Hyperbolic and Lorentzian Geometry: An Introduction...Projective Model The Projective model of the...

Special relativity Lorentz Space Transformations Hyperbolic Plane PSL(2,R) Surfaces

Hyperbolic and Lorentzian Geometry: AnIntroduction

Todd A. Drumm

Howard University

3 November, 2011Centro de Investigacion en Matematicas


Space-time

Three-dimensional space-time

Horizontal, 2 spatial dimensionsVertical, time dimension

Katie’s frame of reference

Maxx traveling at a constant speedKatie creates a spark, light travels out, creating cone in3-dimensionsSlope of cone is speed of light 1/c , set c = 1


Space-time

Maxx’s frame of reference

Katie traveling at a constant speedChange by translation and linear mapEinstein: “The speed of light is the same for Katie and Maxx”

Need transformations that preserve the slope of the light cone.


Three-dimensional Lorentzian space E2,1

With choice of an origin o, identify E2,1 with its tangentspace V2,1: p ↔ (p − o)

V2,1

The Lorentzian inner product: 〈v,w〉 = v1w1 + v2w2 − v3w3

A vector v is called

timelike if 〈v, v〉 < 0lightlike if 〈v, v〉 = 0spacelike if 〈v, v〉 > 0

Light cone: L = {v ∈ V2,1|〈v, v〉 = 0}future pointing light cone: Lf = {v ∈ L v3 > 0}past pointing light cone: Lp = {v ∈ L v3 < 0}

For timelike vectors T = {v ∈ V2,1|〈v, v〉 < 0}, T f and T p aredefined similarly.


Perpendicular planes

Timelike: v⊥ is spacelike

Lightlike: v⊥ is non-degeneratev ∈ v⊥

Tangent to light cone


Perpendicular planes

Spacelike: v⊥ is degenerate

Intersects future light cone in two rays.Future pointing vectors v± ∈ v⊥: chosen so that {v−, v+, v}are a right-handed basis for V2,1

,


Aside: Affine Transformations

γ ∈ Aff(Rn) where γ = (g , v)

g ∈ GL(n,R)

v ∈ Rn

γ(x) = g(x) + v

Lemma

If g does not have 1 as an eigenvalue, then any affinetransformation γ = (g , v) has a fixed point.

Proof.

If g does not have 1 as an eigenvalue then (g − I ) has an inverseand (g − I )−1(−v) is a solution to

g(x) + v = x


Lorentz Transformations

γ ∈ Aff; γ = (g , v) and γ(x) = g(x) + v

g ∈ O(2, 1) is called the linear part of γ.

SOo(2, 1) is the identity componenet of O(2, 1).Conjugacy inside O(2, 1) is determined by trace.

v ∈ V2,1 is called the translational part or γ.

Start with G ⊂ SO(2, 1).The map u : G → V2,1 such that u(gh) = u(g) + gu(h) is acocyle.

The vector space of cocyles is Z 1(G ,V2,1).Defines Γ with elements γ = (g , u(g)).Γ, also u, is called an affine deformation of G .For two affine deformations u1, u2 which are translationallyconjugate by w, u1(g)− u2(g) = w − g(w), is called acoboundary, and form B1(G ,V2,1).H1(G ,V2,1) = Z 1(G ,V2,1)/B1(G ,V2,1).


The Linear Part

An element g ∈ SOo(2, 1) is:

elliptic if it exactly one real and two complex eigenvalues, ( or,tr(g) < 3) ;parabolic if the only eigenvealue is 1 (or nonidentity andtr(g) = 3);

x0(g) is a fixed lightlike eigenvector whose length can only bechosen arbitrarily.

hyperbolic if it has three positive real eigenvalues λ < 1 < λ−1

(or, tr(g) > 3);

x±(g) are the expanding/contracting eigenvector; lightlikevectors chosen with third coordinate = +1.x0(g) is the fixed eigenvector; spacelike vector chosen so that〈x0(g), x0(g)〉 = 1 and {x−(g), x+(g), x0(g)} is aright-handed basis for V2,1.


Nonidentity Components

O(2, 1) has four connected components.

SOo(2, 1)

SO(2, 1) \ SOo(2, 1)

Example: h ∼

1−λ

−λ−1

Exchanges future and past vectors.

O(2, 1) \ SO(2, 1)

Example: h ∼

−1λ

λ−1

Example: h ∼

−1−λ

−λ−1

1 is not an eigenvalue for these matrices.


Hyperbolic Lorentzian Transformations

γ = (g , v), such that g ∈ SO(2, 1) is hyperbolic.

Proposition

If γ = (g , v) is a hyperbolic affine transformation, then there existsa unique line ` parallel to x0(g) such that γ(`) = `. Furthermore,if γ has no fixed points then

` is the only invariant line,

γ acts by translation along `,

E2,1/〈γ〉 is a Lorentzian manifold with exactly one closedgeodesic (the image of ` under the projection).


Hyperboloid Model

The hyperboloid model of the hyperbolic plane lives naturallyinside V2,1

H = {v ∈ T f 〈v, v〉 = −1, v3 > 0}


Hyperboloid Model

Tangent vectors

SpacelikeDefines metric on H,Lenght of vector ‖v‖ =

√〈v, v〉

Angle well defined cos(θ) = 〈v,w〉‖v‖‖w‖

Differentiable paths p : [a, b] 7→ H

Arclength∫ b

a

√‖ dp

dt ‖dt

Geodesics: {planes through the origin} ∩ H.

Boundary: { Null directions} ∼= S1

Orientation preserving isometries: SOo(2, 1)


Projective Model

The Projective model of the hyperbolic plane:

PT = {[v] v ∈ T and v ∼ kv for any k ∈ R \ 0}

Geodesics: Projectivizations of planes through the origin(inside the lightcone)

Boundary: PL ∼= S1

Unifies the hyperboloid model and its negative

Elements of SO(2, 1) \ SOo(2, 1) act on the hyperbolic plane.Reverse orientation


Klein Model

The Klein model of the hyperbolic plane:K = {(x , y) ∈ R2 x2

1 + x22 < 1}

Project H on to plane x3 = 1

Geodesics: Chords of the boundary circle

Boundary: x21 + x2

2 = 1

NOT CONFORMAL


Poincare Disk

The Poincare disk model of the hyperbolic plane:D = {z ∈ C |z | < 1}

Stereographically projection of the hyperboloid model withrespect to (0, 0.− 1).

Identify xy -plane with complex plane

Geodesics: circles perpedicular to unit circle.

Boundary: S1 = {z ∈ C |z | = 1}


Upper half-plane

The upper half-plane model of the hyperbolic plane:H2 = {z ∈ C Im(z) > 0}

Equivalent by Mobius transformation to D

Metric: ds = 1y dz

Geodesics: vertical rays and circles centered on the real line.

Boundary: R ∪ {∞} ∼= S1

Isometries: PSL(2,R) acting by fractional lineartransformations.


PSL(2, R)

PSL(2,R) = SL(2,R)/{±1}

Fractional linear transformation:

[a bc d

](z) = az+b

cz+d

Action extend to boundary.For x ∈ R, if cx + d = 0 then g(x) =∞g(∞) = a/c , or ∞ if c = 0

Classification by trace (all elements of the same trace areconjugate) for g ∈ SL(2,R):

elliptic if | tr(g)| < 2;

Rotation about a fixed point in H2.

parabolic if | tr(g)| = 2;

One fixed point on ∂H2, andall parabolics are conjugate.

hyperbolic if | tr(g)| > 2;

Two fixed points (attracting and repelling) on ∂H2

Axis of g is geodesic whose endpoints are the fixed points.


The Lie algebra

sl(2,R) = Te(SL(2,R))

sl(2,R) = {v ∈ M2 tr(v) = 0}Three-dimensional vector space

SL(2,R) action: gA(v) = gvg−1

Basis:

x1 =

[1−1

], x2 =

[1

1

], x3 =

[−1

1

]Inner product: B(v,w) = 1

2 tr(vw)

B(x1, x1) = B(x2, x2) = 1,. B(x3, x3) = −1For i 6= j , B(xi , xj) = 0

Proposition (A miracle occurs)

As a vector space with its natural inner product, sl(2,R) ∼= V2,1.

γ = (g , v)↔ (g , v)


The Lie algebra

sl(2,R) = Te(SL(2,R))

sl(2,R) = {v ∈ M2 tr(v) = 0}Three-dimensional vector space

SL(2,R) action: gA(v) = gvg−1

Basis:

x1 =

[1−1

], x2 =

[1

1

], x3 =

[−1

1

]Inner product: B(v,w) = 1

2 tr(vw)B(x1, x1) = B(x2, x2) = 1,. B(x3, x3) = −1For i 6= j , B(xi , xj) = 0

Proposition (A miracle occurs)

As a vector space with its natural inner product, sl(2,R) ∼= V2,1.

γ = (g , v)↔ (g , v)


Cyclic Groups

Hyperbolic elements

Conjugate to g =

[λ−1

λ

]( 0 < λ < 1)

Fixed points are 0 and ∞Axis is vertical ray from 0tr(g) = λ+ λ−1

d(i , g(i)) = 2 lnλ−1

For z on the axis of g , | tr(g)| = 2 cosh(

d(z,g(z))2

)


Cyclic Groups

Cylinders

Upper half-plane:

H2/〈g〉 :

Unique closed geodesic, whose length is related to | tr(g)|


Surfaces

The three-holed sphere, or pair of pants

Disk view

After identification with the ends cut off.


Deformations of Surfaces

Three-holed sphere example: Start with our three holed sphere,deform surface by changing generators (all other elements will alsochange)

Change in length and relationships of closed geodesics.


Inifinitesimal Deformations of Surfaces

Paths of elements

gt = g exp(vt + O(t2)) = g + gvt + O(t2)ht = h + hwt + O(t2)

gtht = g(h + hwt + O(t2)) + gvt(h + hw + O(t2)) + O(t2)= (gh + ghw + gvh)t + O(t2)

point tangent vector lie algebra vectorg gv v

h hw w

gh ghw + gh(h−1vh) w + h−1vh


Affine deformations

{ affine deformations of surface group}l

{ infinitesimal deformations of corresponding surfaces}

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