The forward Lorentz transformation is a mathematical expression used in special relativity to relate the coordinates of an event in one inertial reference frame (let's call it the "unprimed frame") to the coordinates of the same event in another inertial reference frame moving at a constant velocity relative to the first frame (let's call it the "primed frame").
The forward Lorentz transformation for time (t') as a function of time (t) and relative velocity (v) between the frames is given by:
t' = γ(t - vx/c^2)
where: t is the time in the unprimed frame (the frame in which the event occurs), t' is the time in the primed frame (the frame moving at velocity v relative to the unprimed frame), v is the relative velocity between the two frames, c is the speed of light in a vacuum, and γ is the Lorentz factor, given by γ = 1 / √(1 - v^2/c^2).
Now, let's consider the scenario you described: a spaceship traveling at close to the speed of light relative to the Earth. We'll take the Earth as the unprimed frame and the spaceship as the primed frame.
If t = 0 in the Earth frame, it means the event being observed (e.g., the spaceship passing by a point on Earth) happens at t = 0 in the Earth's frame. Now, let's examine the situation in the spaceship's frame (primed frame) when t' = 0:
t' = γ(t - vx/c^2) = 0
Since t = 0 and we know v < c (close to the speed of light but not equal to it), the term in parentheses is negative (since -vx is negative) but not zero. Therefore, γ must approach infinity to make the whole expression equal to zero.
This result suggests that, according to the coordinates in the spaceship's frame, the event (e.g., the spaceship passing a point on Earth) occurred infinitely in the past (t' = 0). However, this apparent contradiction with causality (the effect happening before the cause) is a consequence of using the coordinates from two different reference frames that are in relative motion.
It's essential to remember that the Lorentz transformation describes how events and measurements in one frame relate to events and measurements in another frame. These apparent time anomalies arise due to the way time and space are perceived in different frames in special relativity. However, there is no physical time travel or causality violation involved. The laws of causality and physics are upheld within each reference frame independently.
In practical terms, observers on Earth and observers on the spaceship will both experience their respective time correctly according to their own frames, and there will be no actual "time travel" or violation of causality. The Lorentz transformation provides a consistent mathematical framework for describing the relationships between different inertial frames, preserving the fundamental principles of causality and relativity.