Question

events occur at the same place in an inertial reference frame S and are sepa- rated in time by an interval of 4 s. What is the spatial separation between the 2 events in an inertial reference frame in which the events are separated by a time interval of 6 s

Answer 1

The spatial separation between the 2 events is 13.416 × 10⁸ m

In space time-interval, the invariance of line element explains that if there are two inertial reference frames S and S', the spatial separation is invariant in all inference frames.

i.e.

$\mathbf{\Big [ [\Delta x]^2 -[c^2 \Delta t ^2] \Big]_{frame \ 1} = \Big [[\Delta x]^2 -[c^2 \Delta t ^2] \Big]_{frame\ 2} }$

$\mathbf{[\Delta x_1]^2 -[c^2 \Delta t_1 ^2] = [\Delta x_2]^2 -[c^2 \Delta t_2 ^2] }$

where;

• Δx₁ = 0  (since it occurs at same place)
• Δt₁ = 4 s
• Δt₂ = 6 s

$\mathbf{[0]^2 -[c^2 (4) ^2] = [\Delta x_2]^2 -[c^2 (6) ^2] }$

$\mathbf{ [\Delta x_2]^2 =36(c^2) - 16(c^2)]}$

$\mathbf{ [\Delta x_2]^2 =20(c^2)}$

$\mathbf{ \Delta x_2 = \sqrt{20} \ c}$

here;

• c= speed of light = 3 × 10⁸

$\mathbf{ \Delta x_2 = \sqrt{20} \times 3 \times 10^8}$

$\mathbf{ \Delta x_2 =13.416 \times 10^8 \ m}$

Therefore, we can conclude that the spatial separation between the 2 events is 13.416 × 10⁸ m

Learn more about spatial separation here:

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