I have recently reread "A Brief History of Time" by Stephen Hawking and in the book, he states that quantum mechanics and relativity are contradictory to each other and therefore cannot both be correct. I realize that at present physics lacks a unified theory that incorporates both but in what ways are they contradictory to each other? And if they cannot both be correct, how is it that we are able to make incredibly accurate predictions using both? Does it have to do with the presence or lack of singularities within the Universe? Thanks in advance.
Davon2006-09-24T23:20:32Z
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Detailed Answer (Summary at the end if you don't want to read it): Great question, and lots of good responses. I want to point something out: general relativity is well tested, but not to the precision and accuracy of quantum mechanics. Many physicists prefer the quantum view of the universe over the general relativistic view in the sense that one is "more right" than the other. Feynman, for example, didn't like GR all that much from what I hear. But a lot of the black hole jocks (Hawking, Penrose, etc.) are fueling the idea that QFT (Quantum Field Theory - the successor of QM that takes into account Special Relativity but can't handle GR) has some flaws in light of recent discoveries in the theoretical work of blackhole physics. When applying QFT to Black Holes (which are heavily based on GR), they come up with results that indicate some issues with QFT. The QFT guys disagree.
So when Hawking says "both" theories are wrong, he's coming at it from a very biased, but most likely correct, point of view. Keep that in mind.
Black holes represent the limits of GR - as matter becomes more and more dense inside a blackhole (and even in Neutron stars), the energy density gets high enough that QFT is needed to really understand the physics. A blackhole in GR is explained as a singularity - the highly dense matter of the BH shrinks to a infinitesimally small point. Einstein and others didn't like this, of course, because it represents a mathematical "infinity" which are usually considered not physical. So black hole physicists deal with the Event Horizon of the blackhole, not the singularity. This is where QFT comes in. The correct theory of the universe should explain what happens as the star collapses to a black hole's singularity. Does something stop the collapse or does a true singularity really exist in nature?
Now QFT treats the forces of nature as an exchange of particles. Two electrons "feel each other" because they keep sending photons back and forth. Furthermore, the vaccuum of space is constantly "bubbling" with virtual particles that come into and out of existance according to the Uncertainty Principle. As a particle moves in space, it's interacting with this "bubbling brew" of particles. GR, on the other hand, treats space as a very smooth medium that warps and bends as particles with mass/energy pass through it.
Since gravity is caused by this curvature, quantizing gravity to treat it with QFT causes some bad mathematical results - namely infinities. It is, technically speaking, a non-renormalizable theory. That translates into: they can't find a way to ignore the bad results. Furthermore, no one can explain why gravity is so much weaker than the other forces. Think about it, every massive particle has gravity, but it takes entire planet-size masses for gravity to become important. That's not the case with the other forces. As physicists are trying to unify all the forces into one theory/force, gravity being so much weaker makes things a lot harder. It's also the hardest to test on the small scale for the same reason.
Summary: OK, all that long explanation boils down to this: no one has succeeded in quantizing gravity because when they apply the same "techniques" to gravity as they did to the other interactions of nature, the Math yells at them and gives them results that go to infinity. This is the case with the other interactions, but those smart theorists found a way to eliminate the infinities which doesn't work with gravity. It has to do with how space is so "rough" on the small scale (in the sense that particles keep popping in and out, thus causing gravity to fluctuate). Also gravity is so much weaker than the other forces that it is hard to unify it with them (extra dimensions might solve this).
RE: Quantum Mechanics vs. Relativity? I have recently reread "A Brief History of Time" by Stephen Hawking and in the book, he states that quantum mechanics and relativity are contradictory to each other and therefore cannot both be correct. I realize that at present physics lacks a unified theory that incorporates both but in...
The fundamental problem between the two is that relativity assumes that space and time can be infinitely divided -- that there is no "fundamental unit" of space or time. Quantum mechanics assumes (but does not prove) the opposite -- that space and time do have a fundamental unit (i.e., there must be a minimum "quantity" of space and time). In most cases only one of the theories needs to be used to make predictions. However, there are cases where both need to be applied, and that's where the problems arise, because they don't play well together. Most physicists assume that singularities do not really exist; however, they need a more complete theory (string theory, perhaps) to make them go away.
Both can't be correct is a strong statement to make. The problem lies in the fact that relativity is known to work very well in a grand scale like the scale of stars and galaxies and quantum mechanics works in the scale of atoms and electrons. But when one tries to use them in different scales one gets contradictory results. That probably means we need to adjust them to work in all scales but no one has yet figured out how.