Abstract
This chapter begins consideration of foundations for general relativity. This is both a relativistic theory of gravitation and a freeing from absolute or background structures (as Chaps. 8 to 10 explain further). The latter perspective emphasizes Background Independence, a notion widely used in Quantum Gravity programs as well. Given the substantial conceptual and technical differences incurred by Background Independence, we further motivate the first—and only observationally established—seat of this—general relativity—by firstly briefly surveying observational evidence for general relativity. Secondly, by outlining useful features of some of its simpler cosmological and black hole solutions.
We then continue to discuss our preamble’s focal topics—the nature of time, space, spacetime, frames, clocks and ‘rods’—now in the context of general relativity. This chapter works in the spacetime setting (as opposed to the next chapter’s dynamical setting). On the one hand, special and general relativistic spacetimes enjoy some parallels such as similar types of notions of causality and simultaneity. On the other hand, general relativity’s Background Independence produces many difference between general relativity’s rendition of the focal topics and both Newtonian theory’s and special relativity’s. This occurs in particular through general relativity being diffeomorphism invariant, which corresponds to a far larger and more nuanced invariance group than the Lorentz and Poincaré groups of special relativity. In anticipation to Chap. 11’s introduction to Quantum Gravity, we end by outlining how general relativity points to its own limitation in the form of singularity theorems.
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Notes
- 1.
From here on, we assume the reader knows Differential, Affine and Metric Geometry; if not, take a detour to Appendix D. If elsewise unfamiliar with this Chapter’s material on GR, consult [874] as preliminary reading.
- 2.
Partial derivative and connection come with transformation laws with extra compensating portions that cancel out in the covariant derivative’s own tensorial transformation law: Appendix D.3. Indeed the ‘compensating field’ treatment of Gauge Theory in Chap. 6 can be formulated in terms of another type of connection, as per Appendix F.4.
- 3.
\(\mbox{g}\) is the spacetime metric’s determinant, which is a scalar density, \(\Gamma^{(4)}\) is the spacetime metric connection and \(\nabla_{\mu }\) is the spacetime covariant derivatives. While there is a role for an affine connection, it is the metric connection (indeed computable from the metric: Appendix D.4) subsumes this role in GR.
- 4.
In Newtonian Gravitation formulated along the original lines with no mention made of Curved Geometry, the single word ‘Gravitation’ is used in all three of the above senses. These become sharply distinguished upon passing to a geometrical formulation of Newtonian Gravitation. Moreover, this distinction transcends to GR, in which setting the geometry involved is both better-behaved and more standard from a mathematical point of view.
- 5.
This is an SR insight: \(E = mc^{2}\). Moreover, it has a more immediate and significant consequence: since energy gravitates and all particles have energy, everything has to couple to Gravity.
- 6.
Here \(\mathcal{R}_{\mu \nu }\) is the Ricci curvature tensor and ℛ the Ricci scalar curvature.
- 7.
\(\mathcal{I}^{+}\) here denotes future null infinity, \(\mathcal{I}^{-}\) is past null infinity and \(\mbox{i}^{0}\) is spatial infinity. These play a significant role as edges of Penrose diagrams: Fig. 7.1, where the corresponding types of geodesics can begin and end.
- 8.
In this sense, ‘homogeneous’ means the same at each point over a mathematical space (in the present cosmological context physical 3-\(d\) space).
- 9.
Compact astrophysical objects are white dwarfs, neutron stars or black holes. Astrophysical binaries are pairs of objects orbitally bound in close proximity to each other. Such a configuration, where each object is a neutron star or a black hole, is potentially a strong source of gravitational waves. See e.g. [296] for a pedagogical account of source counting for white dwarf stars.
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Anderson, E. (2017). Time and Spacetime in General Relativity (GR). In: The Problem of Time. Fundamental Theories of Physics, vol 190. Springer, Cham. https://doi.org/10.1007/978-3-319-58848-3_7
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