Illinois Geometry Lab

Geodesics on Lorentz Surafaces of Revolution in Minkowski Space Brian Friedin, Rob Halliday, Richard Yoon, Yifei Li, Bill Karr, Stephanie Alexander Illinois Geometry Lab IGL Open House, Spring 2013 Comparison Geometry Triangles are formed by shortest paths called geodesics. Geodesics are solutions to a differential equation in smooth spaces. In spaces of constant curvature (e.g. the sphere, Euclidean space, and Hyperbolic space) geodesics, triangles, and angles have well-known behavior. Comparison geomtry is about comparing the geometry of spaces with curvature bounded above (CAT(κ)) or below (CBB(κ)) by κ to spaces with constant curvature κ. CBB = “fatter” triangles, CAT = “skinnier” triangles. Figure: Two-dimensional spaces of constant positive, negative, and zero curvature, respectively. Lorentz Geometry Lorentz manifolds model spacetime in relativity theory. They have one time dimension and one or more spatial dimensions. Figure: Tangent Space in Lorentz Geometry Mathematically defined with metric that is negative definite in one-dimension and positive definite in the spatial dimensions. No notion of distance, but geodesics defined using metric. Three types of geodesics: timelike, spacelike, lightlike (null). This property is called causal character. IGL Project Make computer visualizations to aid research in Lorentz spaces with curvature bounds. Geodesic equations are difficult to solve numerically. We use MATLAB for fast numerical computation. Focus on surfaces of revolution in Minkowski space. z -axis represents time, angle is spatial position. Constant Curvature (a) Constant Curvature -1 (b) Constant Curvature +1 Left: geodesics of each causal character on surface with constant curvature -1. Right: geodesics of each causal character on surface with constant curvature +1. Bounded Curvature (c)CAT(0) Lorentz Surface (d) CBB(0) Lorentz Surface Left: geodesics of each causal character on CAT(0) surface with r (s )= √ s 2 + 1. Right: geodesics of each causal character on CBB(0) surface with r (s )= log(s - 1). Mathematics Start with warping function r (s ) > 0. Produce unit speed profile curve (r (s ), f (s )) by f (s )= Z s o q r 0 (t ) 2 + 1 dt . Revolve profile curve around z -axis, obtaining F (θ, s )=(r (s ) cos θ, r (s ) sin θ, f (s )). Metric in these coordinates is given by r (s ) 2 0 0 -1 . Geodesic equations are given by ¨ θ + 2 r 0 (s ) r (s ) ˙ θ ˙ s = 0, ¨ s + r 0 (s )r (s ) ˙ θ 2 = 0 Timelike, spacelike, and lightlike are determined by k ˙ γ k 2 < 0, k ˙ γ k 2 > 0, or k ˙ γ k 2 = 0, respectively. Sectional curvature formula given by K (θ, s )= - r 00 (s ) r (s ) . Facts Several statements are true of geodesics and can be seen with our work in MATLAB. All geodesics γ have constant “square speed” in the Lorentzian sense: r (s ) 2 ˙ θ 2 - ˙ s 2 = k ˙ γ k 2 = const. Clairaut’s relation holds: r (s ) 2 ˙ θ = const. One can prove that all (with the exception of equatorial) geodesics on CBB(κ) with κ> 0 fall into big bangs and big crunches in finite time. This is seen in the top right figure. All nonspacelike geodesics on CAT(κ) spaces with κ< 0 have bounded θ , i.e. they do not spiral around indefinitely. Future Work Lorentz surfaces of revolution are warped products of the form (-R) × r S 1 . In the future, we would like to make animations in these spacetimes. Also, replace S 1 with other Riemannian manifolds for more general warped products of the form (-R) × r M . Study geodesics in warped products with M = S 2 , H 2 or other interesting manifolds. These posters are made with the support of University of Illinois at Urbana-Champaign Public Engagement Office

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Geodesics on Lorentz Surafaces of Revolution in Minkowski SpaceBrian Friedin, Rob Halliday, Richard Yoon, Yifei Li, Bill Karr, Stephanie Alexander

Illinois Geometry LabIGL Open House, Spring 2013

Comparison Geometry

Triangles are formed by shortest paths called geodesics.Geodesics are solutions to a differential equation insmooth spaces.In spaces of constant curvature (e.g. the sphere,Euclidean space, and Hyperbolic space) geodesics,triangles, and angles have well-known behavior.Comparison geomtry is about comparing the geometryof spaces with curvature bounded above (CAT(κ)) orbelow (CBB(κ)) by κ to spaces with constant curvature κ.CBB = “fatter” triangles, CAT = “skinnier” triangles.

Figure: Two-dimensional spaces of constant positive, negative, andzero curvature, respectively.

Lorentz Geometry

Lorentz manifoldsmodel spacetime inrelativity theory.They have one timedimension and one ormore spatialdimensions.

Figure: Tangent Space in Lorentz Geometry

Mathematically defined with metric that is negativedefinite in one-dimension and positive definite in thespatial dimensions.No notion of distance, but geodesics defined usingmetric.Three types of geodesics: timelike, spacelike, lightlike(null). This property is called causal character.

IGL ProjectMake computer visualizations to aid research in Lorentzspaces with curvature bounds.Geodesic equations are difficult to solve numerically.We use MATLAB for fast numerical computation.Focus on surfaces of revolution in Minkowski space.z-axis represents time, angle is spatial position.

Constant Curvature

(a) Constant Curvature −1 (b) Constant Curvature +1

Left: geodesics of each causal character on surface with constant curvature −1.Right: geodesics of each causal character on surface with constant curvature +1.

Bounded Curvature

Left: geodesics of each causal character on CAT(0) surface with r (s) =√

s2 + 1.Right: geodesics of each causal character on CBB(0) surface with r (s) = log(s − 1).

MathematicsStart with warping function r (s) > 0. Produce unit speedprofile curve (r (s), f (s)) by

f (s) =∫ s

√r ′(t)2 + 1 dt .

Revolve profile curve around z-axis, obtainingF (θ, s) = (r (s) cos θ, r (s) sin θ, f (s)).

Metric in these coordinates is given by(r (s)2 0

0 −1

Geodesic equations are given by

θ + 2r ′(s)r (s)

θs = 0, s + r ′(s)r (s)θ2 = 0

Timelike, spacelike, and lightlike are determined by‖γ‖2 < 0, ‖γ‖2 > 0, or ‖γ‖2 = 0, respectively.Sectional curvature formula given by

K (θ, s) = −r ′′(s)r (s)

FactsSeveral statements are true of geodesics and can beseen with our work in MATLAB.

All geodesics γ have constant “square speed” in theLorentzian sense:

r (s)2θ2 − s2 = ‖γ‖2 = const.Clairaut’s relation holds:

r (s)2θ = const.One can prove that all (with the exception of equatorial)geodesics on CBB(κ) with κ > 0 fall into big bangs andbig crunches in finite time. This is seen in the top rightfigure.All nonspacelike geodesics on CAT(κ) spaces with κ < 0have bounded θ, i.e. they do not spiral aroundindefinitely.

Future WorkLorentz surfaces of revolution are warped products ofthe form (−R)×r S1.In the future, we would like to make animations in thesespacetimes.Also, replace S1 with other Riemannian manifolds formore general warped products of the form (−R)×r M.Study geodesics in warped products with M = S2,H2 orother interesting manifolds.

These posters are made with the support of University of Illinois at Urbana-Champaign Public Engagement Office