I understand the analogy and have come across it many times with pictures of the two dimensional set up that allude to the three dimensions plus time of the spacetime geometry. It is accompanied by an explanation similar to and sometimes not as good as yours, and it in effect tells us we can't really display it or visualize it because we are just stupid humans, lol. However, when have you ever heard of a physical explanation for how the sun warps the space around it, or of any physical effect to explain the curvature that can be tied to the observations of how objects "roll around the rubber sheet" to move through spacetime?Well, I am not going to be able to set you straight on much of anything relating to physics. I have a long way to go before I understand it myself. But in one conversation I had with Mazulu about Penrose's theory on conformal cyclic cosmology, I proposed a possibility. I can't remember exactly where I got the idea and while I probably have it wrong I will give the explanation again here.
Visualize a 2D space with a sun at the center. This is the rubber sheet analogy which I am sure you are familiar with. Consider an object some distance from the sun and assume it is stationary to start. Consider the area of the space inside a circle centered on the sun with a radius of the distance the object is from the sun. The area of that circle is larger than one would expect from a circle in flat space. This space is non-euclidean. The area outside the circle is flatter than the space inside. So there are more degrees of freedom inside the circle than outside. So consider the object to be small and randomly jittering around. As the magnitude of the jitters is increased they can take the object further inside or outside the circle and so the probability that the jitter goes toward the center of the circle is greater than away from it, simply because there are more possible locations there. Sideways jitters have equal probability so there is no "force" produced in that direction. It is all just probability. Note that I am not saying I believe this, just that it is a concept I read somewhere. You can replace the jitter with a probability wave function or probably other known features of matter. But for this to work the feature of matter that is used where I said jitter, has to be the same for all types of matter, because of the gravitational constant.
So that is one possible example of how geometry could account for gravity. I am sure there are others. I am unsure whether they represent how gravity works in reality. In other words I don't really know enough about the subject to know if even my description of the concept is correct. But that usually does not stop me.
The answer is in the kinds of connections you have been studying; that we human's have been observing the effects of gravity since the days of the Plato's cave analogy. We have slowly improved our quantification of it to account for almost all observed anomalies right up through the EFEs. They are the best answer yet of how to calculate and predict the effects of gravity. But then too, there is an unfortunate evolution from the mathematical work that so nicely produces explanations for the observations and gives reliable predictions, to an insistence by a few that there doesn't have to be anything physical to explain how gravity works because gravity IS geometry and math; no need for any physical connection to explain action at a distance if you believe that geometry and math can physically curve spacetime :shrug:.
But it is all good for me because it means I have a hobby that isn't likely to go away, and as a result, I am one of the few hacks who has a layman explanation for the particulars of a quantum gravity model that is hard to falsify. It is not science and so professionals aren't and shouldn't be interested, but it is layman thinking about what a model might be like when science finally quantifies it. And I know that the great minds who will read that statement (no one here specifially) will be tempted to make reference to wild or idle fantasies that can't be falsified either, but they do seem to have trouble pointing to specifics of my model that they would call outrageous "blue unicorn" thinking,
So think all you want about how to depict spacetime visually or even to make a single scientific statement about how it works physically, and you will see the stimulus for my "aether think" hobby, i.e. there is no physical effect that we know of yet that causes the curvature of spacetime or action at a distance.
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