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5-Point Curvature Detector?

  1. Aug 10, 2009 #1
    I just finished reading Synge's account of a "5-point curvature detector"; Sections 7 and 8 of Chapter XI of his book "Relativity: The General Theory". I re-read it every 5 years or so and always get stuck at the same spots. The object is to estimate the Riemann tensor and the curvature of the world line at a source using trip time measurements (of light pulses) to a collection of mirrors (3 mirrors in Sect. 7 and 4 mirrors in Sect. 8). After a few pages of calculations in Sect. 7 he arrives at some formulae (176 and 177). He then points out in Sect. 8 that these formulae will not do since they involve Fermi distances which are "mere mathematical constructs" not measurable quantities. He states that what is needed is 5-points (4 mirrors and a source) not 4 (3 mirrors and a source; and the mathematical details "can be supplied by the methods of the preceding section".

    This is the first point at which he loses me. If you apply the methods of the preceding section it seems you will end up with the same non-measurable mathematical constructs in the formulae.

    He now gives a new method for a rough "order of magnitude" estimate using the 10 trip-time measurements. The 10 measurements are plugged into a determinant which gives an estimate of deviation from flat space-time. With a little hand waving he comes up with estimate of a typical component of the Riemann tensor and a typical component of the curvature of the world line of the source.

    Even if we fill in all the mathematical details how can 10 trip time measurements determine 10 components of the Riemann tensor and the curvatures of the world line of the source. If we allow the source to follow a geodesic, the curvatures of the world line would be zero and 10 measurements might work, but how?

    Does anyone have references to this type of problem: actual measurement of the Riemann tensor (by optical or other means)?

    Thanks for any help, Skippy
  2. jcsd
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