Conservation of Energy: Missing Features in Its Nature and Justification and Why They Matter

Abstract

Misconceptions about energy conservation abound due to the gap between physics and secondary school chemistry. This paper surveys this difference and its relevance to the 1690s-2010s Leibnizian argument that mind-body interaction is impossible due to conservation laws. Justifications for energy conservation are partly empirical, such as Joule's paddle wheel experiment, and partly theoretical, such as Lagrange's statement in 1811 that energy is conserved if the potential energy does not depend on time. In 1918 Noether generalized results like Lagrange's and proved a converse: symmetries imply conservation laws and vice versa. Conservation holds if and only if nature is uniform. The rise of field physics during the 1860s-1920s implied that energy is located in particular places and conservation is primordially local: energy cannot disappear in Cambridge and reappear in Lincoln instantaneously or later; neither can it simply disappear in Cambridge or simply appear in Lincoln. A global conservation law can be inferred in some circumstances. Einstein's General Relativity, which stimulated Noether's work, is another source of difficulty for conservation laws. As is too rarely realized, the theory admits conserved quantities due to symmetries of the Lagrangian, like other theories. Indeed General Relativity has more symmetries and hence (at least formally) more conserved energies. An argument akin to Leibniz's finally gets some force. While the mathematics is too advanced for secondary school, the ideas that conservation is tied to uniformities of nature and that energy is in particular places, are accessible. Improved science teaching would serve the truth and enhance the social credibility of science.

Spacetime Conventionalism Revisited

“…(2) Pitts (2010Pitts ( , 2016Pitts ( , 2021; Lam (2011); Read (2020); Dewar and Weatherall (2018); Duerr (2019b,a) for discussions of the problem of defining gravitational energy. A further noteworthy paper, along lightly different lines, is Lehmkuhl (2011).…”

“…which is such that a total energy-momentum complex is conserved with respect to the the flat metric's torsion free derivative operator (Pitts 2010(Pitts , 2016(Pitts , 2021. This would eliminate any potential for conventionalism with regard to total energy conservation but at the cost of introducing extra background or auxiliary structures into the theory.…”

Spacetime Conventionalism Revisited

“…This means that the quantity given the flux into a region is still conventional in the sense that it will be different for different observers moving within the same spacetime patch irrespective of the existence or not of Killing vector fields. In this context, one could consider introduction of a gravitational-energy–momentum tensor which is such that a total energy–momentum complex is conserved with respect to the flat metric’s torsion-free derivative operator (Pitts, 2010, 2016, 2021). This would eliminate any potential for conventionalism with regard to total energy conservation but at the cost of introducing extra background or auxiliary structures into the theory.…”

“… 4 See Pitts (2010, 2016, 2021); Lam (2011); Read (2020); Dewar and Weatherall (2018); Duerr (2019a,b) for discussions of the problem of defining gravitational energy. A further noteworthy paper, along slightly different lines, is Lehmkuhl (2011).…”

On Two Slights to Noether’s First Theorem: Mental Causation and General Relativity

“…In the 1910s the 19th century mathematical tradition of analytical mechanics (Lagrange, Hamilton, Jacobi) at last became a standard part of physics, and the ideas were widely understood as applied to continua (whether elastic media or fields such as the electromagnetic and gravitational potentials). I have explained this textbook material (Goldstein, 1980, chapter 12) (Davis, 1970) as clearly as I could (at times working with Alin Cucu) in application to the philosophy of mind (Pitts, 2020a;Cucu and Pitts, 2019;Pitts, 2021), so I will be brief here. One infers the laws from a mathematical function of space and time, now called the Lagrangian density L, by taking a curious kind of derivative of the Lagrangian density with respect to the fields/potentials and their temporal and spatial derivatives (rates of change).…”