A physical mechanism gives (by means of the force f_{m} ) the acceleration am within time tm to the
object of the mass m_{m} in M reference frame. The same physical mechanism applied in N reference
frame gives the acceleration a_{n}=a_{m} at time t_{n}=t_{m}
(by means of the force f_{n}=f_{m} ) to the object of the mass m_{n}= m_{
m} .
(18)
It means, that the same physical actions in different reference frames will cause the same acceleration: The physical actions
mentioned however are not contemporary. Looking from frame N at the object M just accelerating in frame M, we can observe
this action as a con-spacetime event, characterized by the fact that
in general, due to different space and time densities in reference frame M. We obtain:
(19)
or,
(20)
At _{t/m/n} / _{m/n} not changing
with time we obtain:
(21)
The contemporary event's time interval T_{mn} = t_{n} *
_{t/m/n} in M-frame is corresponding to time interval t_{n} in N-frame. If acceleration
a_{n} in N-frame is applied, the object is accelerated to speed vn within time t_{n} .
Looking from frame M, the object is accelerated to speed _{mn} =
A_{mn} * T_{mn} = v_{n }* _{m/n}
within contemporary time T_{mn} . At time t_{m} = t_{n} however,
the speed v_{m} = v_{n} is reached in M-frame: v_{m} = A_{mn} t_{m} = a_{n} t_{m} = v_{n} A_{mn} t_{m} = a_{n} t_{m}
Substituting _{t} for _{n}
, and _{m/n/n} from equation
(12a) for
_{m/n/nm}i n equation
(10), we obtain
where _{nm} now represents an angle in N-frame corresponding to the angle
_{t} in M-frame. Using the equivalent marking
_{t} for this angle, we obtain
(28)
The angle _{t} in M-frame is detected in N-frame as
_{t} and vice versa, in consequence of differrent geometry.
Applying equations (5a) and (5b) we have (see Fig 5):
(29)
(30)
The inequalities (29) show that time and length in a primed (unprimed) frame are contracted (dilated) for the observer from
unprimed (primed) frame at
or, at
The inequalities (30) show that time and length in an unprimed (primed) frame are dilated (contracted) for the observer from
primed (unprimed) frame at
or, at
A bar-stick X_{1}-X_{2} breathing with space, not moving with respect to M-frame, is situated
in this frame at an angle a with reference to a vector of the speed at which M-frame moves with reference to an N-frame (see Fig 6).
The length X_{1}-X_{2} is drawn in units of the N-frame. Fig 6
All points of the bar-stick X_{1}-X_{2} are moving at a speed
_{nm}=v_{m/n} with reference to N-frame, and, the frame N is
moving at a speed (see equation 17)
(_{m/n/} is the spacetime density at an angle
)
with reference to frame M.
At an angle _{nm} (in N-frame) however, all bar-stick points are moving at a
speed _{nm}=v_{m/n}Cos(
_{nm}) in direction of the bar-stick line with reference to N-frame, and frame N is moving at a speed
in this direction with reference to frame M. Spacetime density
_{m/n/(+)}
stands for density at angle + :
The length of the bar-stick in N-frame:
The con-spacetime length of the bar-stick in M-frame:
We can see that L_{mn} < L_{nm} for Cos > 0 .
In Fig6: L_{mn} = MX_{2} = NX_{3} = L_{nm}. This is for
the reason only that the bar-stick X_{1} - X_{2} in M-frame is drawn in units of the N-frame.
In fact (L_{mn} = MX_{2}) < (L_{nm} = NX_{3}).
Space and time densities
Designating M -frame space density as
and taking into consideration that spacetime is symetrical with reference to the axis x, we obtain
(31)
The equation (31) is valid for
(31a)
Substituting Cos_{nm} = 0 for
_{nm} = /2 in equation (10), we can modify
the interval of the validity for equation (31):
(31b)
We also have:
acc. to equation (25),
Designating M- frame linear time density as
and, taking into consideration that spacetime is symetrical with reference to the axis x, we obtain
(32)
where the variable is defined acc. to equations (31a) and (31b).
Obviously the linear time density is not only difficult to understand, but also to measure. The clocks used in our primed
system do not distinguish the anisotropic behaviour of the time flow. This is why we may assume that these clocks are
showing the average time. We may designate the primed frame average time density as
(33)
Solving the integral in eq. (33) we receive
(34)
Length and time contraction and dilation
Assuming N-frame reference point is passing point X_{2} in the very same contemporary moment
when M-frame reference point is passing point M, we can find out, that within contemporary time interval (ending at the same
contemporary moment in both frames)
when the N-frame reference point is just passing point M (still continuing on its way through the bar-stick)
the M-frame reference point is just passing the point X_{2} (just leaving the bar-stick)
The reason is that the bar-stick is breathing with spacetime, and consequently it is dilated in M-frame, since the spacetime
structure in M-frame is dilated due to lower space and time densities.
In fact, if the bar-stick length in N-frame is taken as l_{n} = v_{n}t_{n} =
v_{m/n}t_{n} , then real bar-stick length ( not con-spacetime) in M-frame is the same:
The length l_{m} observed from frame N can be expressed as
(36)
where _{s/m/n} is defined by equation (31) and
is defined by equation (31b).
Applying equation (32), we have
(37)
where _{t/m/n} is defined by equation (32).
Respecting fact, that space and time densities acc. to equations (31) and (32) are lower than 1, the equations (36)and (37) say,
that
The dimensions and time, measured by the instruments breathing with spacetime, are dilated with reference to the frame own
dimensions and time. It means, that the dimensions of the moving object are detected longer from our frame, than from moving
object own frame. And, that the clock situated in a moving object own frame are lagging with respect to the clock in our frame.
As observers from any inertial reference frame, we are detecting, as contemporary, the events from the past of all reference
frames moving with respect to our frame.