The Geometry of Relativitypeople.oregonstate.edu/~drayt/talks/OSUgeomPub.pdf · Introduction Special Relativity General Relativity Applications Cosmology Curvature Acceleration Black

IntroductionSpecial RelativityGeneral Relativity

Applications

The Geometry of Relativity

Tevian Dray

Department of MathematicsOregon State University

http://www.math.oregonstate.edu/~tevian

OSU 4/27/15 Tevian Dray The Geometry of Relativity 1/27

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Books

The Geometry of Special RelativityTevian DrayA K Peters/CRC Press 2012ISBN: 978-1-4665-1047-0http://physics.oregonstate.edu/coursewikis/GSR

Differential Forms andthe Geometry of General RelativityTevian DrayA K Peters/CRC Press 2014ISBN: 978-1-4665-1000-5http://physics.oregonstate.edu/coursewikis/GDF

http://physics.oregonstate.edu/coursewikis/GGR

OSU 4/27/15 Tevian Dray The Geometry of Relativity 2/27

IntroductionSpecial RelativityGeneral Relativity

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Circle GeometryHyperbola GeometryApplications

Trigonometry

ds2 = dx2 + dy2

Φ

r Hr cosΦ, r sinΦL

x2 + y2 = r2

rφ = arclength

4

53

θ

tan θ =3

4=⇒ cos θ =

4

5

OSU 4/27/15 Tevian Dray The Geometry of Relativity 3/27

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Circle GeometryHyperbola GeometryApplications

Measurements

Width:θ

1

θ

1

Apparent width > 1

1cos θ

1cos θ

Slope:

y

xφ

y’

x’

y

x

φθ

m 6= m1 +m2

tan(θ + φ) = tan θ+tanφ1−tan θ tanφ = m1+m2

1−m1m2

OSU 4/27/15 Tevian Dray The Geometry of Relativity 4/27

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Circle GeometryHyperbola GeometryApplications

Rotations

y

B

θ

θ

A

y’

x’

x

OSU 4/27/15 Tevian Dray The Geometry of Relativity 5/27

IntroductionSpecial RelativityGeneral Relativity

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Circle GeometryHyperbola GeometryApplications

Trigonometry

ds2 = −c2 dt2 + dx2

.ρ

ββ ρρ( cosh , sinh )

β

ρβ = arclength

4

5

3

β

tanhβ =3

5=⇒ coshβ =

5

4

(coshβ ≥ 1; tanhβ < 1)

OSU 4/27/15 Tevian Dray The Geometry of Relativity 6/27

IntroductionSpecial RelativityGeneral Relativity

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Circle GeometryHyperbola GeometryApplications

Trigonometry

β

βB

t’

A

x’

t

x

•

β coshρ

β sinhρρ

β

•

OSU 4/27/15 Tevian Dray The Geometry of Relativity 7/27

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Circle GeometryHyperbola GeometryApplications

Length Contraction

x’

t’t

x

x’

t’t

x

ℓ ′ = ℓcoshβ

β β

ℓ

ℓ ′

•

ℓ

ℓ ′

•

OSU 4/27/15 Tevian Dray The Geometry of Relativity 8/27

IntroductionSpecial RelativityGeneral Relativity

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Circle GeometryHyperbola GeometryApplications

Time Dilation

x’

ct’ct

x

OSU 4/27/15 Tevian Dray The Geometry of Relativity 9/27

IntroductionSpecial RelativityGeneral Relativity

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Circle GeometryHyperbola GeometryApplications

Pole & Barn

A 20 foot pole is moving towards a 10 foot barn fast enough thatthe pole appears to be only 10 feet long. As soon as both ends ofthe pole are in the barn, slam the doors. How can a 20 foot polefit into a 10 foot barn?

-20

-10

0

10

20

-20 -10 10 20 30

-20

-10

0

10

20

-10 10 20 30

barn frame pole frame

OSU 4/27/15 Tevian Dray The Geometry of Relativity 10/27

IntroductionSpecial RelativityGeneral Relativity

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Circle GeometryHyperbola GeometryApplications

Relativistic Mechanics

A pion of (rest) mass m and (relativistic) momentum p = 34mc

decays into 2 (massless) photons. One photon travels in the samedirection as the original pion, and the other travels in the oppositedirection. Find the energy of each photon. [E1 = mc2, E2 =

14mc2]

0

0

mc2

Β

E

E1

E2

pc

p1c

p2c

p0c

E0

E0

p0c

Β

Β

ΒΒ

p 0c

sinhΒ

p 0c

sinhΒ

E0c

coshΒ

E0c

coshΒ

0

0

mc2

Β

E1

p1 c

E2

p2 c

p0c

E0

E0

p0c

Β

Β

Β

p0 c sinh

Β

p0 c sinh

Β

E0 c cosh

ΒE

0 c coshΒ

OSU 4/27/15 Tevian Dray The Geometry of Relativity 11/27

IntroductionSpecial RelativityGeneral Relativity

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Circle GeometryHyperbola GeometryApplications

Twin Paradox

One twin travels 24 light-years to star X at speed 2425c; her twin

brother stays home. When the traveling twin gets to star X, sheimmediately turns around, and returns at the same speed. Howlong does each twin think the trip took?

β

•24

25 7

coshβ =25

7

•

7

q

q =7

coshβ=

49

25 49/25

7

24

25

β

Straight path takes longest!

OSU 4/27/15 Tevian Dray The Geometry of Relativity 12/27

IntroductionSpecial RelativityGeneral Relativity

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Circle GeometryHyperbola GeometryApplications

Addition of Velocities

v

c= tanhβ

tanh(α+ β) =tanhα+ tanhβ

1 + tanhα tanhβ=

uc+ v

c

1 + uvc2

Einstein addition formula!

OSU 4/27/15 Tevian Dray The Geometry of Relativity 13/27

IntroductionSpecial RelativityGeneral Relativity

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The MetricDifferential FormsGeodesicsEinstein’s Equation

Line Elements

a

a

dr2 + r2 dφ2 dθ2 + sin2 θ dφ2 dβ2 + sinh2 β dφ2

Black Hole: ds2 = −(

1− 2mr

)

dt2 + dr2

1− 2mr

+ r2 dθ2 + r2 sin2 θ dφ2

Cosmology: ds2 = −dt2 + a(t)2(

dr2

1−kr2+ r2

(

dθ2 + sin2 θ dφ2)

)

s = 0 s = 1

flat Euclidean Minkowskian (SR)curved Riemannian Lorentzian (GR)

OSU 4/27/15 Tevian Dray The Geometry of Relativity 14/27

IntroductionSpecial RelativityGeneral Relativity

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The MetricDifferential FormsGeodesicsEinstein’s Equation

Vector Calculus

ds2 = d~r · d~r

dy ^|

d~r

dx ^

d~r

r d

^

dr ^r

d~r = dx ı+ dy = dr r + r dφ φ

OSU 4/27/15 Tevian Dray The Geometry of Relativity 15/27

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The MetricDifferential FormsGeodesicsEinstein’s Equation

Differential Forms in a Nutshell (R3)

Differential forms are integrands: (∗2 = 1)

f = f (0-form)

F = ~F · d~r (1-form)

∗F = ~F · d~A (2-form)

∗f = f dV (3-form)

Products: F ∧ G = ~F× ~G · d~A

F ∧ ∗G = ~F · ~G dV

Exterior derivative: (d2 = 0)

df = ~∇f · d~r

dF = ~∇× ~F · d~A

d∗F = ~∇ · ~F dV

d∗f = 0

OSU 4/27/15 Tevian Dray The Geometry of Relativity 16/27

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The MetricDifferential FormsGeodesicsEinstein’s Equation

The Geometry of Differential Forms

dx

dx + dy r dr = x dx + y dy

OSU 4/27/15 Tevian Dray The Geometry of Relativity 17/27

IntroductionSpecial RelativityGeneral Relativity

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The MetricDifferential FormsGeodesicsEinstein’s Equation

Geodesic Equation

Orthonormal basis: d~r = σi ei

Connection: ωij = ei · d ej

dσi + ωij ∧ σj = 0

ωij + ωji = 0

Geodesics: ~v dλ = d~r

~v = 0

Symmetry: d~X · d~r = 0

=⇒ ~X · ~v = const

OSU 4/27/15 Tevian Dray The Geometry of Relativity 18/27

IntroductionSpecial RelativityGeneral Relativity

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The MetricDifferential FormsGeodesicsEinstein’s Equation

Einstein’s Equation

Curvature:Ωi

j = dωij + ωi

k ∧ ωkj

Einstein tensor:γ i = −

1

2Ωjk ∧ ∗(σi ∧ σj ∧ σk)

G i = ∗γ i = G ij σ

j

~G = G i ei = G ij σ

j ei

=⇒d∗~G = 0

Field equation: ~G+ Λ d~r = 8π~T

(curvature = matter)

OSU 4/27/15 Tevian Dray The Geometry of Relativity 19/27

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CosmologyCurvatureAccelerationBlack Holes

Cosmological Redshift

a = a(t)

1 + z =a(tR)

a(tE )≈ 1 +

a

a∆s

(redshift ∝ distance)

OSU 4/27/15 Tevian Dray The Geometry of Relativity 20/27

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CosmologyCurvatureAccelerationBlack Holes

Curvature

ds2 = r2(dθ2 + sin2θ dφ2) Tidal forces!

OSU 4/27/15 Tevian Dray The Geometry of Relativity 21/27

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CosmologyCurvatureAccelerationBlack Holes

Gravitational Lensing

EarthSun

star

perceiv

edpath

actual path

OSU 4/27/15 Tevian Dray The Geometry of Relativity 22/27

IntroductionSpecial RelativityGeneral Relativity

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CosmologyCurvatureAccelerationBlack Holes

Rindler Geometry

constant curvature = constant acceleration

.ρ

ββ ρρ( cosh , sinh )

β

Ρ=const

v×

Ρ=0, Α=-¥

Ρ=0,Α=¥

x = ρ coshαt = ρ sinhα

=⇒ ds2 = dρ2 − ρ2 dα2

Can outrun lightbeam!

OSU 4/27/15 Tevian Dray The Geometry of Relativity 23/27

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CosmologyCurvatureAccelerationBlack Holes

From Rindler to Minkowski

Ρ=const

Α=const

x=-t

x=t

v×

Ρ=0, Α=-¥

Ρ=0,Α=¥

v=-¥

u=¥

u v V=0

U=0

U V

u = α− ln ρ, v = α+ ln ρ U = −e−u = −ρ e−α, V = ev = ρ eαds

2 = −dU dV = −d(t− x) d(t+ x)

OSU 4/27/15 Tevian Dray The Geometry of Relativity 24/27

IntroductionSpecial RelativityGeneral Relativity

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CosmologyCurvatureAccelerationBlack Holes

From Schwarzschild to Kruskal

ds2 = −(

1− 2mr

)

dt2 + dr2

1− 2mr

+ r2 dθ2 + r2 sin2 θ dφ2

r=2mHv=-¥L

r=2mHu=¥L

u v

r=0

r=0

r=2mHV=0L

r=2mHU=0L

U V

X

T

ds2 = −32m3

re−r/2m dU dV

OSU 4/27/15 Tevian Dray The Geometry of Relativity 25/27

IntroductionSpecial RelativityGeneral Relativity

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CosmologyCurvatureAccelerationBlack Holes

From Schwarzschild to Kruskal

ds2 = −(

1− 2mr

)

dt2 + dr2

1− 2mr

+ r2 dθ2 + r2 sin2 θ dφ2

r=2mHv=-¥L

r=2mHu=¥L

u v

usthem

BH

WH

ds2 = −32m3

re−r/2m dU dV

OSU 4/27/15 Tevian Dray The Geometry of Relativity 25/27

IntroductionSpecial RelativityGeneral Relativity

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CosmologyCurvatureAccelerationBlack Holes

Wormholes

Constant radius = constant acceleration!

OSU 4/27/15 Tevian Dray The Geometry of Relativity 26/27

IntroductionSpecial RelativityGeneral Relativity

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CosmologyCurvatureAccelerationBlack Holes

Wormholes

OSU 4/27/15 Tevian Dray The Geometry of Relativity 26/27

IntroductionSpecial RelativityGeneral Relativity

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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

OSU 4/27/15 Tevian Dray The Geometry of Relativity 27/27