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Old Jun 23rd 2011, 12:42 PM   #1
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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?
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Old Jun 23rd 2011, 12:44 PM   #2
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Join Date: Jun 2011
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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 $\displaystyle u<c$ 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?
omoplata is offline   Reply With Quote
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