A discrepancy-free transformation of velocity is derived using the Time relativity transformation of coordinates because relativistic transformation of velocity creates a discrepancy. The relativistic transformation of velocity expresses the velocity u2 of an object q in frame 2 in terms of its velocity in frame 1. In frame 2 at time tq, the position of q is xq=u2*tq.
As q departs from the origin of frame 2, we may compute its position by using the time at the origin which is t2o. The so computed position is x2o=u2*t2o. Notice that x2o does not equal xq. So, the relativistic transformation of velocity gives 2 different positions to the object q, which we call the discrepancy of double position. The cause of this discrepancy is that the time at the origin and at the abscissa xq are not equal due to relativity of simultaneity.
Below, we derive a discrepancy-free transformation of velocity. Let us take 2 frames of reference with frame 2 moving at the velocity vo in frame 1. An object q moves at the velocity v2 in frame 2. We will derive the velocity of q in frame 1, which is v1.
The object q moves a distance during a time interval. At the start, q passes by the point x2a in frame 2 which is a fixed point of the x axis of frame 2. x2a coincides with the point x1a which is a fixed point of the x axis of frame 1, see Figure 1. At the end, q passes by the point x2b of frame 2. x2b coincides with the point x1b of frame 1, see Figure 2.
We emphasize that x2a, x2b, x1a and x1b are fixed points on their respective x axes because x2a and x2b move with frame 2 in frame 1, but x1a and x1b stay still in frame 1. x2a, x2b, x1a and x1b do not move with q. Because the point x2a moves with frame 2 at the velocity vo, it arrives at the point x’1a in frame 1 at the end of the time interval, see Figure 2.
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Without length contraction, time relativity transformation solves paradoxes and explains incongruent relativistic experiments, which allows us to build a transformation of coordinates without length contraction. For abscissa transformation, Figure 1 shows a spaceship in the frame of O1, its backend is at O1 and frontend at A1. At time zero the spaceship is stationary, from time zero to time t1, it is accelerated. The trajectories of its backend and frontend are the parallel curves from O1 to O2 and from A1 to A2. At time t1 the spaceship moves at the velocity v, its backend is at O2 and frontend at A2. So, the frame of O2 moves with the spaceship at the velocity v in the frame of O1.
Because the trajectories of the backend and frontend are parallel, the distance O2A2 constantly equals the distance O1A1 which is the proper length of the spaceship. O2A2 is the length of the moving spaceship in the stationary frame O1. So, the length of the moving spaceship constantly equals its proper length.
For Time transformation, The transformation of time from the stationary frame to the mobile frame is like that in special relativity. In Figure 3, at time zero a light signal is sent in the mobile frame from O2 to the mirror M which reflects it back to O2. The time the light signal takes to complete the journey is tO2=2L2/c, with c being the speed of light and L2 the distance from O2 to M. tO2 is also the time of O2. In the stationary frame the light signal is sent when O2 coincides with O1 and goes from O1 to the moving M which reflects it to O2. The time of this journey is tO1=2L1/c, with L1 being the distance from O1 to M. tO1 is also the time of O1. Because L2= L1√(1-v^2/c^2 ), tO1 is related to tO2 by equation (7).
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