Re: neophyte GR question

leithaus wrote:
[GR is based on coordinate charts on the manifold]
Now, what bugs me is that charts don't come for free. From meter rods
to GPS to atomic clocks, no physically useful coordinate machinery
comes without a footprint in the stress-energy tensor. Moreover, i
don't see how this is bootstrapped away in the development of the
equations.

In principle, if you use real rulers and clocks to implement your coordinate charts, then you must include them in your energy-momentum tensor. Note, however, there is no necessity to use such instruments to construct coordinates, and the equations and formalism work just as well if you just arbitrarily assign labels to the points of the manifold, being careful to obey the rules (uniqueness within a chart, using 4-tuples of real numbers for the labels, continuity and differentiability of the labels commensurate with the topology of the manifold, etc.). Such labels, of course, do not carry any mass or energy, and are not entered into the stress-energy tensor. Of course you do have the problem of defining a metric in terms of such coordinates, but if you plan ahead for this, it can be done independently of the test of GR for which the coordinates to be used....

In practice, of course, the rulers and clocks a real experimenter might use are VASTLY smaller than the astronomical objects being observed and used to test GR, so the approximation that neglects their contributions to the stress-energy tensor is an extremely good one. For instance, the satellite of Gravity Probe B is so much smaller than the earth and sun that one can consider it to follow a geodesic path to much better accuracy than one can measure its trajectory (the orientations of its gyroscopes are another matter). The clocks and telescopes on earth clearly do not significantly affect the contribution of the earth to the geometry of the manifold. No rulers or clocks are erected near the satellite's orbit, and coordinates out there are constructed via optical and radio observations of the satellite (and other objects) from earth.


Tom Roberts

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