I apologize for any confusion caused by my previous response. Let's clarify the concept of time dilation related to the curvature of spacetime.
In the theory of general relativity, time dilation is indeed caused by the curvature of spacetime, which is influenced by both gravity and relative motion. The presence of mass or energy causes a curvature in spacetime, and this curvature affects the flow of time.
When we say "the further up you go, the faster time goes," it refers to the phenomenon of time running faster at higher altitudes in a gravitational field. This effect is known as gravitational time dilation. As an observer moves away from a massive object (such as the Earth), which creates a gravitational field, they experience time passing at a slightly faster rate compared to an observer closer to the object. This means that clocks in a weaker gravitational field run faster than clocks in a stronger gravitational field.
However, when we talk about people in space experiencing slower time compared to people on Earth when they are further away, we are referring to a different concept. This situation is related to relative motion and the effects of special relativity.
According to special relativity, when an object is moving relative to an observer, time appears to pass more slowly for the moving object. This effect is called time dilation due to relative velocity.
In the context of astronauts in space, when they are further away from Earth, they are usually in a state of orbital motion or traveling at high speeds relative to Earth. As a result, they experience time dilation due to their relative motion. From their perspective, time appears to pass more slowly compared to an observer on Earth.
So, it's important to differentiate between gravitational time dilation, where time runs slower in stronger gravitational fields, and time dilation due to relative motion, where time appears to pass more slowly for objects in motion relative to a stationary observer. The combination of these effects can create complex situations where time dilation may seem to vary in different scenarios.