Gauge-Invariant Perturbation Theory on the Schwarzschild Background Spacetime Part I: Formulation and Odd-Mode Perturbations
This article is Part I of our series of full papers on a gauge-invariant “linear” perturbation theory on the Schwarzschild background spacetime which was briefly reported in our short papers by the present author in 2021. We first review our general framework of the gauge-invariant perturbation theo...
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| description | This article is Part I of our series of full papers on a gauge-invariant “linear” perturbation theory on the Schwarzschild background spacetime which was briefly reported in our short papers by the present author in 2021. We first review our general framework of the gauge-invariant perturbation theory, which can be easily extended to the “higher-order” perturbation theory. When we apply this general framework to perturbations on the Schwarzschild background spacetime, gauge-invariant treatments of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>l</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn></mrow></semantics></math></inline-formula> mode perturbations are required. On the other hand, in the current consensus on the perturbations of the Schwarzschild spacetime, gauge-invariant treatments for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>l</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn></mrow></semantics></math></inline-formula> modes are difficult if we keep the reconstruction of the original metric perturbations in our mind. Due to this situation, we propose a strategy of a gauge-invariant treatment of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>l</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn></mrow></semantics></math></inline-formula> mode perturbations through the decomposition of the metric perturbations by singular harmonic functions at once and the regularization of these singularities through the imposition of the boundary conditions to the Einstein equations. Following this proposal, we derive the linearized Einstein equations for any modes of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>l</mi><mo>≥</mo><mn>0</mn></mrow></semantics></math></inline-formula> in a gauge-invariant manner. We discuss the solutions to the odd-mode perturbation equations in the linearized Einstein equations and show that these perturbations include the Kerr parameter perturbation in these odd-mode perturbations, which is physically reasonable. In the Part II and Part III papers of this series of papers, we will show that the even-mode solutions to the linearized Einstein equations obtained through our proposal are also physically reasonable. Then, we conclude that our proposal of a gauge-invariant treatment for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>l</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn></mrow></semantics></math></inline-formula>-mode perturbations is also physically reasonable. |
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| spelling | doaj-art-1f1a9a9ff4164eb4bf07d34843df8bb02025-08-20T02:45:41ZengMDPI AGUniverse2218-19972025-01-011123910.3390/universe11020039Gauge-Invariant Perturbation Theory on the Schwarzschild Background Spacetime Part I: Formulation and Odd-Mode PerturbationsKouji Nakamura0Gravitational-Wave Science Project, National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka 181-8588, Tokyo, JapanThis article is Part I of our series of full papers on a gauge-invariant “linear” perturbation theory on the Schwarzschild background spacetime which was briefly reported in our short papers by the present author in 2021. We first review our general framework of the gauge-invariant perturbation theory, which can be easily extended to the “higher-order” perturbation theory. When we apply this general framework to perturbations on the Schwarzschild background spacetime, gauge-invariant treatments of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>l</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn></mrow></semantics></math></inline-formula> mode perturbations are required. On the other hand, in the current consensus on the perturbations of the Schwarzschild spacetime, gauge-invariant treatments for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>l</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn></mrow></semantics></math></inline-formula> modes are difficult if we keep the reconstruction of the original metric perturbations in our mind. Due to this situation, we propose a strategy of a gauge-invariant treatment of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>l</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn></mrow></semantics></math></inline-formula> mode perturbations through the decomposition of the metric perturbations by singular harmonic functions at once and the regularization of these singularities through the imposition of the boundary conditions to the Einstein equations. Following this proposal, we derive the linearized Einstein equations for any modes of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>l</mi><mo>≥</mo><mn>0</mn></mrow></semantics></math></inline-formula> in a gauge-invariant manner. We discuss the solutions to the odd-mode perturbation equations in the linearized Einstein equations and show that these perturbations include the Kerr parameter perturbation in these odd-mode perturbations, which is physically reasonable. In the Part II and Part III papers of this series of papers, we will show that the even-mode solutions to the linearized Einstein equations obtained through our proposal are also physically reasonable. Then, we conclude that our proposal of a gauge-invariant treatment for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>l</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn></mrow></semantics></math></inline-formula>-mode perturbations is also physically reasonable.https://www.mdpi.com/2218-1997/11/2/39black holeSchwarzschild spacetimeperturbation theorygauge-invariance |
| spellingShingle | Kouji Nakamura Gauge-Invariant Perturbation Theory on the Schwarzschild Background Spacetime Part I: Formulation and Odd-Mode Perturbations Universe black hole Schwarzschild spacetime perturbation theory gauge-invariance |
| title | Gauge-Invariant Perturbation Theory on the Schwarzschild Background Spacetime Part I: Formulation and Odd-Mode Perturbations |
| title_full | Gauge-Invariant Perturbation Theory on the Schwarzschild Background Spacetime Part I: Formulation and Odd-Mode Perturbations |
| title_fullStr | Gauge-Invariant Perturbation Theory on the Schwarzschild Background Spacetime Part I: Formulation and Odd-Mode Perturbations |
| title_full_unstemmed | Gauge-Invariant Perturbation Theory on the Schwarzschild Background Spacetime Part I: Formulation and Odd-Mode Perturbations |
| title_short | Gauge-Invariant Perturbation Theory on the Schwarzschild Background Spacetime Part I: Formulation and Odd-Mode Perturbations |
| title_sort | gauge invariant perturbation theory on the schwarzschild background spacetime part i formulation and odd mode perturbations |
| topic | black hole Schwarzschild spacetime perturbation theory gauge-invariance |
| url | https://www.mdpi.com/2218-1997/11/2/39 |
| work_keys_str_mv | AT koujinakamura gaugeinvariantperturbationtheoryontheschwarzschildbackgroundspacetimepartiformulationandoddmodeperturbations |