3 years, 2 months ago
Why does the curvature of spacetime cause matter to experience gravity?
I'm familiar with the usual rubber sheet metaphor used to explain how space curvature causes gravity, but I'd like a more realistic understanding.
In the rubber sheet metaphor, a rubber sheet is stretched out horizontally and two spheres are placed onto the sheet. The weight of the spheres indents the rubber, creating slopes in the sheet that are angled down towards a point between the spheres. The slopes result in the spheres rolling down and towards each other as if there were a force pulling them together.
I've always found this explanation to be inadequate since the attraction experienced by the spheres is not dependent solely on the geometry of the rubber sheet. In addition to curvature, the metaphor requires an external force to pull the spheres "down" against the sheet is so that the spheres will actually move and follow the sheet's curvature. If that external force was not there, even if you somehow granted the spheres the ability to curve the rubber sheet in the same way, the spheres would just sit still because it was the external force that caused the actual acceleration while the curvature just directed it.
So, how can the curvature of spacetime in and of itself cause the acceleration of gravity?
In the rubber sheet metaphor, a rubber sheet is stretched out horizontally and two spheres are placed onto the sheet. The weight of the spheres indents the rubber, creating slopes in the sheet that are angled down towards a point between the spheres. The slopes result in the spheres rolling down and towards each other as if there were a force pulling them together.
I've always found this explanation to be inadequate since the attraction experienced by the spheres is not dependent solely on the geometry of the rubber sheet. In addition to curvature, the metaphor requires an external force to pull the spheres "down" against the sheet is so that the spheres will actually move and follow the sheet's curvature. If that external force was not there, even if you somehow granted the spheres the ability to curve the rubber sheet in the same way, the spheres would just sit still because it was the external force that caused the actual acceleration while the curvature just directed it.
So, how can the curvature of spacetime in and of itself cause the acceleration of gravity?
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M$1 Answer
Like all analogies, there can be some information lost in translation, or for this case, confusion. The confusion being the loss in equivalence of the external force and gravity between the simplified case, and the real-world case.
To make it clearer on the analogy, let's do the experiment in space, where there is no measurable gravity on our 2 hand-held spheres (so small-scale test), nor any gravity between the 2 spheres (which is understandable since they are so small anyways to matter for this thought experiment). We'll put these spheres on a rubber sheet and have them magically charged and have an electromagnetic force such that each sphere can feel a pull from an external EM-force (when enabled), BUT NOT feel a pull towards each other due to their own EM-charge (this is the part that needs to be magical about their EM-charges).
For the first case, if we don't enable the external EM-force below the rubber sheet, the 2 spheres won't do anything; no forces exist between them or around them afterall, so they are just "hovering" above the rubber sheet. If we now enable this EM-force below the sheet, the 2 spheres will be pulled downwards into the sheet, and their degree of depression in it will depend on the magnitude of their charge.
Now it should be easier to see that the spheres will move based on the sole contours of the sheet alone because their goal is to move in a straight line directly to the source of the EM-force, but the rubber sheet is preventing that; if neither charge is large enough to encapsulate another in their depression, then they'll just sit there in their own depressions as they attempt to move as close as possible to the EM-source (whose pull is perpendicular to the rubber sheet across the entire sheet, rather than being a point source pull); consequently, if one sphere's "pull" into the sheet is so much that the contours in the sheet will encapsulate the other, then this 2nd sphere will be pulled towards the 1st sphere. And it is ONLY due to the contours of the sheet and this external force that is causing this interaction of them converging, since there is no other forces between the 2 spheres otherwise.
Similarly, back to the real world on cosmological scales (rather than hand-held), we have the sun and earth, where the sun's pull is obviously large enough to encapsulate earth. Gravity in this case would be equivalent to our magical EM-force, therefore to map this back to the rubber sheet analogy, we MUST PRETEND there is NO FORCE between the sun and earth themselves (this is where your confusion probably arises), but rather an external pull downwards into the sheet of space-time so as to warp it into a canonical shape. The "external force" is none other than gravity in this simplistic view.
The reason why you think our simplistic analogy requires a new "external" force (of gravity), is that you didn't map the original force from the real-world (namely, you didn't change it as a force between objects into an external force acting below the objects). B/c gravity is required in both situations, but also by different mechanisms, it leads to this confusion; look back to the magical EM-force example again to make it clearer.
To make it clearer on the analogy, let's do the experiment in space, where there is no measurable gravity on our 2 hand-held spheres (so small-scale test), nor any gravity between the 2 spheres (which is understandable since they are so small anyways to matter for this thought experiment). We'll put these spheres on a rubber sheet and have them magically charged and have an electromagnetic force such that each sphere can feel a pull from an external EM-force (when enabled), BUT NOT feel a pull towards each other due to their own EM-charge (this is the part that needs to be magical about their EM-charges).
For the first case, if we don't enable the external EM-force below the rubber sheet, the 2 spheres won't do anything; no forces exist between them or around them afterall, so they are just "hovering" above the rubber sheet. If we now enable this EM-force below the sheet, the 2 spheres will be pulled downwards into the sheet, and their degree of depression in it will depend on the magnitude of their charge.
Now it should be easier to see that the spheres will move based on the sole contours of the sheet alone because their goal is to move in a straight line directly to the source of the EM-force, but the rubber sheet is preventing that; if neither charge is large enough to encapsulate another in their depression, then they'll just sit there in their own depressions as they attempt to move as close as possible to the EM-source (whose pull is perpendicular to the rubber sheet across the entire sheet, rather than being a point source pull); consequently, if one sphere's "pull" into the sheet is so much that the contours in the sheet will encapsulate the other, then this 2nd sphere will be pulled towards the 1st sphere. And it is ONLY due to the contours of the sheet and this external force that is causing this interaction of them converging, since there is no other forces between the 2 spheres otherwise.
Similarly, back to the real world on cosmological scales (rather than hand-held), we have the sun and earth, where the sun's pull is obviously large enough to encapsulate earth. Gravity in this case would be equivalent to our magical EM-force, therefore to map this back to the rubber sheet analogy, we MUST PRETEND there is NO FORCE between the sun and earth themselves (this is where your confusion probably arises), but rather an external pull downwards into the sheet of space-time so as to warp it into a canonical shape. The "external force" is none other than gravity in this simplistic view.
The reason why you think our simplistic analogy requires a new "external" force (of gravity), is that you didn't map the original force from the real-world (namely, you didn't change it as a force between objects into an external force acting below the objects). B/c gravity is required in both situations, but also by different mechanisms, it leads to this confusion; look back to the magical EM-force example again to make it clearer.
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M$
Actually: curvature of worldlines would be better, since curved spacetime is a misnomer.
Since spacetime consist of events that combine a position and a time based on the state of an observer, the worldline of the observer is straight (by definition). The observer isn't a person, but the zero point of the frame of reference.
That's a free-falling observer. Seen from there other worldlines are curved, because of gravity.
The reason is - in my eyes- something different and the curvature is an outcome and not the cause.
(Personally I see gravity as an effect of interactions of 'parallel lightcones' that interact and deflect the other one, since there seems to be some kind of rotation with these cones.)