofIvy]

Why is that? I know it's already been shown that adding either the fifth postulate or its negation to the other four postulates will produce a consistent system, but that's not quite the same as what you said...

Euler and Poincare' both developed "models" for non-Euclidean geometries- that is, ways of associating points, lines, etc. in non-Euclidean geometry. For example, in Poincare's "half-plane model" (he also had a disk model) for hyperbolic geometry you draw a line in the Euclidean plane and model points in hyperbolic geometry (a non-Euclidean geometry in which the parallel postulate is replaced by the postulate that "through any point not on a given line, there exist an infinite number of lines through that point, parallel to the given line) as a point on one side of that line (say, above it). Lines in hyperbolic geometry are modeled as either lines perpendicular to the given "bounding" line (vertical lines) or as semi-circles with center in the bounding line. It is then possible to interpret all of the postulates of hyperbolic geometry as statements in the model and prove them from all Euclidean postulates. If it were possible to prove the 5th (parallel) postulate from the others, then it would be possible to prove its analog in the poincare' model, contradicting the hyperbolic parallel postulate already proven.

 

[JesseM]

Euler and Poincare' both developed "models" for non-Euclidean geometries- that is, ways of associating points, lines, etc. in non-Euclidean geometry. For example, in Poincare's "half-plane model" (he also had a disk model) for hyperbolic geometry you draw a line in the Euclidean plane and model points in hyperbolic geometry (a non-Euclidean geometry in which the parallel postulate is replaced by the postulate that "through any point not on a given line, there exist an infinite number of lines through that point, parallel to the given line) as a point on one side of that line (say, above it). Lines in hyperbolic geometry are modeled as either lines perpendicular to the given "bounding" line (vertical lines) or as semi-circles with center in the bounding line. It is then possible to interpret all of the postulates of hyperbolic geometry as statements in the model and prove them from all Euclidean postulates. If it were possible to prove the 5th (parallel) postulate from the others, then it would be possible to prove its analog in the poincare' model, contradicting the hyperbolic parallel postulate already proven.

But Hurkyl said "Any proof of Euclid's fifth postulate is a proof that Euclidean geometry is logically inconsistent"...wouldn't your argument be summed up as "Any proof of Euclid's fifth postulate is a proof that hyperbolic geometry is logically inconsistent" instead? Just a quibble, of course...

 

[Hurkyl]

Why is that? I know it's already been shown that adding either the fifth postulate or its negation to the other four postulates will produce a consistent system, but that's not quite the same as what you said...

Just continue along the line of thought. The easiest is simply taking a contrapositive. The existence of a model proves relative consistency:

Euclidean geometry is consistent ===> Hyperbolic geometry is consistent

and this statement is equivalent to

Hyperbolic geometry is inconsistent ===> Euclidean geometry is inconsistent


Another way to see it is that Euclidean geometry contains a model of Hyperbolic geometry. But if PP can be proven from the other axioms, then Euclidean geometry can prove PP in this model. But since ~PP is true in this model, Euclidean geometry has derived contradictory statements! Therefore, it must also be contradictory.

Because spherical geometry with great circles as lines and their intersections as points satisfies the first four postulates of Euclid but not the fifth.

I wanted to make a note on this: if I recall correctly, you cannot prove some basic things such as:

Let C be a circle. Suppose A lies inside C, and B lies outside C. Then, the line segment AB intersects the circle C.

This can be realized by the fact Q² is a model of Euclidean geometry: it satisfies all 5 axioms. It just has the pesky quality that circles don't have all that many points!


And you also cannot prove things other theorems such as any point P on a line separates the line into two halves. This means that the points H and K are in opposite halves if and only if the line segment HK contains P, I think, even with the parallel postulate. (And you obviously cannot without the PP, because this is false in spherical geometry!)


More modern approaches use Hilbert's axioms, or something similar. These do exclude spherical geometry by virtue of the betweenness axioms. (They can prove the above theorem) You have to weaken these to "separation axioms" to allow spherical geometry.