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Hi Alex, nice ask I will try and answer.
First think of gravity as an aspect of geometry rather than a force. Second realise that the Universe is always trying to be more chaotic and dis-orgainsed (increase its entropy levels - which generally means not pack things into a smaller volume of space in a more ordered fashion).
Now the sheet analogy is a way of trying to visualise a 4D piece of geometry in a 3D world - something which is really hard to do! You are replacing the invisible force with a visual element of geometry meant to represent what really underlies the nature of our reality. Throw a ball or fall of a cliff and you are seeing geometry of spacetime, not a force in action - all very hard to come to grips with.
If I had to answer how do we visualise gravity as a 3d geometry, I'd start by saying remember 2d circular log paper - which looks like an old fashion radar screen? Well imagine that in 3d with a mass being at the epi-centre. So to a distant observer the lines showing how far 1 centimetre of distance is are smaller as you get closer to the point source of mass.
So spacetime is more curved the closer you get to a point of mass. A lightyear of distance could look like a micron to a distant observer of spacetime very close to a black hole's event horizon.
Similarily when you get very, very, very close to the baryonic matter that comprises the nucleus of any atom (protons and neutrons) you'd expect the gravitational curvature of spacetime at distance less than 10 ^ -20 metres to be rather extreme given G is inversely proportional to a 1/d^2 effect.
We don't really undertstand theoretically how the four forces interact at these very close distances to atomic matter. Say that again - our best scientific models can't apply and describe what happens very, very close to the nucleus of an atom. In a certain way a proton may have its own gravitational event horizon - if its density (volume / mass) is above a certain level. We can't model these happenings yet, at best we move from relativist physic (which is exotic at time) to quantum mechanics - which is the equivalent of relativity on LSD. Under QM particles can be in two places at the same time, very often (called tunnelling), and interfere (differaction) with each other, and appear and dissappear etc...
So after my long set up rant, my responce to your question - the volume of space has to be geometrically more distorted closer to the concentrated centre of mass (or energy). So yes you could use a bowling ball on a sheet of foam or you could consider helium balloons released inside a large circus tent. All will want to travel towards the apex.
Rather than lines to or around a point soure of mass, consider space and distance shrink as you get closer to mass.
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PS - where this thought experiment was heading and how it ties into dark energy / dark matter
The loss of mass in a closed system leading to no change in a Newtonian / Kepler framework, but a significant change to a relativistic framework is where I wanted folk to head. Throughtout school and most of our life we see space as a fixed entity that only Captain Kirk (or Piccard) moves through easily, as if he was going down to the corner store, or driving across country.
But space isn't always like that. If on a cosmic scale matter can be created and destoyed without apparently conserving energy or momentum, in finite spots for a finite time, then there is a different framework for studying cosmic expansion other than dark energy or dark matter (or perhaps offering a candidate for them).
Hawking has showed its possible for matter or energy to jump from the cosmic foam (alah the world of the really tiny quantum mechanics that our universe appears to float in) and not necessarily spell doom and gloom. Hawkings initial postulate was a pair of virtual particles (pro and anti to conserve mass and energy) tunnel into our reality then immediately zap each other and annihilate... except if they appear near the event horizon of a black hole, so one escapes and one is sucked in. In this situation you've just added mass and energy to the Universe.
My thought experiment was a simple two body test of what are the effects of a sudden large extract of mass from the universe - answer spacetime uncurves and we can detect this change in an an otherwise unexplained shift in the orbit of planets. The sun is losing mass every day as its converted into energy and it radiates past us. So spacetime around our solar system is uncurving (very, very, very slowly) every second of our lives.
But what if around galactic centres matter is coming into existence? Physicists studying MOND have shown you only need a very small adjustment to explain why galaxies don't fly apart. Calculated as if this force was attributed to mass gain you need about 1 hydrogen atom per cubic metre of space per year. Theoretical physics has shown that empty space should contain about 10 ^ 120 Joules of energy per cubic metre, the greatest whoopsie in the framework of our thinking - or is it? What if that energy is there - but it manifests very, very slowly over all time - not just all at once? If there is that much potential energy, but in only leaks in in the form of 1 Hydrogen atom per year, maybe all parties could reconcile some pretty wierd theories.
But all that at the moment is just pure theoretical spectulation with no solid grounding - yet. But as said the thought experiement was to get folk thinking and have them realise space, the final frontier, is not a constant, but a variable dependent on localised mass and energy.
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