Physics Help Forum Special Relativity: From an Introduction to Modern Astrophysics by Carroll and Ostlie

 Special and General Relativity Special and General Relativity Physics Help Forum

 Jun 23rd 2011, 12:42 PM #1 Junior Member   Join Date: Jun 2011 Posts: 2 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 $\displaystyle u t_{A}$ 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?
 Jun 23rd 2011, 12:44 PM #2 Junior Member   Join Date: Jun 2011 Posts: 2 Latex doesn't seem to work. So here's the code. I put tex tags instead of math tags. 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 $\displaystyle u t_{A}$ 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?

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