Hyperbolic and Lorentzian Geometry: An Introduction...Projective Model The Projective model of the...
Transcript of Hyperbolic and Lorentzian Geometry: An Introduction...Projective Model The Projective model of the...
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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
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Special relativity Lorentz Space Transformations Hyperbolic Plane PSL(2,R) Surfaces
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
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Special relativity Lorentz Space Transformations Hyperbolic Plane PSL(2,R) Surfaces
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.
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Special relativity Lorentz Space Transformations Hyperbolic Plane PSL(2,R) Surfaces
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.
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Special relativity Lorentz Space Transformations Hyperbolic Plane PSL(2,R) Surfaces
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.
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Special relativity Lorentz Space Transformations Hyperbolic Plane PSL(2,R) Surfaces
Perpendicular planes
Timelike: v⊥ is spacelike
Lightlike: v⊥ is non-degeneratev ∈ v⊥
Tangent to light cone
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Special relativity Lorentz Space Transformations Hyperbolic Plane PSL(2,R) Surfaces
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
,
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Special relativity Lorentz Space Transformations Hyperbolic Plane PSL(2,R) Surfaces
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
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Special relativity Lorentz Space Transformations Hyperbolic Plane PSL(2,R) Surfaces
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).
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Special relativity Lorentz Space Transformations Hyperbolic Plane PSL(2,R) Surfaces
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.
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Special relativity Lorentz Space Transformations Hyperbolic Plane PSL(2,R) Surfaces
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.
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Special relativity Lorentz Space Transformations Hyperbolic Plane PSL(2,R) Surfaces
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).
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Special relativity Lorentz Space Transformations Hyperbolic Plane PSL(2,R) Surfaces
Hyperboloid Model
The hyperboloid model of the hyperbolic plane lives naturallyinside V2,1
H = {v ∈ T f 〈v, v〉 = −1, v3 > 0}
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Special relativity Lorentz Space Transformations Hyperbolic Plane PSL(2,R) Surfaces
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)
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Special relativity Lorentz Space Transformations Hyperbolic Plane PSL(2,R) Surfaces
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
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Special relativity Lorentz Space Transformations Hyperbolic Plane PSL(2,R) Surfaces
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
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Special relativity Lorentz Space Transformations Hyperbolic Plane PSL(2,R) Surfaces
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}
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Special relativity Lorentz Space Transformations Hyperbolic Plane PSL(2,R) Surfaces
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.
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Special relativity Lorentz Space Transformations Hyperbolic Plane PSL(2,R) Surfaces
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.
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Special relativity Lorentz Space Transformations Hyperbolic Plane PSL(2,R) Surfaces
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)
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Special relativity Lorentz Space Transformations Hyperbolic Plane PSL(2,R) Surfaces
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)
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Special relativity Lorentz Space Transformations Hyperbolic Plane PSL(2,R) Surfaces
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
)
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Special relativity Lorentz Space Transformations Hyperbolic Plane PSL(2,R) Surfaces
Cyclic Groups
Cylinders
Upper half-plane:
H2/〈g〉 :
Unique closed geodesic, whose length is related to | tr(g)|
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Special relativity Lorentz Space Transformations Hyperbolic Plane PSL(2,R) Surfaces
Surfaces
The three-holed sphere, or pair of pants
Disk view
After identification with the ends cut off.
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Special relativity Lorentz Space Transformations Hyperbolic Plane PSL(2,R) Surfaces
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.
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Special relativity Lorentz Space Transformations Hyperbolic Plane PSL(2,R) Surfaces
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
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Special relativity Lorentz Space Transformations Hyperbolic Plane PSL(2,R) Surfaces
Affine deformations
{ affine deformations of surface group}l
{ infinitesimal deformations of corresponding surfaces}