Special Relativity: From an Introduction to Modern Astrophysics by Carroll and Ostlie 1. The problem statement, all variables and given/known data
Because there is no such thing as absolute simultaneity, two observers in relative motion may disagree on which of two events A and B occurred first. Suppose, however, that an observer in reference frame S measures that event A occurred first and caused event B. For example, event A might be pushing a light switch, and event B might be a light bulb turning on. Prove that an observer in another frame S' cannot measure event B (the effect) occurring before event A (the cause). The temporal order of cause and effect is preserved by the Lorentz transformation equations. Hint: For event A to cause event B, information must have traveled from A to B, and the fastest that anything can travel is the speed of light. 2. Relevant equations
Suppose $\displaystyle (x_{A},y_{A},z_{A},t_{A}),(x_{B},y_{B},z_{B},t_{B} )$ are the coordinates in the coordinate system S, and $\displaystyle (x'_{A},y'_{A},z'_{A},t'_{A}),(x'_{B},y'_{B},z'_{B },t'_{B})$ are the coordinates in the coordinate system S', which is traveling at speed $\displaystyle u<c$ in the positive $\displaystyle x$ direction. Then from the Lorentz transformation equations,
$\displaystyle t'_{A}=\frac{t_{A}(u x_{A}/c^{2})}{\sqrt{1(u^{2}/c^{2})}}$
$\displaystyle t'_{B}=\frac{t_{B}(u x_{B}/c^{2})}{\sqrt{1(u^{2}/c^{2})}}$
$\displaystyle x'_{A}=\frac{x_{A}u t_{A}}{\sqrt{1(u^{2}/c^{2})}}$
$\displaystyle x'_{B}=\frac{x_{B}u t_{B}}{\sqrt{1(u^{2}/c^{2})}}$
$\displaystyle y'_{A}=y_{A}$
$\displaystyle y'_{B}=y_{B}$
$\displaystyle z'_{A}=z_{A}$
$\displaystyle z'_{B}=z_{B}$
Because B happened after A,
$\displaystyle t_{B} > t_{A}$
Because event A caused event B,
$\displaystyle \frac{\sqrt{(x_{B}x_{A})^{2} + (y_{B}y_{A})^{2} + (z_{B}z_{A})^{2}}}{t_{B}t_{A}} \leq c$ 3. The attempt at a solution
I need to prove that $\displaystyle t'_{B}  t'_{A} > 0$
$\displaystyle t'_{B}t'_{A} = \frac{(t_{B}  t_{A})(u/c^{2})(x_{B}x_{A})}{\sqrt{1(u^{2}/c^{2})}}$
What do I do now?
