Faster than ‘c’

Copyright © Rahul Shankar Bharadwaj 2025 

Chapter 4

Faster than ‘c’

A famous conundrum related with the imaginary numbers with ‘i’ is the ‘Lorentz factor’. Since the values obtained for speeds greater than ‘c’ are complex numbers of the form p/qi , hence it has been deduced that the speeds greater than ‘c’ are not possible. However earlier we saw that ‘i’ only denotes inversion or reversal in the orientation of the numerical quantity. So, what do we mean by inversion of quantities at speeds greater than ‘c’?

Before moving ahead with problem of Lorentz factor, let us further try to understand inversion or reversal with the example of cars moving on a highway.

Imagine a highway where all cars except ours are moving at 100 kms/hr. Our car is stationary; therefore, we see the speed of other cars as +100 kms/hr or −100 kms/hr depending on their direction of motion. Another point to be noted is that cars moving in both directions are crossing us with their bumper first and tail latter, which is the normal orientation of cars for us.

As we begin to accelerate in positive direction, till the time our speed is less than +100 kms/hr, the cars moving in positive direction of motion keep crossing us from behind in same orientation and we see their speed as positive though the relative magnitude of their speed is decreasing. As our speed increases beyond +100 kms/hr, we see their speeds as negative, and they begin to cross us from the front but with their tail first and bumper latter. This is the flipping or inversion or reversal of orientation of the cars for us. On the contrary the cars moving in opposite direction with negative speeds still cross us from front with their bumper first and tail latter i.e. the normal orientation.

Coming back to Lorentz factor; the most important assumption in the theory of special relativity is that the speed of light in vacuum i.e. ‘c’ is constant for all observers irrespective of their relative motion. Now which numerical quantity of speed on number loop can appear same to all observers irrespective of their relative motion? I am not sure about ‘c’ but ∞ speed will certainly appear same to all observers, because earlier we have already seen in chapter 1, that at infinite speed, an object can exist at all points in space at the same given time. Hence, the stipulation conveys to mathematics that ‘c’ is ∞. Accordingly, the mathematics yield results assuming ‘c’ to be ∞. Therefore, when we input the value of speed greater than ‘c’, the Lorentz factor yields values in the form p/qi , which can also be written as -(p/q )i. This form of number belongs to the numbers ahead of +∞ on number loop which means that there is reversal or inversion in the orientation of quantity. Thus, as per what we have seen in this book so far, it is not the end of the road; the quantity does not cease to exist but only get reversed and inverted from our perspective. How does this reversal or inversion in orientation play out in case of speeds greater than ‘c’?

We will try to unravel this problem by another thought experiment. Let us assume that we are located at a certain distance from earth and watching the events on earth separated by a certain time in the sequence E1, E2 and onwards as shown in figure below. These events are visible to us in the form of light packets coming to us from the earth at speed ‘c’.

Figure 4.1

Now, if we start accelerating away from earth, the events will be visible to us in the same sequence but the inter se time between two events will increase because of decrease in relative speed of light with us. This is the earth’s time dilation experienced by us when attaining speeds comparable to ‘c’ which confirms to the time dilation obtained through ‘Lorentz factor’.

As we will keep accelerating, we will reach the speed equal to ‘c’. Suppose we attain this speed when E4 event packet just reached us, hence we will start travelling parallel to E4 event packet of light. Thus, E5 event packet will never reach us and as per our observation, life on earth will freeze at event E4. This means, for us, the time on earth will become infinite. The Lorentz factor also gives us the same value i.e.

1/√[1- (c^2/c^2)] = 1/√(0) = 1/0 = ∞

Figure 4.2

As we accelerate further and our speed becomes greater than ‘c’, we will start approaching the event packets E1 to E3 from behind in the sequence E3, then E2 and finally E1. This will make us see the events on earth in reverse order or we will experience negative time on earth. The Lorentz factor yields value in the form −(p/q )i for speed greater than ‘c’. What it entails is that the time dilation on earth from our perspective is now in 'i' dimension. That means, we will see events on earth in reverse order i.e. reversal of time on earth or going back in time denoted by the negative sign in the result obtained from Lorentz factor. Also, since the light is now being received from the direction opposite to earth, so we will see the image of earth in the opposite direction! This is the inversion of orientation denoted by ‘i’.

Figure 4.3

It will be further interesting to note that while moving at speed faster than 'c' away from earth, the earth will appear in front of us and will appear to be moving further away as we advance!

Is faster than ‘c’ speed possible in real world? We will try to explore its feasibility in the next chapter.

Chapter 5: 'i' Dimension Possibilities

Copyright © Rahul Shankar Bharadwaj 2025