It is good to think about the basics, but all is OK for the following reasons. We are using entirely standard methods of special relativity, exactly as in Marion and Thornton chapter fourteen, third edition. The proper time is defined as usual by the Lorentz transform:

c squared tau squared = (c squared – v0 squared) t squared.

So the frame defining the proper time is fixed on the moving particle, which has no velocity with respect to its own frame (left hand side). Here v0 is the non relativistic velocity of the particle. In consequence the Lorentz factor is

gamma = (1 – (vo / c) squared) power minus half

as used in many previous UFT papers. The relativistic momentum of a particle is defined with respect to the laboratory frame of the observer as p = dr / dtau. The relativistic energy / momentum of a free particle is (E / c, p) exactly as in all previous UFT papers. The de Broglie Einstein equations are exactly the same:

E = gamma m c squared = h bar omega

p = h bar kappa = gamma m v0

where v0 is the classical velocity in the frame of the observer. The Minkowski force equation is exactly the same as in UFT238. If the frame is fixed on the particle, the velocity of the frame and the velocity of the particle are the same (v = u) with respect to the observer. The Minkowski four force is simply the derivative of the four momentum with respect to proper time. The formalism being used in UFT376 is the same as in all previous and relevant UFT papers. The difference between ECE2 covariance and Lorentz covariance is that in ECE2 covariance the space contains finite torsion and curvature as you know. The Minkowski space is flat spacetime. The relativistic kinetic energy is

T = (gamma – 1) m c squared

which reduces to T = (1/2) m v0 squared as gamma goes to one. So v0 is clearly the velocity of the particle in the Newtonian limit.

In a message dated 06/05/2017 15:54:21 GMT Daylight Time, writes:

The situation is complicated because bold v (or u in M&T) is the velocity of an object while the Lorentz transform was derived for a relative speed between frames (v). So the proper time relates originally to v but is used here by replacing v with u. It is not clear to me how this relates to the Lagrange equations. I have the impression that variables of both frames are mixed in an inadmissible way. Instead of denoting something a Newtonian velocity we should speak of frames of reference only. I will further think on this on Monday because we have a big meeting of our Munich group tomorrow.

Horst

Am 06.05.2017 um 15:50 schrieb EMyrone:

Yes the derivation is first given in detail in Notes for UFT238, and in Eqs. (6) to (12) of the attached. It is also given in Problem (14.36) of Marion and Thornton, third edition, as part of the Minkowski four-force, found by differentiating the relativistic four momentum (E / c, p) with respect to proper time tau. In problem 14-38 they refer to some other definition, which is not the Minkowski force.

In a message dated 06/05/2017 14:09:25 GMT Daylight Time, mail writes:

Can you give me a reference where the Minkowski force was derived in

detail? In M&T, Problem 14-38, there is the exponent -3/2 which gives

gamma^3, not gamma^4 when moved to the LHS.Horst