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General relativity and quantum mechanics stand out as the pillars of twentiethcentury science, able to describe almost all known phenomena from the scale of subatomic particles all the way up to the rotations of galaxies and even the history of the universe itself. Despite this grand success, which includes stunning agreement with experiment, these two theories represent physics at a crossroads one that is plagued with crisis and controversy. The problem is that at rst sight, these two theories are at complete odds with each other. The general theory of relativity (GR), Einstein s crowning achievement, describes gravitational interactions, that is, interactions that occur on the largest scales that we know. But it not only stands out as Einstein s greatest contribution to science but it also might be called the last classical theory of physics. That is, despite its revolutionary nature, GR does not take quantum mechanics into account at all. Since experiment indicates that quantum mechanics is the correct description for the behavior of matter, this is a serious aw in the theory of general relativity. We don t think about this under normal circumstances because quantum effects only become important in gravitational interactions that are extremely strong or taking place over very small scales. In the situations where we might apply general relativity, say to the motion of the planet mercury around the sun or the motion of
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the galaxies, quantum effects are not important at all. Two places where they will be important are in black hole physics and in the birth of the universe. We might also see quantum effects on gravity in very high energy particle interactions. On the other hand, quantum mechanics basically ignores the insights of relativity. It basically pretends gravity doesn t exist at all, and pretends that space and time are not on the same footing. The notion of space-time does not enter in quantum mechanics, and although special relativity plays a central role in quantum eld theory, gravitational interactions are nowhere to be found there either.
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A Quick Overview of General Relativity
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This isn t a book on GR, but we can give a very brief overview of the theory here (see Relativity Demysti ed for details). The central ideas of general relativity are the notion that geometry is dynamic and that the speed of light limits the speed of all interactions, including gravity. We start with the notion of the metric, which is a way of describing the distance between two points. In ordinary three-dimensional space the metric is ds 2 = dx 2 + dy 2 + dz 2 (1.1)
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This metric follows from the pythagorean theorem by making the distances involved in nitesimal. Note that this metric is invariant under rotations. Something that is key to relativistic thinking is focusing on those quantities that are invariant. To move up to a relativistic context, we extend the notion of a measure of distance between two points to a notion of distance between two events that happen in space and time. That is, we measure the distance between two points in space-time. This is done with the metric ds 2 = c 2 dt 2 + dx 2 + dy 2 + dz 2 (1.2)
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This metric extends the idea of geometry to include time as well. But not only that, it also extends the notion of a distance measure between two points that is invariant under rotations to one that is also invariant under Lorentz transformations, that is, Lorentz boosts between one inertial frame and another. While adding time to the mix certainly extends the notion of geometry into an unfamiliar realm, we still have a xed geometry that does not take into account gravitational elds. To extend the metric in a way that will do this, we have to enter the domain of non-euclidean geometry. This is geometry which does not require
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