Question

Spacetime interval: What is the interval between two events if in some given inertial reference frame the events are separated by: (a) 7.5 x 10 m and 3s? (b) 5x10 m and 0.58? (c) 5x 10"m and 58?

Answer 1

• a.
  `\Delta s ^2 = 8.0888 \ 10^(17) m^2`

• b.
  `\Delta s ^2 = 3.0234 \ 10^(16) m^2`

• c.
  `\Delta s ^2 = 3.0234 \ 10^(20) m^2`

Step-by-step explanation:

The spacetime interval
`\Delta s^2` is given by

`\Delta s ^2 = \Delta (c t) ^ 2 - \Delta \vec{x}^2`

please, be aware this is the definition for the signature ( + - - - ), for the signature (- + + + ) the spacetime interval is given by:

`\Delta s ^2 = - \Delta (c t) ^ 2 + \Delta \vec{x}^2`.

Lets work with the signature ( + - - - ), and, if needed in the other signature, we can multiply our interval by -1.

a.

`\Delta \vec{x}^2 = (7.5 \ 10 \ m)^2`

`\Delta \vec{x}^2 = 5,625 m^2`

`\Delta (c t) ^ 2 = (299,792,458 (m)/(s) \ 3 \ s)^2`

`\Delta (c t) ^ 2 = (899,377,374 \ m)^2`

`\Delta (c t) ^ 2 = 8.0888 \ 10^(17) m^2`

so

`\Delta s ^2 = 8.0888 \ 10^(17) m^2 - 5,625 m^2`

`\Delta s ^2 = 8.0888 \ 10^(17) m^2`

b.

`\Delta \vec{x}^2 = (5 \ 10 \ m)^2`

`\Delta \vec{x}^2 = 2,500 m^2`

`\Delta (c t) ^ 2 = (299,792,458 (m)/(s) \ 0.58 \ s)^2`

`\Delta (c t) ^ 2 = (173,879,625.6 \ m)^2`

`\Delta (c t) ^ 2 = 3.0234 \ 10^(16) m^2`

so

`\Delta s ^2 = 3.0234 \ 10^(16) m^2 - 2,500 m^2`

`\Delta s ^2 = 3.0234 \ 10^(16) m^2`

c.

`\Delta \vec{x}^2 = (5 \ 10 \ m)^2`

`\Delta \vec{x}^2 = 2,500 m^2`

`\Delta (c t) ^ 2 = (299,792,458 (m)/(s) \ 58 \ s)^2`

`\Delta (c t) ^ 2 = (1.73879 \ 10^(10) \ m)^2`

`\Delta (c t) ^ 2 = 3.0234 \ 10^(20) m^2`

so

`\Delta s ^2 = 3.0234 \ 10^(20) m^2 - 2,500 m^2`

`\Delta s ^2 = 3.0234 \ 10^(20) m^2`