Tolman’s Paradox

Physics binds the happenings in our Universe with limitations. Number of physical quantities are limited to a certain extent, which leads to “impossibilities”. The best example of such limitations, is velocity. Nothing in this universe, can travel faster than speed of light.

But why?

The statement “speed of light is the limit”, is not a claim, but rather a fact. It’s as good as saying one can’t go any more north than the North pole. The definition of speed itself is maxed out by the speed of light. To explore how, let us consider a miracle vehicle which can travel with the speed of light. It possesses the ability to generate constant thrust and accelerate for enough amount of time to reach the speed of light. Now, although the passenger in the vehicle will experience the constant acceleration, for the observer on Earth, the vehicle will appear as if its acceleration decreases asymptotically towards zero, which is explained by Time Dilation. This asymptomatic speed limit is independent of the acceleration we choose. This limit represents that for the observer on Earth, the clocks on the vehicles have stopped, and the passengers do not experience time.

However, what if a particle did exist, which can travel faster than light?

In September 2011, it was reported that a tau neutrino had travelled faster than the speed of light in a major release by CERN; however, later updates from CERN on the OPERA project indicate that the faster-than-light readings were resultant from “a faulty element of the experiment’s fibre optic timing system”. This release caused a stir among physicists all over the globe, and the possibilities with the existence of such a particle were discussed. The hypothetical particle was named “Tachyon”. One such possibility, was a device that could be used to communicate with signals sent at a speed faster than light, named “Tachyonic Antitelephone”.

1

One of the consequences of such a device is known as “Tolman’s Paradox” (although originally the thought experiment was done by Einstein and Sommerfield, and is known as “Causality Paradox”). A two-way variation of the original thought experiment is discussed below.

Consider two passengers A and B travelling in space with a relative velocity of 0.8c. At some point of time, they pass right next to each other, and that point is referred as x=0, t=0(A’s frame of reference) and x’=0, t’=0(B’s frame of reference). In their respective frames of reference, both A and B are stationary at x=x’=0.

In A’s frame of reference, B moves with 0.8c in positive X direction, where as A moves with 0.8c in negative X direction in B’s frame of reference. Both have devices capable of transmitting signals with a speed of (say) 2.4c with respect to the respective senders.

When A’s clock shows that it has been 200 days since they both met, he decides to send a message saying “I am bored” to B, using the Tachyonic Antitelephone. After 300 days in A’s frame, the message should reach B ( [0.8c x (200+t)] = 2.4c x t).

Now, since B is aging slowly than A because of time dilation (by a factor of (1-v^2/c^2)^1/2=0.6 here), B receives the message 0.6×300=180 days after meeting A. As soon as B receives the message, he immediately sends a message back to A saying “Let’s play a game”.

The message, in B’s time frame, should reach A after 180+90=270 days. Now, since time dilation is symmetrical for both observers, A will receive the message only after 0.6×270=162 days since they met.

The message, has indeed travelled back in time, and has reached A even before she sent the first message. Now, since A plays a game, she would not be bored, and would never feel the need to send a message in the first place. Thus, the causality of the event has been violated. The numbers can be double checked using Lorentz Transformations as well.

2

 

3

We conclude, that not only is it impossible for a particle to travel faster than light, existence of such a particle would result in some fascinating and mind-bending consequences.

Who knows, what surprises the Universe holds for us!

Anand Pathak
Second Year, Mechanical Engineering

 

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

%d bloggers like this: