By Abraham A. Ungar

This can be the 1st ebook on analytic hyperbolic geometry, totally analogous to analytic Euclidean geometry. Analytic hyperbolic geometry regulates relativistic mechanics simply as analytic Euclidean geometry regulates classical mechanics. The e-book provides a unique gyrovector house method of analytic hyperbolic geometry, totally 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 response to the gyroparallelogram legislations simply as vectors are equivalence sessions of directed segments that upload in keeping with the parallelogram legislations. within the ensuing “gyrolanguage” of the e-book 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's 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 booklet percentage. The scope of analytic hyperbolic geometry that the publication offers is cross-disciplinary, regarding nonassociative algebra, geometry and physics. As such, it really is obviously appropriate with the targeted conception of relativity and, really, with the nonassociativity of Einstein pace addition legislation. besides analogies with classical effects that the e-book emphasizes, there are extraordinary disanalogies besides. therefore, for example, in contrast to Euclidean triangles, the edges of a hyperbolic triangle are uniquely made up our minds by way of its hyperbolic angles. dependent formulation for calculating the hyperbolic side-lengths of a hyperbolic triangle by way of its hyperbolic angles are awarded within the publication. The booklet starts with the definition of gyrogroups, that is absolutely analogous to the definition of teams. Gyrogroups, either gyrocommutative and nongyrocommutative, abound in workforce thought. strangely, the probably structureless Einstein pace addition of exact 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. by way of hybrid recommendations of differential geometry and gyrovector areas, it's proven that Einstein (Möbius) gyrovector areas shape the surroundings for Beltrami–Klein (Poincaré) ball versions of hyperbolic geometry. ultimately, novel purposes of Möbius gyrovector areas in quantum computation, and of Einstein gyrovector areas in specific relativity, are offered.

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77) again we have Proof. 97) for all a , b E G. 15, then the resulting equation can be written as gyr[a;I. 92). 92). 93). 94) is equivalent to the first one. 94) follows from the first (third) by replacing a by -a (or, alternatively, by replacing b by -b). 0 We are now in a position to find that the left gyroassociative law and the left loop property of gyrogroups have right counterparts. 28 have Let (G, @) be a gyrogroup. Then, for a n y a , b, c E G we (i) (a@b)@c= a@(b@gyr[b,a]c) Right Gyroassociative Law (ii) gyr[a,b] Right Loop Property = gyr[a, b@a] Gyrogroups 43 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. 48) +) and any automorphism +) we have by the left gyroassociative law, + (Aa Ab) + Agyr[a, b + + + + ] =~ A((a b) gyr[a, bJz) = A(.

Formula a69(8a@b) = b (&a) m a = b (b E a)@a = b Terminology Left Cancellation Law (First) Right Cancellation Law (Second) Right Cancellation Law Source Eq. 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.