Absolute Objects, Counterexamples and General Covariance

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Pitts, J. Brian
conference paper description
The Anderson-Friedman absolute objects program has been a favorite analysis of the substantive general covariance that supposedly characterizes Einstein's General Theory of Relativity (GTR). Absolute objects are the same locally in all models (modulo gauge freedom). Substantive general covariance is the lack of absolute objects. Several counterexamples have been proposed, however, including the Jones-Geroch dust and Torretti constant curvature spaces counterexamples. The Jones-Geroch dust case, ostensibly a false positive, is resolved by noting that holes in the dust in some models ensure that no physically relevant nonvanishing timelike vector field exists there, so no absolute object exists. The Torretti constant curvature spaces case, allegedly a false negative, is resolved by testing an irreducible piece of the metric, the conformal metric density of weight -2/3, for absoluteness; this geometric object is absolute. A new counterexample is proposed involving the orthonormal tetrad said to be necessary to couple spinors to a curved metric. The threat of finding an absolute object in GTR + spinors is overcome by the use of an alternative spinor formalism that takes a symmetric square root of the metric (with the help of the matrix diag(-1,1,1,1)), eliminating 6 of the 16 tetrad components as irrelevant. The importance of eliminating irrelevant structures, as Anderson emphasized, is clear. The importance of the choice of physical fields is also evident. A new counterexample due to Robert Geroch and Domenico Giulini, however, finds an absolute object in vacuum GTR itself, namely the scalar density $g$ given by the metric components' determinant. Thus either the definition of absoluteness or its use to analyze GTR's substantive general covariance is flawed. Anderson's belief that all absolute objects are nonvariational (that is, not varied in a suitable action principle) and vice versa is also falsified by the Geroch-Giulini counterexample. However, it remains plausible that all nonvariational fields are absolute, so adding nonvariationality as a necessary condition for absoluteness, as Hiskes once suggested, would likely leave no useful work to the Anderson-Friedman condition of sameness in all models. Simply having only variational fields in an action principle (suitably free of irrelevant fields) might be a satisfactory analysis of substantive general covariance, if one exists. This proposal also resembles the suggestion that GTR is "already parameterized," if one decides to parameterize theories by defining the nonvariational fields in terms of preferred coordinates called clock fields. More questions need to be addressed. Which fields should be tested for absoluteness: only primitive fields (which ones?), or all or some (which?) of their concomitants also? Geroch observes that some kinds of geometric objects, such as tangent vectors, scalar densities, and tangent vector densities of non-unit weight, satisfy the condition of sameness in all models if they merely fail to vanish. If these "susceptible" geometric objects can hardly help being absolute, to what degree are they, or the theories harboring them, responsible for this absoluteness? The answer to this question helps to determine the significance of the Geroch-Giulini counterexample.