By Abraham A. Ungar
This can be the 1st e-book on analytic hyperbolic geometry, absolutely analogous to analytic Euclidean geometry. Analytic hyperbolic geometry regulates relativistic mechanics simply as analytic Euclidean geometry regulates classical mechanics. The booklet provides a unique gyrovector house method of analytic hyperbolic geometry, absolutely analogous to the well known vector area method of Euclidean geometry. A gyrovector is a hyperbolic vector. Gyrovectors are equivalence periods of directed gyrosegments that upload in accordance with the gyroparallelogram legislations simply as vectors are equivalence periods of directed segments that upload based on the parallelogram legislations. within the ensuing "gyrolanguage" of the ebook one attaches the prefix "gyro" to a classical time period to intend the analogous time period in hyperbolic geometry. The prefix stems from Thomas gyration, that is the mathematical abstraction of the relativistic influence often called Thomas precession. Gyrolanguage seems to be the language one must articulate novel analogies that the classical and the trendy during this publication share.The scope of analytic hyperbolic geometry that the ebook provides is cross-disciplinary, related to nonassociative algebra, geometry and physics. As such, it really is clearly appropriate with the detailed idea of relativity and, quite, with the nonassociativity of Einstein speed addition legislations. in addition to analogies with classical effects that the booklet emphasizes, there are notable disanalogies in addition. hence, for example, not like Euclidean triangles, the edges of a hyperbolic triangle are uniquely decided by way of its hyperbolic angles. stylish formulation for calculating the hyperbolic side-lengths of a hyperbolic triangle when it comes to its hyperbolic angles are awarded within the book.The ebook starts off with the definition of gyrogroups, that's totally analogous to the definition of teams. Gyrogroups, either gyrocommutative and nongyrocommutative, abound in crew conception. unusually, the doubtless structureless Einstein pace addition of detailed relativity seems to be a gyrocommutative gyrogroup operation. Introducing scalar multiplication, a few gyrocommutative gyrogroups of gyrovectors develop into gyrovector areas. The latter, in flip, shape the atmosphere for analytic hyperbolic geometry simply as vector areas shape the surroundings for analytic Euclidean geometry. through hybrid options of differential geometry and gyrovector areas, it really is proven that Einstein (Möbius) gyrovector areas shape the atmosphere for Beltrami-Klein (Poincaré) ball types of hyperbolic geometry. eventually, novel purposes of Möbius gyrovector areas in quantum computation, and of Einstein gyrovector areas in unique relativity, are offered.
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Additional resources for Analytic Hyperbolic Geometry: Mathematical Foundations and Applications
5 (Gyrogroups). A groupoid (G,@) is a gyrogroup if its binary operation satisfies the following mioms. I n G there is at least one element, 0 , called a left identity, satisfying (GI) O@a= a f o r all a E G. There is an element 0 E G satisfying axiom (Gl) such that f o r each a E G there is an element e a E G, called a left inverse of a, satisfying (W e m u =o Moreover, f o r any a , b, c E G there exists a unique element gyr[a, b]c E G such that the binary operation obeys the left gyroassociative law (G3) a@(b@c)= (a@b)@gyr[a, b]c The map gyr[a, b] : G -+ G given by c H gyr[a, b]c is an automorphism of the groupoid (G, @), gyr[a, bl E Aut(G,@) (G4) and the automorphism gyr[a, b] of G is called the gyroautomorphism of G generated by a , b E G.
38) Eq. 40) Eq. 16 (The Cogyrotranslation Theorem). Let (G, @) be a gyrogroup. 44) ( a @ z )E (b@z)= a E b = a e b Proof. 42) follows by a right cancellation. 4). Let (G,+) be a gyrogroup. 46) G. Proof. 45) that (a El b) + ( b E c ) = ( a El b) + ( b - gyr[b, c]c) =a + gyr[a,bl(-gyr[b,). 18 (Left and Right Gyrotranslations). Let (G, @) be a gyrogroup. 15, gyrotranslations are bijective. 19 For any two elements a, b of a gyrogroup (G, +) and any automorphism A of (G, +), A E Aut(G, +), Agyr[a, b] = gyr[Aa, Ab]A Proof.
24 Let (G,+) be a gyrogroup, let a , b E G be any two elements of G, and let YEAut(G) be any automorphism of (G,+). 69) Proof. 39), b = z X u . 69). 75) (gyr[a, bl1-l = gyr-l [a,bl for the inverse gyroautomorphism. 25 (Gyrosum Inversion, Gyroautomorphism Inversion). 77) gyr-'[a; b] = gyr[-b, -a] Proof. Being a group, the product of two elements of the gyrosemidirect product group G XAuto(G) has a unique inverse. It can be calculated in two different ways. 80) for all a, b E G. 82) yields -b - a = -gyr[-b, -a](.
Analytic Hyperbolic Geometry: Mathematical Foundations and Applications by Abraham A. Ungar