Abstract
In this chapter the key topics covered by the thesis are explored in more technical detail. We follow the theoretical steps in formulating a numerical evolution, and the background to the specific problems studied. Discussion of the implementation aspects of the numerical evolution are left to the following chapter in which the code which was developed is described. As in Chap. 1, we divide this chapter into three sections, GR, NR and Scalar Fields.
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Notes
- 1.
- 2.
The pseudo in pseudo-Riemannian means that the metric is not positive definite, i.e. \(g_{ab} V^a V^b \ngtr 0\) \(\forall ~ \vec {V}\), which is obviously very important physically as it is due to the minus sign associated with the time direction, but does not make a big difference to our discussion here of geometric properties.
- 3.
Why this contraction between the first and third indices, rather than others? One can show that any other contraction is either equal to zero or \(\pm R_{ab}\) due to the symmetries of the Riemann tensor, so it is in effect the only one possible. We will also see in Sect. 2.1.2 why this contraction is relevant in relation to tidal forces.
- 4.
Although we are cheating a bit since the \(\nabla \) in the Newtonian case is the 3 dimensional spatial gradient and not a four dimensional quantity. We should really show that the time components do not contribute in some chosen frame and then generalise from a tensor equation.
- 5.
This will be correct if the change in the metric \(\delta g^{ab}\) and its derivatives go to zero at infinity.
- 6.
This is equivalent to the projection of the covariant derivative into the spatial slice \(D_a = P^b_a \nabla _b\) for the derivative of a scalar or a purely spatial tensor, but when acting on a general four tensor, one must also project the indices of the tensor itself into the spatial slice.
- 7.
This is true for the definition of K used here, but an opposite sign convention for \(K_{ab}\) is possible and used by some authors. In addition, as will be discussed later, in more general cases the volume growth may be a gauge effect, rather than due to the physical expansion of the space.
- 8.
When we start specifying scalar field data on the initial slice, the gauge parameters will appear in the constraint equation. This is a consequence of the fact that the time derivatives of the scalar field are usually specified with reference to coordinate time and not for the (gauge independent) normal geodesic observer. Their appearance in the constraints is then to remove their effect from the gauge dependent quantities, rather than because the constraints depend on the gauge.
- 9.
If the former condition, real eigenvalues, is met but not the latter, a complete set of eigenvectors, it is only weakly hyperbolic.
- 10.
Alternatively the desensitised lapse \(\alpha /\sqrt{\gamma }\) can be specified as a function of space and time, but we do not take this approach in this work. One can see that in any case the slicing conditions we use results in a similar behaviour to specifying a constant desensitised lapse, with the lapse becoming smaller in regions in which the normal observer volume is shrinking.
- 11.
Note that many texts also use a \(\chi \) which is equivalent to our \(\chi ^2\). One should thus take care with conventions when comparing results.
- 12.
Note that it is also possible to add terms to the evolution equations proportional to the difference between the evolved \(\tilde{\Gamma }^i\) and that calculated from the derivatives of the metric on the slice, in order to stabilise the evolution, as in [24], but we do not use this “constraint damping” method in the work presented here.
- 13.
It is also common to use the reduced Planck mass \(M_{ \text{ Pl }}^{reduced} = \sqrt{\hbar c/8\pi G}\) which eliminates some factors of \(8 \pi \) in the equations. However, since in our GR work we tend to keep the \(8\pi \)’s explicit, we also keep them here.
- 14.
Note that the use of t and \(\tau \) is the opposite convention to many standard Cosmology texts. The aim of using them in this way is to make a connection with the NR work described above. In cosmology conformal time t is the “unphysical” coordinate time whereas the time \(\tau \) is the proper time for a comoving observer, so it seems more consistent with the other material presented here to use them in this way.
- 15.
However, note that in Chap. 4 we will set \(a=1\) at the start of inflation rather than at the current time, for numerical convenience.
- 16.
What constitutes a “natural” initial state for the universe can be more a question of philosophy than physics. If we had a better understanding of what happened at higher energies, for example, a quantum theory of gravity, we would be better placed to comment on what is “natural” in this context. However, it is generally true in physics that randomness is more natural than a very ordered state.
- 17.
Note that the additional conformal time is additional coordinate time, and does not necessarily correspond to a large amount of additional proper time being experienced by a comoving observer.
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Clough, K. (2018). Technical Background. In: Scalar Fields in Numerical General Relativity. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-92672-8_2
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