Can a "density" of points create curvature?

[friend]

Spacetime shrinks in a gravitational field. As I understand it, objects falling into a black hole will appear to contract in size and run slower as they approach the horizon. This is similar to how things contract and slow down when traveling close to the speed of light because they are traveling through more space points than the same process at zero speed.

So perhaps in some sense space is accumulating near a black hole so that processes occur in a shorter distance when compared to distant observers. So if there is something causing space to bunch up and accumulate in gravitational fields, will this produce the curvature we see in general relativity?

 

[Nugatory]

This is similar to how things contract and slow down when traveling close to the speed of light because they are traveling through more space points than the same process at zero speed.

That's not what's going on, and if you consider the way that time dilation is symmetrical you'll see that it can't be. If A and B are moving relative to one another, then A's "physical processes" are slowed relative to B's, but B's are also slowed relative to A. That can't be explained by either of them "travelling through more space points" because they can't both be the one that's traveled through more points. (But don't confuse time dilation with the differential aging that appears in the twin paradox - that's a different phenomenon and it is not necessarily symmetrical).

So perhaps in some sense space is accumulating near a black hole so that processes occur in a shorter distance when compared to distant observers. So if there is something causing space to bunch up and accumulate in gravitational fields, will this produce the curvature we see in general relativity?

That's not what's going on. The curvature is produced by the stress-energy tensor, as described by the Einstein Field Equations. For more, you can google for "Scwarzschild solution".

 

[friend]

Thank you. In the SR situation there would be more "spacetime points" for a traveling object with respect to the observer. So A will say that B is traveling with respect to A. And B will say that A is traveling with respect to B. In either case, the traveling object will appear to be going through more points with respect to the observers reference frame. So time will appear to run slower for the traveler as viewed by the observer. I'm not saying that there is ontologically more points in the region of the traveller. It only appears that way to the observer.

My question arises because we have a similar effect in a gravitational field. Things shrink and time dilates. But in the case of GR there IS an actual change in the metric in a gravitational field. Correct me if I'm wrong, but doesn't that exactly mean that we have more points in that region, that we can fit more yardsticks there than otherwise (with so many points per yardstick)? Doesn't that mean that there are more points in that region? I think the only question left if why should points "bunch up" in a gravitational field?

 

[DrGreg]

Image credit: Daniel R. Strebe. CC-BY-SA-3.0

@friend, here is a 2-dimensional analogy to what's happening in 4-dimensional spacetime. Would you say that the map above shows that space is stretched near the South Pole, and that there are fewer "space points" near the pole?

 

[friend]

Image credit: Daniel R. Strebe. CC-BY-SA-3.0

@friend, here is a 2-dimensional analogy to what's happening in 4-dimensional spacetime. Would you say that the map above shows that space is stretched near the South Pole, and that there are fewer "space points" near the pole?

The points on the real globe map to a stretched space from our perspective.

Again, more yardsticks can be fit into a region with a heavy gravitational field than with no gravitational field. Viewed from far away the gravitational field bunches up more space points. What I'm really trying to get at is whether a gravitational field can be viewed as a bunching up of space. Perhaps the question is better asked, if there is a spherically symmetric change in the metric so that things appear contracted from far away, does this lead to a curvature? Or does there have to be a gradual change in the metric from uncontracted to more contracted near a point to give us a curvature? Perhaps this is just another way of specifying a curvature.