Solution Found!
Suppose that a point P in spacetime with coordinates x =
Chapter 15, Problem 15.39(choose chapter or problem)
Suppose that a point P in spacetime with coordinates x = (x, x4) lies inside the backward light cone as seen in frame S. This means that x - x < 0 and x4 < 0 at least in frame S. Prove that these two conditions are satisfied in all frames. Since this means that all observers agree that t < 0, this justifies calling the inside of the backward light cone the absolute past.
Questions & Answers
QUESTION:
Suppose that a point P in spacetime with coordinates x = (x, x4) lies inside the backward light cone as seen in frame S. This means that x - x < 0 and x4 < 0 at least in frame S. Prove that these two conditions are satisfied in all frames. Since this means that all observers agree that t < 0, this justifies calling the inside of the backward light cone the absolute past.
ANSWER:Step 1 of 3
If \(x \cdot x<0\) in frame \(\mathcal{S}\), then \(x^{\prime} \cdot x^{\prime}<0\) in any other frame \(\mathcal{S}^{\prime}\), since \(x \cdot x\) has the same value in all frames.
The condition \(x \cdot x=\mathbf{x}^{2}-x_{4}^{2}<0\) implies that \(|\mathbf{X}|<\left|x_{4}\right|\).