# Understand result of Lorentz transform

Assume an event that happens at P=(ct:2,x:4) in some inertial frame of reference S. Assume a second inertial frame S' in standard configuration and $$\beta=4/5$$ ($$\gamma=5/3$$). The Lorentz transform is:

$$P'=\frac{5}{3}\begin{pmatrix} 1 & -4/5\\ -4/5 & 1\end{pmatrix}\begin{pmatrix} 2\\ 4\end{pmatrix}=\begin{pmatrix} -2\\ 4\end{pmatrix}$$.

That means that the event is perceived in S' before t=0. When S' pass by (ct:0,x:0) it could "say" to S the information about the event. In this way, S knows its future.

An error in my calculations or an apparent paradox ?

The mistake in your reasoning is to assume that because an event happens somewhere in S' before t'=0, it means that someone at the origin of S' would know about it at t'=0, and could pass that information to someone in S when the origins of the two frames pass. In both S and S', the event is so far from the origin that no information about the event could reach the origin in 2 seconds.

Information can never travel faster than light. You can draw a light cone eminating from the origin and the point $$P$$, both before and after the transform. Since $$P$$ is outside the lightcone of the origin it can't communicate anything with the origin.

In relativity causality means that if a point $$P$$ is outside a light cone it is impossible to get it inside the light cone with a Lorentz transformation.

Note that OP is a spacelike vector, where O=(0,0).

The temporal ordering of events O and P are not preserved by the boost.

This is how causality gets preserved in special relativity, even when you mess with space and time. Timelike seperated events have a well-preserved time ordering. Spacelike-seperated events do not. Since your event is spacelike seperated from the origin, whether it happens before or after $$t=0$$ is not preserved by a boost.

This is also the heart of the "you can't travel faster than the speed of light" thing -- faster than light travel is moving along a spacelike line, and there are observers that would parse this travel as moving into the past. The line is not time-orderable. Travelling faster than light isn't just "hard" or "forbidden", it is meaningless in special relativity -- "superluminal motion" is the same thing in the theory as "a spatial seperation in some reference frame".

An event being simultaneous to observer S' doesn't mean that the observer perceives it at that instant in time; for S' to become aware of it, a signal of some sort has to travel from the event to S', and this can only happen at the speed of light, or slower.