Special theory of relativity

Now, we set out to compare the values of second and third row to see the effect of change in the position of event while keeping the time of event same. In the relativistic transformation, we see that time value changes just because there is spatial change in stationary reference. The time values are 0.0000052393 s and 0.0000051366 s for x = 2 m and x = 100 m respectively. This is yet another dimension of relativistic kinematics. This suggests that space and time are entangled. Individual measurements of event parameters do not only change because of relative speed (It is a foregone conclusion in relativistic kinematics). The additional point here is that time value changes simply because of change in space value (x) even when the relative velocity is kept constant.

We conclude thus :

1: Spatial and temporal values in the inertial references are different on account of relative velocity.

2: The temporal (t) values are dependent on spatial values (x,y,z). The space and time specifications of an event are entangled.

The spatial dependence of temporal parameter, as a matter of fact, is also evident from the relativistic time relation :

$$t\prime =\gamma \left(t-\frac{vx}{c^2}\right)$$

Note the presence of spatial parameter (x) on the right hand side of the equation. Clearly, spatial and temporal values of an event are entangled.

Velocity addition

Let us consider a scenario of police car chase. The police personnel fires a shot in the direction of the criminal’s car speeding ahead. What is the velocity of the bullet? It depends on the observer. The bullet moves from the gun, which is stationary in the reference of police car. The velocity of bullet will be as per the specification of the gun. This will be the same velocity as when fired from ground. Let this velocity be u'. Clearly, this velocity is inherent to the gun irrespective of its motion. If “v” be the velocity of the police car, then according to Galilean transformation, the velocity of bullet in the ground reference, "u", is obtained by addition of two velocities :

u = v + u’

When a javelin is thrown, the velocity of javelin is sum of the velocities of the javelin thrower and the javelin itself with respect to thrower. The very idea of thrower to throw javelin while running is to leverage his/her velocity towards increasing the velocity of javelin in ground reference. Also, consider motion of two cars which are moving towards each other with velocities u and w along a straight line. The velocity of approach for any of two cars is u+w. These are well established results which are outcome of non-relativistic Newtonian kinematics.

The seemingly well defined algorithm about algebraic addition of velocity runs into serious problem when we extend this concept to high speed cases. Let us do a bit of “Gedankenexperimenten”. Let two spaceships are approaching with a velocity c/2 and 3c/4. What is the velocity of approach? Clearly, it is c/2+3c/4 = 5c/4. This is a speed which exceeds the speed of light. Take another example. Let us consider that we move to a light source like a bulb in our house. Of course, we imagine that there is no atmosphere. The speed of approach here is speed of light plus our speed of approach – again exceeding the speed of light and also rendering it a variable dependent on our motion.

On the other hand, electromagnetic theory specifies the speed of light in vacuum to be exactly c. Not only that, Michelson's experiment concluded that speed of light is a constant in all directions. Lorentz experiment proved that speed of light in vacuum is independent of motion of source emitting it. We, therefore, deduce Galilean or non-relativistic addition of Velocity is not true at high speed or in the relativistic context. This is one aspect of relativistic consideration. The other aspect emerges from special theory of relativity which embodies Lorentz transformation. Let us explore Lorentz factor :

$$\gamma =\frac{1}{\sqrt{\left(1-\frac{v^2}{c^2}\right)}}$$

When v->c, γ ->∞. For v=0.999999999c, γ= 22360. When v=c, γ is undefined (a ratio with zero as denominator). When v>c, γ is imaginary. Clearly, speed of light (electromagnetic wave) in vacuum is the highest speed in nature. Matter can not achieve the speed of light. There is no question of exceeding it. This aspect of speed limit for matter has been verified experimentally, wherein it is found that a particle acquires greater relativistic mass instead of gaining speed even when it is accelerated to achieve or exceed speed of light (by continuously imparting energy). We shall describe this experiment after we have discussed about relativistic mass. We should, however, treat this violation of addition of velocity as one of the key experimental evidence that gave strong credence to special theory of relativity in the initial years.