That spaceships can escape Earth’s gravity makes gravity sound more like a force that can be opposed by a greater force rather than a property of spacetime geometry.
In: Physics
Both force and curvature are just two different ways of describing the attractive nature of mass. You can look at it in a similar way to electromagnetism, like how magnets attract via exchange of photons as the force carrier, which is where the idea for the graviton (as the hypothetical force carrier for gravity) came from. Or, you can look at it in terms of geodesics – think of two people standing near each other on the equator facing one of the poles as an example. If they both started walking in an exactly parallel path, they will eventually meet and you could argue that there was an attractive force between them. Geodesics are a property of a higher dimension; from the perspective of those people, they are walking on a flat 2D surface yet the Earth is of course 3D and curved. That curvature is a useful analogue to describe curved spacetime.
When modelling trajectories, escape velocities and such, it’s most simple to look at it in terms of forces. The acceleration of the rocket is greater than the acceleration due to gravity, hence it can gain velocity to the point where it becomes greater than the escape velocity of Earth (where the craft’s velocity will only be reduced to 0 at a point an infinite distance away). If you want to look at it in terms of curvature, think of a marble in a shallow bowl, where the middle is the centre of mass. A little push will cause the marble to follow a curved trajectory and return to the centre. A bit harder and it will loop around the bowl before decelerating, and at some point, you can push it so hard that it flies out of the bowl. You’ve basically modelled escape velocity in that instance.
Gravity curves space. More mass gives more curvature.
The Earth is curved, but this doesn’t keep us from sailing around it. Yes, it’s more work to escape a gravity well than just coast between points in empty space, but that’s what rocket engines are for.
Gravity is a force of attraction.
The mass of an object creates a greater gravitational force which bends spacetime.
Acceleration is able to overcome the gravitational pull as the force of the accelwration is greater than the atraction force being applied to it.
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