Gravitational Lensing of Stationary Spacetimes:A Finsler Geometry Approach

Nishanth Gudapati

Center of Mathematical Sciences and Applications,Harvard University

BHI Colloquium,December 04, 2018

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Outline of the Talk

I Basis of Finsler GeometryI Structures on Finsler manifoldsI Optical GeometryI Gravitational Lensing of Static and Stationary SpacetimesI Joint with Marcus Werner (Kyoto University)

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Outline of the Talk

I Basis of Finsler GeometryI Structures on Finsler manifoldsI Optical GeometryI Gravitational Lensing of Static and Stationary SpacetimesI Joint with Marcus Werner (Kyoto University)

2 / 15

Outline of the Talk

I Basis of Finsler GeometryI Structures on Finsler manifoldsI Optical GeometryI Gravitational Lensing of Static and Stationary SpacetimesI Joint with Marcus Werner (Kyoto University)

2 / 15

Outline of the Talk

I Basis of Finsler GeometryI Structures on Finsler manifoldsI Optical GeometryI Gravitational Lensing of Static and Stationary SpacetimesI Joint with Marcus Werner (Kyoto University)

2 / 15

Outline of the Talk

I Basis of Finsler GeometryI Structures on Finsler manifoldsI Optical GeometryI Gravitational Lensing of Static and Stationary SpacetimesI Joint with Marcus Werner (Kyoto University)

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‘Riemannian’ Geometry

Recall that in Riemannian geometry, a fundamental role is played by thefollowing (scalar) structure on the tangent bundle of a (smooth) manifoldM

F = F (x1, x2, · · · ; dx1, dx2, · · · dxn) (1)

1. F is the ‘arc length’; positive2. Homogeneous of degree 13. F 2 gives the metric tensor

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‘Riemannian’ Geometry

Recall that in Riemannian geometry, a fundamental role is played by thefollowing (scalar) structure on the tangent bundle of a (smooth) manifoldM

F = F (x1, x2, · · · ; dx1, dx2, · · · dxn) (1)

1. F is the ‘arc length’; positive2. Homogeneous of degree 13. F 2 gives the metric tensor

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Finsler Geometry

We can extend this construction. We define that a pair (M,F ) is a Finslermanifold if F : TM → [0,∞) is such that

Definition1. F (x , y), x ∈ M, y ∈ Tx M is non-negative (0 iff y = 0); smooth in

TM \ 02. F is homogeneous of degree 1 i.e., F (x , λy) = λF (x , y), λ > 03. The Hessian of F is convex i.e., 1

2(F 2)y i y j is positive-definite ∀ TM \ 0

Examples:1. Randers spaces F =

√aijy i y j + bi y i

2. Minkowski spaces F (y) (independent of x)3. Riemannian geometry

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Finsler Geometry

We can extend this construction. We define that a pair (M,F ) is a Finslermanifold if F : TM → [0,∞) is such that

Definition1. F (x , y), x ∈ M, y ∈ Tx M is non-negative (0 iff y = 0); smooth in

TM \ 02. F is homogeneous of degree 1 i.e., F (x , λy) = λF (x , y), λ > 03. The Hessian of F is convex i.e., 1

2(F 2)y i y j is positive-definite ∀ TM \ 0

Examples:1. Randers spaces F =

√aijy i y j + bi y i

2. Minkowski spaces F (y) (independent of x)3. Riemannian geometry

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Finsler Geometry

We can extend this construction. We define that a pair (M,F ) is a Finslermanifold if F : TM → [0,∞) is such that

Definition1. F (x , y), x ∈ M, y ∈ Tx M is non-negative (0 iff y = 0); smooth in

TM \ 02. F is homogeneous of degree 1 i.e., F (x , λy) = λF (x , y), λ > 03. The Hessian of F is convex i.e., 1

2(F 2)y i y j is positive-definite ∀ TM \ 0

Examples:1. Randers spaces F =

√aijy i y j + bi y i

2. Minkowski spaces F (y) (independent of x)3. Riemannian geometry

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Structures on Finsler Geometry IFundamental TensorThe fundamental tensor gij is defined as gij = 1

2(F 2)y i y j on TM \ 0. Thistensor has some analogy with the metric tensor as discussed below.Lemma

gij(y)y i y j = F 2(y); gijy i

Fy j

F = 1

Proof.

gij = FFy i y j + Fy i Fy j . Thus,

gijy i y j = FFy i y j y i y j + y i Fy i y jFy j = F 2 (2)

where we have used the Euler’s theorem for homogeneous functions:

y i Fy i = F

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Structures on Finsler Geometry IFundamental TensorThe fundamental tensor gij is defined as gij = 1

2(F 2)y i y j on TM \ 0. Thistensor has some analogy with the metric tensor as discussed below.Lemma

gij(y)y i y j = F 2(y); gijy i

Fy j

F = 1

Proof.

gij = FFy i y j + Fy i Fy j . Thus,

gijy i y j = FFy i y j y i y j + y i Fy i y jFy j = F 2 (2)

where we have used the Euler’s theorem for homogeneous functions:

y i Fy i = F

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Structures on Finsler Geometry IINotions of ‘Angle’, ‘Connection’ and ‘Curvature’

I Landsberg angles.Indicatrix: Σx := {y ∈ Tx M |F (x , y) = 1}Landsberg angle for a surface:

dθ :=√gF 2 (y1dy2 − y2dy1)

and

L :=∫S1

√gF 2 (y1dy2 − y2dy1) (3)

I There are various notions of connection in Finsler geometry due toBerwald, Rund, Cartan, Hashiguchi and Chern. We shall use theChern connection

Aijk :=12F ∂yk gij , (Cartan tensor) (4)

∇i v j =∂i v j + Γjikvk (5)

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Structures on Finsler Geometry IINotions of ‘Angle’, ‘Connection’ and ‘Curvature’

I Landsberg angles.Indicatrix: Σx := {y ∈ Tx M |F (x , y) = 1}Landsberg angle for a surface:

dθ :=√gF 2 (y1dy2 − y2dy1)

and

L :=∫S1

√gF 2 (y1dy2 − y2dy1) (3)

I There are various notions of connection in Finsler geometry due toBerwald, Rund, Cartan, Hashiguchi and Chern. We shall use theChern connection

Aijk :=12F ∂yk gij , (Cartan tensor) (4)

∇i v j =∂i v j + Γjikvk (5)

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‘Gauss-Bonnet’ Theorem for Finsler geometry

A standard reference for Finsler geometry is the book by Bao-Chern-Shen(2000)

Theorem (Chern (1990); Bao-Chern(1996))If (M,F ) is a compact (w/o boundary), connected Landsberg surface, then

1L

∫M

K√gdx1 ∧ dx2 = χ(M) (6)

where K is the ‘Gauss curvature’ of (M,F ) and χ is theEuler-Characteristic of M.

(more later)

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Optical Reference Geometry IAbramowicz-Carter-Lasota (1988) and the static caseNow let us discuss an application of the previously discussed constructions.This is in the field of optical geometry. As a special case, consider theSchwarzschild black hole spacetime

g = −Φdt2 + Φ−1dr2 + r2dω2 (7)

then one can define a projection π : (M, g)→ (Mo, go) to a (Riemannian)optical metric (Mo, go) such that

go := Φ−2dr2 + Φ−1r2dω2 (8)

which can in turn be represented in a conformally flat form as

go = σ2(dr2 + r2dω2) (9)

where σ = (1 + M2r )3(1− M

2r )−1.

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Optical Reference Geometry IAbramowicz-Carter-Lasota (1988) and the static caseNow let us discuss an application of the previously discussed constructions.This is in the field of optical geometry. As a special case, consider theSchwarzschild black hole spacetime

g = −Φdt2 + Φ−1dr2 + r2dω2 (7)

then one can define a projection π : (M, g)→ (Mo, go) to a (Riemannian)optical metric (Mo, go) such that

go := Φ−2dr2 + Φ−1r2dω2 (8)

which can in turn be represented in a conformally flat form as

go = σ2(dr2 + r2dω2) (9)

where σ = (1 + M2r )3(1− M

2r )−1.

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Optical Reference Geometry IIUseful properties

I The crucial advantage of the optical metric is that the null geodesicsof the physical metric (M, g) correspond to the (Riemannian)geodesics of (Mo, go) (Fermat’s principle). Supposeγ : [a, b]→ (M, g)

Dγ γ = 0, g(γ, γ) = 0, =⇒ ∇γ|π γ|π = 0, (V .Perlick, 1990)

I Consider the optical metric at the equatorial plane θ = π/2, (Mo, go)has a ‘throat’ at the trapped photon sphere:

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Optical Reference Geometry IIUseful properties

I The crucial advantage of the optical metric is that the null geodesicsof the physical metric (M, g) correspond to the (Riemannian)geodesics of (Mo, go) (Fermat’s principle). Supposeγ : [a, b]→ (M, g)

Dγ γ = 0, g(γ, γ) = 0, =⇒ ∇γ|π γ|π = 0, (V .Perlick, 1990)

I Consider the optical metric at the equatorial plane θ = π/2, (Mo, go)has a ‘throat’ at the trapped photon sphere:

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Lensing and Angle of Deflection IGibbons-Werner (2008)

Gibbons-Werner have introduced a geometric method, based on theGauss-Bonnet theorem, to establish a relation between the angle ofdeflection (lensing) and curvature using geometric and topologicalproperties of the optical metric (Mo, go). Gauss-Bonnet for the(Riemannian) optical geometry∫

DKdS +

∫∂Dκdt +

∑iαi = 2πχ(D)

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Lensing and Angle of Deflection IIGibbons-Werner(2008)

∫AL

K +∫∂AL

Kγ + 2π = θs + θo (10)

I The inner boundary curve is taken to be the photon sphere, thusKγ = go(∇γ γ, γ) = 0 at the inner boundary

I The Gauss curvature of (Mo, go) at θ = π/2 is K = −2mr3 (1− 3m

2r )I Non-trivial topology χ = 1− g = 0 necessary for image multiplicityI In the other case with asymptotic limit:∫

A∞KdS +

∫ π

−α

1R Rdφ+ π

2 · 2 = 2π

gives the relation

α = −∫

A∞KdS = 4m

b +O(m2) (11)

for Schwarzschild11 / 15

Lensing and Angle of Deflection IIGibbons-Werner(2008)

∫AL

K +∫∂AL

Kγ + 2π = θs + θo (10)

I The inner boundary curve is taken to be the photon sphere, thusKγ = go(∇γ γ, γ) = 0 at the inner boundary

I The Gauss curvature of (Mo, go) at θ = π/2 is K = −2mr3 (1− 3m

2r )I Non-trivial topology χ = 1− g = 0 necessary for image multiplicityI In the other case with asymptotic limit:∫

A∞KdS +

∫ π

−α

1R Rdφ+ π

2 · 2 = 2π

gives the relation

α = −∫

A∞KdS = 4m

b +O(m2) (11)

for Schwarzschild11 / 15

Lensing and Angle of Deflection IIGibbons-Werner(2008)

∫AL

K +∫∂AL

Kγ + 2π = θs + θo (10)

I The inner boundary curve is taken to be the photon sphere, thusKγ = go(∇γ γ, γ) = 0 at the inner boundary

I The Gauss curvature of (Mo, go) at θ = π/2 is K = −2mr3 (1− 3m

2r )I Non-trivial topology χ = 1− g = 0 necessary for image multiplicityI In the other case with asymptotic limit:∫

A∞KdS +

∫ π

−α

1R Rdφ+ π

2 · 2 = 2π

gives the relation

α = −∫

A∞KdS = 4m

b +O(m2) (11)

for Schwarzschild11 / 15

Lensing and Angle of Deflection IIGibbons-Werner(2008)

∫AL

K +∫∂AL

Kγ + 2π = θs + θo (10)

I The inner boundary curve is taken to be the photon sphere, thusKγ = go(∇γ γ, γ) = 0 at the inner boundary

I The Gauss curvature of (Mo, go) at θ = π/2 is K = −2mr3 (1− 3m

2r )I Non-trivial topology χ = 1− g = 0 necessary for image multiplicityI In the other case with asymptotic limit:∫

A∞KdS +

∫ π

−α

1R Rdφ+ π

2 · 2 = 2π

gives the relation

α = −∫

A∞KdS = 4m

b +O(m2) (11)

for Schwarzschild11 / 15

Stationary SpacetimesI (M, g) is stationary if M = Σ× R, Σ is asymptotically flat and ∃ a

Killing isometry T i.e., LT g ≡ 0 such that T is ‘asymptotically’timelike.

I Steady state solutions of black holesI

g = X−1q + X (dt + Ai dx i )2 (Weyl) (12)

I (Q) What about the lensing of stationary spacetimes and rotatingblack holes?

I Werner(2012), Jusufi-Werner et. al. (2017), lensing for Kerrspacetimes osculating Riemannian geometry

I Ono, Ishihara and Asada (2017, 18) - the physical null geodesics aretreated as non-geodesic curves in the spatial metric (not opticalmetric)

I Our aim is to realize lensing for stationary spacetimes in Finslergeometry and topoogy.

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Stationary SpacetimesI (M, g) is stationary if M = Σ× R, Σ is asymptotically flat and ∃ a

Killing isometry T i.e., LT g ≡ 0 such that T is ‘asymptotically’timelike.

I Steady state solutions of black holesI

g = X−1q + X (dt + Ai dx i )2 (Weyl) (12)

I (Q) What about the lensing of stationary spacetimes and rotatingblack holes?

I Werner(2012), Jusufi-Werner et. al. (2017), lensing for Kerrspacetimes osculating Riemannian geometry

I Ono, Ishihara and Asada (2017, 18) - the physical null geodesics aretreated as non-geodesic curves in the spatial metric (not opticalmetric)

I Our aim is to realize lensing for stationary spacetimes in Finslergeometry and topoogy.

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Stationary SpacetimesI (M, g) is stationary if M = Σ× R, Σ is asymptotically flat and ∃ a

Killing isometry T i.e., LT g ≡ 0 such that T is ‘asymptotically’timelike.

I Steady state solutions of black holesI

g = X−1q + X (dt + Ai dx i )2 (Weyl) (12)

I (Q) What about the lensing of stationary spacetimes and rotatingblack holes?

I Werner(2012), Jusufi-Werner et. al. (2017), lensing for Kerrspacetimes osculating Riemannian geometry

I Ono, Ishihara and Asada (2017, 18) - the physical null geodesics aretreated as non-geodesic curves in the spatial metric (not opticalmetric)

I Our aim is to realize lensing for stationary spacetimes in Finslergeometry and topoogy.

12 / 15

Stationary SpacetimesI (M, g) is stationary if M = Σ× R, Σ is asymptotically flat and ∃ a

Killing isometry T i.e., LT g ≡ 0 such that T is ‘asymptotically’timelike.

I Steady state solutions of black holesI

g = X−1q + X (dt + Ai dx i )2 (Weyl) (12)

I (Q) What about the lensing of stationary spacetimes and rotatingblack holes?

I Werner(2012), Jusufi-Werner et. al. (2017), lensing for Kerrspacetimes osculating Riemannian geometry

I Ono, Ishihara and Asada (2017, 18) - the physical null geodesics aretreated as non-geodesic curves in the spatial metric (not opticalmetric)

I Our aim is to realize lensing for stationary spacetimes in Finslergeometry and topoogy.

12 / 15

Stationary SpacetimesI (M, g) is stationary if M = Σ× R, Σ is asymptotically flat and ∃ a

Killing isometry T i.e., LT g ≡ 0 such that T is ‘asymptotically’timelike.

I Steady state solutions of black holesI

g = X−1q + X (dt + Ai dx i )2 (Weyl) (12)

I (Q) What about the lensing of stationary spacetimes and rotatingblack holes?

I Werner(2012), Jusufi-Werner et. al. (2017), lensing for Kerrspacetimes osculating Riemannian geometry

I Ono, Ishihara and Asada (2017, 18) - the physical null geodesics aretreated as non-geodesic curves in the spatial metric (not opticalmetric)

I Our aim is to realize lensing for stationary spacetimes in Finslergeometry and topoogy.

12 / 15

Stationary SpacetimesI (M, g) is stationary if M = Σ× R, Σ is asymptotically flat and ∃ a

Killing isometry T i.e., LT g ≡ 0 such that T is ‘asymptotically’timelike.

I Steady state solutions of black holesI

g = X−1q + X (dt + Ai dx i )2 (Weyl) (12)

I (Q) What about the lensing of stationary spacetimes and rotatingblack holes?

I Werner(2012), Jusufi-Werner et. al. (2017), lensing for Kerrspacetimes osculating Riemannian geometry

I Ono, Ishihara and Asada (2017, 18) - the physical null geodesics aretreated as non-geodesic curves in the spatial metric (not opticalmetric)

I Our aim is to realize lensing for stationary spacetimes in Finslergeometry and topoogy.

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Finsler-Randers optical metricKerr-Randers optical metric due to Gibbons-Herdeiro-Warnick-Werner(2009)

aijy i y j =Σ2

H

(dr2

∆ + ∆ sin2 θ

H dφ2)

(13)

b =− 2mra sin2 θ

H dφ (14)

on the equatorial plane and outside the ergo region (|a| < M).

LemmaLet (M, g) be an asymptotically-flat stationary Lorentzian spacetime,there there exists a projection π : (M, g)→ (M, g) to optical metric ofFinsler-Randers type outside the ergo-region (if it exists) and isasymptotically Riemannian.

- Critical quantity is b: the condition |b| < 1 accounts for positivity andconvexity.

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Finsler-Randers optical metricKerr-Randers optical metric due to Gibbons-Herdeiro-Warnick-Werner(2009)

aijy i y j =Σ2

H

(dr2

∆ + ∆ sin2 θ

H dφ2)

(13)

b =− 2mra sin2 θ

H dφ (14)

on the equatorial plane and outside the ergo region (|a| < M).

LemmaLet (M, g) be an asymptotically-flat stationary Lorentzian spacetime,there there exists a projection π : (M, g)→ (M, g) to optical metric ofFinsler-Randers type outside the ergo-region (if it exists) and isasymptotically Riemannian.

- Critical quantity is b: the condition |b| < 1 accounts for positivity andconvexity.

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Ongoing work...I Finslerian Gauss-Bonnet: Itoh-Sabau-Shimada (2010)∫

D

1LK√gdx1 ∧ dx2 +

∫∂D

1L

K Nγ

σ(t)dt +n∑

i=1

1Lαi = χ(D) (15)

(simplified case presented here; extends to the lensing context,piecewise (C∞) boundary)

I In Isothermal coordinates: aijy i y j = e2νδijy i y j

I The ‘Gauss-Curvature’ K in orthonormal frameI T-parallel vs N-parallel. Consider the geodesic equation (critical point

of Finslerian arc-length)

d2

dt2σi + Γi

jkddt σ

j ddt σ

k = ddt (log F (σ(t))) d

dt σi (16)

which corresponds only to the T−auto parallel transport and not N.Contrast with Riemannian geometry! Sabau-Shibuya (2015; Griffithsformulation)

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Ongoing work...I Finslerian Gauss-Bonnet: Itoh-Sabau-Shimada (2010)∫

D

1LK√gdx1 ∧ dx2 +

∫∂D

1L

K Nγ

σ(t)dt +n∑

i=1

1Lαi = χ(D) (15)

(simplified case presented here; extends to the lensing context,piecewise (C∞) boundary)

I In Isothermal coordinates: aijy i y j = e2νδijy i y j

I The ‘Gauss-Curvature’ K in orthonormal frameI T-parallel vs N-parallel. Consider the geodesic equation (critical point

of Finslerian arc-length)

d2

dt2σi + Γi

jkddt σ

j ddt σ

k = ddt (log F (σ(t))) d

dt σi (16)

which corresponds only to the T−auto parallel transport and not N.Contrast with Riemannian geometry! Sabau-Shibuya (2015; Griffithsformulation)

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Ongoing work...I Finslerian Gauss-Bonnet: Itoh-Sabau-Shimada (2010)∫

D

1LK√gdx1 ∧ dx2 +

∫∂D

1L

K Nγ

σ(t)dt +n∑

i=1

1Lαi = χ(D) (15)

(simplified case presented here; extends to the lensing context,piecewise (C∞) boundary)

I In Isothermal coordinates: aijy i y j = e2νδijy i y j

I The ‘Gauss-Curvature’ K in orthonormal frameI T-parallel vs N-parallel. Consider the geodesic equation (critical point

of Finslerian arc-length)

d2

dt2σi + Γi

jkddt σ

j ddt σ

k = ddt (log F (σ(t))) d

dt σi (16)

which corresponds only to the T−auto parallel transport and not N.Contrast with Riemannian geometry! Sabau-Shibuya (2015; Griffithsformulation)

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Thank you for your attention, glad to be here!

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