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The Heuristic Power of Theory Classification, the Case of General Relativity

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Abstract

In this article, we explore the heuristic power of the theoretical distinction between framework and interaction theories applied to the case of General Relativity. According to the distinction, theories and theoretical elements can be classified into two different groups, each with clear ontological, epistemic and functional content. Being so, to identify the group to which a theory belongs would suffice to know a priori its prospects and limitations in these areas without going into a detailed technical analysis. We make the exercise here with General Relativity, anticipate its ontological, epistemic and functional content and show afterwords that such expectations are justified in this case, being consistent with formal issues of General Relativity. With this, we attempt to make a case for the use of the distinction as a powerful tool for scientific and philosophical analysis.

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Notes

  1. There are several places discussing interpretations of Special Relativity where the distinction between framework and interaction theories (or principle and constructive theories) have a role. A largely discussed example is the so-called Brown-Janssen debate (see [26,27,28,29,30,31,32]), however, there is an important difference between our approach to the problem and how the Brown-Janssen debate has been framed. The Brown-Janssen debate is articulated in terms of the kinematical/dynamical distinction (opposing the kinematical nature of SR and the dynamical approach of Lorentz’s theory), and it is mostly focused on the explanatory power of each theory. Certainly, the kinematical nature of SR is related to the fact that SR is a framework theory, while Lorentz’s dynamical theory corresponds to an interaction theory, however, when approaching the problem from the framework/interaction distinction, the focus is no longer restricted to the epistemic (explanatory) dimension, but it also highlights ontological and functional differences which are relevant for our purposes. In this sense, our view is much closer to Camp [31], who while analysing the Brown-Janssen dispute highlighted the framework functional role of SR following DiSalle [33].

  2. This criterion of momentum transference is particularly apt to theories within physics, see [47,48,49]

  3. Here we refer to the epistemic content of the distinction in a different way as proposed in Flores [25]. He had in mind the ‘context of discovery’ when talking about the epistemic dimension of the distinction, however, he argues (as a conclusion of his work) that each theory has a related form of explanation, namely bottom-up or top-down, and we believe this is a much rich epistemic content of the distinction than the context of discovery, therefore we restrict our attention to this point.

  4. The principles alone are not enough to uniquely derive GR, extra elements (such as simplicity) has to be taken into account [50], therefore most of what is said in this article about GR is also valid for a larger family of theories, of which GR is just one. Nonetheless, due to its relevance, the discussion will only refer to GR.

  5. Although our goal is to highlight the constraining role of GR upon interactions well-defined in Minkowsky spacetime, it should be noticed that the extent to which GR is strictly reducible to SR locally have been questioned. For instance, Read et al. [52] argues that higher derivatives of the metric don’t vanish at a point under certain formulations of the Equivalence Principle, and claim that this entail consequences for the relationship between GR and SR.

  6. This recipe, also known as the Minimal Coupling scheme, is nonetheless not the unique way of generalising theories from flat space. Note that the inclusion of other curvature terms in the equations reveals ambiguities, as it has been widely warned (see Goenner [54], Tino et al. [55]). When passing from SR to GR it should be taken into account that further constraints on the behaviour of matter in the presence of spacetime curvature might appear using different schemes.

  7. For further detail about the derivation of all the expression presented here and in the examples of the following footnotes, see [53, pp. 179–190].

  8. The Lagrangian density corresponds to \(\mathcal {L}=-\rho\), with \(\rho\) the proper density (which is the energy in a local rest frame of the fluid). For such Lagrangian density, after changing the derivatives for covariant derivatives, the energy-momentum tensor becomes:

    $$\begin{aligned} T_{\mu \nu }=(\rho + p) u_{\mu }u_{\nu }+pg_{\mu \nu } \end{aligned}$$
    (3)

    with ‘p’ the isotropic pressure, and u the four-velocity that characterise the fluid.

  9. For example, the Lagrangian for the electromagnetic field

    $$\begin{aligned} L=-\frac{1}{4}F_{\alpha \beta }F^{\alpha \beta }=-\frac{1}{4}g^{\alpha \beta }g^{\mu \nu }F_{\mu \alpha }F_{\nu \beta } \end{aligned}$$
    (4)

    , from it, after changing the derivatives for covariant derivatives, the energy momentum tensor reduces to

    $$\begin{aligned} T_{\mu \nu }=F^{\alpha }_{\mu }F_{\alpha \nu }-\frac{1}{4} g_{\mu \nu }F_{\alpha \beta }F^{\alpha \beta } \end{aligned}$$
    (5)

    which again satisfy Einstein field equation 2.

  10. There is more than one formulation of the equivalence principle, for a discussion of the relationship between different versions, see [57].

  11. The postulates are:

    1. 1.

      Spacetime is endowed with a symmetric metric.

    2. 2.

      The trajectories of freely falling test bodies are geodesics of that metric.

    3. 3.

      In local freely falling reference frames, the non-gravitational laws of physics are those written in the language of special relativity.”

  12. The idea also appears in textbooks, although perhaps less frequently, see Carroll [59, p. 178]. On another perspective in this regard, Mukohyama and Uzan [60] argues that the Lorentzian structure is an effective property that emerges due to fields propagation in an effective metric.

  13. We here refer to what Lehmkuh [57] call ‘Einstein’s Equivalence Principle’ (EEP) –it is not possible to distinguish between being in a gravitational field and noninertial frames–, and not to what he called ‘Weak Equivalence Principle’ (the numerical equivalence between inertial and gravitational mass) nor ‘Strong Equivalence Principle’(SEP) (the possibility of re-obtaining Special Relativity from GR ).

  14. The claim “a universe in which the laws of physics are covariant” seems at first sight as a physically loaded statement, however, as Carroll says “It is more or less content-free”

    “...it is always possible to write laws in a coordinate-independent way, if the laws are well-defined to begin with. A physical system acting in a certain way doesn’t know which coordinate system you are using to describe it; consequently, anything deserving of the name “law of physics” (as opposed to some particular statement of that law) must be independent of coordinates. An insistence on explicit coordinate-independence says nothing about the adaptation of laws to curved spacetime; as we have seen, manifestly tensorial equations take on the same form regardless of the geometry. [...] making things “tensorial” or “generally covariant” is a simple matter of logical necessity, not a physical principle that one could imagine disproving by experiment ”[59, p. 178]

    .

  15. ** Different versions and the actual role of the equivalence principle has been discussed by [57], who concluded that the equivalence principle played rather a heuristic role during the construction of GR (a sort of bridge between theories), not being strictly necessary for the theory. If this were not the case, this is, if the equivalence principle played a necessary role within GR, then the title of this section would have been literal, and not just a provocative statement meant to call the reader’s attention.

  16. This brings us to Brown-Janssen debate (see references in Footnote 1). Of course, Feynman’s theory is not renormalizable and therefore has ‘problems’ in its own terms, problems that would prevent Feynman to consider it as anything more than a pedagogical exercise. Despite this fact, the leading idea motivating Feynman’s efforts in the construction of an alternative that opposes the geometric interpretation of GR aligns to Brown’s defence of Lorentz’s ether theory as opposed to the so-called ‘kinematical’ approach to SR defended by Janssen.

  17. We thanks an anonymous referee for calling our attention to this set of experiments and for the subtleties involved in this problem.

  18. Against these intuitions, Marletto and Vedral [81] designed an experiment with a single particle that gets entangled with itself by an accelerated frame. However, the proposal assumes that the metric is non-classical and that it can create entanglement, this is, they assume spacetime substantivalism and, on top of that, assume that such an entity is non-classical. Both assumptions go beyond the pure formalism of GR. Besides, the proposal also takes EEP as a premise and imposes it in order to connect accelerated frames of reference and gravitational fields responsible for the entanglement (a framework strategy).

  19. Please keep in mind that we refer to EEP as explained in Footnote 13

  20. These conclusions goes as far as current physical theories go, this is, accepting the stage of development reached by the frameworks of Quantum Mechanics (QM) and General Relativity to date. It is known that QM is not a final theory and some people also question GR’s fundamentality, therefore, it might be the case that what looks like an interaction at this scale (if a positive result of the experiment is obtained) corresponds to an unavoidable feature of a more fundamental framework combining GR and QM, just as Universal Gravitation as an interaction dissolves in GR. Our ignorance concerning future theories should prevent us, therefore, to draw any conclusion. Nevertheless, this kind of reasoning would prevent any form of conclusion in any possible arena, a price too high to be paid (and a too boring situation to be bear). All our conclusion should include the phrase ‘considering scientific theories’ state of the art’, but we will prevent you from such suffering.

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Acknowledgements

We would like to thank Federico Benitez, Carlos Rubio and Radouane Gannouji, for reading previous versions of this manuscript and for giving enormously fruitful feedback and comments.

Funding

This work was supported by Pontificia Universidad Caólica de Valparaíso, Grant No. 039.374/2021.

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    Diego Maltrana & Nicolás Sepúlveda-Quiroz

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Maltrana, D., Sepúlveda-Quiroz, N. The Heuristic Power of Theory Classification, the Case of General Relativity. Found Phys 52, 94 (2022). https://doi.org/10.1007/s10701-022-00614-5

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