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With this value of [P.sup.2], we can exclude [P.sup.[mu]] from MPTD equations, obtaining closed system with second-order equation for [x.sup.[mu]] (so we refer to the resulting equations as Lagrangian form of MPTD equations).
Adding the consequences found above to the MPTD equations (2)-(4), we have the system
The ultrarelativistic behavior of MPTD particle in an arbitrary gravitational field will be analyzed by estimation of three-acceleration as v [right arrow] c.
Behavior of Ultrarelativistic MPTD Particle and the Rainbow Geometry Induced by Spin
Using (8) and (13), we present MPTD equations (9)-(11) in the following form:
The second problem is that acceleration of MPTD particle grows up in the ultrarelativistic limit.
In resume, assuming that MPTD particle sees the original geometry [g.sub.[mu]v], we have a theory with unsatisfactory behavior in the ultrarelativistic limit.
The same observation has been made from analysis of MPTD particle in specific metrics [49-53].
Remarkably, this leads to MPTD equations; see Section 13.2 below.
MPTD Particle as the Spinning Particle without Gravimagnetic Moment.
Comparing the systems, we see that our spinning particle has fixed values of square of spin and canonical momentum, while for MPTD particle these quantities represent constants of motion.
We also found that the risk of premature birth increases with exposure to local traffic-generated pollutants, and this risk was strongest for VPTD followed by MPTD and PTD.