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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

Δs²=−(cΔt)²+Δx²+Δy²+Δz²

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.

Thank you.

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    $\begingroup$ The metric you are considering isn't applicable in the vicinity of a black hole, you would need the Schwarzschild metric there. The Minkowski metric only works when no masses are there to bend the spacetime. $\endgroup$
    – Photon
    Apr 15 at 12:45
  • $\begingroup$ Then what about a rocket speeding by at the moon moving at 2.9728E8 meters per sec the time dilation factor is still the same? $\endgroup$ Apr 15 at 13:22
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" How would the spacetime interval Δs²=−(cΔt)²+Δx²+Δy²+Δz² reflect the distance between the space station and the earth in this scenario? "

This is one of the misconceptions you have. A spacetime interval is not calculated between 2 objects . It is calculated between 2 events

So, you do not calculate the spacetime interval between earth and space station.

You calculate the spacetime interval between
event 1 = the space station at a certain moment
& event 2 = the earth at a certain moment

For example, it could be
event 1 = the space station emitting a signal

event 2 = the earth receiving the signal

In this case, the spacetime interval would be 0 .

Or event 1 = space station emitting a signal

event 2 = the earth controller guy calling his boss to tell him that he has received the signal.

In this case, the spacetime interval would be less than 0. Hence, there is a cause effect relation between event 1 and 2.

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  • $\begingroup$ Thank for the clarification. However if a signal was sent from earth to the space station (plugging in the given numbers) how would the interval reflect that? Also, so the interval at its crux is the duration that it takes a signal to get from A to B? It's about how far light has to travel? $\endgroup$ Apr 15 at 13:40
  • $\begingroup$ @PythonHouse " Also, so the interval at its crux is the duration that it takes a signal to get from A to B? " No. space time interval is not the duration. The duration part is only the first term i.e. the ct term. The rest of the equation is also important $\endgroup$ Apr 15 at 13:44
  • $\begingroup$ So if ct is the duration of the signal. What does the rest signify? Δs²=−(cΔt)²+Δx²+Δy²+Δz². The Δs² doesn't me distance? $\endgroup$ Apr 15 at 13:47
  • $\begingroup$ @PythonHouse can u come to this chatroom ? chat.stackexchange.com/rooms/122572/twins-and-triplets-paradox $\endgroup$ Apr 15 at 13:49

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