Session H10: History of Physics Contributed Papers

10:45 AM–12:33 PM, Sunday, February 14, 2010
Room: Maryland B

Sponsoring Unit: FHP
Chair: David Cassidy, Hofstra University

Abstract ID: BAPS.2010.APR.H10.1

Abstract: H10.00001 : Did Minkowski Change his Mind about Noneuclidean Symmetry in Special Relativity?

10:45 AM–10:57 AM

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  Felix T. Smith
    (SRI International)

Minkowski (M.) observed in 1907 that the symmetry of relativistic velocity space is the same as that of noneuclidean geometry. He withheld this text from publication, but Sommerfeld published it 6 years after he died, Ann. Phys. (4)\textbf{ 47}, 927 (1915), [1]. Six weeks later in a long, careful article, G\"{o}tt. Nach. (1908) 53, [2], M. made only a much weaker statement about the noneuclidean parallel. In [3], Phys. Zeitschr.\textbf{ 10}, 104 (1909), he avoided the issue entirely. M's reasons for the changes have never been known. I now show that a key equation in [1] had an error in sign, undetected by Sommerfeld or other commentators, which M. evidently soon saw. This error had led to omitting the factor $\beta =\left( {1-v^2/c^2} \right)^{-1/2}$ in the relativistic velocity ${\rm {\bf u}}={\rm {\bf p}}/m_0 =\beta {\rm {\bf v}}=\beta {\rm {\bf \dot {x}}}$. With the error corrected, it became clear that while velocity is constrained to a negative-curvature 3-space, space-time is a flat 4-space. The changes between [1], [2] and [3] will be discussed in the light of M's evolving understanding, his different intended audiences, and the possibility that he chose to defer the noneuclidean aspects of velocity and of space-time for later treatment.

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