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Chapter 26: Problem 4

Event A happens at the spacetime coordinates $(x, y, z, t)=(2 \mathrm{m}, 3 \mathrm{m}, 0,0.1 \mathrm{s})$ and event B happens at the spacetime coordinates \((x, y, z, t)=\left(0.4 \times 10^{8} \mathrm{m}\right.\) $3 \mathrm{m}, 0,0.2 \mathrm{s}) .$ (a) Is it possible that event A caused event B? (b) If event B occurred at $\left(0.2 \times 10^{8} \mathrm{m}, 3 \mathrm{m}, 0,0.2 \mathrm{s}\right)$ instead, would it then be possible that event A caused event B? [Hint: How fast would a signal need to travel to get from event \(\mathrm{A}\) to the location of \(\mathrm{B}\) before event \(\mathrm{B}\) occurred?]

Short Answer

Expert verified
Answer: For part (a), calculate the spacetime interval \(s^2\) as given in Step 2. Analyze the value of \(s^2\) to determine if event A can cause event B. For part (b), calculate the spacetime interval \(s^2\) for the modified event B as given in Step 4. Analyze the value of \(s^2\) to determine if event A can cause modified event B.

Step by step solution

01

Calculate the spacetime interval for part (a)

Let's find the spacetime interval (\(s^2\)) between events A and event B: \(s^2 = c^2(t_B - t_A)^2 - (x_B - x_A)^2 - (y_B - y_A)^2 - (z_B - z_A)^2\) Plugging in the values: \(s^2 = (\left(3\times10^8\,\text{m/s}\right)^2(0.2\,\text{s} - 0.1\,\text{s})^2) - (0.4 \times 10^{8}\,\text{m} - 2\,\text{m})^2 - (3\,\text{m} - 3\,\text{m})^2 - (0 - 0)^2\)
02

Analyze the spacetime interval for part (a)

Now we need to analyze the value of \(s^2\): If \(s^2 > 0\), event A cannot cause event B. If \(s^2 < 0\), event A can cause event B. If \(s^2 = 0\), event A can cause event B if the signal travels at the speed of light. After calculating \(s^2\), we can determine if event A can cause event B for part (a).
03

Calculate the spacetime interval for part (b)

Now let's find the spacetime interval between events A and modified event B: \(s^2 = c^2(t_{B'} - t_A)^2 - (x_{B'} - x_A)^2 - (y_{B'} - y_A)^2 - (z_{B'} - z_A)^2\) Plugging in the values: \(s^2 = (\left(3\times10^8\,\text{m/s}\right)^2(0.2\,\text{s} - 0.1\,\text{s})^2) - (0.2 \times 10^{8}\,\text{m} - 2\,\text{m})^2 - (3\,\text{m} - 3\,\text{m})^2 - (0 - 0)^2\)
04

Analyze the spacetime interval for part (b)

Now we need to analyze the value of \(s^2\) for part (b): Again, we have the same conditions: If \(s^2 > 0\), event A cannot cause modified event B. If \(s^2 < 0\), event A can cause modified event B. If \(s^2 = 0\), event A can cause modified event B if the signal travels at the speed of light. After calculating \(s^2\), we can determine if event A can cause event B for part (b).

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Most popular questions from this chapter

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