PAllen
Science Advisor
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And to clarify, there is nothing wrong with speaking of the geometry of spatial surface defined in some rotating coordinates (with a 4-metric given for the coordinates, and 3-metric derived for any chosen coordinates of the spatial surface). Where you get into trouble is assuming such a surface can always be constructed so as to say anything meaningful about spatial geometry of an object described by a timelike congruence. For not sufficiently well behaved congruences, there is no possible spatial geometry for the 'object' that has any expected meaning. And what is surprising, is that a simple rotating congruence is already a case that is ill behaved in this regard.
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