## Abstract

This paper studies space-time theories from the perspective of the Semantic View of theories. Set-theoretic models are used to reconstruct several non-quantum space-time theories and to characterize their mutual relationships. Further, the Semantic View is adopted to discuss the question of what a space-time theory is to begin with. While the space-time theories incorporated in Newtonian theories, on the one hand, and in Einstein’s General theory of relativity (GTR), on the other hand, are markedly different, GTR and many rival theories of gravitation do not differ on their space-time theory, but only on the way the structure of a space-time is explained.

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## Notes

- 1.
“Model” is a technical term in this context, the main idea being that models satisfy axioms. See [20] for a brief introduction to model theory.

- 2.
See [34] for a formulation and discussion of the Received View.

- 3.
- 4.
- 5.
It seems attractive of this view that it characterizes theories in terms of real stuff as it were. A different approach that nevertheless provides model-theoretic reconstructions of theories is advocated by G. Ludwig; see [23] for a recent outline.

- 6.
- 7.
For BMS, the sets \(S_i\) are merely not

*necessarily*mathematical; they may contain physical objects, but need not. But in this way, BMS allow for many models that are irrelevant for the purposes of representing physical systems. I will not follow them in this regard. See their pp. 20–23 for a discussion. - 8.
See BMS, p. 8 for a precise definition of typification.

- 9.
Data may of course also be richer than a theory allows. But this is just to say that a theory is restricted to certain aspects of a class of systems. See BMS, Sec. II.7 for details about the idealized empirical claim of a theory.

- 10.
Since theoreticity is theory-relative according to BMS (see below), the empirical claim is only empirical in a theory-relative sense. But below, we will often use the term “empirical” in the absolute sense of “subject to empirical scrutiny.”

- 11.
See BMS, Sec. VII.2.3.

- 12.
See e.g., BMS, Sec. VII.2.

- 13.
There is an alternative way to associate claims with a theory. The idea is that a theory describes real-world systems not because the latter are among a theory’s models, but rather because they are

*isomorphic*to models constitutive of the theory (see e.g., [39], pp. 43–44; see [12] for a weakening). But the notion of isomorphism is a mathematical one, and to apply it to real-world systems, we have to formalize them in terms of models. We have to build up set-theoretical constructions of real-world stuff. If this is so, why not start with real-world models and then say that they can be embedded in models constitutive of a theory? - 14.
See [36], Ch. XII or BMS, p. 15 for a definition.

- 15.
E.g., [2], p. 50.

- 16.
Cf. [17], pp. 278–279.

- 17.
This is a counterpart to Quine’s claim that theories commit us to assume the existence of those things to which quantified variables refer [26]. Note though that proponents of the Semantic View need not assume that theories are, or should be, literally true. In fact, an influential proponent of the Semantic View [39], does not take serious parts of the models that stretch beyond the observable.

- 18.
See [15], Chs. 2–3 for a discussion of classical space-times. Newtonian space-time is defined on his pp. 33–34.

- 19.
- 20.
The term “coordinate charts” is well-known from differential geometry and anticipates the terminology from differential geometry used later in this paper. In the present context, we have one coordinate chart for space points and one for instances of time.

- 21.
See [29], p. 222 for a discussion.

- 22.
Costa et. al. [11] construct points (in their case, space-time points) from underlying events, but we have then to posit the events.

- 23.
One could try to introduce distances though through equivalence classes of pairs of objects.

- 24.
In differential geometry, distances are conceptualized using metric tensor fields (see below). Tensor fields are only defined on manifolds, and our space-time is not yet assumed to be a manifold. This is not a problem though because we are here not yet assuming the full mathematical formalism, but rather talking about physical distances.

- 25.
See e.g., [36], §12.3 for the notion of a representation theorem.

- 26.
See BMS, Def. DII-4 on p. 61 for a formal definition of (abstract) links.

- 27.
- 28.
Equation (3) fixes the Euclidean nature of space using coordinate charts. Alternatively, one could try to do without them. This would avoid some descriptive fluff. For instance, we could demand that the triangle inequality, Eq. (4) holds. Even this equation is not free of descriptive fluff, but at least it dispenses with coordinate charts. However, it is much easier to use coordinate charts to fix the properties of a space ([29], p. 50).

- 29.
The status of other space points not picked using physical objects remains peculiar anyway. In GTR, this peculiarity is much discussed in debates about the hole argument. The latter is often taken to show that the assumption of space-time points in empty space has untenable consequences [13, 21]. But the hole argument does not raise any problem peculiar to our approach. First, most formulations of GTR quantify over all points of the manifold, regardless how they are identified empirically, so we do not have an objection specific to our approach. Second, what the hole argument shows is only that space-time points identified independently of any physical events are problematic. If we understand it that the elements of the space-time do not have identity apart from the roles that they take in the models, our formalization does not seem problematic.

- 30.
It can be shown that the equation holds for all representations if it holds for one pair of representations (\(c_T,c_S\)).

- 31.
- 32.
- 33.
See [40], pp. 35–36 for details about compatibility, consult ibid., pp. 20–21 for contraction and ibid., p. 19 for the dual.

- 34.
Note though that we are here talking not about trajectories of real particles, but rather about hypothetical world lines.

- 35.
See [40], p. 17 for technical details.

- 36.
For a discussion on whether the unparameterized geodesics in a space-time uniquely fix the metric see [24].

- 37.
See [40], p. 67.

- 38.
See e.g., [41], Sec. 2.3.

- 39.
Some proponents of the Semantic View, e.g., BMS and [3] part company with me at this point because they include particular coordinate charts in their models. A potential reason is that, in practice, scientific work done with a theory often uses specific coordinates, and that a change in coordinates can have nontrivial consequences for this work. See [22], pp. 95–100 for illustrations.

- 40.
Nevertheless, some actual models of a theory may not match the actual world. The actual models are solutions to the equations of a theory, but these solutions may not describe our real world, e.g., due to unrealistic initial conditions.

- 41.
See BMS, Sec. II.3 for two formalizations of theoreticity with respect to a theory.

- 42.
Cf. BMS, pp. 51–52.

- 43.
BMS reconstruct CPM using local models; cf. our discussion below in the next section.

- 44.
[29], pp. 52, 66–68.

- 45.
Absolute objects are contrasted to dynamical objects. See [18], pp. 64–70 for the distinction.

- 46.
[29], pp. 52, 66–68.

*S*and*T*cannot be part of the kernel if there are models that have only portions of the space-time as components. - 47.
See e.g., [40], p. 215.

- 48.
Because extendability involves the manifold and the metric, a requirement of inextendability would have law-like status according to BMS.

- 49.
One can rebut this objection though by saying that, even though every model of a space-time theory contains the whole space-time, the models need not be

*used*to describe the whole space-time. For instance, formally, the Schwarzschild solution of GTR is a solution for the whole space-time, but it is often used as an idealization to understand subsystems of the Universe only. GTR would then still be a space-time theory. - 50.
Here, the metric structure does not only mean the metric itself, but also physical relationships about events on world lines incorporated in the metric field tensor.

- 51.
According to the definition, the link is a two-place relation; it can be generalized to an

*n*-place relation easily if there are*n*general relativistic dynamic theories. If there is no matter of a particular type, e.g., no perfect fluid component, the pertinent part of \(T_{tot}\) equals zero. - 52.
See [40], pp. 437–439 for technical details about diffeomorphisms. We can relax the conditions in the definition because space-time manifolds that are identical up to certain structure-preserving maps are not supposed to picture distinct physical possibilities.

- 53.
This is effectively an amendment of Bartelborth’s Def. 12.

- 54.
- 55.
Alternatively, we could regard BDD as inter-theoretical as we did with GTR before. However, this would complicate matters too much.

- 56.
Often, the Brans–Dicke theory is defined more narrowly with a zero potential.

- 57.
A conformal transformation maps the metric

*g*to \(\Omega ^2 g\), where \(\Omega \) is a strictly positive smooth function. \(\Omega ^2 g\) yields the same angles as does*g*, but other geometric features are not invariant under conformal transformations. See [40], p. 445. - 58.
See [29], Sec. V.1, particularly pp. 175–177 for a general account of asymptotic reduction, and ibid., Sec. V.2 for limiting case reduction.

- 59.
For the relationship between Newton’s Theory of Gravitation and GTR see [30], Ch. VIII.

- 60.
GTR is not just a specialization of BDT because the latter has additional degrees of freedom, viz., the scalar field (see BMS, Def. DIV.1 on p. 170 for specialization).

- 61.
Scheibe’s comparison between a classical space-time and Minkowski space-time is simpler because both manifolds are homogeneous.

- 62.
I here refer to BMS because their account of approximate reduction seems simpler than that by Scheibe.

- 63.
This does not require that a distance measure between models of both theories be defined, but it does require that there is a least a topology over models in one of the theories (cf. BMS, Ch. VII, particularly Sec. VII.3.1). At this point, BMS differ from Scheibe, who demands that a common space be defined in which the models of both theories are included.

- 64.
See BMS, Defs. VI-5 and VI-6 on p. 277 for direct and indirect reduction. Scheibe’s limiting case reduction is only indirect in the terms of BMS.

- 65.

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## Acknowledgements

I’m grateful for very helpful and constructive criticism by Dennis Lehmkuhl and Erhard Scholz. Thanks also to Raphael Bolinger for his comments.

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Beisbart, C. (2017). A Model-Theoretic Analysis of Space-Time Theories. In: Lehmkuhl, D., Schiemann, G., Scholz, E. (eds) Towards a Theory of Spacetime Theories. Einstein Studies, vol 13. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-3210-8_7

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