Abstract
We now return to the discussion of general relativity. Equipped with the preceding discussions of both the quantisation of field theories, and the geometrical interpretations of gauge transformations, it is time to set about formulating what will eventually become a theory of dynamical spacetime obeying rules adapted from quantum field theory. But before we get there we must cast classical GR into a form amenable to quantisation.
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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.
For what follows, it will be helpful to recall some aspects of symplectic geometry. In the symplectic formulation of classical mechanics a system consists of a phase space in the form of an even-dimensional manifold \(\Gamma \) equipped with a symplectic structure (anti-symmetric tensor) \(\Omega _{\mu \nu }\). Given any function \(f:\Gamma \rightarrow \mathbb {R}\) on the phase space, and a derivative operator \(\nabla \), there exists a vector field associated with f, given by \(X_f^\alpha = \Omega ^{\alpha \beta } \nabla _\beta f\). Given two functions f, g on \(\Gamma \), the Poisson brackets between the two can be written as \(\{f,g\} = \Omega ^{\alpha \beta }\nabla _\alpha \,f \nabla _\beta g \) which can also be identified with \(-\mathcal {L}_{X_f} g = \mathcal {L}_{X_g} f\)—the Lie derivative of g along the vector field generated by f or vice-versa. Thus in this picture, the Poisson bracket between two functions tells us the change in one function when it is Lie-dragged along the vector field generated by the other function (or vice-versa). For more details see [6, Appendix B].
- 9.
The terms “metric formulation” and “connection formulation” will be defined in Sect. 4.5.
- 10.
So long as the manifold is continuous, not discrete. This is an important point to keep in mind for later.
- 11.
The similar word vielbein (“any legs”) is used for the generalisation of this concept to an arbitrary number of dimensions (e.g. triads, pentads).
- 12.
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.
- 13.
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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Vaid, D., Bilson-Thompson, S. (2017). Expanding on Classical GR. In: LQG for the Bewildered. Springer, Cham. https://doi.org/10.1007/978-3-319-43184-0_4
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