Depends on context, what is the dynamical problem?

If no force, gravity or acceleration, p is conserved. But maybe you could have some trade off between the v in m, and the multiplying v?

It is a very interesting question. As I know the paradox of the twins is also dealt in the Minkowski plane and one explains the lastin change of the one twin that during its smooth movement there is a little part which is no longer uniform, namely when he turns back to the Earth with his rocket.

Hi,

The Minkowski space four-vector for space–time is represented by,the group velocity represents the propagation velocity of a modulation of a pure sine .... Therefore the sensible thing to do is to consider well controlled cases and .... independent variable τ by one of the four coordinates xμ, preferably the time t = −ix4 ⧸c. it is reasonable to consider a variable velocity in a Minkowskian space-time..

Dear Readers,

Thanks for your replies.I would like to explain my question again.

The Minkowskian geometrical structure governs an inertial frame. The geodesic equations of the Minkowskian geometrical structure follow the law of inertia.Therefore , considering a force which in turn implies a variable velocity violates the geodesic equations of the Minkowskian structure and hence, it is necessary to define another geometrical structure which reduces to the Minkowskian structure in the absence of the force under consideration.The geodesic equations of the new geometrical structure will follow the path of the particle under the action of the force under consideration and also, this geometrical structure may not governs an inertial force.

Regards

Ramesh Chandra

Dear Dr. Chandra,

I agree with you. I think when the paradox of the twins was been worked out the theory of general relativity has not existed yet. Maybe even this kind of problems have led to it.

If you want to keep your frame inertial, you avoid inertial forces so you try avoid accelerations (or variable velocity)

This does not mean you cannot handle equations like F=dp/dt using the relativistic p, but there is some debate about how to handle force in SR (see some discussion in Herbert Goldstein, Clasical Mechanics.

Dear J. Weisz,

I am still unable to understand how one may consider the force equations in a Minkowskian space-time without violating the the geodesic equations of the space-time and when one violates the geodesic of the Minkowskian space-time to consider the force equations, the geometrical structure of the space-time will be different from the Minkowskian structure.

Regards

Ramesh Chandra

Well, the answer could be to consider the system nearly inertial, so you have only real forces in it, and consider that SR is a very near variation of the Newtonian universe, and you use the relativistic p only because you know that this slightly improves your answer.

I agree that if you think your way, there are problems with this. My best answer is to try to understand Minkowski force in the SR setting. Usualy advanced sources discuss this.

It is obvious that real motion does not follow geodesics, ie. if you launch a projectile the result will depend on the initial condition, angle and speed, so the so called bending of space-time is fairly irrelevant in these cases.

Dear J.Weisz,

I agree with your view provided, we are able to define a Minkowskian structure different from the Minkowskian space-time of the special relativity in which the force or the acceleration may be derived from the geodesic equations of the Minkowskian structure. Of course, this new structure should reduce to the Minkowskian space-time structure in the absence of the force.

Regards

Ramesh Chandra

Dear J.Wetsz,

I am eagerly waiting for your reply.

Regards

Ramesh Chandra

Dear Readers,

To explain my answer of this question ,I presented my work ' Lorentz transformation via Minkowskian Space-Time', DOI:10.13140/RG.2.2.22314.06080.

Regards

Ramesh Chandra

as far as I know without real forces particles or waves would just follow geodesics, but in SR there is no real curvature, so you just have straight lines at constant speed

Curvature forces inertial forces to appear.

The metric of SR is always simple -1,1,1,1 on the diagonal.

regards, Juan

I think this case is similar to the limit between the Newtonian mechanics and the SR. As long as the velocities are small we can use the usual classical mechanics to describe the events in the spacetime. So if the forces are small e. g. the curvature of the lines are small we can use the SR. Maybe one can define a Taylor series in anyway and take into account the several orders.