Abstract
The current early stage in the investigation of the stability of the Kerr metric is characterized by the study of appropriate model problems. Particularly interesting is the problem of the stability of the solutions of the Klein–Gordon equation, describing the propagation of a scalar field of mass \(\mu \) in the background of a rotating black hole. Rigorous results prove the stability of the reduced, by separation in the azimuth angle in Boyer–Lindquist coordinates, field for sufficiently large masses. Some, but not all, numerical investigations find instability of the reduced field for rotational parameters \(a\) extremely close to \(1\). Among others, the paper derives a model problem for the equation which supports the instability of the field down to \(a/M \approx 0.97\).
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Notes
If not otherwise indicated, the symbols \(t,r,\theta ,\varphi \) denote coordinate projections whose domains will be obvious from the context. In addition, we assume the composition of maps, which includes addition, multiplication and so forth, always to be maximally defined. For instance, the sum of two complex-valued maps is defined on the intersection of their domains. Finally, we use Planck units where the reduced Planck constant \(\hbar \), the speed of light in vacuum \(c\), and the gravitational constant \(G\), all have the numerical value \(1\).
See also the Section 5.1 on ‘Damped wave equations’ in [5].
Since the solutions are scalable in \(M\) we will only present results for \(M=1\).
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Acknowledgments
H.B. is thankful for the hospitality and support of the ‘Department of Gravitation and Mathematical Physics’, (ICN, Miguel Alcubierre), Universidad Nacional Autonoma de Mexico, Mexico City, Mexico and the ‘Division for Theoretical Astrophysics’ (TAT, K. Kokkotas) of the Institute for Astronomy and Astrophysics at the Eberhard-Karls-University Tuebingen. This work was supported in part by CONACyT grants 82787 and 167335, DGAPA-UNAM through grant IN115311, SNI-México, and the SFB/Transregio 7 on “Gravitational Wave Astronomy” of the German Science Foundation (DFG). M.M. acknowledges DGAPA-UNAM for a postdoctoral grant.
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Appendix
Appendix
Lemma 6.1
Let \(\mu \in [0,\infty )\). By
for every \(\lambda \in \mathbb{R } \times (-\infty ,0)\), where \((\mu ^2 - \lambda ^2)^{1/2}\) denotes the square root with strictly positive real part, there is defined a biholomorphic map
with inverse
given by
for every \(z \in (\mathbb{C } \setminus B_{\mu }(0)) \cap ((0,\infty ) \times \mathbb{R })\). In addition,
for every \(z \in (\mathbb{C } \setminus B_{\mu }(0)) \cap ((0,\infty ) \times \mathbb{R })\) (Fig. 5).
Proof
First, we note that \(g\) is well-defined. For this, let \(\lambda \in \mathbb{R } \times (-\infty ,0)\) and \(\lambda _1 := \mathrm Re (\lambda ), \lambda _2 := \mathrm Im (\lambda ) < 0\). Then
Hence \(\mu ^2 - \lambda ^2\) is real iff \(\lambda _1 = 0\). In the latter case,
Therefore, \(\mu ^2 - \lambda ^2 \in \mathbb{C } \setminus ((-\infty ,0] \times \{0\})\), and there is precisely one square root of \(\mu ^2 - \lambda ^2\) with strictly positive real part. Further,
In particular, if
is such that \(|z| \le \mu \), then
Hence
and
The latter implies that
and hence that \(\mathrm Re (z) \le 0\). As a consequence, \(|z| > \mu \).
For the second step, let \(z \in (\mathbb{C } \setminus B_{\mu }(0)) \cap ((0,\infty ) \times \mathbb{R })\), \(x := \mathrm Re (z),\) and \(y := \mathrm Im (z)\). Then
and
Since
the latter implies that
In particular, we conclude that by
for every \(z \in (\mathbb{C } \setminus B_{\mu }(0)) \cap ((0,\infty ) \times \mathbb{R })\), there is defined a map
such that
for every \(z \in (\mathbb{C } \setminus B_{\mu }(0)) \cap ((0,\infty ) \times \mathbb{R })\). Therefore, \(g\) is surjective.
Further, if \(\mu = 0\),
and if \(\mu \ne 0\),
for every \(\lambda \in \mathbb{R } \times (-\infty ,0)\). Therefore, \(g\) is injective and altogether bijective with inverse \(h\). Finally, it follows for \(\lambda \in \mathbb{R } \times (-\infty ,0)\) that
and hence for every \(z \in (\mathbb{C } \setminus B_{\mu }(0)) \cap ((0,\infty ) \times \mathbb{R })\) that
\(\square \)
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Beyer, H.R., Alcubierre, M., Megevand, M. et al. Stability study of a model for the Klein–Gordon equation in Kerr space-time. Gen Relativ Gravit 45, 203–227 (2013). https://doi.org/10.1007/s10714-012-1470-0
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DOI: https://doi.org/10.1007/s10714-012-1470-0