I have a low level understanding of the spacetime interval, which is the invariant distance between two points in spacetime as measured by two observers at both ends of Δs².
However, the formulation of the interval is a bit confusing and I'm failing to fully understand it both conceptually and intuitively.
If it was just a basic Pythagorean theorem it wouldn't be an issue but the −(cΔt)² term and what it means in reality eludes me.
So I thought a thought experiment would hopefully help someone here to help better articulate its inherent meaning.
It goes as follows.
If the sun, at a distance of 149.6 billion meters, were converted into a black hole having a Schwarzschild radius of 2.95E+3 meters and there was a space station hover above the event horizon at 3.00E+3 meters.
Which would give it an escape velocity from there equaling 2.9628E+8 meters a second and thus a time dilation factor of 1/sqrt(1-(2.9728E8/299792458)^2) = 7.7403.
How would the spacetime interval reflect the distance between a signal sent to the space station from the earth in this scenario using the Schwarzschild metric. (See formula below)
Or a scenario where there's a rocket speeding by near the moon and sends a signal to earth moving at 2.9728E8 meters per second so the time dilation factor is still the same using
To further help, I understand that s>0 (space-like) more space in between than light can cross in the time => no causal relation.
s=0 (light-like) exactly on the "light cone"
s<0 (time-like) less space in between than light can cross in the time => A causal relation is possible.
In this scenario or any other. How does the interval determine if something is space-like, light-like or time-like?
Please help me understand this better.
Also if these thought experiments are inappropriate for explaining the spacetime interval I apologize and ask that an appropriate thought experiment be presented in it place.