Saturday, 16 May 2009

General Relativity and Nonlinearity

One of the difficulties set up by the structure of Einstein's general theory of relativity is the tension between GR's requirement that there be no prior geometry, and the assumption that the geometry must necessarily reduce to the flat fixed geometry of special relativity's Minkowski metric as a limiting case over small regions.

Although it's not news that GR shouldn't presume a prior geometry (GR's fields are not superimposed on a background metric, they define the metric), this is one of those irritating principles that's easier to agree with in principle than it is to actually implement.
It's only human nature when attacking a problem to want to start off with some sort of fixed point or known property that everything else can be defined in relation to. It's like starting a jigsaw by identifying the four corner pieces first. We tend to start off by imagining the shape of the environment and then imagining placing a test object within it ... but the act of placing an observer itself modifies the shape and characteristics of the metric, and means that the signals that the observer intercepts might have different characteristics to those that we might otherwise expect to have passed though the particle's track, if the particle hadn't actually been there, or if it had been moving differently. Although the basic concept of a perfect "test particle" isn't especially valid under relativity theory, we like to assume that the shape of spacetime is largely fixed by large background masses, and that the tiny contribution of our observer-particle won't change things all that much (we like to assume that our solutions are insensitive to small linear "perturbations" of the background field).

Unfortunately, this assumption isn't always valid. Even though the distortion caused by adding (say) a single atom with a particular state of motion to a solar system may well be vanishingly small, and limited to a vanishingly-tiny region of spacetime compared to the larger region being looked at, every observation that the atom and solar system make of each other will be based on the properties of exchanged signals that all have to pass through that teensy-weensy distorted region. So if we build a theory on mutual observation and the principle of relativity, even a particle-distortion or gravitomagnetic distortion that's only significant in over a vanishingly small speck of spacetime surrounding the atom still has the potential to dramatically change what the atom sees, and how outsiders see the atom. It changes the properties of how they interact, and by doing that, it also changes the characteristics of the physics. Although a star isn't going to care much whether an individual distant atom makes a tiny distortion in spacetime or not, our decision as to whether to model that distortion or not can change the functional characteristics of our theory, and change the way that we end up modelling the star, and some of the predictions that we make for it. It also has the potential to wreck the validity of the frame-based approach that people often use with general relativity – if we take nonlinearity seriously, we should probably be talking about the relativity of object views, rather than the relativity of "frames".

Field components aren't always guaranteed to combine linearly, they can twist and impact and writhe around each other in fascinating ways, and generate new classes of effect that didn't exist in any of the individual components. For instance, if we take a bowling ball and a trampoline, and place the ball on the trampoline, their combined height is less than the sum of the two individual heights, and the trampoline geometry has some new properties that aren't compatible with its original Euclidean surface. The surface distorts and the rules change. [Ball+ Trampoline] <> [Ball] + [Trampoline].
Or, place a single bowling ball on an infinite trampoline surface and it settles down and then stays put. But place two bowling balls on the surface, reasonably near to each other, and the elastic surface will push them towards each other in an attempt to minimise its stresses and surface area, producing relative motion. A one-ball model is static, a two-ball model is dynamic, so the rules just changed again.
The result of assuming a background field and simply overlaying particles isn't guaranteed to be the same as a more realistic model in which the particles are intrinsically part of the background field. Nonlinear behaviour generates effects that often can't be generated by simple overlay superimpositions.

Einstein's special theory of relativity rejects the idea of any such interaction between a particle and its surrounding spacetime, so this class of nonlinear effect is incompatible at the particle level with our current general theory of relativity (which is engineered to reduce to SR). While we understand that perhaps a fully integrated model of physics can't be broken up into self-consistent self-contained pieces that can be modelled individually and then assembled into a whole, we try it anyway, because it's easier to tackle smaller bite-size theories than to try to create the full Theory of Everything from scratch. And when we work on these isolated theories, and try to make them internally consistent without taking into account external factors, we end up with a series of theoretical building blocks built on different principles that don't fit together properly.

For Einstein's general theory of relativity, we say that the theory must reduce to the flat-spacetime physics of special relativity over small regions, which makes the theory pretty much incompatible with attempts to model particle-particle interactions as curvature effects. But if what we understand as "physics" is the result of particle-observers communicating through an intermediate medium, and the geometrical properties of those particles on the metric is an intrinsic part of how they interact – if physics is about nonlinear interactions between geometrical features – then by committing to special relativity as a full subset of GR, we might have guaranteed that our general theory can never describe the problem correctly, because any solution with a chance of being right will be ruled out for being in conflict with special relativity. Since deep nonlinearity (which GR1915 doesn't have) seems to be the key to reproducing QM behaviour in a classically-based model, it's not surprising that serious attempts to try to find a way to combine GR and QM have tended to run into the nonlinearity issue:

Albert Einstein, 1954
at the present time the opinion prevails that a field theory must first, by "quantization", be transformed into a statistical theory of field probabilities ... I see in this method only an attempt to describe relationships of an essentially nonlinear character by linear methods.
Roger Penrose, 1976, quoted by Ashtekar:
... if we remove life from Einstein's beautiful theory by steam-rollering it first to flatness and linearity, then we shall learn nothing from attempting to wave the magic wand of quantum theory over the resulting corpse.
Some GR researchers did try to move general relativity beyond a reliance on a fixed initial geometry and dimensionality (see John Wheeler's work on pregeometry), but the QM guys were better at analysing where their "perturbative" and "nonperturbative" approaches differed than the GR guys were at identifying the artefacts that special relativity might have introduced into their model.

In order to work out what parts of current GR might be artefacts of our approach, it's helpful to look at non-SR solutions to the general principle of relativity, and compare the results with those of the usual SR-based version.
The two approaches give two different sorts of metric. If we embrace nonlinearity, we get a relativistic acoustic metric and a general theory that supports Hawking radiation, classically. The second approach (where we start by assuming that a particle's own distortion is negligible and doesn't play a role in what the particle sees) gives us standard classical theory, Minkowski spacetime, the current version of general relativity, and a deep incompatibility with Hawking radiation and quantum mechanics.

So I'd suggest that perhaps we shouldn't be trying to reconcile "current GR" with quantum theory ... we should instead be trying to replace our current crippled version of general relativity with something more serious, that didn't rely on that additional SR layer. There seem to have been two different routes available to us to construct a general theory of relativity, and it's possible that we might have chosen the wrong one.

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