The use of the split-quaternions of norm one ( " qq " * = 1 ) for hyperbolic motions of the Poincar?disk model of hyperbolic geometry is one of the great utilities of the algebra.
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The four hyperbolic motions that produced z above can each be inverted and applied in reverse order to the semicircle centered at the origin and of radius \ sqrt { z } to yield the unique hyperbolic line perpendicular to both ultraparallels p and q.
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The reason is that uniform acceleration describes a hyperbola in spacetime ( see hyperbolic motion ), and anything beyond the asymptotes ( which cross at a distance of c 2 / a ) can't possibly catch up with you without exceeding c.
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Then by means of hyperbolic motions one can measure distances between points on semicircles too : first move the points to " Z " with appropriate shift and dilation, then place them by inversion on the tangent ray where the logarithmic distance is known.
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1911 : Max von Laue derived in the first edition of his monograph " Das Relativit�tsprinzip " the transformation for three-acceleration by differentiation of the velocity addition, then the rest acceleration, and eventually the formulas for hyperbolic motion which corresponds to ( ):
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Max Born ( 1909 ) subsequently coined the term " hyperbolic motion " ( ) for the case of constant magnitude of four-acceleration, then provided a detailed description for charged particles in hyperbolic motion, and introduced the corresponding " hyperbolically accelerated reference system " ( ).
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Max Born ( 1909 ) subsequently coined the term " hyperbolic motion " ( ) for the case of constant magnitude of four-acceleration, then provided a detailed description for charged particles in hyperbolic motion, and introduced the corresponding " hyperbolically accelerated reference system " ( ).
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This article exhibits these examples of the use of hyperbolic motions : the extension of the metric d ( a, b ) = \ vert \ log ( b / a ) \ vert to the half-plane, and in the location of a quasi-sphere of a hypercomplex number system.
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Instead, the relativistic equation of motion for an object with constant proper acceleration is x ^ 2-c ^ 2t ^ 2 = c ^ 4 / \ alpha ^ 2, as per the article Hyperbolic motion ( relativity ) . talk ) 19 : 41, 16 November 2012 ( UTC)
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For instance, equations of motion and acceleration transformations were developed in the papers of Hendrik Antoon Lorentz ( 1899, 1904 ), Henri Poincar?( 1905 ), Albert Einstein ( 1905 ), Max Planck ( 1906 ), and four-acceleration, proper acceleration, hyperbolic motion have been analyzed by Hermann Minkowski ( 1908 ), Max Born ( 1909 ), Gustav Herglotz ( 1909 ), Arnold Sommerfeld ( 1910 ), von Laue ( 1911 ) ( see section on history ).
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