Is it really impossible that gravity can be modeled non-geometrically? I read the following in Weinberg paper Gravity: "An alternative way to conceive of gravity would of course be to follow the lead of other theories, and regard the gravitational field as simply a distribution of properties (the field strenghts) in flat spacetime. What ultimately makes this unattractive is that the distinctive properties of this spacetime would be completely unobservable, because all matter and fields gravitate. In particular, light rays would not lie on the "light cone" in a flat spacetime, once one incorporated the influence of gravity. It was ultimately the unobservability of the inertial structure of Minkowski space that led Einstein to eliminate it from his theory of gravitation and embrace the geometric approach." I'd like to know: 1. Is gravity in flat spacetime means the same as force based gravity or is force based gravity another method where there is no spacetime but fixed space and time? If so, this means gravity in flat spacetime is fields based gravity in contrast to force based gravity? 2. How come light rays won't lie on the "light cone" in a flat spacetime gravity theory? 3. What is meant by the "unobservability of the inertial structure of Minkowski" that makes impossible gravity based on flat spacetime? 4. Is it totally impossible to model gravity that is not based on spacetime curvature? Maybe there is another way or is Gravity Geometry forever? Thanks.
It might be helpful to read Einstein's description of rulers on a heated slab http://www.bartleby.com/173/24.html Einstein points out that being able to tile a surface with squares that don't overlap at all is possible only on a plane. Einstein doesn't specifically mention the surface of a sphere as a counterexample, but you can imagine trying to do it, and realize that it won't work - for instance, the circumference of the earth at the equator (0 degrees lattitude) won't equal the circumference of the Earth a short distance above it (say 1 minute of an arc above the equator). The point is that with the actual rulers we use, observable rulers, the geometry of space-time is measurably curved - at least according to General Relativity (and light bending experiments agree with this prediction). We can't tile space with perfect cubes that fit perfectly together, nor can we tile space-time with perfect hypercubes. This happens because space-time isn't flat (and spatial slices of consant Schwarzschild time aren't flat either). It turns out you can make such a "heated ruler" theory to describe gravity. You wind up with imaginary rulers and clocks that perfectly cover an unobservable flat background space-time with squares, like the marble slab, and real rulers that expand and contract and clocks that speed up and slow down due to "extra fields" that affect all matter uniformly (like the heated rulers), so that actual rulers can't tile the geometry (with hypercubes for the example of space-time). Note that in a space-time geometry, clocks play the role of rulers, in that they measure "distances in time". More formally, one actually uses the Lorentz interval of special relativity than the usual concept of distance, but it probably won't be too confusing to gloss over this point. There are some limits to this approach, that Weinberg didn't mention. For instance, you can't make a flat background spacetime have wormholes, because the topology isn't the same. You also tend to run into problems trying to model black holes (a black hole, fully extended with the Kruskal extentions, is equivalent to a wormhole, so the topology is basically different).
1) What do you mean by "force" based gravity? If you mean "forces" as conceived by Newton, then that is untenable due to special relativity. Forces (conceived thusly) are instantaneous and SR tells us no information can travel faster than the speed of light. This is the reason we use field theories. The "forces" are mediated by fields and so one can have non-instantaneous transmission of "forces". 2) If we assumed all space-time is flat, then the gravitational effect would have to be modeled non-geometrically. We know from observation; however, that light rays bend in the presence of matter. In this sense, light wouldn't follow the straight-line geodesics which define a light cone in flat space-time. 3) Because all matter (massless or massive) are affected by gravity, there is no way to "shield" the effects of gravity. This means that even if you used light, you cannot grid out the straight line grids of a flat space-time in any way, because the light rays themselves would bend. There is no way to "map out" a flat grid when you are in the presence of matter, and therefore, you can't "see" this flat background space-time whenever you have matter around (which is always, since no matter = no interactions=no observations). 4) I don't think it's "totally impossible". One certainly can come up with different models for gravity, but the appeal of a geometric approach, as Weinberg points out, is that it gets rid of a lot of unnecessary assumptions like some flat background space-time which we can never observe experimentally.
Because of the symmetry in the theory. One can model gravity as physical field or mathematical spacetime geometry and curvature. I know our physics now is such that we only accept models and accept the map for the territory. From symmetry, the following is so: Gravity as physical field Gravity as spacetime geometry and curvature They are equivalent by symmetry. But are they totally equivalent? No. It's like asking this. A car as a physical object A car as modeled as curve and geometry in the graphics program Are they equivalent? Maybe by symmetry, but not or instead of selling you an actual car, I may as well sell you the software for the autocad graphics program. Bottom line is. If one models gravity as a physical field. There may be a way to shield gravity. In General Relativity, there is no way to shield it. So there is the limitation of GR. When we focus too much on GR, we would become limited by what is possible and beyond.
I stated "there is no way to 'shield' the effects of gravity" not based off the theory but based off experiment. If you can show some experiment that shows a potential of "gravitational shielding", then please share. A theory is not "limited" just because it prohibits something. A theory is only "limited" if it is unpredictive or experimentally unfalsifiable or overly narrow in its application.
Perhaps this can happen 100 years later or if done kept hidden from public (Black Project) to avoid loss of revenue of expensive jet fuel and profit. What I'm saying is that there is no mechanism in General Relativity to shield gravity while in gravity as field based, it is possible. So GR is limiting. In fact, so limiting that it makes physicists sure no shielding can occur.. but note GR is just a model that we mustn't mistake for the territory.
Here are my 2cts: As such models have been published, evidently this is not impossible. I suspect that this is more a matter of notation than a fundamental difference: in "flat spacetime", light rays bend around the Sun. Some people have attempted a different description; I don't know how successful these attempts were/are. If I can find back a recent one, I'll add it as illustration.
What models? pls mention them. Anyway I wrote this thread about 2 weeks ago. I learnt that string theory as a theory of quantum gravity can pull off the non-geometric thing even explain the dynamics of black holes.. all without curved spacetime.. but as spin-2 field on flat spacetime below the planck scale and quantum gravity near the planck scale or black hole (I assume black hole is tied up to planck scale.. isn't it.. or is it because of the unique geometry that's why it can't be described in flat spacetime?). But the sun has mass, won't it be enough to attract the photons classically? I think the argument is that it has no mass. But they say this can be modelled on flat spacetime. So what makes massless light bend around the sun in flat spacetime (what argument do proponents of this use?)? A field based approach is more logical. The geometry thing may be due simply to certain symmetry inherent in it and doesn't mean gravity is geometry. It's like saying my car can be modelled in graphics program.. hence my car is geometry. So please share all field based models. Thanks.
Here's something with essentially no spacetime: Geroch's "Einstein Algebras" projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.cmp/1103858122&page=record
Interesting, but pls explain first how light can be bent by the sun if it has no mass.. unless there is an unmeasured third polarization and light has a tiny mass like neutrinos? If not. What is the reason massless light can be bent by the sun if one won't take the geometric approach to gravity as a priori?
Even in Newtonian gravity light can be bent by the sun. The Newtonian gravitational force on a massless object is 0, but "bending" is not force, and it does not require force to accelerate massless objects in Newtonian mechanics. Bending is acceleration, and [itex]a=GM/r^2[/itex] regardless of m. However, the fact that a is independent of m is precisely the feature that allows you to express gravity geometrically.
Then why did they announce in 1919 that the bending of the light by the sun is proof that General Relativity was right when they could just announce Newton was right?!
So using spin-2 field on flat spacetime (which is equivalent to GR covered by harmonic coordinates as convinced to me by atyy and other people in other threads). How does one explain this extra factor of 2 thing (without using the geometry of General Relativity as a priori)?
I have no motivation to even attempt that (try to derive a GR result without using GR). It sounds like a very difficult task for little or no benefit.
This paper summarizes (in a biased way) field theory gravity (FTG). http://arxiv.org/abs/gr-qc/9912003 There is a Lagrangian and the interaction term is [tex] \Phi^{\mu\nu}T_{\mu\nu} [/tex] which predicts spin 2 and spin 0 interactions.
Sorry, must search... As a matter of fact, Einstein first reduced GR to a flat spacetime approximation and then used the Huygens construction for the calculation. His argument was thus what also is called "gravitational lensing". You can read it here (not far from the end, starting with p.198 in the English translation): http://www.alberteinstein.info/gallery/gtext3.html You may be surprised to read in the above-mentioned overview that Einstein regarded GR as a field theory; the geometry was for him a mathematical toolbox.
No, classical gravity as a field in flat spacetime is the same as classical gravity as curved spacetime geometry restricted to harmonic coordinates. If it is impossible in one framework, it is impossible in the other framework.
So in flat spacetime, how does light bend around the sun... via the Huygen's path Harrylin mentions? How else? Harmonic coordinates mean near the singularities as you mentioned. Does this mean inside the event horizon (say 10 light years across) or most inner part of it near the center (near planck scale)? Whatever, in quantum gravity which goes beyond the planck scale, it can address the issues inside the event horizon or just near the singularities? This is because it is not possible to address black holes in flat spacetime. Then would quantum gravity of spin-2 field in flat spacetime be able to address black holes whose event horizon (say 10 light years across) is still much below the planck scale size? How? Note it is only near the singularity that planck scale physics address. Hope you get what I'm asking or I'd have to rewords this again. Thanks.