Question

Use equation

(Δs)²=c²(Δt)²−[(Δx)²+(Δy)²+(Δz)²]
to calculate the square of the spacetime interval between these two events.

Two civilizations are evolving on opposite sides of a galaxy whose diameter is 10⁵ light years. At time t=0 in the galaxy frame of reference, civilization A launches its first interstellar spacecraft. Civilization B launches its first spacecraft 50,000 years later.

Answer 1

The square of the spacetime interval between the launch events of the two civilizations is approximately
`2.2407 * 10^41` meters².

Two civilizations, A and B, separated by half the diameter of a 10⁵ light-year galaxy, launch their first spaceships 50,000 years apart (in the galaxy's frame of reference). We need to find the square of the spacetime interval between these two events.

1. Define constants:

- Galaxy diameter (d): 10⁵ light years = 10⁵ * 9.461 * 10^12 meters

- Time difference (Δt): 50,000 years = 50,000 * 3.1557 * 10^7 seconds

- Speed of light (c): 299,792,458 meters per second

2. Calculate spatial distance:

- Half the galaxy diameter represents the spatial distance between the civilizations: d/2 =
`5 * 10⁵ * 9.461 * 10^12` meters

3. Apply the spacetime interval equation:

-
`(Δs)² = c²(Δt)² - [(Δx)² + (Δy)² + (Δz)²]`

- Since the civilizations are on opposite sides of the galaxy, only the spatial distance x is relevant.
`y` and z are negligible.

- Substitute the values:
`(Δs)² =`
`(299,792,458)² * (50,000 * 3.1557 * 10^7)² - (5 * 10⁵ * 9.461 * 10^12)²`

4. Solve for the square of the spacetime interval:

`- (Δs)² ≈ 2.2407 * 10^41 meters²`