...United States of America JULY 2005 VOLUME 2 CONTENTS S. J. Crothers On the Geometry of the General Solution for the Vacuum Field of the Point-Mass. . ... ...ed from private donations. July, 2005 PROGRESS IN PHYSICS Volume 2 On the Geometry of the General Solution for the Vacuum Field of the Point-Mass Ste... ...wing unjustifed assumptions, ta- citly or otherwise; S J. Crothers. On the Geometry of the General Solution for the Vacuum Field of the Point-Mass 3 ... ...ired by Einstein’s theory. Mere assumption is not permissible. 2 The basic geometry of the simple point-mass The usual metric g s of the spacetime man... ...chanics 91 Volume 2 PROGRESS IN PHYSICS July, 2005 Hydrodynamic Covariant Symplectic Structure from Bilinear Hamiltonian Functions Salvatore Capozzie... ...riant vector, bivector or tensor felds, it is possible to derive a general symplectic structure which leads to holonomic and anholonomic formulations ... ...as, for example, the connection Γ γ αβ . Despite of this lack, a covariant symplectic structure can be identifed also in the framework of General Rela... ...l theories. This statement does not still mean that the identifcation of a symplectic structure immediately leads to a full quantization but it could ... ...paper is to show that a prominent role in the identifcation of a covariant symplectic structure is 92 S. Capozziello, S. De Martino, S. Tzenov. Hydrod...

...lved and Unsolved Problems, Conjectures, and Theorems in Number Theory and Geometry (edited by M. L. Perez), Xiquan Pub- lishing House, Phoenix, 2000.... ...itional potential energy which manifests the fractal and nondifferentiable geometry. 2 Non relativistic quantum mechanics 2.1 Quantum potential In the... ...established as a mere manifestation of the nondifferen- tiable and fractal geometry instead of being deduced from a postulated Schr¨ odinger equation.... ...e energy- momentum tensor, one solves the Einstein feld (i. e. space- time geometry) equations for the metric potentials, then one writes the geodesic... ...loses its meaning. Even in the case of only one “particle”, the space-time geometry is determined by the particle itself and by its motion, so that th... ... , are also invariant! The pure- acceleration boosts transformations are “symplectic”. It can be shown also that areas greater (smaller) than the Pla... ...ariables the classical Newtonian dynamics is going to be modifed since the symplectic two-formω μν in Phase Space will have additional non-vanishing e... ... , are also invariant! The pure- acceleration boosts transformations are “symplectic”. It can be shown also that areas greater (smaller) than the Pla... ...the Standard Model [85], hyper-matrices and hyper-determinants [88], multi-symplectic mechanics, the de Donde-Weyl formula- tions of QFT [82], to cite...