Abstract
Retention of structure across theory change has been invoked in support of a ‘structural’ alternative to more traditional entity-based scientific realism. In that context the transition from Newtonian mechanics to the Special Theory of Relativity is often regarded as a very significant instance of structural preservation, or retention, associated with correspondence-based recovery. The joint derivation, from a small set of elementary and ontologically neutral assumptions, of both the Galilei and the Lorentz transformation exemplifies the virtues of structural approaches to the foundations of physical theories. The common origination of the resulting two relativistic frameworks sheds light on both the successes and the limitations of correspondence claims. However, the cognitive-operational character of the basic assumptions lends no support to the structural realist’s ‘inference to the best explanation’.
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Notes
If it can be said that “a theory’s mathematical formulation ‘encodes’ the structure of that theory’s target domain”, Frigg and Votsis (2011, 243) note that “the notion of encoding structure is far from clear.” Attempts (e.g. by Frigg and Votsis 2011, 229) to produce a formal definition of what the proponents of structural realism mean by structure amount to rehearsing familiar mathematical concepts: morphisms, equivalence classes etc. While those concepts can contribute to the identification of intratheoretical or intertheoretical relationships, such precision does not license the widespread use of expressions like ‘the structure of the world’, or help shed light on what it would precisely mean, beyond the metaphor, for certain parts of physical theories to ‘latch on’ to the structure in question.
Although not perhaps in the strict sense of the word, as discussed by Votsis (2011).
“The simple statement that quantum mechanics reduces to classical mechanics in the limit where the principal quantum number n approaches infinity, while found in many textbooks, is not true in general” (Liboff 1984, 50); neither does this limit generally coincide with the outcome of neglecting the magnitude of Planck’s constant against the typical ‘action’ of a system (this is often expressed as the nonsensical ‘limit’ h→0).
For example, ‘the structural realist simply asserts that, in view of the [quantum] theory’s enormous empirical success, the structure of the universe is (probably) something like quantum–mechanical’ (Worrall 1989, 123; emphasis added).
Ontic Structural Realism branches into Eliminativism, which rejects the existence of anything that is ‘merely’ structured (objects, events…), retaining ‘structure only’ as the legitimate focus of a realist perspective; and Non-eliminativism, which allows entities to occupy a lower-status position in a structural hierarchy. There, some ‘relational structure of the world’ (Ladyman & Ross 2007, 130) would reign at the top, by virtue of its ‘not supervening on the intrinsic properties of a set of individuals’ (ibid.).
That the Lorentz metric is generally valid ‘even’ in the absence of electromagnetic fields, whereas the speed of light appears to be a specific feature of electromagnetism, has often been pointed out, e.g. by Hawking and Ellis (1973, 91).
To the above and similar derivations of the general form of \(T_{{R \to R^{\prime}}}\), Feigenbaum (2008) objects that their one-dimensional character trivializes isotropy, unjustifiably reducing it to parity: “This is inappropriate, because both the results and the arguments they rely upon are not sustainable in higher dimensions without some modifications” (Feigenbaum 2008, 2). Basic derivations take it for granted that the corresponding axes of two inertial frames, e.g. (Ox) in R and (O′x′) in R′, are parallel. This is a “highly non-trivial detail” (ibid., 5), which Feigenbaum addresses with a technically more demanding three-dimensional treatment. One noteworthy outcome of the latter is that group-theoretical composition simply follows from the joint assumptions of isotropy and homogeneity. Feigenbaum emphasizes the importance of so-called Wigner rotations, describing his approach as “a lengthy constructive proof of the proposition that if this (Wigner) rotation is always the identity then the theory is of Galilean-Newtonian invariance, while should it ever be non-trivial for any value of its arguments, then the theory is of Lorentz invariance.” (ibid., 3). Wigner rotations are relevant because the transitivity of parallelism cannot simply be taken for granted, and it affects the possibility of composing velocities additively. Their importance notwithstanding, Feigenbaum admits that “these [Wigner] rotations are barely utilized in the construction”. His three-dimensional approach agrees with more pedestrian accounts in that isotropy and homogeneity jointly suffice to determine the form, and through it all the conceptual significance of a RF based upon the neutral character of inertial frames. The resulting transformation must be linear, or more accurately affine, providing that spatial coordinates and time in two relative inertial frames are one-to-one pointwise related.
‘Light-free’ methods for synchronizing clocks, e.g. involving sound or slow transportation, are discussed in Arons (1965, 882–87). Mermin (1984, footnote 6) also suggests ways of obviating the use of light signals to work out the empirical value of κ. To be fair, it appears to be remarkably difficult to dispense with the propagation of light as a suitably uniform process for exchanging signals and comparing records made at distinct locations. On this basis, Drory (2015) argues for the indispensability of the speed of light as a foundation for STR as a physical theory, over and above the SRF it embodies.
The variation of energy dE that is associated with an infinitesimal displacement dr is by definition such that \(dE = \frac{\text{dp}}{dt} \cdot {\text{dr}}\), or \(d\left( {\frac{E}{\kappa}} \right)\) d(κt)−dpdr = 0, where the conversion factor κ is introduced so that κt and r have the same dimensions, and so do \(\frac{E}{\kappa}\) and p. Restricting as before to relative uniform motion in the x direction, then \(d\left( {\frac{E}{\kappa}} \right)\) d(κt)−dpdx = 0 (p is the x component of momentum, as given in frame R). Interpreting this condition as a zero scalar product and comparing with the invariant quadratic form x 2−κ 2 t 2 suggests that the dynamical quantities E and p stand with respect to each other in the same relativistic relationship as x and t do. The suggestion can be further motivated by the fundamental connection Noether’s theorem establishes between x and p on the one hand, and t and E on the other: conservation of momentum is a formal expression of invariance under spatial translation, whereas conservation of energy connects to invariance under time shifts. Substituting \(\frac{E}{{\kappa^{2} }}\) for t and p for x then yields a dynamical counterpart of T v,κ :
$$p^{\prime} = \gamma \left( {p - \frac{v}{{\kappa^{2} }}E} \right)\quad {\text{and}}\quad E^{\prime} = \gamma \left( {E - vp} \right).$$If a body is at rest \(\left( {p^{\prime} = 0} \right)\) in R′, then \(p - \frac{v}{{\kappa^{2} }}E = 0\) and given p = γmv its energy in R is E = γmκ 2. Since E = γE′, the energy of the body in frame R′ is E′ = mκ 2, where m is the ‘rest mass’ of the body. Just as in the kinematical case, a non-Euclidean invariant \(p^{2} - \frac{{E^{2} }}{{\kappa^{2} }} = p^{\prime2} - \frac{{E^{\prime2} }}{{\kappa^{2} }}\) holds between the related quantities and E 2 = p 2 κ 2 + m 2 κ 4.
According to the discussion in Sect. 2, Einstein consciously ‘retained’ something essential from (the structure of) Poisson’s equation, in order to derive his own field equations. The connection is indeed visible upon close inspection of the two equations when those are written
$$\nabla^{2} {\text{U = 4}}\pi {\text{G}}\rho \quad {\text{and}}\quad\frac{\kappa^{4}}{2}\left({{R} _{\mu \nu} - \frac{1}{2}g_{\mu \nu} R + \varLambda g_{\mu \nu} } \right) = 4\pi GT_{\mu \nu} .$$However, that sense of retention is not what the structural realist is after.
To the focus of this paper on STR, it might be objected that STR has been superseded by Einstein’s 1916 ‘General Theory’—a fact that would greatly lessen the import of any philosophical claims involving STR. This objection is based, however, on a widespread misconception, which certainly owes much to the characterization, by Einstein himself, of his gravitation theory as a ‘general theory of relativity’. ‘Special’ Relativity, qua SRF, actually yields a ‘superlaw’ in the sense of Wigner (1995), based upon invariance under transformations of the Poincaré group (Galilean relativity provides a no less general, alternative superlaw, derived from the same assumptions but related to invariance under transformations constitutive of the Galilei group). Allowing frames to accelerate can lead to no further generalization: since accelerated frames are readily identified by observable effects, they lack the neutral character of inertial frames that makes it possible to develop a relativistic programme. The starting point of STR is the idea that gravitational effects can be locally cancelled by accelerated motion. A connection—in the sense of differential geometry—must then be established between all those local cancellations for gravity to be a genuine physical interaction. Gravity can thus be regarded as a local gauge symmetry induced by the Lorentz or Poincaré transformation. The object of GTR is not the abstract equivalence of ‘generalized’ reference frames but a local account of gravity that complies with the accepted version of a RF. GTR cannot therefore be invoked to minimize the relevance or the impact of arguments involving STR in its (S)RF function.
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Delhôtel, JM. Retaining Structure: A Relativistic Perspective. J Gen Philos Sci 48, 239–256 (2017). https://doi.org/10.1007/s10838-016-9348-6
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DOI: https://doi.org/10.1007/s10838-016-9348-6