Question

Which quantity is invariant-that is, has the same value-in all reference frames? a) time interval, \(\Delta t\) d) space-time interval, b) space interval, \(\Delta x\) \(c^{2}(\Delta t)^{2}-(\Delta x)^{2}\) c) velocity, \(v\)

Short answer

Answer: b) space-time interval

Step-by-step solution

Definition of Lorentz Transformation

Lorentz transformation relates the space and time coordinates of an event as observed from two different inertial reference frames. One frame is moving at constant velocity, v, with respect to the other. The formula for Lorentz transformation of space and time coordinates is given by: \(x' = \gamma (x - vt)\) \(t' = \gamma (t - \frac{v}{c^2}x)\) where \(x'\) and \(t'\) are the space and time coordinates in the moving frame, \(x\) and \(t\) are the coordinates in the stationary frame, and \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\) is the Lorentz factor.

Time Interval

In different reference frames, the time interval between two events can have different values due to time dilation. It is not invariant under Lorentz transformation. \(\Delta t' = \gamma \Delta t\)

Space Interval

Similarly, space interval (distance) between two events can have different values in different reference frames due to length contraction. It is not invariant in different reference frames. \(\Delta x' = \gamma \Delta x\)

Velocity

The velocity of an object is different in different reference frames due to Lorentz transformation, it is not invariant. Transforming velocities from one frame to another require the Velocity-Addition Formula, a consequence of Lorentz transformation.

Space-time Interval

The space-time interval is given by the formula: \(s^2 = c^2 \Delta t^2 - \Delta x^2\) The space-time interval between two events is an invariant quantity under Lorentz transformation, which means it has the same value in all inertial reference frames. \(s'^2 = c^2 \Delta t'^2 - \Delta x'^2 = s^2\) In conclusion, the space-time interval (option d) is the quantity that is invariant or having the same value in all reference frames.