Relativity - Can the order of event be changed - Causality question

Let us consider a rest frame where two events A and B happened at the same point. The event A happens earlier than event B i.e., mathematically,  (\( t_A\) < \(t_B  \)).

The question is: Can you have a frame moving with velocity \( v \) where event B happens earlier than event A i.e.  (\( t_A\) > \(t_B  \))?

(An exam question from a European university where A. Einstein studied and H. Minkowski

 was the professor, and called Einstein a "lazy dog" 😄)

I will answer in three different ways: (1) Melting of ice (2) Lorentz transformations (3) Minkowski diagram

Here is my answer 1, by Gedanken experiment - melting of ice:

Let us have a thought experiment (gedanken experiment) where a ice cube in a glass melts into water. Here we define event A when water exist as ice and event B when all the ice melts. So in rest frame  \( t_A\) < \(t_B  \)). 

Do you think you can have a frame where the event B happens before event A? i.e., water turning into ice cube? Obviously not, it would violate second law of thermodynamics, which says disorder (entropy) should always increase.

Here is a standard answer 2, by Lorentz transformations:

Lorentz transformations for time is:

$$ t'_A = \gamma (t'_A-v x_0/c^2) \quad t'_B = \gamma (t'_B-v x_0/c^2) $$

Where the position \( x_0 \) is same for both events A and B

Subtracting them we get:

$$ t'_A - t'_B = \gamma (t'_A-t'_B) $$

Therefore if \( t_A-t_B < 0 \) implies \( t'_A-t'_B < 0 \). i.e., If event A happens before event B in rest frame at the same place, then the same happens in all other frames. The time interval between A and B changes by factor of \( \gamma \), but the order cannot be changed!

Here is answer 3, by Minkwoski diagram:

Here I will only prove that the order of event A and B can be changed!! if the two event happend at different position. It will be for you to prove that if we graduall make the position closer the order of event cannot be changed. I will take the diagram from this great web site: https://tikz.net/relativity_minkowski_diagram/ that has very many Minkowski diagrams. 

The above diagram shows indeed in some circumstances the order of the event can be can changed in the moving frame w.r.t. a rest frame, but only when two event happens are seperated by fairly large distance. It will never happen if the two events are brought closer so that they happen at the same point as seen in rest frame. 

Hope you have enjoyed the question! 😊

THE END