Two frames S and S' are in relative motion. The frame S' is moving with speed c/2, where c is the speed of light. In the frame S two events occur at (\(x_{1}, t_{1}\)) and (\(x_{2}, t_{2}\)). In the frame S', these events occur simultaneously. Then the value of \((x_{2} - x_{1})/(t_{2} - t_{1})\) is :
MH SET - 2024, SET Exam - 2024,
Answer: B
The ratio is \(2c\).
By the Lorentz transformation, the time interval in S' is:
\[ \begin{aligned} \Delta t' &= \gamma\left(\Delta t - \frac{v\,\Delta x}{c^{2}}\right) \\ &= 0 \\ \frac{\Delta x}{\Delta t} &= \frac{c^{2}}{v} \\ &= \frac{c^{2}}{c/2} \\ &= 2c \end{aligned} \]
A ratio greater than \(c\) is expected: events simultaneous in another frame must be space-like separated in S. \(c/2\) is the frame speed itself, and \(c/4\) and \(4c\) do not satisfy \(\Delta t' = 0\).
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