Abstract
Contemporary discussions of physical determinism that engage with modern spacetime and gravitational theory have typically focused on the question of the global uniqueness of solutions for initial-value problems. In this chapter I investigate the violation of local uniqueness, found in examples like Norton’s dome, which are not typically considered in light of spacetime theory. In particular, I construct initial trajectories for massive particles whose worldlines are not uniquely determined from initial data, both for a charged particle in special relativistic electromagnetism and for a freely falling particle in pure general relativity. I also show that the existence of such examples implies the violation of the Strong Energy Condition, and consider their implications for the interpretation of spacetime theory.
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Notes
- 1.
Thanks to Dennis Dieks and the audience in Budapest for the Fifth Budapest-Krakow Research Group Workshop on May 23–24, 2016 for their helpful comments.
- 2.
Note that since 0 ≤ θ ≤ 90∘, 0 ≤ dh∕dr ≤ 1, hence 0 ≤ r ≤ g 2∕b 4 and 0 ≤ h ≤ 2b 2∕3g 4: the dome must have a finite height.
- 3.
From the present perspective, the stress-energy tensor T ab is not an independent object for a relativistic spacetime once the metric has been specified, since the Riemann tensor R bcd a associated with the Levi-Civita connection determines the Ricci tensor R ab and curvature scalar R, which in turn determine T ab through Einstein’s equation, T ab = (1∕8π)(R ab − (1∕2)Rg ab ).
- 4.
One may apply the formulas of Wald (1984, p. 446) to calculate these explicitly, although doing so does not seem to give any obvious insight into the nature of the indeterminism this spacetime exhibits.
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Fletcher, S.C. (2017). Indeterminism, Gravitation, and Spacetime Theory. In: Hofer-Szabó, G., Wroński, L. (eds) Making it Formally Explicit. European Studies in Philosophy of Science, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-319-55486-0_10
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