Answer:
Explanation:
The spacetime interval [tex]\Delta s^2[/tex] is given by
[tex]\Delta s ^2 = \Delta (c t) ^ 2 - \Delta \vec{x}^2[/tex]
please, be aware this is the definition for the signature ( + - - - ), for the signature (- + + + ) the spacetime interval is given by:
[tex]\Delta s ^2 = - \Delta (c t) ^ 2 + \Delta \vec{x}^2[/tex].
Lets work with the signature ( + - - - ), and, if needed in the other signature, we can multiply our interval by -1.
[tex]\Delta \vec{x}^2 = (7.5 \ 10 \ m)^2[/tex]
[tex]\Delta \vec{x}^2 = 5,625 m^2[/tex]
[tex]\Delta (c t) ^ 2 = (299,792,458 \frac{m}{s} \ 3 \ s)^2[/tex]
[tex]\Delta (c t) ^ 2 = (899,377,374 \ m)^2[/tex]
[tex]\Delta (c t) ^ 2 = 8.0888 \ 10^{17} m^2[/tex]
so
[tex]\Delta s ^2 = 8.0888 \ 10^{17} m^2 - 5,625 m^2[/tex]
[tex]\Delta s ^2 = 8.0888 \ 10^{17} m^2[/tex]
[tex]\Delta \vec{x}^2 = (5 \ 10 \ m)^2[/tex]
[tex]\Delta \vec{x}^2 = 2,500 m^2[/tex]
[tex]\Delta (c t) ^ 2 = (299,792,458 \frac{m}{s} \ 0.58 \ s)^2[/tex]
[tex]\Delta (c t) ^ 2 = (173,879,625.6 \ m)^2[/tex]
[tex]\Delta (c t) ^ 2 = 3.0234 \ 10^{16} m^2[/tex]
so
[tex]\Delta s ^2 = 3.0234 \ 10^{16} m^2 - 2,500 m^2[/tex]
[tex]\Delta s ^2 = 3.0234 \ 10^{16} m^2[/tex]
[tex]\Delta \vec{x}^2 = (5 \ 10 \ m)^2[/tex]
[tex]\Delta \vec{x}^2 = 2,500 m^2[/tex]
[tex]\Delta (c t) ^ 2 = (299,792,458 \frac{m}{s} \ 58 \ s)^2[/tex]
[tex]\Delta (c t) ^ 2 = (1.73879 \ 10^{10} \ m)^2[/tex]
[tex]\Delta (c t) ^ 2 = 3.0234 \ 10^{20} m^2[/tex]
so
[tex]\Delta s ^2 = 3.0234 \ 10^{20} m^2 - 2,500 m^2[/tex]
[tex]\Delta s ^2 = 3.0234 \ 10^{20} m^2[/tex]
