Abstraction

After examining a rotating clock with the time displayed as the angle of an arm rotating in a plane perpendicular to its constant speed motion in an inertial reference frame, the displayed time of the clock (i.e. the angle of the arm) is found to be an invariant of Lorentz Transformation. This can be further generalized to all physical clocks including atomic clocks. Since the time of Special Relativity is not an invariant of Lorentz Transformation, the displayed time of any physical clock can never be used as the time of Special Relativity and can never show time dilation.

Introduction

In Special Relativity, the time of a reference frame and the displayed time of a clock attached to the reference frame are two completely different things. A clock itself is a system of motion that should follow the laws of motion and Lorentz Transformation, rather than a pure time keeper which can record time directly. Instead, a clock can only record the result of a physical process (e.g. the angle of a rotating arm, the number of cycles of oscillation, etc) during a period of time. The result of a process is usually the multiplication of time and the speed of the process (e.g., the rotating speed of the arm or the frequency of oscillation, etc). After Lorentz Transformation, though time will increase by a factor, the speed of the process will decrease by the same factor, which makes the final multiplication unchanged after Lorentz Transformation.

Derivation

Assume a clock moving at a constant speed v along the x-direction of an inertial reference frame called Frame A while the frame attached to the clock is called Frame B. In the following, all variables of Frame B will carry apostrophe ( ' ) to distinguish them from those of Frame A. The clock uses the angle of its arm to represent its displayed time, which rotates at a constant speed ω' in the plane perpendicular to the clock's moving direction.

At the event that the clock passes the origin of Frame A, the location of the clock is at

(1) x_{1}= x’_{1}= 0

where x_{1 }and x'_{1}are the coordinate of the clock in Frame A and Frame B respectively, and the arm of the clock points at 0 degree:

(2) α_{1}= α'_{1}= 0

which represents zero time of both Frame A and Frame B:

(3) t_{1}= t’_{1}= 0

At a new event:

(4) t_{2 }= τ

(5) x_{2 }= vτ

(6) α_{2 }= ωτ

in Frame A, the corresponding variables in Frame B after Lorentz Transformation:

(7) x’_{2}=γ(x_{2}– vt_{2}) = 0

(8) t’_{2 }=γ(t_{2}– vx_{2}/c^{2}) = τ/γ

(9) y'_{2}= y_{2}

(10) z'_{2}= z_{2}

whereγ= 1/(1– v^{2}/c^{2})^{1/2 }and (y_{2}, z_{2}) and (y'_{2}, z'_{2}) are the coordinates of the tip of the arm in Frame A and B respectively. Here are the relationships between the angle and the coordinates:

(11) tan(α_{2}) = y_{2}/z_{2}

(12) tan(α'_{2}) = y'_{2}/z'_{2}

From Equation (9), (10), (11) and (12), we have:

(13) α'_{2 }= α_{2}

Since the displayed time of the clock has been calibrated to the angle of the arm of the clock, Equation (13) tells that the displayed time of a rotating clock is an invariant of Lorentz Transformation. Therefore, we will never see time dilation shown by the displayed time of a rotating clock. Many people including Einstein believe that a moving clock will display a time different from the displayed time of a static clock.This mistake is caused by the belief of people including Einstein that all clocks can directly record time. The fact is that any physical clock can only record the result of a process during a period of time, but not time itself. The result of a process is usually the multiplication of time and the speed of the process. After Lorentz Transformation, time changes by a factor, but the speed of the process will also change which will exactly cancel the effect of the change of time and make the final multiplication unchanged after Lorentz Transformation. This can be illustrated in the following:

Since

(14) α'_{2 }= ω't'_{2}

from (4), (6), (8),(13) and (14), we get:

(15) ωτ = ω'τ/γ

That is

(16) ω' =γω

Equation (8) shows there is a time expansion after Lorentz Transformation from Frame B to Frame A, but Equation (16) shows there is a slowdown of the rotating speed of the arm of the clock from Frame B to Frame A. The angle of the arm (i.e. the multiplication of time and the rotating speed) remains unchanged after LorentzTransformation. Since the angle of the arm of the clock represents the displayed time of a clock, therefore, the displayed time (for example, 30 degrees: one o'clock, 60 degrees: two o'clock, 90 degrees: three o'clock, etc) remains unchanged after Lorentz Transformation.

Discussion

Since any clock can display its time as the angle of an arm through either mechanical gears or a digital converter, the above derivation can be logically extended to any physical clock, even to an atomic clock on which the displayed time is the number of cycles of oscillation (i.e. the multiplication of time and frequency of the oscillation) and remains unchanged after Lorentz Transformation because the expansion of time cancels the slowdown of the frequency in the multiplication. Therefore, time dilation will never be noticed on all clocks in Special Relativity.

Some people argue that Special Relativity has been mathematically proved without any contradiction.This may be true. The problem of Special Relativity is not in its mathematical derivation. The problem of Special Relativity is that it is not a mathematical theory, but a theory of physics that has to be connected to the physical world. Einstein made an assumption that the time in Special Relativity is the time measured by a clock, which has led to the contradiction as I illustrated above.

Conclusion

Though Special Relativity is beautiful in mathematical formulation, it remains a mathematical theory without any way to land on the physical world as its time can't be measured by any clock. All its predictions are pure imaginations. There is no such thing called time dilation in the physical world. People will never be able to travel to the past or future.