Agreed, the proper time must appear in the Euler Lagrange equations, so there exist relativistic Euler Lagrange equations which seem to have been inferred for the first time in UFT376. There must also exist a relativistic Hamilton Principle of Least Action. Eq. (17) of Note 376(6) is the gravitational equivalent of the Coulomb law, the mass density is defined by Eq. (21) in terms of X and Y. The method used to find the orbits is a brilliantly effective one, no one has done this kind of work before. It was apparently thought that special relativity does not produce precession. Using kappa bold as an input parameter means solving Eqs. (15), (16) and (17) simultaneously for the orbit. This is equivalent to solving Eqs. (15) and (16) and looking at Eq. (17) as a constraint. One could also use kappa and the mass density as two input parameters. There are many possibilities, and the method has been built up over several years, so it is now very efficient. I cannot see how any honest scientist can ignore eighty three refutations of a theory, plus all of Steve Crothers’ refutations. In other words, EGR is completely obsolete.

To: EMyrone@aol.com

Sent: 05/05/2017 22:04:52 GMT Daylight Time

Subj: Re: The Role of the Spin Connection in Direction of PrecessionThe method used in UFT 375 is the same as in UFt 328, except that in 328 polar coordinates were used and and the code was programmed by hand. The “scatter plot” was nothing else than graphing the orbit from the time functions r(t) and phi(t) as I do routinely now from the solutions X(t) and Y(t). However the equations solved are not identical to the Minkowski equation. The Lagrangian is the same but for UFT 328/376 in the Euler-Lagrange equations the t parameter was used while it should be correctly the tau parameter as described in the note. In so far the method of UFT 328/375 can be an approximation at best and differences are expected to the solution of the Minkoswki equation. I will check the code, the relativistic momentum must be constant otherwise there must be an error.

Horst

PS: It is difficult to say what the role of kappa is in eq.(17) of the note. Eq.(17) seems to be a kind of constraint. The solutions of the orbit X(t) and Y(t) follow from (15) and (16) alone, otherwise the equation system would be overdetermined. When X,Y are known, eq.(17) gives one equation for both components kappa_X, kappa_Y. On can define one of them and then compute the other.

What does eq.(21) mean? Is rho_m the denstity of the orbiting mass? It must be a function rho_m(t), not a field rho_m(bold r(t), t), because X and Y are the orbital trajectories.Am 05.05.2017 um 15:52 schrieb EMyrone:

The work so far for UFT376 is very important and has made great progress. We need to make cross checks at this stage. With reference to Note 376(6) it is already known from your first class numerical work in UFT375 that orbital precession in the usual direction is given by the lagrangian (1). The Minkowski force equation for the orbit is obtained from this lagrangian, so the two methods must give the same result self consistently, the same precession and same direction of precession. This is a rigorous test of the code. Both the Minkowski force equation and the lagrangian are well known in special relativity as you know. The relativistic lagrangian is actually designed to give the relativistic momentum as in Marion and Thornton, using the canonical equation (2) of the note, which is the same as Marion and Thornton, third edition, chapter 14, Eq. (14.107). The relativistic momentum comes from conservation of momentum and the Lorentz transform. The Minkowski force is dp / d tau where tau is the proper time and where p is the relativistic momentum. So the fact that you have already obtained the orbital precession from the relativistic lagrangian in a previous paper (UFT375) means that exactly the same precession must emerge from the Minkowski force equation, the correctly relativistic Newton equation. The potential energy remains the same throughout. This assumes that the field Eq. (17) is decoupled from Eqs. (15) and (16). If Eqs. (15) to (17) are solved simultaneously (as they should be), and if kappa is used as an input parameter, one could possibly get precessions of all kinds, depending on a model for kappa. In addition, the precession from Eq. (1) must be the same as the result obtained in UFT328, using the scatter plot method and simultaneous solution of the same relativistic lagrangian and hamiltonian. This is another cross check. As you know, the difference between ECE2 and special relativity is that kappa appears in ECE2 together with the field equation (17). The kappa input parameter does not occur in special relativity. The precession cannot be different from the lagrangian and force equation, so the only possibility is that the code needs to be checked. Some Runge Kutta procedure may need a finer mesh or similar. Similarly, the non relativistic Newton equation is derivable from the non relativistic lagrangian, and both methods must give the same results.