Travelling together with an object X of the mass m moving in frame M at a speed v_{x/m} ,
we cannot detect speed v_{x/m}. Our speed with respect to spacetime structure is now
v_{m/x}. Thus the linear momentum of the object X of the rest mass m, motionless with respect
to frame X, but moving at a speed v_{m/x} with respect to the spacetime structure of the frame M may be
defined as p_{m/x}. It is the same linear momentum, as if the object X of a non-relativistic mass m
was moving at a speed v_{x/m} with respect to frame X :
(47)
Looking from frame M, we can see our own frame M moving at a contemporary speed
It means, v_{x/m} represents galileian projection in M-frame, corresponding to the speed
v_{m/x} in X-frame. Further we obtain (see also equation (17)):
(47a)
Consequently,
(47b)
Equations (47) and (47b) define the linear momentum of the object in its own reference frame. In spite of the fact, that object is
without motion with reference to its own frame, it moves with respect to spacetime structure of any other reference frame, and all
kinds of interactions influencing motion of the object, are realized by affecting of this relative motion.
With reference to frame M :
where
dp_{x/m}
stands for the linear momentum differential of the object X in M-frame, and,
M_{mx} and dM_{mx}
stand for galileian projection of the mass and its differential from X to M-frame.
Substituting for M_{mx} from equation (46a), we can derive
(48)
In case when dv_{x/m} is in line with v_{x/m}, we obtain
(49)
The galileian projection of the linear momentum from M- into N-frame:
Substitution for M_{n(mx)} from equation (42), and for
yields :
(50)
Applying equation (12), we can express
d_{m/n} / (_{m/n}
)^{2} in equation (50) :
(50b)
It is dv_{m/n} = 0 at v_{m/n} = const., if M-frame moves without acceleration, and
without rotation with reference to N-frame, and, d = 0 if the acceleration of the
object X in M-frame is in line with the speed v_{x/m} . Under these circumstances, we receive:
(50c)
Substitution for
in equation (50c), yields:
(50d)
Applying equation (49), we have in common case:
(51)
(52)
In common motion configuration X/M/N, we may write
(53)
Under the circumstances that M-frame moves without acceleration and without rotation with reference to N-frame, and that
acceleration of the object X in M-frame is in line with the speed v_{x/m} , we receive
(54)
The force is defined as
(55)
Dividing equation (47) by time differential dt_{x}, we receive the force responsible for acceleration of the
object in its own frame.
(56)
Dividing equation (47b) by time differential dt_{x} yields
Since
we have
(56a)
Comparing equations (56) and (56a), we receive:
(56b)
The force acting on the object moving in M-frame, derived from equation (48):
(57)
If acceleration is acting in direction of the speed, equation (57) gives :
(57a)
If acceleration is acting in opposite to direction of the speed, equation (57) gives :
(57b)
The galileian projection of the force from M- to N-frame, derived from equation (50) :
(58)
Applying equation (50b), we obtain :
(58a)
where (in equations 58 and 58a)
F_{n(x/m)}
stands for the galileian projection of the force from M- to N-frame. Direction of the force is defined
by the angle _{nm} according to equation (10), giving
galileian projection of the angle from M- to N-frame resulting from vectors' superposition in equation (58),
a_{x/m}
stands for acceleration of the object X in M-frame,
a_{m/n}
stands for acceleration of the frame M with reference to N-frame,
v_{x/m}
stands for speed of the object X in M-frame at the angle a from v_{m/n},
v_{m/n}
stands for speed of the frame M with reference to frame N,
_{m/n}
stands for spacetime density at angle a in M- with reference to N-frame, and,
stands for angular speed of the speedv_{x/m} vector in M-frame
Under the circumstances that M-frame moves without acceleration and without rotation with reference to N-frame,
and that acceleration of the object X in M-frame is in line with the speed v_{x/m}, we receive(see
equation 50d) :
(58b)
Applying equations (51) and (52) :
(59)
(60)
In comon motion configuration X/M/N :
(62)
And finally, under the circumstances that M-frame moves without acceleration and without rotation with reference to
N-frame, and that acceleration of the object X in M-frame is in line with the speed v_{x/m}, we receive:
(62)
Remark: Direction of the vector a_{x/m} is defined by the angle
_{nm} according to equation (10).