In trying to understand why matter curves space i thought up a model that has the curiosity of suggesting the concept of maximum gravity, suggesting that as the mass of an object increases gravity increases as well, until the mass reaches a point at which the gravity starts to increase more slowly until it actually levels off at some maximum level, after which an increase in mass will not further increase the corresponding gravity. Has any data or observations suggested such a phenomenon? I realize it may be hard to verify this claim.
I don't think so, in fact the data suggests just the opposite. If you believe in black holes, and the data is pretty strong these days, then the strength of gravity has no upper limit. In the Newtonian approximation, the strength of the field is linear in the mass. Once you introduce GR, the gravitational field gets all kinds of new sources. Moreover, the field can actually contribute to itself (this is the nonlinearity of Einstein's equations). So rather than leveling off in very massive systems, the gravitational actually gets amplified by the nonlinearity in the system. This is why you only have to squeeze a finite amount of stuff into a small enough region to produce a black hole, a gravitational object where the gravitational field (curvature) actually diverges. All that being said, it may be that true gravitational singularities do not exist. The issue is muddled because the heart of a black hole is precisely one of the places where quantum gravity should be important, and as you no doubt are aware, a satisfactory theory has yet to be invented. Nevertheless, many people suspect that quantum effects will smear out the singularity, so maybe in some ultimate way your idea will turn out to be right. Perhaps quantum effects limit the ultimate strength of gravity.
That was helpful, thanks. I have another question now, what is the main difficulty in trying to explain quantum gravity with GR?
Just thought that I should point out there is no real singularity at [itex]r=0[/itex] of a Schwarzschild metric; we usually get one because of the choice of co-ordinate system. A more suitable co-ordinate system eliminates the divergence. A good discussion can be found in Carroll.
masudr, I think you may be a bit confused. There definitely is a spacetime singularity at [tex] r = 0 [/tex], the curvature invariant [tex] I = R_{\alpha \beta \gamma \delta} R^{\alpha \beta \gamma \delta} [/tex] diverges at [tex] r = 0 [/tex]. I believe you are referring to the coordinate singularity at [tex] r = 2m [/tex]. This coordinate singularity can indeed be removed by a different choice of coordinates (though the surface [tex] r = 2m [/tex] is still special even if the curvature doesn't diverge there).
Einstein's curved-space model for gravity seems to suggest the concept of maximum gravity. This stems from the observation that if an object is very massive then it will "sink" space-time so much, that the space near and around the object is perpendicular to "regular" space-time outside the influence of this object's gravitational field, so any further distortion of space won't produce a stronger gravity. It also suggests that initially, as an object nears such a massive object, its acceleration due to gravity increases with the distance from the massive object. Gradually, as this object gets nearer the massive object the acceleration does not increase as fast, but at a certain point it reaches a maximum level, beyond which it will no longer be accelerated by the gravity of the massive object.
I guess there's certainly the speed of light limit to acceleration. If you can't go faster than the speed of light than the maximum acceleration would be the one taking you from 0 to c in the shortest of time units. This is not what i was suggesting though when i was talking about a maximum acceleration due to gravity.
The speed of light is a limit on velocity, but not on acceleration. It would take an infinite acceleration (from the object's own perspective) to get something up to the speed of light in a finite time, anything short of that will just get it closer and closer to the speed of light without reaching it.
I did a search on maximum gravity and found the following number: 5x10^18 kg/m^2 The source is http://www.metaresearch.org/msgboard/topic.asp?TOPIC_ID=324 , but it may be under the LeSage model, which i don't find particularly attractive (even if the idea is kind of cool).
Reaching [itex]c[/itex] or not is irrelevant, that acceleration used for a shorter period of time, can reach 1 m/s and stop as well. However, a greater acceleration than the one that *would* accelerate something to [itex]c[/itex] in [itex]t_p[/itex] (Planck time) is implausible, isn't it?
I'm sorry but this is not correct. Are you basing your understanding of GR on the common "sheet-visualization" to illustrate the curvature around a massive object? That's only a flawed visualization and cannot be used to investigate the properties of GR seriously. The GR geodesic equations themselves directly contradict your statements. There is no limit in GR of "acceleration due to gravity", and no limit on the strength of the gravitational field.