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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
IntroductionSpecial RelativityGeneral Relativity
Applications
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
Applications
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
IntroductionSpecial RelativityGeneral Relativity
Applications
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
IntroductionSpecial RelativityGeneral Relativity
Applications
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
Applications
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
Applications
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
IntroductionSpecial RelativityGeneral Relativity
Applications
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
Applications
Circle GeometryHyperbola GeometryApplications
Time Dilation
x’
ct’ct
x
OSU 4/27/15 Tevian Dray The Geometry of Relativity 9/27
IntroductionSpecial RelativityGeneral Relativity
Applications
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
Applications
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
Applications
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
Applications
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
Applications
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
Applications
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
IntroductionSpecial RelativityGeneral Relativity
Applications
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
IntroductionSpecial RelativityGeneral Relativity
Applications
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
Applications
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
Applications
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
IntroductionSpecial RelativityGeneral Relativity
Applications
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
IntroductionSpecial RelativityGeneral Relativity
Applications
CosmologyCurvatureAccelerationBlack Holes
Curvature
ds2 = r2(dθ2 + sin2θ dφ2) Tidal forces!
OSU 4/27/15 Tevian Dray The Geometry of Relativity 21/27
IntroductionSpecial RelativityGeneral Relativity
Applications
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
Applications
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
IntroductionSpecial RelativityGeneral Relativity
Applications
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
Applications
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
Applications
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
Applications
CosmologyCurvatureAccelerationBlack Holes
Wormholes
Constant radius = constant acceleration!
OSU 4/27/15 Tevian Dray The Geometry of Relativity 26/27
IntroductionSpecial RelativityGeneral Relativity
Applications
CosmologyCurvatureAccelerationBlack Holes
Wormholes
OSU 4/27/15 Tevian Dray The Geometry of Relativity 26/27
IntroductionSpecial RelativityGeneral Relativity
Applications
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