The Geometry of Relativity

The Geometry of Relativity Tevian Dray Department of Mathematics Oregon State University http://www.math.oregonstate.edu/~tevian PNWMAA 4/11/15 Tevian Dray The Geometry of Relativity 1/25

Books The Geometry of Tevian Dray A K Peters/CRC Press 2012 ISBN: 978-1-4665-1047-0 http://physics.oregonstate.edu/coursewikis/gsr Differential Forms and the Geometry of Tevian Dray A K Peters/CRC Press 2014 ISBN: 978-1-4665-1000-5 http://physics.oregonstate.edu/coursewikis/gdf http://physics.oregonstate.edu/coursewikis/ggr PNWMAA 4/11/15 Tevian Dray The Geometry of Relativity 2/25

Trigonometry Circle Geometry Hyperbola Geometry Applications x 2 +y 2 = r 2 ds 2 = dx 2 +dy 2 Φ r r cosφ, r sinφ 5 3 θ 4 rφ = arclength tanθ = 3 4 = cosθ = 4 5 PNWMAA 4/11/15 Tevian Dray The Geometry of Relativity 3/25

Measurements Circle Geometry Hyperbola Geometry Applications Width: θ 1 1 cosθ 1 cosθ θ Slope: y Apparent width > 1 y y 1 φ x m m 1 +m 2 tan(θ +φ) = tanθ+tanφ 1 tanθtanφ = m1+m2 1 m 1m 2 φ θ x x PNWMAA 4/11/15 Tevian Dray The Geometry of Relativity 4/25

Rotations Circle Geometry Hyperbola Geometry Applications y y A θ θ B x x PNWMAA 4/11/15 Tevian Dray The Geometry of Relativity 5/25

Trigonometry Circle Geometry Hyperbola Geometry Applications ds 2 = c 2 dt 2 +dx 2 ρ. ( ρcosh β, ρsinh β ) 4 3 β β 5 tanhβ = 3 5 = coshβ = 5 4 ρβ = arclength (coshβ 1;tanhβ < 1) PNWMAA 4/11/15 Tevian Dray The Geometry of Relativity 6/25

Trigonometry Circle Geometry Hyperbola Geometry Applications t t A ρ ρ sinh β β β B x x β ρ cosh β PNWMAA 4/11/15 Tevian Dray The Geometry of Relativity 7/25

Length Contraction Circle Geometry Hyperbola Geometry Applications t t t t x x x x l = l coshβ l l β l β l PNWMAA 4/11/15 Tevian Dray The Geometry of Relativity 8/25

Time Dilation Circle Geometry Hyperbola Geometry Applications ct ct x x PNWMAA 4/11/15 Tevian Dray The Geometry of Relativity 9/25

Pole & Barn Circle Geometry Hyperbola Geometry Applications A 20 foot pole is moving towards a 10 foot barn fast enough that the pole appears to be only 10 feet long. As soon as both ends of the pole are in the barn, slam the doors. How can a 20 foot pole fit into a 10 foot barn? 20 20 10 10-20 -10 0 10 20 30-10 0 10 20 30-10 -10-20 barn frame -20 pole frame PNWMAA 4/11/15 Tevian Dray The Geometry of Relativity 10/25

Relativistic Mechanics Circle Geometry Hyperbola Geometry Applications A pion of (rest) mass m and (relativistic) momentum p = 3 4 mc decays into 2 (massless) photons. One photon travels in the same direction as the original pion, and the other travels in the opposite direction. Find the energy of each photon. [E 1 = mc 2, E 2 = 1 4 mc2 ] pc p0c sinhβ 0 p 2 c E 2 Β p 0 c Β E 0 E0c coshβ p0c sinhβ p0c p2c 0 Β E2 Β E0 E mc 2 0 E 0 Β E0c coshβ mc 2 0 p0c Β Β E0 E0c coshβ E0c coshβ Β Β p 0 c p0c sinhβ p 1 c E 1 p0c sinhβ p1c E1 PNWMAA 4/11/15 Tevian Dray The Geometry of Relativity 11/25

Twin Paradox Circle Geometry Hyperbola Geometry Applications One twin travels 24 light-years to star X at speed 24 25c; her twin brother stays home. When the traveling twin gets to star X, she immediately turns around, and returns at the same speed. How long does each twin think the trip took? 24 25 7 β coshβ = 25 7 q 7 25 q = 7 coshβ = 49 25 49/25 Straight path takes longest! β 24 7 PNWMAA 4/11/15 Tevian Dray The Geometry of Relativity 12/25

Addition of Velocities Circle Geometry Hyperbola Geometry Applications v c = tanhβ tanh(α+β) = tanhα+tanhβ u 1+tanhαtanhβ = c + v c 1+ uv c 2 Einstein addition formula! PNWMAA 4/11/15 Tevian Dray The Geometry of Relativity 13/25

Line Elements The Metric Differential Forms Geodesics Einstein s Equation a a dr 2 +r 2 dφ 2 dθ 2 +sin 2 θdφ 2 dβ 2 +sinh 2 βdφ 2 Black Hole: ds 2 = ( ) 1 2m r dt 2 + dr2 +r 2 dθ 2 +r 2 sin 2 θdφ 2 1 2m r ( Cosmology: ds 2 = dt 2 +a(t) 2 dr 2 1 kr +r 2( dθ 2 +sin 2 θdφ 2)) 2 s = 0 s = 1 flat Euclidean Minkowskian (SR) curved Riemannian Lorentzian (GR) PNWMAA 4/11/15 Tevian Dray The Geometry of Relativity 14/25

Vector Calculus The Metric Differential Forms Geodesics Einstein s Equation ds 2 = d r d r Ý Ö Ö Ü ß Ö Ö Ö d r = dx î+dy ĵ = drˆr+r dφ ˆφ PNWMAA 4/11/15 Tevian Dray The Geometry of Relativity 15/25

The Metric Differential Forms Geodesics Einstein s Equation Differential Forms in a Nutshell (R 3 ) Differential forms are integrands: ( 2 = 1) f = f F = F d r F = F da f = f dv (0-form) (1-form) (2-form) (3-form) Exterior derivative: (d 2 = 0) df = f d r df = F da d F = FdV d f = 0 PNWMAA 4/11/15 Tevian Dray The Geometry of Relativity 16/25

The Metric Differential Forms Geodesics Einstein s Equation The Geometry of Differential Forms dx +dy r dr = x dx +y dy dx PNWMAA 4/11/15 Tevian Dray The Geometry of Relativity 17/25

Geodesic Equation The Metric Differential Forms Geodesics Einstein s Equation Orthonormal basis: Connection: d r = σ i ê i ω ij = ê i dê j dσ i +ω i j σ j = 0 ω ij +ω ji = 0 Geodesics: v dλ = d r v = 0 Symmetry: d X d r = 0 = X v = const PNWMAA 4/11/15 Tevian Dray The Geometry of Relativity 18/25

Einstein s Equation The Metric Differential Forms Geodesics Einstein s Equation Curvature: Ω i j = dω i j +ω i k ω k j Einstein tensor: γ i = 1 2 Ω jk (σ i σ j σ k ) G i = γ i = G i j σ j G = G i ê i = G i j σ j ê i = d G = 0 Field equation: G+Λd r = 8π T (curvature = matter) PNWMAA 4/11/15 Tevian Dray The Geometry of Relativity 19/25

Curvature Curvature Rindler Geometry Kruskal Geometry ds 2 = r 2 (dθ 2 +sin 2 θdφ 2 ) Tidal forces! PNWMAA 4/11/15 Tevian Dray The Geometry of Relativity 20/25

Acceleration Curvature Rindler Geometry Kruskal Geometry constant curvature = constant acceleration β ρ. ( ρcosh β, ρsinh β ) Ρ 0,Α v Ρ 0,Α Ρ const x = ρ coshα t = ρ sinhα Can outrun lightbeam! = ds 2 = dρ 2 ρ 2 dα 2 PNWMAA 4/11/15 Tevian Dray The Geometry of Relativity 21/25

From Rindler to Minkowski Curvature Rindler Geometry Kruskal Geometry x t Α const Ρ 0,Α v x t Ρ const Ρ 0,Α u U 0 v u v V 0 U V u = α lnρ, v = α+lnρ ds 2 = dudv U = e u = ρe α, V = e v = ρe α PNWMAA 4/11/15 Tevian Dray The Geometry of Relativity 22/25

From Schwarzschild to Kruskal Curvature Rindler Geometry Kruskal Geometry ds 2 = ( 1 2m r ) dt 2 + dr2 1 2m r +r 2 dθ 2 +r 2 sin 2 θdφ 2 r 2m u T r 0 r 2m U 0 r 2m v X u v r 2m V 0 U V r 0 ds 2 = 32m3 r e r/2m dudv PNWMAA 4/11/15 Tevian Dray The Geometry of Relativity 23/25

From Schwarzschild to Kruskal Curvature Rindler Geometry Kruskal Geometry ds 2 = ( 1 2m r ) dt 2 + dr2 1 2m r +r 2 dθ 2 +r 2 sin 2 θdφ 2 r 2m u BH them us r 2m v u v WH ds 2 = 32m3 r e r/2m dudv PNWMAA 4/11/15 Tevian Dray The Geometry of Relativity 23/25

Wormholes Curvature Rindler Geometry Kruskal Geometry Constant radius = constant acceleration! PNWMAA 4/11/15 Tevian Dray The Geometry of Relativity 24/25

Wormholes Curvature Rindler Geometry Kruskal Geometry PNWMAA 4/11/15 Tevian Dray The Geometry of Relativity 24/25

SUMMARY http://relativity.geometryof.org/gsr http://relativity.geometryof.org/gdf http://relativity.geometryof.org/ggr Special relativity is hyperbolic trigonometry! Spacetimes are described by line elements! Curvature = gravity! Geometry = physics! THE END PNWMAA 4/11/15 Tevian Dray The Geometry of Relativity 25/25

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