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Schwarzschild Perturbations in the Lorenz Gauge
Leor Barack (Soton)Carlos Lousto (UTB)
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Talk Plan
Why Lorenz Gauge?
Schwarzschild perturbations reformulated in the Lorenz gauge: Now both odd- and even- parity modes.
Code for time evolution of Lorenz-gauge perturbations from a point particle: Circular orbits in Schwarzschild.
Future work: Self-force calculations, generic orbits, Kerr black hole.
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Gauge freedom in Perturbation Theory
hgg )bckgrnd( 10 functions, but only 2 “physical” dof
In particular, represent the same physical perturbation,
for any (“small”, differentiable) displacement field
);(2and hh
Useful gauges in black hole perturbation theory
Regge-Wheeler gauge: 0)4,3,2,1()()( ),(),( iiii
lmi
lmilmi hYhh tr
Radiation gauge: 0or0
nhlh
Lorenz (“harmonic”) gauge: 0;
h
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Problem: Standard BH perturbation techniques give h in other gauges.
Solutions attempted:
Transform h (RW/Radiation) h (Lorenz), and evaluate contribution of difference to SF
Transform hS (Lorenz) hS (RW/Radiation), and try make sense of SF in RW/Rad gauges
Our approach: Step back; reformulate BH perturbation theory in Lorenz gauge.
Grav. self-force and gauge freedom
Lorenz gauge
Lorenz gauge
)03Whiting&Detweiler(lim
)97MSTQW(lim
S
dirself
hh
hhF
prtclx
prtclx
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Why Lorenz gauge? (I)
Lorenz gauge:
Particle singularities look like particle singularities:
Pointlike and isotropic
Regge-Wheeler gauge:
Particle singularities are not isotropic. Field depends on direction of approach to singularity
Radiation gauge:
Typically, particle singularities are not even isolated!
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RW/Zerilli/Moncrief/Teukolsky variables are discontinuous at the particle. The (mode decomposed) Lorenz-gauge MP is continuous. No -function derivatives in the source term of the field equations.
No need to resort to complicated MP reconstruction procedures, as when working with Moncrief/Teukolsky variables.
In particular, no need to take derivatives of numerical integration variables – advantage in numerical implementation.
Field equations manifestly hyperbolic.
Price to pay: Field equations remain coupled.
Why Lorenz gauge? (II)
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Schwarzschild perturbations in the Lorenz gauge: formulation
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1. Linearized Einstein equations in the Lorenz gauge
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2. Tensor-harmonic decomposition
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3. Make sure variables are suitable for numerical time-evolution
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4. Get separated equations for the h(i)’s
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ODD PARITY
EVEN PARITY
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5. Write gauge conditions in mode-decomposed form,
… and use them to reduce the system of field equations:
0
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EVEN PARITY
ODD PARITY
Circular equatorial orbit, Axially-symmetric (m=0) modes
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r/M
rp
fEH lr
Analytic solution for the axially-symmetric, odd-parity modes in the case of a circular orbit
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MP “reconstruction”
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Code for time evolution of the MP Eqs
Circular geodesics in Schwarzschild
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Grid for 1+1d numerical evolution
Grid size: At least 3-4 Torb
(>300M for r0=6M)
Resolution:
~1 grid pt/M2 for fluxes extraction
~106 grid pts/M2 (near particle) for SF extraction
Event
Hor
izon Null Infinity
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Finite difference scheme (2nd order convergent)
0
hh
hh
[Demonstrated here for a scalar field – works similarly for our coupled MP Eqs]
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Results: Scalar field
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Results: Scalar field
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Results: Scalar field
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Results: Scalar field
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Results: Scalar field
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---------------------------------------- [F_r(r0+)-F_r(r0-)]/(2L): ("A_r") -2.16430508258478D-002 -2.16430501533300D-002 -2.16430495556174D-002 ---------------------------------------- [F_r(r0+)+F_r(r0-)]/2: ("B_r") -1.07310613335618D-002 -1.07310600978665D-002 -1.07310599479544D-002 ---------------------------------------- ---------------------------------------- |F_r(r0+)-F_r(r0-)|/2-|A|*L: ("C_r") 6.76214094093648D-005 6.76204006326203D-005 6.76195040637838D-005 ---------------------------------------- F_r(REG): -9.90421617212026D-005 -9.90409250297095D-005 -9.90407736144393D-005 ---------------------------------------- F_t(r0): 1.09151357977453D-004 1.09149895091873D-004 1.09148592668721D-004 ----------------------------------------
Results: Scalar force (sample output, screen capture)
[leor@hercules scalar]$ ./a.out l=1 Evolution time (# of orbits)=5 Initial Resolution (steps per M in r*,t)=5 ITERATION # 1 ITERATION # 2 ITERATION # 3 ---------------------------------------- Phi(r0): Cycle # 2 : 0.136805155563096 Cycle # 3 : 0.136805162960639 Cycle # 4 : 0.136805163931519 ---------------------------------------- F_r(r0+): Cycle # 2 : -4.31956375723335D-002 Cycle # 3 : -4.31956353278614D-002 Cycle # 4 : -4.31956342813805D-002 ---------------------------------------- F_r(r0-): Cycle # 2 : 2.17335149052099D-002 Cycle # 3 : 2.17335151321284D-002 Cycle # 4 : 2.17335143854717D-002 ----------------------------------------
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Sample results: Metric Perturbation (odd parity)
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Sample results: Metric Perturbation (Even parity,l=m=2, rp=7M)
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Sample results: MP (even parity,l=m=2, rp=7M)
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Tests of Code
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Test #1: 2nd-order Numerical Convergence
l=2,m=1r0=7M
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l=2,m=1r0=7M
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l=m=2
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l=m=2
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Test #2: conservation of the gauge condition
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500
500500
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500
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Test #3: Flux of energy radiated to infinity
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lmE
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What’s next?
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Gravitational Self Force
• Getting the self force is now straightforward:
No need to take 3 derivatives, just one – as in scalar case
Modes l=0,1 given (almost) analytically, in the harmonic gauge
Most crucial: no gauge problem; mode-sum implemented as is:
• Two options:
Either decompose Ffull in Scalar harmonics & use the “standard” RP;
Or decompose Ffull in Vector harmonics and re-derive the RP accordingly.
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Eccentric orbits in Schwarzschild
• Generalization is easy, as code is time-domain
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Orbits in Kerr
N. Hernandez (MSc Thesis, Brownsville 2005) shows promise of this approach