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A Metric Tensor of the New General Lorentz Transformation Model

Starting with a new General Lorentz Transformation model (GLT-model)in (2) the general line element has been derived. This line element is a function of two free parameters that determine observation signal velocities in systems K and K', where the system K' is moving relative to the system K with a velocity v. From the general line element we obtained a corresponding general metric tensor of GLT - model. Generally, the free parameters of the GLT-model are functions of the space-time coordinates; i.e. they are functions of the state of the energetic potential of the fields in which observation signals propagate. In order to identify two free parameters of GLT-model in a gravitational field, the general line element is transformed into spherical polar coordinates, which are appropriate to the problem. The next step was the diagonalisation of the line element. Thus, we obtained the Schwartzschild's like form of the line element for a spherically symmetric non-rotating body. The identification of that line element with the Schwartzschild's spherically symmetric vacuum solution gives the quantities of the free parameters of the general line element. These quantities are equal to the quantities of the same parameters, which have been obtained by Novakovic in (2), employing the well-known gravitational redshift experiment and the energy equations, derived from a null component of a four-momentum vector of GLT-model. In the case of the weak gravitational field (like in our Solar system) we can neglect quadratic term of the gravitational potential in the solution of the free parameters of GLT-model, what leads to the solution exactly equal to the Schwartzschild’s metrics of the line element. On that way GLT-model is verified in a gravitational field. Since GLT-model satisfies the Lorentz-Einstein coordinate transformations in the Special Relativity and the Einstein field equations in the General Relativity (1), one can conclude that GLT-model is valid both in Special and General Theory of Relativity. Moreover, it is shown that there exists a simple coordinate transformation procedure that transforms a general line element into diagonal one, with metric components (-1, 1, 1, 1), what is equal to the metrics of the line element in SR. Four - vectors for position, velocity, momentum and acceleration as well as the energy-momentum tensor of perfect fluid have been derived as functions of two free parameters of GLT-model. It means that two free parameters of GLT-model can unify some elements in SR and GR. Since the GLT-model satisfies the Maxwell's equations in the covariant sense, the electromagnetic tensor and the energy momentum tensor of the electromagnetic field are also described as functions of two free parameters of GLT-model. Thus, the electromagnetic tensor includes (on the natural way) the influences of gravitation to the electromagnetic field. Consequently, the Maxwell's equations transformed by GLT-model describe electromagnetic field in a gravitational field. Is that an unification of electromagnetic and gravitational fields? Since the elements of the general metric tensor are functions of the free parameters of GLT-model, the Christoffel symbols, the geodesic equations, the Riemann and Ricci Tensors, the energy-momentum tensor and the Einstein field equations in GR can also be described as a functions of the free parameters of GLT-model. It enables coordinate transformations in GR as we do it in SR. Furthermore, the free parameters of GLT-models can be the functions of the light frequencies in K and K’ systems, as it is shown in (2). Thus, the Maxwell's equations transformed by GLT-model and the Einstein field equations in GR can become the functions of the light frequencies in K and K’ systems. Consequently, it seems that the possibilities of an unification of Einstein's Special and General Theories of Relativity, as well as a new unification of electromagnetic and gravitational fields are opened. This unification should be done for multivariable gravitational field (two-body or n-body gravitational problems) as it has been suggested by the references (3,4). It can be shown that the free parameters of GLT-model can help in this unification, because they can include the motions in multivariable gravitational fields. A definition of the free parameters of GLT-model includes the possibilities that an observation signal can be the light signal, as well as superluminal signal (like tachyons signal) and subluminal signals (like sonar signals). In that sense, it seems that GLT-model can include gravitational influences to a Dual Relativity (5), or to the Tachyons Region of motions (6). References (1) Einstein Albert (1989). The Collected Papers of Albert Einstein. Vol. 2: The Swiss years: writings, 1900-1909. Princeton, NJ. (2) Novakovic Branko, Novakovic Dario and Novakovic Alen (2000). A New General Lorentz Transformation model. In the book "Computing Anticipatory Systems", Ed. by D.M. Dubois, Publish. by AIP - American Institut of Physics, Melville, New York, 2000, pp. 437-450. (3) Peterson I. (1994). A New Gravity? Challenging Einstein's General Theory of Relativity. In Science News, Vol. 146, p. 376-378, Dec. 3, 1994. (4) Yilmaz H. (1992). Toward a Field Theory of Gravitation. In Il Nuovo Cimento, Vol. 107B, No. 8, p. 941-960, Aug. 1992. (5) Dubois Daniel, and Nibart Gilles (1999). Toward to a Computational Derivation of a Dual Relativity with Forward-Backward Space-Time Shifts. Abstract book of the CASYS’99, Third International Conference on Computing Anticipatory Systems, Simp. 10, pp. 15-19, Ed. by D.M. Dubois, Publ. by CHAOS asbl, HEC LIEGE, Belgium, Aug. 9-14, 1999. (6) Nibart Gilles (1999). Do Tachyions Violate the Causality Principle? Abstract book of the CASYS’99, Third International Conference on Computing Anticipatory Systems, Simp. 10, p. 24, Ed. by D.M. Dubois, Publ. by CHAOS asbl, HEC LIEGE, Belgium, Aug. 9-14, 1999.

New General Lorentz Transformations; General Line Element; Metric Tensor; Special Relativity; General R elativity

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