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The Penrose inequality for the perturbed Schwarzschild initial data

• J. Tafel

University of Warsaw

Jurekfest 2019 joint work with J. Kopi´ nski

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 1 / 14

The Penrose inequality

The surface area of the Kerr horizon |Sh| = 8πm(m +

• m2 − a2),

hence (the Penrose inequality) m ≥

• |Sh|

16π .

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 2 / 14

The Penrose inequality

The surface area of the Kerr horizon |Sh| = 8πm(m +

• m2 − a2),

hence (the Penrose inequality) m ≥

• |Sh|

16π . Stronger version E2 − p2 ≥ |Sh| 16π + 4πJ2 |Sh| , where E, p, J are, respectively, the total energy, momentum and angular momentum.

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 2 / 14

Consider the Cauchy data with a horizon. If a future singularity is surrounded by the event horizon (physically realistic data, the cosmic censorship conjecture) configuration tends to a stationary state then the no-hair theorem etc. (almost) imply that the end state is the Kerr metric with E∞ and S∞

h

E∞ ≤ E (because of radiation) |S∞

h | ≥ |Sh| (BH thermodynamics)

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 3 / 14

Consider the Cauchy data with a horizon. If a future singularity is surrounded by the event horizon (physically realistic data, the cosmic censorship conjecture) configuration tends to a stationary state then the no-hair theorem etc. (almost) imply that the end state is the Kerr metric with E∞ and S∞

h

E∞ ≤ E (because of radiation) |S∞

h | ≥ |Sh| (BH thermodynamics)

Conclusion: the Penrose inequality should be satisfied on the initial surface

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 3 / 14

Vacuum initial data with a horizon

Constraints on g′

ij, K ′ ij

∇i

• K ′i

j − H′δi j

• = 0

R′ + H′2 − K ′2 = 0 where H′ = K ′i

i and K ′2 = K ′ ijK ′ij and R′ is the Ricci scalar of g′ ij.

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 4 / 14

Vacuum initial data with a horizon

Constraints on g′

ij, K ′ ij

∇i

• K ′i

j − H′δi j

• = 0

R′ + H′2 − K ′2 = 0 where H′ = K ′i

i and K ′2 = K ′ ijK ′ij and R′ is the Ricci scalar of g′ ij.

Horizon: compact surface with vanishing expansion of outer null rays (marginal outer trapped surface - MOTS) H′ − K ′

nn + h = 0 ,

where ni is the unit normal vector and h = ∇ini is the mean curvature

• f the surface.
• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 4 / 14

Vacuum initial data with a horizon

Constraints on g′

ij, K ′ ij

∇i

• K ′i

j − H′δi j

• = 0

R′ + H′2 − K ′2 = 0 where H′ = K ′i

i and K ′2 = K ′ ijK ′ij and R′ is the Ricci scalar of g′ ij.

Horizon: compact surface with vanishing expansion of outer null rays (marginal outer trapped surface - MOTS) H′ − K ′

nn + h = 0 ,

where ni is the unit normal vector and h = ∇ini is the mean curvature

• f the surface.

If H′ = h = 0 then the Penrose inequality follows from the Hamiltonian constraint (Geroch, ...., Huisken and Ilmanen).

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 4 / 14

The conformal approach

g′

ij = ψ4gij,

K ′i

j = ψ−6Ai j + 1

3H′δi

j ,

where Ai

i = 0.

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 5 / 14

The conformal approach

g′

ij = ψ4gij,

K ′i

j = ψ−6Ai j + 1

3H′δi

j ,

where Ai

i = 0.

The momentum constraint ∇iAi

j = 2

3ψ6∇jH′ .

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 5 / 14

The conformal approach

g′

ij = ψ4gij,

K ′i

j = ψ−6Ai j + 1

3H′δi

j ,

where Ai

i = 0.

The momentum constraint ∇iAi

j = 2

3ψ6∇jH′ . The Hamiltonian constraint (the Lichnerowicz equation) △ψ = 1 8Rψ − 1 8AijAijψ−7 + 1 12H′2ψ5 .

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 5 / 14

The conformal approach

g′

ij = ψ4gij,

K ′i

j = ψ−6Ai j + 1

3H′δi

j ,

where Ai

i = 0.

The momentum constraint ∇iAi

j = 2

3ψ6∇jH′ . The Hamiltonian constraint (the Lichnerowicz equation) △ψ = 1 8Rψ − 1 8AijAijψ−7 + 1 12H′2ψ5 . The MOTS condition ni∂iψ + 1 2hψ − 1 4Annψ−3 + 1 6H′ψ3 = 0 on Sh .

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 5 / 14

The conformal approach

g′

ij = ψ4gij,

K ′i

j = ψ−6Ai j + 1

3H′δi

j ,

where Ai

i = 0.

The momentum constraint ∇iAi

j = 2

3ψ6∇jH′ . The Hamiltonian constraint (the Lichnerowicz equation) △ψ = 1 8Rψ − 1 8AijAijψ−7 + 1 12H′2ψ5 . The MOTS condition ni∂iψ + 1 2hψ − 1 4Annψ−3 + 1 6H′ψ3 = 0 on Sh . Existence theorems under extra conditions (D. Maxwell, Anglada).

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 5 / 14

Approximate constraint equations

Assumption: the preliminary metric gij is flat and Sh is the sphere with radius m/2.

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 6 / 14

Approximate constraint equations

Assumption: the preliminary metric gij is flat and Sh is the sphere with radius m/2. If K ′

ij = 0 then

ψ = ψ0 = 1 + m 2r and the conformal transformation leads to the Schwarzschild initial metric on t = const g′ = dr ′2 1 − 2m

r ′

+ r ′2(dθ2 + sin2 θdϕ2) The Penrose inequality is saturated.

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 6 / 14

Approximate constraint equations

Assumption: the preliminary metric gij is flat and Sh is the sphere with radius m/2. If K ′

ij = 0 then

ψ = ψ0 = 1 + m 2r and the conformal transformation leads to the Schwarzschild initial metric on t = const g′ = dr ′2 1 − 2m

r ′

+ r ′2(dθ2 + sin2 θdϕ2) The Penrose inequality is saturated. Let ǫ be a small parameter and ψ = ψ0 + ψ1 + ψ2 + ... Aij = Bij + ... H′ = B + ... where ψn ∼ ǫn and Bij, B ∼ ǫ.

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 6 / 14

The approximate constraints ∇iBi

j = 2

3ψ6

0∇jB

△ψ = −1 8BijBijψ−7 + 1 12B2ψ5

0 .

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 7 / 14

The approximate constraints ∇iBi

j = 2

3ψ6

0∇jB

△ψ = −1 8BijBijψ−7 + 1 12B2ψ5

0 .

Remarks: The momentum constraint is a linear underdetermined system for Bij and B.

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 7 / 14

The approximate constraints ∇iBi

j = 2

3ψ6

0∇jB

△ψ = −1 8BijBijψ−7 + 1 12B2ψ5

0 .

Remarks: The momentum constraint is a linear underdetermined system for Bij and B. A relation between total energy and area of horizon should follow from the Lichnerowicz equation.

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 7 / 14

The approximate constraints ∇iBi

j = 2

3ψ6

0∇jB

△ψ = −1 8BijBijψ−7 + 1 12B2ψ5

0 .

Remarks: The momentum constraint is a linear underdetermined system for Bij and B. A relation between total energy and area of horizon should follow from the Lichnerowicz equation. ψ1 is harmonic and ψ2 satisfies the Poisson equation with the Neumann type boundary condition.

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 7 / 14

Integration of the Lichnerowicz equation

• S∞

ψ,rdσ =

• Sh

ψ,rdσ + S∞

Sh

(−1 8BijBijψ−7 + 1 12B2ψ5

0)d3x

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 8 / 14

Integration of the Lichnerowicz equation

• S∞

ψ,rdσ =

• Sh

ψ,rdσ + S∞

Sh

(−1 8BijBijψ−7 + 1 12B2ψ5

0)d3x

The ADM total energy E = − 1 2π

• S∞

∂rψdσ.

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 8 / 14

Integration of the Lichnerowicz equation

• S∞

ψ,rdσ =

• Sh

ψ,rdσ + S∞

Sh

(−1 8BijBijψ−7 + 1 12B2ψ5

0)d3x

The ADM total energy E = − 1 2π

• S∞

∂rψdσ. The ultimate surface area of Sh |Sh| =

• Sh

ψ4dσ = 16

• Sh

(1 + 2ψ1 + 2ψ2 + 3 2ψ2

1)dσ .

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 8 / 14

Integration of the Lichnerowicz equation

• S∞

ψ,rdσ =

• Sh

ψ,rdσ + S∞

Sh

(−1 8BijBijψ−7 + 1 12B2ψ5

0)d3x

The ADM total energy E = − 1 2π

• S∞

∂rψdσ. The ultimate surface area of Sh |Sh| =

• Sh

ψ4dσ = 16

• Sh

(1 + 2ψ1 + 2ψ2 + 3 2ψ2

1)dσ .

Integration of the Lichnerowicz equation over spherical coordinates 1 r 2 ∂r(r 2∂rψ) = −1 8BijBijψ−7 + 1 12B2ψ5 where f = f sin θdθdφ. An expression for |Sh| follows.

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 8 / 14

An approximate formula for PI = E2 − |Sh|

16π

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 9 / 14

An approximate formula for PI = E2 − |Sh|

16π

PI = m2 16π2 1 32Brr − 4 3B2

h − 3m2

8π ψ2

1h

+m π ∞

m 2

r 2(1 − m 2r )1 8BijBijψ−7 − 1 12B2ψ5

0dr ,

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 9 / 14

An approximate formula for PI = E2 − |Sh|

16π

PI = m2 16π2 1 32Brr − 4 3B2

h − 3m2

8π ψ2

1h

+m π ∞

m 2

r 2(1 − m 2r )1 8BijBijψ−7 − 1 12B2ψ5

0dr ,

where △ψ1 = 0 and ∂rψ1 + 1 mψ1 = 1 32Brr − 4 3B on Sh .

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 9 / 14

An approximate formula for PI = E2 − |Sh|

16π

PI = m2 16π2 1 32Brr − 4 3B2

h − 3m2

8π ψ2

1h

+m π ∞

m 2

r 2(1 − m 2r )1 8BijBijψ−7 − 1 12B2ψ5

0dr ,

where △ψ1 = 0 and ∂rψ1 + 1 mψ1 = 1 32Brr − 4 3B on Sh . Arbitrary sign of PI for unrestricted fields Bij, B.

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 9 / 14

Axially symmetric perturbations

One of the momentum constraints: Bϕθ = ω,r sin θ , Bϕr = − ω,θ r 2 sin θ .

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 10 / 14

Axially symmetric perturbations

One of the momentum constraints: Bϕθ = ω,r sin θ , Bϕr = − ω,θ r 2 sin θ . Contribution to PI PJ = 2 ∞

m 2

dr̺r 1

−1

(B2

ϕθ + r 2B2 ϕr)

dz 1 − z2 , where z = cos θ and ̺ = m(1 − m

2r )

4r 3ψ7 .

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 10 / 14

Axially symmetric perturbations

One of the momentum constraints: Bϕθ = ω,r sin θ , Bϕr = − ω,θ r 2 sin θ . Contribution to PI PJ = 2 ∞

m 2

dr̺r 1

−1

(B2

ϕθ + r 2B2 ϕr)

dz 1 − z2 , where z = cos θ and ̺ = m(1 − m

2r )

4r 3ψ7 . From regularity conditions ω = f sin4 θ + J(cos3 θ − 3 cos θ) , J = ang.mom.

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 10 / 14

Axially symmetric perturbations

One of the momentum constraints: Bϕθ = ω,r sin θ , Bϕr = − ω,θ r 2 sin θ . Contribution to PI PJ = 2 ∞

m 2

dr̺r 1

−1

(B2

ϕθ + r 2B2 ϕr)

dz 1 − z2 , where z = cos θ and ̺ = m(1 − m

2r )

4r 3ψ7 . From regularity conditions ω = f sin4 θ + J(cos3 θ − 3 cos θ) , J = ang.mom. Simple consequence PJ ≥ J2 4m2 ⇒ PJ ≥ 4π |Sh|J2 .

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 10 / 14

Other 2 momentum constraints (for 4 functions) (r 3Brr sin θ),r + (rBrθ sin θ),θ = 2 3r 3ψ6

0B,r sin θ ,

(Bθθ sin2 θ),θ + (r 2Brθ),r sin2 θ + r 2Brr sin θ cos θ = 2 3r 2ψ6

0B,θ sin2 θ .

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 11 / 14

Other 2 momentum constraints (for 4 functions) (r 3Brr sin θ),r + (rBrθ sin θ),θ = 2 3r 3ψ6

0B,r sin θ ,

(Bθθ sin2 θ),θ + (r 2Brθ),r sin2 θ + r 2Brr sin θ cos θ = 2 3r 2ψ6

0B,θ sin2 θ .

Contribution to the volume integral in PI ˜ PI = ∞

m 2

dr̺r 1

−1

[2r 2B2

rθ + 2(Bθθ + 1

2r 2Brr)2 + 3 2r 4B2

rr − 2

3r 4ψ12

0 B2]dz

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 11 / 14

Other 2 momentum constraints (for 4 functions) (r 3Brr sin θ),r + (rBrθ sin θ),θ = 2 3r 3ψ6

0B,r sin θ ,

(Bθθ sin2 θ),θ + (r 2Brθ),r sin2 θ + r 2Brr sin θ cos θ = 2 3r 2ψ6

0B,θ sin2 θ .

Contribution to the volume integral in PI ˜ PI = ∞

m 2

dr̺r 1

−1

[2r 2B2

rθ + 2(Bθθ + 1

2r 2Brr)2 + 3 2r 4B2

rr − 2

3r 4ψ12

0 B2]dz

= m 2 ∞

m 2

dr r 2ψ0 1

−1

• (1 − m

2r )W 2 − X △s Y

• dz + 1

4 1

−1

X 2

h dz ,

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 11 / 14

Other 2 momentum constraints (for 4 functions) (r 3Brr sin θ),r + (rBrθ sin θ),θ = 2 3r 3ψ6

0B,r sin θ ,

(Bθθ sin2 θ),θ + (r 2Brθ),r sin2 θ + r 2Brr sin θ cos θ = 2 3r 2ψ6

0B,θ sin2 θ .

Contribution to the volume integral in PI ˜ PI = ∞

m 2

dr̺r 1

−1

[2r 2B2

rθ + 2(Bθθ + 1

2r 2Brr)2 + 3 2r 4B2

rr − 2

3r 4ψ12

0 B2]dz

= m 2 ∞

m 2

dr r 2ψ0 1

−1

• (1 − m

2r )W 2 − X △s Y

• dz + 1

4 1

−1

X 2

h dz ,

where ∂z

• (1 − z2)W
• =

1 1 − m

2r

(z2 − 1)∂z(rX,r + 1 2ψ0 △s Y + (ψ0 − 3m rψ0 )Y) .

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 11 / 14

Expansions into the Legendre polynomials Pn

X = ΣXnPn , Y = ΣYnPn .

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 12 / 14

Expansions into the Legendre polynomials Pn

X = ΣXnPn , Y = ΣYnPn . Then W = ΣWn(Xn, Yn)˜ Pn , n ≥ 2 , where polynomials ˜ Pn = 1 n + 2(nPn − 2 n − 1Pn−1,z) are orthogonal.

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 12 / 14

Expansions into the Legendre polynomials Pn

X = ΣXnPn , Y = ΣYnPn . Then W = ΣWn(Xn, Yn)˜ Pn , n ≥ 2 , where polynomials ˜ Pn = 1 n + 2(nPn − 2 n − 1Pn−1,z) are orthogonal. Hence ˜ PI = Σ ∞

m 2

Fn(Xn, Yn)dr .

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 12 / 14

Expansions into the Legendre polynomials Pn

X = ΣXnPn , Y = ΣYnPn . Then W = ΣWn(Xn, Yn)˜ Pn , n ≥ 2 , where polynomials ˜ Pn = 1 n + 2(nPn − 2 n − 1Pn−1,z) are orthogonal. Hence ˜ PI = Σ ∞

m 2

Fn(Xn, Yn)dr . Each Fn has global minimum at Yn = αnXn,r + βnXn.

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 12 / 14

Crucial estimation

˜ PI ≥ p2 + 1 2Σ X 2

nh

(2n + 1)(n2 + n + 1) Here Xnh are multipole moments of the boundary term ( 1

32Brr − 4 3B)h.

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 13 / 14

Crucial estimation

˜ PI ≥ p2 + 1 2Σ X 2

nh

(2n + 1)(n2 + n + 1) Here Xnh are multipole moments of the boundary term ( 1

32Brr − 4 3B)h.

Taking into account inequalities for PJ, ˜ PI and boundary terms in PI yields E2 − |Sh| 16π ≥ 4π |Sh|J2 + p2 + Σ∞

2

(n − 1)(n + 2)X 2

nh

2(2n + 1)3(n2 + n + 1) .

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 13 / 14

Crucial estimation

˜ PI ≥ p2 + 1 2Σ X 2

nh

(2n + 1)(n2 + n + 1) Here Xnh are multipole moments of the boundary term ( 1

32Brr − 4 3B)h.

Taking into account inequalities for PJ, ˜ PI and boundary terms in PI yields E2 − |Sh| 16π ≥ 4π |Sh|J2 + p2 + Σ∞

2

(n − 1)(n + 2)X 2

nh

2(2n + 1)3(n2 + n + 1) . If any of Xnh with n ≥ 2 is different from zero (generic case) then the Penrose inequality is satisfied.

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 13 / 14

Crucial estimation

˜ PI ≥ p2 + 1 2Σ X 2

nh

(2n + 1)(n2 + n + 1) Here Xnh are multipole moments of the boundary term ( 1

32Brr − 4 3B)h.

Taking into account inequalities for PJ, ˜ PI and boundary terms in PI yields E2 − |Sh| 16π ≥ 4π |Sh|J2 + p2 + Σ∞

2

(n − 1)(n + 2)X 2

nh

2(2n + 1)3(n2 + n + 1) . If any of Xnh with n ≥ 2 is different from zero (generic case) then the Penrose inequality is satisfied. If Xn≥2 = 0 (nongeneric case) then higher order terms decide about validitity of the Penrose inequality.

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 13 / 14

Summary

Assumptions: axially symmetric conformally flat initial data with MOTS which corresponds to the sphere in the flat space initial exterior curvature is proportional to a small parameter ǫ and all fields can be expanded in ǫ

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 14 / 14

Summary

Assumptions: axially symmetric conformally flat initial data with MOTS which corresponds to the sphere in the flat space initial exterior curvature is proportional to a small parameter ǫ and all fields can be expanded in ǫ Results: the Penrose inequality E2 − p2 ≥ |Sh|

16π + 4πJ2 |Sh| is satisfied up to ǫ2 in

generic case for maximal data the Penrose inequality is satisfied up to ǫ4 in nongeneric case

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 14 / 14

Summary

Assumptions: axially symmetric conformally flat initial data with MOTS which corresponds to the sphere in the flat space initial exterior curvature is proportional to a small parameter ǫ and all fields can be expanded in ǫ Results: the Penrose inequality E2 − p2 ≥ |Sh|

16π + 4πJ2 |Sh| is satisfied up to ǫ2 in

generic case for maximal data the Penrose inequality is satisfied up to ǫ4 in nongeneric case Perspectives: dependence on φ, nonspherical MOTS, ...

• J. Tafel (University of Warsaw)

The Penrose inequality for the perturbed Schwarzschild initial data Jurekfest 2019 joint work with J. Kopi´ nski 14 / 14