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

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.

[tex]\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} }[/tex]

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

where;

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

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

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

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

[tex]\mathbf{ \Delta x_2 = \sqrt{20} \ c}[/tex]

here;

c= speed of light = 3 × 10⁸

[tex]\mathbf{ \Delta x_2 = \sqrt{20} \times 3 \times 10^8}[/tex]

[tex]\mathbf{ \Delta x_2 =13.416 \times 10^8 \ m}[/tex]

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

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