The last part of the thesis is about the wave equation inside a black hole. This result is again motivated by the investigation of the supposed instability of the interior of our clear black hole solutions.
Introduction
- The maximal globally hyperbolic development in the global the-
- Why another proof?
- Sketch of the proof given by Choquet-Bruhat and Geroch
- Outline of the proof presented in this chapter
- Schematic comparison of the two proofs
Our proof for the case of the Einstein equations will then naturally arise by analogy. Moreover, it is precisely this idea of global uniqueness that is crucial to the existence of the MGHD.
The basic definitions and the main theorems
Given two GHD M and M0 of the same initial data, there exists a GHD M˜ that is an extension of M and M0. Given the initial data, there exists a GHD M˜ that is an extension of any other GHD of the same initial data.
Proving the main theorems
The existence of the maximal common globally hyperbolic de-
Given two GHDs M and M0 with the same initial data, there exists a unique CGHD U of M and M0 with the property that if V is another CGHD of M and M0, then U is an extension of V. By Corollary 2.3.2, this is well-defined and it is clear that ψ is an isometric immersion that respects the embedding of M and the time orientation.
The maximal common globally hyperbolic development does not
In this case, the above lemma ensures the existence of a 'spatial' part of the boundary - only those parts are suitable to restart the local uniqueness argument. Since the strong causality condition atp holds, we can find a causally convex neighborhood W of p whose closure is compact and fully contained in V.34. Let q ∈ I+(p) be a point with the property that J−(q)∩Uc ∩J+ ι(M).
Finishing off the proof of the main theorems
From here one follows the remaining argument from point 4 in the proof of theorem 2.2.8. Characterization of the energy of Gaussian rays on Lorentzian manifolds - with applications to black hole times. This is roughly the state of the art on Gaussian beams (see e.g. [61]).
The main new result of Part I of this chapter is to provide a geometrical characterization of the time behavior of the localized Gaussian beam energy. There exist (actual) solutions of the wave equation (3.1.1) whose N-energy is localized along a given zero geodesic along γ and behaves approximately as −g(N,γ)˙.
A brief historical review of Gaussian beams
The important ideas are first introduced in Section 3.3.1 by the example of the Schwarzschild and Reissner-Nordstr¨om family, whose simple metric form allows an uncomplicated representation. The main purpose of the first part of the appendix is to compare the Gaussian beam approximation with the much older geometric optics approximation. This proof used the geometric optics approximation, and we explain why it does not transfer directly to general Lorentzian manifolds.
We conclude in Appendix 3.C by giving a sufficient criterion for the formation of caustic, i.e. a decomposition criterion for solutions of the eikonal equation. The earliest application of the Gaussian beam method was for the construction of quasimodes, see for example the 1976 paper [60] by Ralston.
Gaussian beams and the energy method
The philosophy behind the energy method is to first derive estimates of an appropriate energy (and higher order energies47) and then to establish pointwise estimates using Sobolev embeddings. A general procedure is to construct an energy from a foliation of spacetime by spacelike slices Στ together with a timelike vector field N, see (3.1.8) in section 3.1.5. Such solutions naturally constitute an obstacle to certain uniform statements about the temporal behavior of the wave energy.
47A first-order energy controls the first derivatives of the wave and is referred to hereafter as just "energy". 50 The right-hand side of (3.1.3) is called the N-energy of the zero geodesic.
Gaussian beams are parsimonious
The breakdown by caustic forms a severe limitation of the range of applicability of the geometric optical approximation—a limitation not shared by the Gaussian beam approximation. By considering a highly oscillating wave that 'gives rise to only one photon', one recovers the characterization of the energy of a Gaussian beam, (3.1.3), given in this chapter. The leading component inλ of the renormalized54 stress energy tensorT(uλ) of the waveλ =a·eiλφ in the geometric optical limit is then given by.
In particular, using the conservation law (3.1.6), it is not difficult55 to prove a geometric characterization of the wave energy in the optical geometric limit, analogous to what we prove in this chapter for Gaussian rays. Characterizing the energy of Gaussian beams is more difficult since (3.1.6) is replaced only by the approximate conservation law56.
Notation
Thus, the wave energy, measured by a local observer with world line σ, is determined by the energy component −g(˙γ,σ) of the ˙ momentum 4-vector ˙γ. In the physics literature (see for example the classic [47], Chapter 22.5), this description is justified using the geometric optics approximation. Moreover, it provides a rigorous justification of the time behavior of the local observer photon energy, which also applies to photons along whose trajectory the caustic would form.
The understanding (3.1.8) of the N-energy of a function is especially useful when we have adequate knowledge of 2u, because one can then derive detailed information about the behavior of the N-energy (cf. Hence the stress-energy tensor ( 3.1.7) together with the concept of N-energy is particularly useful for solutions u of the wave equation.
Part I: The theory of Gaussian beams on
Solutions of the wave equation with localised energy
The function in the theorem is the Gaussian radius, the approximate solution of the wave equation (3.1.9) that we need to construct. In Section 3.2.2 we construct the functions n andφ such that uλ satisfies the following three conditions: The first condition is. This says that as the Gaussian beam becomes more and more oscillating (ie for larger and larger λ), the closer it becomes to the correct solution to the wave equation.
Here we make use of the fact that the Lorentzian manifold (M, g) is globally hyperbolic and thus allows for a well-posed initial value problem for the wave equation. But in section 3.2.3 we will derive more information about the approximate solution u˜ and then (3.2.10) will tell us about the temporal behavior of the localized energy of v, cf.
The construction of Gaussian beams
Specifically, while J induces a vector field on M, V depends on the choice of coordinates. However, we can change this splitting into a geometric one by using the splitting Tγ˙[(s)(T∗M) in the vertical and horizontal subspaces, which is induced by the Levi-Civita connection. In this approach, the second covariant derivative f is considered instead of the partial derivatives in (3.2.26) and thus we obtain the ODE for ∇∇φ.
The background for the reduction of the nonlinear ODE for ∇∇φ to a linear second-order ODE is provided by equation (3.C.3) in Appendix 3.C. This follows either from (3.2.29) or by a direct calculation that makes use of the Jacobi equation and the symmetry properties of the Riemannian curvature tensor.
Geometric characterisation of the energy of Gaussian beams
We remind you again that the solution to the wave equations in Theorem 3.2.43 can also be chosen for a real value. A very robust method for proving the decay of solutions of the wave equation was given by Dafermos and Rodnianski in [21] (but see also [46]). In particular, this method requires an Integrated Local Energy Decay (ILED) statement (preferably with derivative loss), i.e. statement in the form (3.2.50).
The reader might have noticed that whether or not an ILED theorem of the form (3.2.50) exists depends very much on the choice of the time function. On the other hand, whether an ILED statement is useful or not also depends a lot on the choice of the time function.
Part II: Applications to black hole spacetimes
Applications to Schwarzschild and Reissner-Nordstr¨ om black holes 67
The extreme Reissner-Nordstr¨om black hole is given by the choice m = e of the parameters in (3.3.1). 3.A A sketch of the construction of localized solutions to the wave equation using the geometric optical approximation. The construction of the phase function φ is exactly the same as for the wave equation in Section 3.2.2.
2, then the wave energy along a zero hypersurface intersecting the Cauchy horizon is finite. 102 This is Stokes' orientation to the energy evaluation (4.5.3) in the proof of the next lemma. A proof of uniform bounds of solutions to the wave equation in slowly rotating Kerr backgrounds.
The decay of solutions of the external initial boundary value problem for the wave equation.
The wave equation in the black hole interior and the mass inflation
Around 1980 the following results were obtained on the occurrence of the blueshift effect in linear theory: In [45] McNamara showed that for the wave equation on a sub-extremal Reissner-Nordström background there exist initial data on the event horizon (or also on the past zero infinity I-) so that at the fission sphere in the interior of the black hole. The later papers [33] and [11] argue that on a sub-extremal Reissner-Nordström background one has pointwise inflated the transverse derivative on the bifurcation sphere, even if the initial data are compactly supported on the event horizon. . Results on the manifestation of the blue shift effect in the nonlinear theory are known only in spherical symmetry.
In their analysis, they assumed a late polynomial decay of the incoming radiation across the event horizon. Recall from the discussion of the results obtained for the linear theory that even perturbations that are compactly supported at the event horizon are expected to burst pointwise at the Cauchy horizon.
The interior of a Reissner-Nordstr¨ om black hole and notation
To gain a better understanding of this question, we outline the heuristics underlying the mass inflation scenario in the Einstein-Maxwell scalar field model. So if we ignore the feedback of the scalar field on the geometry, we see that an infinite energy of the scalar field along an outgoing zero hypersurface intersecting the Cauchy horizon. However, in this first-order perturbative analysis, the pointwise inflation of ∂vψ does not allow us to make any predictions about whether or not mass inflation occurs.
Extrapolating this result to the nonlinear theory suggests that compactly supported perturbations for a wide range of black hole parameters m and e do not cause mass inflation. In Penrose's schematic representation of section 3.3.1 in Chapter 3, the Lorentzian manifold with boundary (M∪ H+∪ CH+, g) corresponds to region II with the two right boundaries of the diamond attached.
The main theorem
The proof of the main theorem
However, for us the most interesting aspect of statement (4.5.9) is the energy flux limit through Cu0, i.e., the first term in (4.5.9). Next we construct a time-translation invariant vector field N in a neighborhood r of the horizon such that KN(ψ). 17] Dafermos, M. Stability and instability of the Cauchy horizon for the Einstein-Maxwell-scalar symmetric spherical field equations.
Decay for solutions of the wave equation on Kerr outer spacetimes I-II: Examples of |a| M or axial symmetry. Decay for solutions of the wave equation on Kerr outer spacetimes III: the full subextreme case |a|< M.
