Minkowski & Einstein: Hyperbolic Geometry Breakthrough?

[Grimble]

<Moderator’s note: forked from https://www.physicsforums.com/threa...-the-twin-paradox.915212/page-13#post-6215675 >

Thank you, that is very interesting and I can understand much of it.

But can someone tell me if it was the application of hyperbolic geometry that was Minkowski's breakthrough in depicting Einstein's theory?

 

[Nugatory]

Thank you, that is very interesting and I can understand much of it.

But can someone tell me if it was the application of hyperbolic geometry that was Minkowski's breakthrough in depicting Einstein's theory?

Hyperbolic geometry was discovered and studied during the 19th century. Minkowski’s contribution was making the connection between Einstein’s work and what had previously been an interesting abstract mathematical concept with no known practical application.

 

[Grimble]

I thought that might have been the case - but how did Einstein respond to his old teachers discovery with regard to his theory?

 

[Nugatory]

I thought that might have been the case - but how did Einstein respond to his old teachers discovery with regard to his theory?

Enthusiastically, once he recognized the power of Minkowski’s approach (although he was initially skeptical). And of course it was an essential step on the way to General Relativity.

 

[Michael Price]

Enthusiastically, once he recognized the power of Minkowski’s approach (although he was initially skeptical). And of course it was an essential step on the way to General Relativity.

In fact Minkowski was working on special relativity, indpendently, when Einstein beat him to publication. Minkowski was shocked when Einstein's papers hit the world, especially since he'd regarded Einstein as a "lazy dog" - but it did mean he was ready with his own reworking of SR with the development of "spacetime" as a unified entity. (Not anything to do with hyperbolic geometry, BTW. Minkowski spacetime is strictly flat.)

 

[epenguin]

I thought that might have been the case - but how did Einstein respond to his old teachers discovery with regard to his theory?

Was it about that that he said "I no longer recognise my own theory! " or was it something else?

 

[Grimble]

So

Minkowski spacetime is strictly flat

How does that fit with the insistence on describing everything in terms of hyperbolic geometry?

 

[Michael Price]

So
How does that fit with the insistence on describing everything in terms of hyperbolic geometry?

It doesn't fit. Minkowski space, and spacetime, are both flat. Hyperbolic is curved, as is hyperspherical.

 

[Grimble]

Hyperbolic geometry was discovered and studied during the 19th century. Minkowski’s contribution was making the connection between Einstein’s work and what had previously been an interesting abstract mathematical concept with no known practical application.

So Minkowski made the connection between hyperbolic geometry and special relativity - so far, so good...

but it did mean he was ready with his own reworking of SR with the development of "spacetime" as a unified entity. (Not anything to do with hyperbolic geometry, BTW. Minkowski spacetime is strictly flat.)

It doesn't fit. Minkowski space, and spacetime, are both flat. Hyperbolic is curved, as is hyperspherical.

So what then is the connection between Special Relativity, Minkowski space, Minkowski diagrams and hyperbolic geometry?

 

[pinball1970]

Was it about that that he said "I no longer recognise my own theory! " or was it something else?

"Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore."

Quoted in P A Schilpp, Albert Einstein, Philosopher-Scientist (Evanston 1949).

 

[Michael Price]

So Minkowski made the connection between hyperbolic geometry and special relativity - so far, so good...So what then is the connection between Special Relativity, Minkowski space, Minkowski diagrams and hyperbolic geometry?

The first three are connected - but not with hyperbolic geometry.

 

[Grimble]

Very good so just where does hyperbolic geometry fit in?

 

[pinball1970]

Very good so just where does hyperbolic geometry fit in?

General, all the Riemann stuff.
I struggled with it
@Michael Price can expand I think. The maths is tough

 

[cosmik debris]

The Lorentz transform is a hyperbolic rotation of Minkowski spacetime.

Cheers

 

[Michael Price]

There is confusion over the use of "hyperbolic" here, because the term has two completely unrelated meanings.
1) Hyperbolic space or hyperbolic geometry refers to space with a negative curvature. This has nothing to do with special relativity or Minkowski spacetime which has zero curvature ( "flat"). Curved spacetime is part of general relativity, not special relativity.
https://en.m.wikipedia.org/wiki/Hyperbolic_geometry2) Hyperbolic functions, such as sinh, cosh, tanh, which are analogous to the trigonometric functions. These functions are used in special relativity to express Lorentz transformations.
https://en.m.wikipedia.org/wiki/Hyperbolic_function