How Does Curved Spacetime Influence the Path of Light and Objects in Space?

[Nim]

The explanation I've heard for gravity as depicted by relativity, is that it's the consequence of mass curving spacetime. This means that the reason why the light from a star behind Sun is bent is because evening though the light is traveling a straight line, since space itself is bent, traveling a straight line will cause the light to bend.

If this is the reason why we are stuck to the Earth, then what is happening when something leaves Earth? Is it no longer traveling a straight line through the curved space?

 

[LURCH]

That is essentially true. When something leaves Earth, it must undergo some acceleration to get away. While accelerating, the object is no longer traveling in a strait line.

 

[selfAdjoint]

The explanation I've heard for gravity as depicted by relativity, is that it's the consequence of mass curving spacetime. This means that the reason why the light from a star behind Sun is bent is because evening though the light is traveling a straight line, since space itself is bent, traveling a straight line will cause the light to bend.

If this is the reason why we are stuck to the Earth, then what is happening when something leaves Earth? Is it no longer traveling a straight line through the curved space?

First of all, it's spacetime that's curved not just space. That's important as we will soon see.

Second, not just mass but all momentum and energy curves spacetime. Mass is a part of that because it's a form of energy (Einstein's famous equation), but there are other forms too. Light in particular has no mass but it does have momentum (which in its case modifies its frequency).

Third, a truly straight line cannot exist in a curved geometry. What you have instead are curves that are as stright as possible (this can be defined mathematically). They are called geodesics (gee-oh-dessiks is the way I say it; YMMD), and these are the curves that go through the curved geometry of spacetime without being deviated by outside forces.

Now as an example of something that has left Earth, let's take a space station in orbit. Its motion in space is close to an ellipse, but we have to consider its track in spacetime. As it goes around one orbit, time passes and it comes back not to its original position but to that position at a later time. In spacetime it describes a kind of elliptical helix, and this can be shown to be the unique geodesic curve that goes from its original spoacetime position (arbitrariy, (0,0,0,0) ) to its new one (0,0,0,t).

And everything that is acted on only by gravity (i.e. moves through spacetime without any other constraint than the geometry itself) follows a geodesic.