Shelly Yiran Shi

Noether’s first theorem demonstrates that continuous symmetries give rise to conserved quantities. This fact tempts many to hold that symmetry principles explain conservation laws. Yet there is a puzzle: the derivation goes both ways. Why does symmetry explain conservation when the derivation is bidirectional? Lange (2007, 2009) provides an answer: symmetry principles are meta-laws, and meta-laws explain first-order laws just as first-order laws explain facts. Using a “non-standard” Lagrangian, Smith (2008) claims that conservation of angular momentum can hold without rotational symmetry, providing a counter-example to Lange. In this paper, I show that Smith’s non-standard Lagrangian fails to serve as a counterexample. However, that doesn’t leave Lange’s account unchallenged. I argue that the debate between Lange and Smith revolves around an ambiguity which, once clarified, leads to a dilemma. Which symmetry principle explains? Is it the symmetry of the action or the symmetry of equations of motion? If the former, then the symmetry is no more stable than conservation laws. Hence, we lose the desired explanatory direction. If the latter, the symmetry lacks explanatory relevance and fails to exhibit greater stability than conservation laws. However one disambiguates “symmetry”, it remains mysterious why symmetry principles explain conservation laws.

One challenge to relationism in general relativity is that the metric field is underdetermined by the stress–energy tensor. This is manifested in the existence of distinct vacuum solutions to Einstein’s equations. In this paper I reformulate the problem of underdetermination as a problem from vacuum solutions. I call this the vacuum challenge and identify the gravitational degrees of freedom (associated with the Weyl tensor) as the “source” of the challenge. The Weyl tensor allows for gravitational effects that something outside a system exerts on the system. I provide a relationist response to the vacuum challenge.