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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
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
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.
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.
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.
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
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
,
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
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).
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.
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.
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).
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}
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)
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
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
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}
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.
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.
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)
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)
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
)
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)|
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.
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.
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
Special relativity Lorentz Space Transformations Hyperbolic Plane PSL(2,R) Surfaces
Affine deformations
{ affine deformations of surface group}l
{ infinitesimal deformations of corresponding surfaces}
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