Hi Ken,
So the difficulties between the quantum world and general relativity are actually more profound that just the uncertainty principle and the whole problem hinges on finding the much sort quantum theory of gravity. Let me explain…
First up, quantum mechanics. QM as governed by the Schrodinger equation is non-relativistic. We can see that because in this equation we differentiate twice with respect to position, but just once with respect to time, In relativity space and time are treated equally and that clearly isn’t the case with Schrodinger’s equation. Now the relativistic version of QM is quantum field theory. This is quite a change of quantum mechanics- in QFT underlying fields are fundamental objects not particles- and when these fields are stimulated under certain conditions, they give rise to particles- in much the same way as striking the chord A minor on a piano keyboard generate the musical sound of A minor.
Now QFT when applied to condensed matter physics is very accurate- there are problematic infinities in the equations (in order words, instead of a finite numerical value you get an infinity) but these can be renormalized away quite safely with predictions that agree very accurately with results from condensed matter physics experiments. The Fourier transform is a key part of this as it describes how the field works in terms of annihilation and creation operators. Excite the field with the creation operator and you get particles from the field, you can de-excite it until you reach the ground state (also known as the vacuum state). Since the particle is written in terms of annihilation and creation operators the particle is uniquely defined. The theory is also relativistic and can be applied to Maxwell’s equations to give a relativistic version of electromagnetism. So far so good- now to General Relativity.
General Relativity (GR) is a classical field theory and are governed by the Einstein Field Equations. On the left hand side we have quantities like the Reimann tensor which describe the curvature of spacetime, and on the right hand side we have the energy-stress tensor- this represents the matter and energy. In other words then, if we have something heavy like a star or a black hole, its matter-energy is represented by the stress tensor, and this tells spacetime how to distort and curve. Solutions to Einstein’s equations are metric tensors- these tell us (with the aid of a line element) how to measure distances in points between spacetime. The flat spacetime of special relativity is one example, the Schwarzschild spacetime is another. There are also cosmological solutions like the Robertson Walker spacetime and de Sitter spacetime solutions. These are all vacuum solutions though, and this means that outside the event horizons, the stress tensor is zero. Also the stress tensor doesn’t care what the matter and energy is made of, and has no quantum description.
So, this brings us to Hawking radiation. Clearly, all objects in the Universe have to obey all the laws of physics, not just some of them. in particular, they all have to obey the second law of thermodynamics: entropy increases in a closed system. If an object has entropy then it has heat and is coming into thermal equilibrium with its surroundings. As it turns out, black holes have a measure of entropy- this is their event horizons. As a result, black holes must have a temperature and this was what Hawking discovered. How does this help? Well thermodynamics are statistical physical processes which are governed by quantum phenomena- so in the vicinity of the event horizon, QFT and GR have to agree and work together.
The first attempts to do this properly is to put QFT into GR. This is done by replacing the stress tensor with something called the expectation value of the stress tensor- now it is composed of quantum fields rather than a generic ‘lump of matter’. This is where the problems really come in. Firstly, the vacuum (or ground state). The crux of this is the Fourier transform as I mentioned earlier. If you put the Schwarzschild back hole into QFT instead of flat spacetime, you find that there is more than one vacuum state! So the annihilation operator which de-excites you field before the star collapses to become a black hole, might no longer do so after the black holes has formed. As a result the notion of a particle is now deeply ambiguous. Worse still, the infinities!!! In flat spacetime we can renormalize them away, but energy contributes to the stress tensor- it is a potential source of gravity in general relativity and so we can’t just renormalize it! There is a process for doing so in one dimension but there is no general procedure for doing it 4 dimensions, and the processes in 4 dimension involve a mathematical technique called ‘point splitting’ which may or may not be Lorentz invariant!
So where does this leave us? Stuck, frankly! There are two approaches: string theory which has taken up a lot of money and time but delivered very little. We thought a few vacuum states was bad enough but in string theory there are something like 10^235 vacuum states and it is not clear which (if indeed any) are physically meaningful. The other approach is loop quantum gravity- here the idea is that spacetime is made up of tiny course grains down at the Planck level- however great problems remain here, for example it is not possible to recover ordinary flat space results in a low energy limit as one would expect.
The real paradigm shift will be the solution to the problems I have outlined here, they are conceptual and not just computational, and they require a deep understanding of the mathematical and physical tools we use in physics. I suspect at some point, someone will come up with a new understanding of space, time matter and energy- as far removed from general relativity as Newtonian mechanics is from special relativity. There are also many dep problems with quantum mechanics which have yet to be resolved and these do carry over to QFT. To sum up- the physics we have is very successful- it allows to build semi conductor devices and sat navs but there still remain many deep problems to solve.
Cheers,
-Paul
