Can a "density" of points create curvature?

In summary, objects falling into a black hole will appear to contract in size and run slower as they approach the horizon due to the accumulation of space near the black hole. However, this is not the cause of the curvature we see in general relativity. The curvature is produced by the stress-energy tensor and the change in the metric. The analogy of the 2-dimensional map near the South Pole demonstrates how space can appear stretched or contracted depending on the observer's perspective. In a similar way, a gravitational field can be viewed as a bunching up of space, but this is not the exact cause of the curvature. The curvature is a result of the change in the metric from uncontracted to more contracted near a point.
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friend
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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?
 
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  • #2
friend said:
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".
 
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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?
 
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283px-Mercator_projection_SW.jpg

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?
 
  • #5
DrGreg said:
283px-Mercator_projection_SW.jpg

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.
 
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1. Can a high concentration of points create a gravitational field?

Yes, the theory of general relativity states that mass and energy can cause curvature in the fabric of space-time, resulting in the creation of a gravitational field. This means that a high density of points, representing a large mass or energy, can indeed create a gravitational field.

2. How does the density of points affect the curvature of space-time?

In general relativity, the curvature of space-time is directly proportional to the distribution of mass and energy. This means that a higher density of points will result in a stronger curvature of space-time.

3. Can a density of points create curvature in other dimensions besides space and time?

In general relativity, space and time are considered to be the only dimensions in which curvature can occur. However, some theories in physics, such as string theory, suggest the existence of additional dimensions where curvature may also be possible.

4. Is there a limit to how dense a collection of points can be for curvature to occur?

Theoretically, there is no limit to how dense a collection of points can be for curvature to occur. However, as the density increases, the effects of curvature become more significant and may require more complex mathematical models to accurately calculate.

5. Can a density of points create negative curvature?

Yes, according to general relativity, a high concentration of negative mass or energy can create negative curvature in space-time. This is known as anti-gravity and is still a topic of ongoing research in physics.

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