Relativistic dynamics: force, mass, kinetic energy, gravitation and dark matter

Special relativity does not deal with acceleration, general relativity does not deal with non gravitational acceleration, which leave the theory of relativity imperfect. We will demonstrate some relativistic dynamical laws that specify relativistic acceleration, force and kinetic energy. Also, based on equivalence principle does gravitational mass vary with inertial mass?

Newtonian kinematics defines motions of objects with velocity and acceleration, Newtonian dynamics defines force with acceleration and mass, which makes Newtonian mechanics the most complete theory in physics. Special and general relativity are extremely successful, but they lack the capability of dealing with acceleration and force. For relativity mass increases to infinity when u=c, which makes energy and momentum to become incorrectly infinity. Also, general relativity is based on equivalence principle according to which inertial mass is equivalent to gravitational mass. Then does gravitational mass increases when inertial mass increases? So, relativity needs new laws to deal with acceleration and force.

In previous studies of relativity [1][2][3][4][5], we have already correctly treated acceleration, inertial mass, kinetic energy. Below we will demonstrate the laws that describe them. For setting the demonstrations on a strong base, we begin with rigorously proving the equality of differential momentum in 2 relatively moving frames of reference.

In relativity, a change of velocity has different value in relatively moving frames. However, differential momentum has the same value in such frames, which we call equality of differential momentum and have explained in « Velocity, mass, momentum and energy of an accelerated object in relativity » [2].

For rigorously proving this equality, let us take 2 identical objects labeled a and b. The object b moves at the velocity u with respect to a, the frame of reference of the object b is labeled frm. b, see Figure 1. If the object b gets an infinitesimal impulse,it gets a differential momentum and a differential change of velocity labeled du_b.

Notice that in the frame frm. b the velocity of b is constantly zero. Then, how can its change of velocity du_b be nonzero? In fact, du_b is with respect to an inertial frame, not to frm. b. For defining du_b we have to create a new type of inertial frame that coincides with b. We name this type of frame “Proper inertial frame”.

For example, the object b moves at the instant velocity u_t at a given time t. At this time we create the proper inertial frame of b labeled Ref. b which moves at constant velocity that equals u_t. The trajectory of Ref. b is a straight line while that of b is a curve, see Figure 2. After the infinitesimal impulse b moves at the instant velocity u’_t and the change of velocity equals du_b= u’_t– u_t with u_t being the velocity of the inertial frame Ref. b. In the same way the proper inertial frame of the object a is labeled Ref. a.