Lectures on black-hole perturbation theory Leor Barack University of Southampton Kavli-RISE Summer School on Gravitational Waves Kavli-RISE Summer School on GWs () Black-hole perturbation theory 1 / 73
Plan Overview Types of perturbative expansions in GR Applications of black-hole perturbation theory PART I: Basics of perturbation theory in GR Metric perturbations and gauge freedom Perturbations via the Newman–Penrose formalism PART II: Methods of BH perturbation theory Lorenz-gauge formulation Regge-Wheeler-Zerilli formalism Teukolsky equation & metric reconstruction PART III: EMRIs and self-force theory EMRIs as sources of gravitational waves Self-force theory Self-force calculations Kavli-RISE Summer School on GWs () Black-hole perturbation theory 2 / 73
A few bits left for you to work out . . . Problem 0 (EXAMPLE) • ◦ ◦ Draw me a sheep. • ◦ ◦ a ∼ 1-line calculation • • ◦ a ∼ 1-paragraph calculation • • • a ∼ 1-page calculation Kavli-RISE Summer School on GWs () Black-hole perturbation theory 3 / 73
Types of perturbative expansions in GR 4 main systematic perturbative frameworks for solving Einstein’s Field Equations: Post-Newtonian theory: expands about Newtonian gravity, in powers of v/c ◮ example: large-separation compact-object binary Post-Minkowskian theory: expands about flat spacetime, in powers of G ◮ examples: radiation at scri; ultrarelativistic scattering particles Black-hole perturbation theory: expands about Kerr spacetime, in magnitude of small metric perturbation ◮ example: large mass-ratio binary; post-merger ringing FLRW perturbation theory: expands about FLRW cosmological spacetime, in powers of density fluctuation Kavli-RISE Summer School on GWs () Black-hole perturbation theory 4 / 73
Overlapping expansions in the binary problem Kavli-RISE Summer School on GWs () Black-hole perturbation theory 5 / 73
Applications of black-hole perturbation theory (Historical) Stability of the BH/event horizon Stability/development of internal structure; strong cosmic censorship Semi-classical BH theory Interaction with radiation (“pure tones”, power-law decay, universality) Post-merger ringing Compact object in a tidal environment EMRIs, self-force, “problem of motion in GR” Kavli-RISE Summer School on GWs () Black-hole perturbation theory 6 / 73
PART I: PERTURBATION THEORY IN GR – Metric perturbations and gauge freedom – Perturbations via the Newman–Penrose formalism Kavli-RISE Summer School on GWs () Black-hole perturbation theory 7 / 73
Metric perturbation equations We want to solve G µν [ g αβ ( ǫ )] = 8 π T µν ( ǫ ) for ǫ ≪ 1 where g αβ and T µν depend smoothly on a dimensionless parameter ǫ (e.g., binary mass ratio), so that g αβ (0) is a known spacetime [e.g., Kerr, with T µν (0) = 0]. Think of g αβ ( ǫ ) as a 1-parameter family of spacetimes. Kavli-RISE Summer School on GWs () Black-hole perturbation theory 8 / 73
Metric perturbation equations We want to solve G µν [ g αβ ( ǫ )] = 8 π T µν ( ǫ ) for ǫ ≪ 1 where g αβ and T µν depend smoothly on a dimensionless parameter ǫ (e.g., binary mass ratio), so that g αβ (0) is a known spacetime [e.g., Kerr, with T µν (0) = 0]. Think of g αβ ( ǫ ) as a 1-parameter family of spacetimes. Taylor-expand 2 ǫ 2 d 2 g αβ ǫ =0 + 1 g αβ (0) + ǫ dg αβ � � ǫ =0 + · · · =: g (0) αβ + h (1) αβ + h (2) g αβ ( ǫ ) = αβ + · · · � � d ǫ d ǫ 2 � � 2 ǫ 2 d 2 T µν T µν (0) + ǫ dT µν ǫ =0 + 1 � � ǫ =0 + · · · =: T (0) µν + T (1) µν + T (2) T µν ( ǫ ) = µν + · · · � � d ǫ d ǫ 2 � � Regard h ( n ) αβ and T ( n ) µν as tensor fields on the “background” spacetime g (0) αβ . By convention, indices are raised and lowered using g (0) αβ . E.g., h αβ (1) = g αµ (0) g βν (0) h (1) µν . Kavli-RISE Summer School on GWs () Black-hole perturbation theory 8 / 73
Metric perturbation equations Problem 1 •◦◦ β , show g αβ = g αβ Given g αβ = g (0) αβ + h (1) αβ + O ( ǫ ) and g αλ g λβ = δ α (0) − h αβ (1) + O ( ǫ ). Kavli-RISE Summer School on GWs () Black-hole perturbation theory 9 / 73
Metric perturbation equations Problem 1 •◦◦ β , show g αβ = g αβ Given g αβ = g (0) αβ + h (1) αβ + O ( ǫ ) and g αλ g λβ = δ α (0) − h αβ (1) + O ( ǫ ). What equations do the perturbations h ( n ) αβ satisfy? At 0-th order we simply have G (0) µν [ g (0) αβ ] = 8 π T (0) (= 0 for Kerr) , µν where G (0) is the Einstein operator with derivatives ∇ (0) α compatible with g (0) αβ . Kavli-RISE Summer School on GWs () Black-hole perturbation theory 9 / 73
Metric perturbation equations Problem 1 •◦◦ β , show g αβ = g αβ Given g αβ = g (0) αβ + h (1) αβ + O ( ǫ ) and g αλ g λβ = δ α (0) − h αβ (1) + O ( ǫ ). What equations do the perturbations h ( n ) αβ satisfy? At 0-th order we simply have G (0) µν [ g (0) αβ ] = 8 π T (0) (= 0 for Kerr) , µν where G (0) is the Einstein operator with derivatives ∇ (0) α compatible with g (0) αβ . At orders ≥ 1 we have a little complication: G µν involves ∇ α , not ∇ (0) α , acting on tensors defined in g (0) αβ . We need to express ∇ α in terms of ∇ (0) α . Kavli-RISE Summer School on GWs () Black-hole perturbation theory 9 / 73
Derivation of the metric perturbation equations Problem 2 ••• Follow through the steps below to arrive at Eq. (1) (next page) Step 1: Define h αβ := g αβ − h (0) αβ and β ) w α = C α ( ∇ β − ∇ (0) βγ w γ for any w α . βγ − Γ (0) α βγ transforms as a tensor in g (0) Show C α βγ = Γ α βγ , and that C α αβ . Kavli-RISE Summer School on GWs () Black-hole perturbation theory 10 / 73
Derivation of the metric perturbation equations Problem 2 ••• Follow through the steps below to arrive at Eq. (1) (next page) Step 1: Define h αβ := g αβ − h (0) αβ and β ) w α = C α ( ∇ β − ∇ (0) βγ w γ for any w α . βγ − Γ (0) α βγ transforms as a tensor in g (0) Show C α βγ = Γ α βγ , and that C α αβ . Step 2: Show [hint: work in a local inertial frame, where Γ (0) α = 0 ] βγ βγ = 1 2 g αµ � � C α ∇ (0) β h γµ + ∇ (0) γ h βµ − ∇ (0) µ h βγ Kavli-RISE Summer School on GWs () Black-hole perturbation theory 10 / 73
Derivation of the metric perturbation equations Problem 2 ••• Follow through the steps below to arrive at Eq. (1) (next page) Step 1: Define h αβ := g αβ − h (0) αβ and β ) w α = C α ( ∇ β − ∇ (0) βγ w γ for any w α . βγ − Γ (0) α βγ transforms as a tensor in g (0) Show C α βγ = Γ α βγ , and that C α αβ . Step 2: Show [hint: work in a local inertial frame, where Γ (0) α = 0 ] βγ βγ = 1 2 g αµ � � C α ∇ (0) β h γµ + ∇ (0) γ h βµ − ∇ (0) µ h βγ βγδ w β = ( ∇ γ ∇ δ − ∇ δ ∇ γ ) w α , next show Step 3: Using R α R α βγδ ( g ) = R α (0) βγδ ( g (0) ) + 2 ∇ (0) [ γ C α δ ] β + 2 C α µ [ γ C µ δ ] β . Kavli-RISE Summer School on GWs () Black-hole perturbation theory 10 / 73
Derivation of the metric perturbation equations Problem 2 ••• Follow through the steps below to arrive at Eq. (1) (next page) Step 1: Define h αβ := g αβ − h (0) αβ and β ) w α = C α ( ∇ β − ∇ (0) βγ w γ for any w α . βγ − Γ (0) α βγ transforms as a tensor in g (0) Show C α βγ = Γ α βγ , and that C α αβ . Step 2: Show [hint: work in a local inertial frame, where Γ (0) α = 0 ] βγ βγ = 1 2 g αµ � � C α ∇ (0) β h γµ + ∇ (0) γ h βµ − ∇ (0) µ h βγ βγδ w β = ( ∇ γ ∇ δ − ∇ δ ∇ γ ) w α , next show Step 3: Using R α R α βγδ ( g ) = R α (0) βγδ ( g (0) ) + 2 ∇ (0) [ γ C α δ ] β + 2 C α µ [ γ C µ δ ] β . Step 4: Express this in terms of h αβ , and obtain δ R αβ := R αβ − R (0) αβ and, finally, � R αβ − 1 � 2 g αβ g µν R µν δ G αβ = δ . Kavli-RISE Summer School on GWs () Black-hole perturbation theory 10 / 73
Metric perturbation equations: 1st & 2nd order Now keeping only terms of O ( ǫ ), obtain the linearized Einstain’s equations (here specialized to vacuum background, R (0) µν [ g (0) αβ ] = 0 for simplicity): δ G αβ = − 1 β ) µ − 1 2 � (0) ¯ h (1) αβ − R µ ν (0) ¯ µν + ∇ (0) (0) ¯ h (1) 2 g (0) ν ¯ (1) = 8 π T (1) h (1) ( α ∇ µ αβ ∇ (0) µ ∇ (0) h µν (1) α β αβ where � (0) g µν (0) ∇ (0) µ ∇ (0) := “D’Alambertian” operator (GWs!) ν αβ − 1 h (1) = − h (1) ) h (1) h (1) 2 g (0) ¯ αβ h (1) “trace-reversed ”perturbation (since ¯ := αβ h (1) g αβ (0) h (1) trace of h (1) := αβ αβ Kavli-RISE Summer School on GWs () Black-hole perturbation theory 11 / 73
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