4 Relativistic Addition of Velocities

It appears that the Euclidean approach as used in the previous Section does not yield the same equation for relativistic addition of velocities as used in special relativity. Although this particular point may be a serious obstacle to the acceptation of this proposal, it obviously is necessary to point it out.

Figure 3 depicts a situation with three reference frames: a stationary unprimed frame $x$ , a moving primed frame $x'$ and a third, double primed frame $x''$ of an object that moves relative to both other frames, $x$ and $x'$ . Each frame has dimensional axes rotated relative to the other frames as a result of the relative motion.

Figure 3: Relativistic addition of velocities in three reference frames, each with rotated dimensional axes relative to each other.

• Vector $\mathbf{V}$ of magnitude $v$ is the spatial velocity of an observer with rest frame $x'$ as measured by an observer with rest frame $x$ .
• Vector $\mathbf{W}$ of magnitude $w$ is the spatial velocity of a third object as measured by the observer with rest frame $x$ .
• Vector $\mathbf{U}$ of magnitude $u$ is the spatial velocity of that same object but now as measured by the observer with rest frame $x'$ .

When $u$ , $v$ , and $w$ are parallel, the classical relation between them is:

$$w=\frac{u+v}{1+uv/c^2}$$ (18)

If we apply the approach as used consistently until now it yields the expression:

$$w = c\cos (-\alpha )=c\sin (\frac{1}{2} \pi +\alpha )$$

$$= c\sin( \beta +\varphi )=c( \cos \varphi \sin \beta +\cos \beta \sin \varphi )$$

$$= u\sqrt {1-v^2/c^2} +v\sqrt {1-u^2/c^2}$$ (19)

This expression is not nearly similar to the classical expression in Eq. (18).

Like Eq. (18), Eq. (19) still limits the speeds as measured by both observers to the maximum of $c$ , which is also clear by inspection of the Figure. Some remarks will be made now on the probability of either of the equations to be the right one:

1. Equation (18) is in fact based on the universality of light speed and the basis for reasoning is that an object, e.g. a photon, having speed $c$ for an observer in frame $x$ will still have that same speed $c$ for an observer in frame $x'$ . This is one of Einstein's original postulates and also in this Euclidean approach it will still be maintained as a valid postulate, which essentially means that the photons velocity vector, as measured from the moving frame, must have rotated along with that frame. The third object, having speed $w$ , as measured from frame $x$ , is not a photon but a mass-carrying particle for which such a rotation apparently does not apply. It must therefor be emphasized that Eq. (19) for now may only be applied to mass-carrying particles.

2. Equation (18) shows a discontinuity that is unusual in physics. In Fig. 4, Eq. (18) is plotted for the situation where $u$ always equals $v$ .

   Figure 4: Graph of classical equation for relativistic addition of velocities.

   

   With $u$ and $v$ nearing $c$ , the resulting $w$ will also near $c$ , which is in accordance with the classical view. But if (as a matter of mathematical experiment) the range of $u$ and $v$ is extended beyond the maximum value of $c$ then the plot looks like depicted in Fig. 5.

   Figure 5: Classical graph for relativistic addition of velocities with hypothetical (superluminal) extensions.

   

   The part from Fig. 4 can still be recognized but it is clear now that this actually forms part of a continuous function that extends beyond $c$ . The part beyond $u=v=c$ may not be used, solely because the classical function is not defined, nor ever shown to be valid, for such superluminal extensions (actually the space-like quadrants in the classical light cone). This fact strongly suggests that the graph from Fig. 4 is an approximation of the real function.

   Finally, both Eqs. (18) and (19) are plotted together in Fig. 6.

   Figure 6: Classical [Eq. (18)] and newly derived graph [Eq. (19)] for relativistic addition of velocities plotted together for $u=v$ .

   

   Equation (19) is almost identical for speeds below about $c/2$ but begins to deviate at higher speeds. The top of Eq. (19) corresponds to $u=v=c/\sqrt{2}$ . From the circle diagram in Fig. 3 it shows that the time-speed of the object, as measured from frame $x$ , then becomes zero. Equation (19) further shows decreasing values for $w$ in situations where the values of $u$ and $v$ go beyond $c/\sqrt{2}$ (the frame of the moving object then rotates beyond $\pi/2$ relative to frame $x$ ). It turns out that in that case the corresponding time-speed for the object becomes negative. (This situation might be related to anti-particles, running 'backwards in time'.).

   The situation where $u$ equals $v$ gives the maximum possible deviation relative to the classical graph. Other ratios between $u$ and $v$ give (much) smaller deviations and the tops of Eq. (19) will shift outwards towards $c$ as can be seen in Fig. 7 where the ratio between $u$ and $v$ equals 3:1. At a ratio 10:1 both plots are practically identical. Virtually all practical situations that require the velocity addition formula to be used exist under such circumstances, which indicates that a deviation from the classical graph is likely to remain unnoticed.

   Figure 7: Classical [Eq. (18)] and newly derived graph [Eq. (19)] for relativistic addition of velocities plotted together for $u=v/3$ .

   

3. Some interpretations of Fizeau's experiment give rise to doubt concerning the correctness of Eq. (18). If Eq. (19) is used in the analysis of Fizeau's experiment done by Renshaw [7], it yields better results than Eq. (18), although still not within the margins as claimed by Michelson.

   The vast majority of experimental set-ups that are aimed at verification of relativity theory are using two reference frames. These experiments are not suitable for the verification of the velocity addition formula. One would have to use a set-up with three reference frames. At speeds on the order of $10^4$ m/s the difference in resulting values between Eqs. (18) and (19) is on the order of $10^{-5}$ m/s, which might be noticeable using adequately accurate measuring devices.

A hypothetical case will now be used to show that Eq. (19) does not necessarily lead to causality conflicts as a result of the negative time-speeds that can occur.

A spaceship travels relative to Earth at speed $v_s=0.9c$ and heads toward an asteroid that is at rest relative to Earth. The ship launches a missile at the asteroid at $v_m=0.9c$ relative to the ship. An observer on the ship watches the missile destroy the asteroid. According to Eq. (19), an observer on Earth would see the missile traveling at only $0.7846c$ so the missile's spatial speed is lower than that of the spaceship. It seems therefor that this observer would see the ship hit the asteroid before the missile.

The explanation of this paradox can be found in the comparison of the proper times of all objects involved. We call the proper time for the spaceship $\tau_s$ and for the missile $\tau_m$ . For simplicity we set the space-time event of the launch at $t = \tau_m = \tau_s = 0$ and the distance between the spaceship and the asteroid at that moment at $0.9$ light second (as measured by the observer on Earth).

The observer on Earth calculates time-coordinates of the impact (against the asteroid) using his own time $t$ for the spaceship: $t_s=1$ s; and for the missile: $t_m=0.9/0.7846=1.147$ s, so it seems as if the spaceship reaches the asteroid first. In 4D Euclidean space-time however the observer measures the time-speed $\chi_s$ of the spaceship as: $\chi_s = \sqrt{c^2-v_s^2} = \sqrt{c^2-(0.9c)^2} = 0.4359c$ .

According to this observer the absolute value of the timespeed $\chi_m$ of the missile is $\chi_m=\sqrt{c^2-(0.7846c)^2} = 0. 62c$ , but from the circle diagram (Fig. 3) it shows that we must now take the negative root so its value is $\chi_m=-0. 62c$ . Note that the cyclic nature of $\gamma$ now also implies that in this situation $\gamma$ has a negative value in $\tau_m=t_m/\gamma=t_m\chi_m/c$ for the missile.

We calculate the proper times at the moment of impact according to the observer on Earth for the spaceship: $\tau_s = t_s\chi_s/c = 0.4359$ s; and for the missile: $\tau_m = 1.147(-0. 62) = -0. 7111$ s.

In proper time the missile hits the asteroid before the spaceship does despite its lower spatial speed. Causality is therefor not violated. The missile runs backwards in proper time.