Introduction to Numerical Relativity I Intro &...

Introduction to Numerical Relativity IIntro & Geometry

S. Husa

University of the Balearic Islands, [email protected]

August 5, 2010

~S0;M0; Ja0~S1;M1; Ja1~S2;M2; Ja2

S. Husa (UIB) Numerical Relativity ESI Vienna, August 2010 1 / 30

Outline

1 Initial value problem

2 3+1 Decomposition

3 Conformal approach to solve the constraints

4 Black Holes

5 Asymptotics


Outline


2 3+1 Decomposition


4 Black Holes

5 Asymptotics


Outline


2 3+1 Decomposition


4 Black Holes

5 Asymptotics


Outline


2 3+1 Decomposition


4 Black Holes

5 Asymptotics


Outline


2 3+1 Decomposition


4 Black Holes

5 Asymptotics


Initial value problem

General Relativity I

General relativity is the modern (classical) theory of gravitation,

formulated in terms of spacetime geometry,

described by the Einstein equations:

Gab[gcd ] = Rab −1

2Rc

cgab = 8πκTab[gcd , φA]

, Rbd = Rabad .

Rabcd = Γa

bd,c − Γabc,d + Γm

bdΓamc − Γm

bcΓamd , [∇a,∇b]v c = Rc

dabvd ,

Γik` =

1

2g im(gmk,` + gm`,k − gk`,m).

Written in terms of coordinate components and partial derivatives the EEcorrespond to a very complex system of coupled nonlinear PDEs.

Add equations for the dynamics of the matter fields φA – and solveconsistently, do not prescribe gab and then compute Tab!

Case Tab = 0 is highly nontrivial: black holes & gravitational waves!








2Rc

cgab = 8πκTab[gcd , φA], Rbd = Ra

bad .

Rabcd = Γa


bdΓamc − Γm


dabvd ,

Γik` =

1












2Rc


bad .

Rabcd = Γa


bdΓamc − Γm


dabvd ,

Γik` =

1












2Rc


bad .

Rabcd = Γa


bdΓamc − Γm


dabvd ,

Γik` =

1







General Relativity II

While GR can be formulated in a very compact and elegant way, fullyunderstanding the physical content of the theory remains a major challenge:

What is the solution space and what is its physical interpretation?

The complexity/nonlinearity of GR severly limits the physical relevance ofmost exact solutions, exceptions in 3+1 e.g. Kerr, FRW, (A)dS (+ BH).

What does GR tell us about our universe? What effects can we observe?

Quantum theory, singularity theorems ⇒ how should we modify GR?

Can observations help to find a better (quantum) theory of gravity?

Mathematical problems & exact solutions dominated GR for much of its history:

Deep insights gained: positive mass theorem, singularity theorems, nonlinearstability of Minkowski, . . .

GR research has often been decoupled from other areas of physics.











































































Why Numerical Relativity?

Astrophysics, cosmology, general understanding of the solution space of theEE require approximate solutions – analytical and numerical!

Numerical solutions allow to study the equations with a minimum ofsimplifying physical assumptions, and allow mathematical control over theconvergence of the approximation!

Typically, numerical analysis provides theorems that a certainalgorithm will converge to the correct solution for some finite time ata certain order for sufficient resolution.

Nonlinear PDEs need to be studied with non-perturbative quantitativemethods – and in particular also by numerical methods.

The Einstein equations pose a number of fundamental problems/challengesfor treatment with numerical methods, but standard finite difference methodsare often an excellent choice!



























NR comes in many flavours . . .

physical model:

• black holes: formation, ringdown, binary dynamics & GWs, . . .• ’Astrophysical’ matter: supernovae, neutron stars, BH–NS, . . .• ’Fundamental’ matter: hairy BH’s, critical collapse, string dynamics . . .• Cosmology!

technical:

slicing: spacelike or nullasymptotics: cutoff or compactifiedaim: toy or tool

mathematical understanding or astrophysics problems?computational: requires parallelism? mesh-refinement? multiple patches?

dynamical regime:

test fields on fixed background – matter dynamics dominated – rest massdominated – kinetic energy dominated – waves dominated



Is NR difficult?

The art: make continuum (physical!) features manifest in discrete system.

The ideal numerical relativist is an expert (at least) in the geometric arts of GR,PDE theory, numerics, scientific programming, high performance computing, blackhole physics and astrophysics/cosmology!

Fear not! Don’t get confused! But note first hints of trouble:

GR can be viewed as gauge theory of diffeomorphism group → constraints!

→ Computational d.o.f. physical d.o.f.

Construct spacetimes – physics is in the geometry, not particular metriccomponents! Numerical solution procedures require “gauge fixing”.

Nonlinear PDEs typically require case-by-case study – which PDE system dothe EEs actually correspond to?

Complicated nonlinear structure of GR makes it hard to carry over techniquesaddressing related issues in Electrodynamics or Yang-Mills.



Methods options for NR

Discretizing GR can be approached in very different ways:

direct discretization of the geometry (e.g. Regge calculus: triangulation intosimplices, spacetime curvature expressed in terms of deficit angles).

“discrete differential forms” – a standard method in computationalelectrodynamics.

PDE problem, constrained or free evolution, at least 2 “hyperbolic” eqs.!)

“mimetic” discretization (more directly model the underlying physics thanwith PDE system), but some no-go theorems for manifest solution ofconstraints.

Seems fair to say that so far only the standard PDE approach has led to thesolution of real problems.

In the absence of matter fields, the EE do not lead to the formation of shocks(but happen for coordinates) or turbulence ⇒ standard high order finite differenceor spectral methods work very well!



Methods options for NR

Discretizing GR can be approached in very different ways:

direct discretization of the geometry (e.g. Regge calculus: triangulation intosimplices, spacetime curvature expressed in terms of deficit angles).

“discrete differential forms” – a standard method in computationalelectrodynamics.

PDE problem, constrained or free evolution, at least 2 “hyperbolic” eqs.!)

“mimetic” discretization (more directly model the underlying physics thanwith PDE system), but some no-go theorems for manifest solution ofconstraints.

Seems fair to say that so far only the standard PDE approach has led to thesolution of real problems.

In the absence of matter fields, the EE do not lead to the formation of shocks(but happen for coordinates) or turbulence ⇒ standard high order finite differenceor spectral methods work very well!



How?

Gab[gcd ] = 8πκTab[gcd , φA].

EEs are instrinsically 4-dimensional – how/where should we specify boundaryconditions = select the physical solution?

Written in any coordinate system, EEs become underdetermined set of couplednonlinear PDEs (and overdetermined because we have to solve constraints).

First sort out what can be chosen and what is then determined by equations →algorithmic method of solution.

Q: Does the theory have an initial value formulation? Yes! Many!

Unique (modulo diffeomorphisms) solution of EEs equation can be specified byinitial data on a spacelike hypersurface!

Predictability! Important source of physical intuition!



Start simple: Maxwell

4-dimensional:dF = 0, d ∗ F = 4π ∗ J.

space+time – split: introduce electric and magnetic field

E a = Fabnb, Bc =1

2Fab

3εabc .

∂aE a = 4πρ, ∂aBa = 0, ∂tEa = εabc∂

bBc − 4πja, ∂tBa = −εabc∂bE c .

Initial value problem makes sense! Working with Aa additional gauge issuesappear!

Numerical ED is difficult, but well understood (analytical formulation, numericalalgorithms, comparison with experiment)!

Curved background:

LnDiEi = −KDiE

i , LnDiBi = −KDiB

i

In collapsing case (K < 0) ⇒ instability of constraints!

Solution for Maxwell: use√

gE a,√

gBa. GR ??



Start simple: Maxwell

4-dimensional:dF = 0, d ∗ F = 4π ∗ J.

space+time – split: introduce electric and magnetic field

E a = Fabnb, Bc =1

2Fab

3εabc .

∂aE a = 4πρ, ∂aBa = 0, ∂tEa = εabc∂

bBc − 4πja, ∂tBa = −εabc∂bE c .

Initial value problem makes sense! Working with Aa additional gauge issuesappear!

Numerical ED is difficult, but well understood (analytical formulation, numericalalgorithms, comparison with experiment)!

Curved background:

LnDiEi = −KDiE

i , LnDiBi = −KDiB

i

In collapsing case (K < 0) ⇒ instability of constraints!

Solution for Maxwell: use√

gE a,√

gBa. GR ??S. Husa (UIB) Numerical Relativity ESI Vienna, August 2010 10 / 30

3+1 Decomposition

Fast track “3+1”

Simplest way to get PDEs from EEs: chose coordinates x i , t (i = 1, 2, 3)

“read off” metric in the form

ds2 = −α2dt2 + hab(dxa + βadt)(dxb + βbdt), (1)

hab is a positive definite matrix (Riemannian 3-metric), and α 6= 0 (problemsfoliating BH spacetimes).

4 functions α(x i , t) and βj(x i , t) freely specifiable, steer coordinate systemthrough spacetime as time evolution proceeds – physical result is independent ofthis choice ⇔ diffeo invariance.

PDEs resulting from this ansatz are of 2nd order for h, split into 2 parts: 4constraints (no second time derivatives) & 6 evolution equations.


3+1 Decomposition

Projections and the Induced Metric

Given foliation: write all tensors in terms of “horizontal” and “vertical” parts!

Let na = −α∇at denote the future timelike unit normal to Σ & define

Nab := −nanb, ha

b := δab − Nab,

check they are in fact the desired projection operators (exercise!):

habhb

c = hac , Na

bNbc = Na

c , habNb

c = 0,

hab projects onto the tangential, and Na

b onto the normal directions,

habnb = 0, Na

bnb = na.

Apply first to metric → induces a tensor field hab by

hab = gab + nanb = gcdhcahd

b.

hab is purely horizontal, positive definite and nondegenerate for horizontal vectorfields (exercise!) → natural Riemannian metric on Σ.


3+1 Decomposition




Nab := −nanb, ha

b := δab − Nab,


habhb

c = hac , Na

bNbc = Na

c , habNb

c = 0,



habnb = 0, Na

bnb = na.



b.



3+1 Decomposition




Nab := −nanb, ha

b := δab − Nab,


habhb

c = hac , Na

bNbc = Na

c , habNb

c = 0,



habnb = 0, Na

bnb = na.



b.



3+1 Decomposition




Nab := −nanb, ha

b := δab − Nab,


habhb

c = hac , Na

bNbc = Na

c , habNb

c = 0,



habnb = 0, Na

bnb = na.



b.



3+1 Decomposition




Nab := −nanb, ha

b := δab − Nab,


habhb

c = hac , Na

bNbc = Na

c , habNb

c = 0,



habnb = 0, Na

bnb = na.



b.



3+1 Decomposition




Nab := −nanb, ha

b := δab − Nab,


habhb

c = hac , Na

bNbc = Na

c , habNb

c = 0,



habnb = 0, Na

bnb = na.



b.



3+1 Decomposition

Induced Curvatures

2 curvatures associated with embedding of Σ in M:

intrinsic: Riemann tensorextrinsic: describes how Σ bends in M.

Natural derivative operator Da associated with hab:

DcT a1...arb1...bs

:= hcc′h

a1

a′1. . . har

a′rhb′

1

b1. . . h

b′s

bs∇c′T

a′1...a′r

b′1...b

′s

(2)

→ define Riemann tensor of 3Rabcd [hef ].

Extrinsic curvature:

Kab := hcahd

b∇cnd =1

2Lnhab = “velocity”,

Relation of the intrinsic and extrinsic curvatures of Σ to the curvature of M isgiven by two crucial geometric identities, the Gauss-Codazzi Eqs.:

3Rabcd

= haa′hb

b′hc

c′hd′dRa′b′c′

d′− KacKb

d + KbcKad , (3)

DaK ab − DbK a

a = Rcdndhcb. (4)


3+1 Decomposition

Threadings of spacetime

Consider changes of tensors T ...... along the integral curves of a time flow vector ta,

given by Lie derivative:

T ...... := LtT

...... =

∂

∂tT (xα, t) in adapted coordinatesxα, t.

Spacetime engineering: “threading” ta dynamically “steers” spacetime evolution.Decompose ta into a normal and a tangential component:

ta = αna + βa, βana = 0, ta timelike if βaβa − α2 < 0. (5)

α µn

βµ

Σ0

Σt

nµ t

µ

“Lapse” α determines “how fast timeelapses”.

“Shift vector” βa shifts spatialcoordinate points with time evolution.

Lapse and shift are not determined by EEs ⇒ plenty of rope to shoot yourself inthe foot!


3+1 Decomposition

Projecting Gab = 0

Now use EEs – compute all projections of Gab = κTab using Gauss-Codazzi (3,4)!

Projections with na yield:

0 = Gbcnchba = hb

aRbc = DaK ab − DbK a

a (6)

0 = Gabnanb =1

2

(3R + (K a

a)2 − KabK ab)

(7)

No time derivatives of Kab – they are relations between the initial data h and K ,which cannot be freely specified – the constraint equations of GR!

Gabhcahd

b yields another evolution equation (have Lnhab = 2Kab):

Kab = −DaDbα + βcDcKab + KcbDaβc − KbcDaβ

c

+ α(

3Rab + KccKab

)(8)

Bianchi-Id. (∇aGab = 0) ⇒ Constraints propagate!(What happens for small initial violations?)

Mid 70’s: let’s just code them up and collide black holes!S. Husa (UIB) Numerical Relativity ESI Vienna, August 2010 15 / 30

Conformal approach to solve the constraints

Conformal approach to constraints: time symmetry

Standard formalism to solve the constraints: conformal approach developed byLichnerowicz, York, O Murchadha and others: geometrically intuitive, elegant andthe most practical framework known (actually: thin sandwich!).

Mathematically natural or physically meaningful?

Consider the case of “time symmetry”: Kab = 0:

R[hab] = 0, 1 equation for 6 variables!

Idea: express hab from ’base metric’ hab and a conformal factor ψ as

hab = ψ4hab, ψ > 0. (9)

Scalar curvature transforms as

Rh = ψ−5 (Rhψ − 8∆hψ) ⇒ −∆hψ +1

8Rhψ = 0 elliptic!



Conformal approach to constraints: general case

Aab := K ab − 1

3habK trace-free part of extrinsic curvature.

RescaleAab = ψ10Aab

[Aab = ψ2Aab

]→ DaAab = ψ−10DaAab. (10)

Decompose the traceless symmetric Aab as

Aab = AabTT + (LW )ab,

divergence-free (transverse) traceless part AabTT and vector potential W a,

(LW )ab := DaW b + DbW a − 2

3habDcW c .

Insertion AabTT = Aab − (LW )ab into constraints yields coupled elliptic PDEs

Da(LW )ab = −DaAab +2

3ψ6DbK + 8πψ10jb, (11)

−4hψ +1

8Rhψ −

1

8ψ−7(Aab − (LW )ab)2 +

1

12ψ5K 2 = 2πψ5ρ. (12)

Freely specifiable: hab, K , symmetric tracefree Aab.




Aab := K ab − 1



[Aab = ψ2Aab






3habDcW c .



3ψ6DbK + 8πψ10jb, (11)

−4hψ +1

8Rhψ −

1

8ψ−7(Aab − (LW )ab)2 +

1

12ψ5K 2 = 2πψ5ρ. (12)





Aab := K ab − 1



[Aab = ψ2Aab






3habDcW c .



3ψ6DbK + 8πψ10jb, (11)

−4hψ +1

8Rhψ −

1

8ψ−7(Aab − (LW )ab)2 +

1

12ψ5K 2 = 2πψ5ρ. (12)





Aab := K ab − 1



[Aab = ψ2Aab






3habDcW c .



3ψ6DbK + 8πψ10jb, (11)

−4hψ +1

8Rhψ −

1

8ψ−7(Aab − (LW )ab)2 +

1

12ψ5K 2 = 2πψ5ρ. (12)





Aab := K ab − 1



[Aab = ψ2Aab






3habDcW c .



3ψ6DbK + 8πψ10jb, (11)

−4hψ +1

8Rhψ −

1

8ψ−7(Aab − (LW )ab)2 +

1

12ψ5K 2 = 2πψ5ρ. (12)

Freely specifiable: hab, K , symmetric tracefree Aab.S. Husa (UIB) Numerical Relativity ESI Vienna, August 2010 17 / 30


Conformal flatness, Bowen-York, and puncture data - I

Typical simplifying assumptions for constraints:

constant mean curvature (CMC) slices: K = const., AsymptoticallyEuclidean (K = 0) or hyperboloidal (K 6= 0).

Boundary condition for asymptotically Euclidean data: limr→∞ ψ = 1.

Spatial conformal flatness: “seed” metric hab is flat.

CMC appears not to be a strong restriction, but conformal flatness is (at least forstrong data): no conformally flat slices in Kerr.

NR simulations typically use CMC data – decouple Hamiltonian and momentumconstraints.

Conformally flat data are very commonly used in NR, e.g. for neutron stars &even BHs.


Black Holes

Conformal flatness, Bowen-York, and puncture data - II

An application of conformal compactification!

~S0;M0; Ja0~S1;M1; Ja1~S2;M2; Ja2 ~S

S i

“puncture ID”: wormhole topology & each asymptotic end is compactified to asingle point at the price of a coordinate singularity.

Kab = 0 ⇒ ∆ψ = 0 ψ = 1 +∑

imi

2|~r−~ri | .

ID for N BHs: add extra ∞’s → enforce minimal surfaces → trapped surfaces.

Good enough until recently: conformal flatness: hab = δab → analytic solution ofmomentum constraint for spinning and boosted BHs (“Bowen-York” Kab).

Straightforward specification of momenta and spins – great flexibility!


Black Holes

Conformal flatness, Bowen-York, and puncture data - III

Exact solutions exist to the momentum constraint for conformally flat initial data,Bowen-York family describes BH with angular momentum S i & momentum P i :

Aij =3

2r 2

[Pinj + Pjni − (δij − ninj)Pknk

]+

3

r 3

[εkilS

lnknj + εkjlSlnkni

].

By linearity of the momentum constraint, we can superpose solutions to describen BHs with total angular momentum J i =

∑a S i

a and momentum∑

a P ia.

Hamiltonian constraint then takes the simple form

−4 ψ =1

8ψ−7AabAab −

1

12ψ5K 2.

Variations of the conformal formalism to solve constraints are known as(conformal) thin sandwich formalism, there lapse α and shift βi appear explicitlyin the construction.


Black Holes

What is a black hole?

Trapped surface (TS): spacelike2-surface where ingoing and outoingwavefronts decrease in area.

Light cones “tip over” → outflowboundaries → “BH excision.

Under very general conditions: TS ⇒spacetime singularity.

Marginally TS: expansionΘ± = qab∇al±b = 0(q: metric on 2-surface, l±al

a± = 0)

Apparent horizon: the outermost “marginally” trapped surface.

Event horizon: the boundary in the spacetime between points that can send lightrays to infinity, and those that can not. Global concept!


Black Holes

Physical properties of BHs in a given spacetime

The EH is a teleologically defined null surface – local objects more natural in NR.

apparent horizon – outermost future marginally trapped surface, Θ+ = 0.

dynamical horizon – spacelike surface foliated by marginally trapped surfaces.

isolated horizon – null surface foliated by marginally trapped surfaces –“dynamical horizon with no matter or radiation falling in”.

Laws of BH mechanics can be generalized for isolated and dynamical horizons,horizon properties can be computed locally on a spacelike cross-section S , e.g.angular momentum

J(ϕ) = − 1

8πG

∫S

ϕar aKabd2V .

ϕa is a Killing vector on S ,r a is the unit radial normal to S in M.


Black Holes

Locating BHs in a given spacetime

Locating EH requires evolution at least to a stationary state – then track nullsurfaces backward in time, reasonable initial guesses will converge onto the EH.

Describe EH by level surfaces:

F (t, x i ) = 0,

g ab∇a∇b = 0.

∂t f = − g ti

g tt∂i f

+

√(gt i∂i f )2g ttg ij∂i f ∂j f

g tt

For AH/IH/DH look for surfaces of constant null expansion:

Θ± = K − r arbKab ±Dar a convertible to elliptic equation for level set surface.

Solve directly as an elliptic equation or find them with a parabolic “flow”.S. Husa (UIB) Numerical Relativity ESI Vienna, August 2010 23 / 30

Black Holes

Find apparent horizons in 2D

Represent axially symmetric 2-surface as level set surface F (ρ, z) = 0, spacelikeunit vector to the 2-surface, r a, becomes

r a =∂aF√

g ab∂aF∂bF,

Brill wave ansatz for axisymmetric 3-metric:

ds2 = ψ4[e2q(r ,ϑ)

(dr 2 + r 2dϑ2

)+ r 2 sin2 ϑdϕ2

].

Simplest case: assume r(ϑ), choose F (r , ϑ) = r − f (ϑ), and setting Θ = 0 in thecase of time symmetry then yields a single nonlinear ODE boundary value problemof second order with singular boundaries (r ′ := dr/dϑ):

r ′′ =

(1 +

(r ′

r

)2)[

r 2

(qr + 4

ψr

ψ

)− r ′

(cotϑ+ qϑ + 4

ψϑψ

)+ 2r

]+

r ′2

r.

Solve ODE directly, or minimize Θ[F ] for some an ansatz for F (ρ, z).Minimal surfcaces (Kab = 0) and more general AHs are very smooth!


Black Holes

Find and analyse apparent and event horizons in 3D

Overview: J. Thornburg. Living Rev. Rel. 10:3, 2007

AH: Solve for the surface as an elliptic equation or a parabolic flow method:

∂λx i = −Θini .

Generalizations achieve faster convergence, e.g. Gundlach’s “fast flow”.

Performance: use spectral methods, but off-surface evaluations expensive.

Parabolic eq. may require small time steps in numerical evolution!

Flow methods are very robust w.r.t. choice of initial surface!

What is efficient in higher dimensions? Try minimization or flows first?

Location (existence) of AHs depends onthe choice of foliation!

Pathological foliations of a BH may nothave AHs!

EH is gauge-independent, the topologyof spatial cross-sections is not.


Black Holes

Find and analyse apparent and event horizons in 3D

Overview: J. Thornburg. Living Rev. Rel. 10:3, 2007

AH: Solve for the surface as an elliptic equation or a parabolic flow method:

∂λx i = −Θini .

Generalizations achieve faster convergence, e.g. Gundlach’s “fast flow”.

Performance: use spectral methods, but off-surface evaluations expensive.

Parabolic eq. may require small time steps in numerical evolution!

Flow methods are very robust w.r.t. choice of initial surface!

What is efficient in higher dimensions? Try minimization or flows first?

Location (existence) of AHs depends onthe choice of foliation!

Pathological foliations of a BH may nothave AHs!

EH is gauge-independent, the topologyof spatial cross-sections is not.


Black Holes

Modelling BHs numerically

The interior structure of a black hole is supposed to be very complicated – mayonly be interested in the exterior, actually!

Problems:

How to avoid simulating the interior structure of a BH?

Need hyperbolic analysis to check for pure outflow, unphysical characteristicscan create problems (what happens for constrained evolutions?).

Numerical dispersion may create outgoing modes with high frequency.

Ping-pong feedback is possible due to reflections at outer boundary orrefinement boundaries.

Waves that travel toward BH get blue-shifted.

When BHs travel across the grid, information is coming out of the BH!


Asymptotics

Boundary conditions for evolving EEs

Required on a noncompact manifold or one with boundaries:1 BCs should be consistent with the EEs – preserve constraints.

2 BCs should yield well-posed initial boundary value problem.

3 Implement the correct physics, e.g. outgoing waves.

Can typically not be achieved without modeling a global spacetime, e.g. nolocal notion of GWs, nonlinear backscattering of GWs.

Conditions (1+2+3) are easily achieved with symmetry BCs.

Conditions (1) and (2) have only first been met by Friedrich and Nagy, 1999.

“Practical” result for (1)+(2) Kreiss & Winicour 2006, more recent work.

In practice: often use causal isolation of the boundaries from region ofinterest, easy in 1+1 and 2+1 dimensions, harder in 3+1.

Modeling of global spacetime by compactification methods – open probleme.g. for BBH spacetimes.


Asymptotics












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Asymptotics

Isolated systems as models of sources of GW

Key concepts to describe “astrophysical” processes in GR: essential independenceof the large-scale structure of the universe, “radiation leaves system”.

→ physical idealization: isolated system – geometry “flattens at large distances”or approaches some cosmological background geometry.

GR: mass, momentum, emitted gravitational radiation can not be definedunambiguously in local/quasilocal way – only make sense in asymptotic limits.

Formalize as asymptotically flat or asymptotically de Sitter/AdS spacetimes

AF spacetimes usually used to model sources of GWs.

Beware: there are 3 directions towardinfinity: timelike / spacelike / null!

Compactification in these 3 directionsbehaves very differently.

Key idea: use conformal rescalings(Penrose).


Asymptotics

Conformal Compactification

Using conformal rescaling to an unphysical spacetime, we can discuss asymptoticsin terms of local differential geometry.

gab = Ω−2gab, M = p ∈M|Ω(p) > 0, “∞′′ = ∂M = p ∈M|Ω(p) = 0.Einstein’s vacuum equations in terms of Ω & gab:

Gab[Ω−2g ] = Gab[g ]− 2

Ω(∇a∇bΩ− gab∇c∇cΩ)− 3

Ω2gab (∇cΩ)∇cΩ.

singular for Ω = 0, multiplication by Ω2 → degenerate principal part @ Ω = 0.

asymptotically flat: null infinity is null.asymptotically de Sitter: null infinity is space-like.asymptotically anti-de Sitter: null infinity is time-like – boundary datarequired.

Different approaches to compactified equations:

Evolution along null surfaces – works well, but not general due to caustics.Compactification along spacelike directions: standard method to constructinitial data, underresolved blue-shifted waves in evolution.K = const. – spacelike “hyperboloidal” surfaces reaching null infinity(t2 − ~x2 = const. in Minkowski).Friedrich’s conformal field equations: completely regularize rescaled EEs byjudicious introduction of extra variables.current approach NR: first adapt gauge to null infinity, then regularize.


Asymptotics




Gab[Ω−2g ] = Gab[g ]− 2




asymptotically flat: null infinity is null.

asymptotically de Sitter: null infinity is space-like.asymptotically anti-de Sitter: null infinity is time-like – boundary datarequired.




Asymptotics




Gab[Ω−2g ] = Gab[g ]− 2




asymptotically flat: null infinity is null.asymptotically de Sitter: null infinity is space-like.

asymptotically anti-de Sitter: null infinity is time-like – boundary datarequired.




Asymptotics




Gab[Ω−2g ] = Gab[g ]− 2








Asymptotics




Gab[Ω−2g ] = Gab[g ]− 2








Asymptotics




Gab[Ω−2g ] = Gab[g ]− 2





Different approaches to compactified equations:Evolution along null surfaces – works well, but not general due to caustics.

Compactification along spacelike directions: standard method to constructinitial data, underresolved blue-shifted waves in evolution.K = const. – spacelike “hyperboloidal” surfaces reaching null infinity(t2 − ~x2 = const. in Minkowski).Friedrich’s conformal field equations: completely regularize rescaled EEs byjudicious introduction of extra variables.current approach NR: first adapt gauge to null infinity, then regularize.


Asymptotics




Gab[Ω−2g ] = Gab[g ]− 2





Different approaches to compactified equations:Evolution along null surfaces – works well, but not general due to caustics.Compactification along spacelike directions: standard method to constructinitial data, underresolved blue-shifted waves in evolution.

K = const. – spacelike “hyperboloidal” surfaces reaching null infinity(t2 − ~x2 = const. in Minkowski).Friedrich’s conformal field equations: completely regularize rescaled EEs byjudicious introduction of extra variables.current approach NR: first adapt gauge to null infinity, then regularize.


Asymptotics




Gab[Ω−2g ] = Gab[g ]− 2





Different approaches to compactified equations:Evolution along null surfaces – works well, but not general due to caustics.Compactification along spacelike directions: standard method to constructinitial data, underresolved blue-shifted waves in evolution.K = const. – spacelike “hyperboloidal” surfaces reaching null infinity(t2 − ~x2 = const. in Minkowski).

Friedrich’s conformal field equations: completely regularize rescaled EEs byjudicious introduction of extra variables.current approach NR: first adapt gauge to null infinity, then regularize.


Asymptotics




Gab[Ω−2g ] = Gab[g ]− 2





Different approaches to compactified equations:Evolution along null surfaces – works well, but not general due to caustics.Compactification along spacelike directions: standard method to constructinitial data, underresolved blue-shifted waves in evolution.K = const. – spacelike “hyperboloidal” surfaces reaching null infinity(t2 − ~x2 = const. in Minkowski).Friedrich’s conformal field equations: completely regularize rescaled EEs byjudicious introduction of extra variables.

current approach NR: first adapt gauge to null infinity, then regularize.


Asymptotics




Gab[Ω−2g ] = Gab[g ]− 2





Different approaches to compactified equations:Evolution along null surfaces – works well, but not general due to caustics.Compactification along spacelike directions: standard method to constructinitial data, underresolved blue-shifted waves in evolution.K = const. – spacelike “hyperboloidal” surfaces reaching null infinity(t2 − ~x2 = const. in Minkowski).Friedrich’s conformal field equations: completely regularize rescaled EEs byjudicious introduction of extra variables.current approach NR: first adapt gauge to null infinity, then regularize.


Asymptotics

Radiation “lives” at null infinity

Taking appropriate limit in M,worldlines of increasingly distantgeodesic observers converge to nullgeodesic generators of J + (propertime → Bondi time)!

Compactification at i0 leads to “pilingup” of waves, at J + this effect doesnot appear – waves leave the physicalspacetime through the boundary J +.

Observers situated at ”astronomical” distances are modeled @ J +.

E.g. computing the signal at a GW detector, J more realistically corresponds toan observer sufficiently far way from the source to treat the radiation linearly, butnot so far away that cosmological effects have to be taken into account.


Date post:	16-Aug-2018
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Introduction to Numerical Relativity I Intro &...

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