Abstract
The title of this book—Towards a Theory of Spacetime Theories—is an attempt at false modesty. Or, rather, maybe: an act of an unreasonable raising of the chin in the face of a task supposedly impossible to master. After all, we do not even have a comprehensive map of the solution space of general relativity; by far the most established and most investigated spacetime theory; how then are we supposed to draw a map of the space of spacetime theories in which general relativity itself is but one little point? It seems a daunting and impossible task. And still, we cannot afford not to take it on.
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Notes
- 1.
- 2.
See Goenner [9], section 4.3. Einstein’s main engagement with Eddington’s affine approach was to deliver the field equations for the affine connection, see especially the recently published Volumes 13 and 14 of the Collected Papers of Albert Einstein.
- 3.
Einstein’s own notion of ‘geometry’ was very unrestrictive. Indeed, he argued that an arbitrary vector \(v^\mu \) is not more nor less ‘geometrical’ than a metric tensor \(g_{\mu \nu }\). See [13] for details.
- 4.
- 5.
The notion of ‘geometric object’ they draw on here is that of Trautman [18]; which is designed to “include nearly all the entities needed in geometry and physics”. They are less explicit about what they mean by a ‘4-dimensional spacetime manifold’ rather than a ‘4-dimensional manifold’; I would argue that a spacetime manifold needs at least conformal structure (an equivalence class of metrics defined on it) in order to be called a spacetime manifold, for only then can we distinguish between spatial and temporal dimensions.
- 6.
See Thorne et al. [17], p. 18, for qualifications regarding what kind of binary star system they are talking about, and how close the theory has to come to Kepler’s laws.
- 7.
I am using geometrized units here, in which the speed of light is 1, and the Newtonian potential dimensionless. Cf. Will [21], p. 87.
- 8.
The Nordvedt effect would obtain if the ratio of inertial and gravitational mass would be different from 1 for sufficiently large, sufficiently self-gravitating bodies. Thus it would show that while test bodies move on geodesics (the weak equivalence principle), not all massive gravitating bodies do, even if their spin is neglected. The most important experimental realisation were lunar laser tests of the Earth–Moon system in the 1960s, in which no Nordvedt effect was discovered. However, it remains possible that more massive bodies (black holes in particular) would exhibit a Nordvedt effect. See Nordtvedt [15] and Will [21], section 8.1, for details.
- 9.
See Ferreira [8] for details on the PPF framework.
- 10.
Of course, the conviction that the Friedmann solution adequately represents the era that we currently live in goes back to Hubble’s discovery of the redshift of galaxies. But only the discovery of the Cosmic Microwave Background gave convincing evidence of the big bang theory, i.e. the idea that the Friedmann solution applies to the beginning of the universe too; indeed that there was a beginning in the first place. See Smeenk [16] for details.
- 11.
Just a few preliminary results on this: A counterpart to Birkhoff’s theorem exists for the Einstein–Maxwell solutions: the unique spherically symmetric solution to the Einstein–Maxwell solutions is the (likewise static and asymptotically flat) Reissner–Nordström solution. There are some candidates for Birkhoff counterparts in some scalar-tensor theories, but typically extra assumptions are necessary to prove uniqueness [12]. There are also generalisations of Birkhoff’s theorem for some but not all theories with torsion [14], and for Lovelock gravity [22].
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Lehmkuhl, D. (2017). Introduction: Towards A Theory of Spacetime 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_1
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