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2 Physics in a Curved Spacetime

There is an extensive literature on special and general relativity, and the spacetime-based view2 of the laws of physics. For the student at any level interested in developing a working understanding we recommend Taylor and Wheeler [109] for an introduction, followed by Hartle’s excellent recent text [51] designed for students at the undergraduate level. For the more advanced students, we suggest two of the classics, “MTW” [80Jump To The Next Citation Point] and Weinberg [117Jump To The Next Citation Point], or the more contemporary book by Wald [114Jump To The Next Citation Point]. Finally, let us not forget the Living Reviews archive as a premier online source of up-to-date information!

In terms of the experimental and/or observational support for special and general relativity, we recommend two articles by Will that were written for the 2005 World Year of Physics celebration [122Jump To The Next Citation Point121Jump To The Next Citation Point]. They summarize a variety of tests that have been designed to expose breakdowns in both theories. (We also recommend Will’s popular book Was Einstein Right? [119] and his technical exposition Theory and Experiment in Gravitational Physics [120Jump To The Next Citation Point].) To date, Einstein’s theoretical edifice is still standing!

For special relativity, this is not surprising, given its long list of successes: explanation of the Michaelson–Morley result, the prediction and subsequent discovery of anti-matter, and the standard model of particle physics, to name a few. Will [122Jump To The Next Citation Point] offers the observation that genetic mutations via cosmic rays require special relativity, since otherwise muons would decay before making it to the surface of the Earth. On a more somber note, we may consider the Trinity site in New Mexico, and the tragedies of Hiroshima and Nagasaki, as reminders of E = mc2.

In support of general relativity, there are Eötvös-type experiments testing the equivalence of inertial and gravitational mass, detection of gravitational red-shifts of photons, the passing of the solar system tests, confirmation of energy loss via gravitational radiation in the Hulse–Taylor binary pulsar, and the expansion of the Universe. Incredibly, general relativity even finds a practical application in the GPS system: If general relativity is neglected, an error of about 15 meters results when trying to resolve the location of an object [122Jump To The Next Citation Point]. Definitely enough to make driving dangerous!

The evidence is thus overwhelming that general relativity, or at least something that passes the same tests, is the proper description of gravity. Given this, we assume the Einstein Equivalence Principle, i.e. that [122Jump To The Next Citation Point121120]

If the Equivalence Principle holds, then gravitation must be described by a metric-based theory [122]. This means that

  1. spacetime is endowed with a symmetric metric,
  2. the trajectories of freely falling bodies are geodesics of that metric, an
  3. in local freely falling reference frames, the non-gravitational laws of physics are those of special relativity.

For our present purposes this is very good news. The availability of a metric means that we can develop the theory without requiring much of the differential geometry edifice that would be needed in a more general case. We will develop the description of relativistic fluids with this in mind. Readers that find our approach too “pedestrian” may want to consult the recent article by Gourgoulhon [49], which serves as a useful complement to our description.

 2.1 The metric and spacetime curvature
 2.2 Parallel transport and the covariant derivative
 2.3 The Lie derivative and spacetime symmetries
 2.4 Spacetime curvature

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