Space Bending: Questions Answered

[UfoPilot]

please help me understand this, space is flat. space bends when its near matter. When matter moves from point A to point B, the space at point A returns to being flat. Where does the energy come from to bend the space back to flat come from? Why doesn't the space at point A stay bent?

 

[Boing3000]

please help me understand this, space is flat.

The geometry of space-time curvature would indeed be flat if no energy was present anywhere.

space bends when its near matter

This geometry indeed change with respect to the many component of that matter/energy (See stress energy tensor)

When matter moves from point A to point B, the space at point A returns to being flat.

That's correct, the curvature is mostly dependent on distance.

Where does the energy come from to bend the space back to flat come from?

No energy is used to affect that curvature. Only in very extreme cases (like black holes merging) some energy is indeed transferred to that curvature and in the forms of gravitational waves.

Why doesn't the space at point A stay bent?

For the same reason it was flat in the first place.
Also, that curvature is different for every object depending on their relative motion. For example take a snapshot of two identical satellite nearly at the same point in space. If one is going at great speed horizontally, it will "follow" and orbital (but also free fall) geometry. If one is dropped at zero velocity there (also with respect to earth) it will make the apple free fall dive. So despite (nearly) being at the same place (in space), they both will experience a totally different (4 dimensional) geometry (Nothing happens to space itself).

That's why it is also said that in GR space time curvature guide object into free fall along their "straight line" geodesics

 

[PeterDonis]

space is flat

What "space" are you talking about? Are you talking about our universe as a whole? That is spatially flat, yes (according to your best current model). But "space" without qualification is too vague.

space bends when its near matter.

No, spacetime is curved when matter and energy is present. Spacetime is not the same thing as space.

When matter moves from point A to point B, the space at point A returns to being flat.

No; when matter moves from point A to point B, spacetime is more curved in the 4-dimensional region that includes the "world tube" between A and B that the matter occupies, and less curved far away from that region. Nothing "returns to being flat"; spacetime, as a 4-dimensional manifold, does not "change", it just is. What we think of as "change" in our everyday world, in the 4-d spacetime model is just the geometry being different in some regions vs. others.

Where does the energy come from to bend the space back to flat come from?

It doesn't have to come from anywhere. See above.

 

[PeterDonis]

that curvature is different for every object depending on their relative motion.

This is not true. The Riemann curvature tensor, which describes spacetime curvature, is a tensor, i.e., a covariant geometric object. It is the same tensor for every observer, whatever their state of motion.

take a snapshot of two identical satellite nearly at the same point in space. If one is going at great speed horizontally, it will "follow" and orbital (but also free fall) geometry. If one is dropped at zero velocity there (also with respect to earth) it will make the apple free fall dive. So despite (nearly) being at the same place (in space), they both will experience a totally different (4 dimensional) geometry

This is not correct. Both satellites are moving through the same 4-d geometry. They are just moving on different geodesics in that same 4-d geometry.