Abstract
This chapter covers the reformulation of classical general relativity in both the Lagrangian and Hamiltonian frameworks. In the first case this involves the Einstein-Hilbert action, which is shown to be a function of the metric and its derivatives. This also introduces the idea of a sum-over-geometries, describing the dynamics of spacetime as a kind of path integral, analogous to the description of particle and field dynamics in quantum field theory. The Arnowitt-Deser-Misner (ADM) splitting of spacetime into a foliation of hypersurfaces is introduced, and the Hamiltonian, diffeomorphism, and Gauss constraints derived. The technical difficulty of writing the constraints in operator form is used to motivate an alternative formulation more similar to a quantum field theory. Tetrads, the spin connection, and an appropriate choice of gauge group are discussed, and the Palatini formulation is described.
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Notes
- 1.
This definition of the energy-momentum tensor may seem to come out of thin air, and in many texts it is simply presented as such. To save space we will follow suit, but the reader who wishes to delve deeper should consult [1], in which \(T_{\mu \nu }\) is referred to as the dynamical energy-momentum tensor, and it is proven that it obeys the conservation law \(\nabla _\mu T^{\mu \nu }=0\) (as one would hope, since energy and momentum are conserved quantities), as well as being consistent with the form of the electromagnetic energy-momentum tensor.
- 2.
The intrinsic metric will be introduced properly very shortly, specifically in Eq. (4.16) and the associated discussion.
- 3.
The term “fiducial” refers to a standard of reference, as used in surveying, or a standard established on a basis of faith or trust.
- 4.
Generally one assumes that our 4-manifolds can always be foliated by a set of spacelike 3-manifolds. For a general theory of quantum gravity the assumption of trivial topologies must be dropped. In the presence of topological defects in the 4-manifold, in general, there will exist inequivalent foliations in the vicinity of a given defect. This distinction can be disregarded in the following discussion for the time being.
- 5.
From this expression we can also see that \(g_{00} = -N^2 + N^a N_a\) is a measure of the local speed of time evolution and hence is a measure of the local gravitational energy density.
- 6.
The notation \({}^3\Sigma \) is sometimes used to denote that these are three-dimensional hypersurfaces, however this is redundant in our present discussion.
- 7.
A derivation of which can be found in Appendix 1.3 of [4].
- 8.
The terms “metric formulation” and “connection formulation”will be defined in Sect. 4.3.1.
- 9.
We will from time to time commit the cardinal sin of conflating a vector with its components.
- 10.
So long as the manifold is continuous, not discrete. This is an important point to keep in mind for later.
- 11.
The components can be regarded a internal to the “laboratory frame” tangent space, and hence the choice of indices \(I,\,J\ldots \) is appropriate.
- 12.
The similar word vielbein (“any legs”) is used for the generalisation of this concept to an arbitrary number of dimensions (e.g. triads, pentads).
- 13.
A Riemannian metric by definition always assigns lengths greater than zero to distinct points in a manifold. This is in contrast to the Minkowski metric, which as per footnote 2 of Chap. 2, may not.
- 14.
In constructing a theory of quantum gravity we may find that this is not strictly true, as spacetime may not be viably treated as continuous at all length scales, and in fact the concept of a background spacetime may not be valid at all. But these are subtleties to dwell upon in the latter parts of this book. For now, we’ll focus on classical theories of spacetime structure.
- 15.
It should be borne in mind that the isomorphism is between the algebras, not the groups. The group SU(2), with its relationship to rotations, is compact. This reflects the fact that rotating an object through a finite number of finite rotations can return it to its starting orientation. However the Lorentz group is non-compact, reflecting the fact that even an arbitrarily-large number of finite boosts cannot accelerate an object to the speed of light, and so boosts may be parametrised by a “rapidity” which takes values between negative and positive infinity.
- 16.
If we use two copies of the curvature tensor then we get Yang-Mills theory (\(F \wedge F\)). But that doesn’t include the tetrad.
- 17.
In D dimensions, the rotation group has \( D(D-1)/2 \) degrees of freedom corresponding to the number of independent elements of an antisymmetric \( D\times D \) matrix.
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Bilson-Thompson, S. (2024). Expanding on Classical GR. In: Loop Quantum Gravity for the Bewildered. Springer, Cham. https://doi.org/10.1007/978-3-031-43452-5_4
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