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Chapter 33: Problem 51

A light beam is emitted at event A and arrives at event B. Show that the spacetime interval between the two events is zero.

Short Answer

Expert verified
The spacetime interval between two events involving the emission and arrival of light is zero.

Step by step solution

01

Recall the spacetime interval formula

In special relativity, spacetime interval, \(I\), between two events is given by the equation, \(I^2 = \Delta x^2 - c^2 \Delta t^2\), where \(\Delta x\) is the spatial separation, \(c\) is the speed of light, and \(\Delta t\) is the time separation.
02

Apply light travel condition

For a light beam being emitted from event A and arriving at event B, the spatial separation, \(\Delta x\), is exactly equal to the product of speed of light and time taken i.e., \(\Delta x = c \Delta t\). This is because in this case, light is moving in space so the distance travelled by light is the speed of light times the time it travels.
03

Substitute into the spacetime interval equation

Substitute \(\Delta x = c \Delta t\) into the spacetime interval equation, we get \(I^2 = (c \Delta t)^2 - c^2 \Delta t^2 = 0\). Therefore, the spacetime interval between two events involving light's travel is indeed zero.

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