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IntroductionSpecial RelativityGeneral Relativity
The Geometry of Relativity
Tevian Dray
Department of Mathematics
Oregon State University
http://www.math.oregonstate.edu/~tevian
Tevian Dray The Geometry of Relativity
IntroductionSpecial RelativityGeneral Relativity
Books
The Geometry of Special Relativity
Tevian DrayA K Peters/CRC Press 2012ISBN: 978-1-4665-1047-0http://physics.oregonstate.edu/coursewikis/GSR
Tevian Dray The Geometry of Relativity
IntroductionSpecial RelativityGeneral Relativity
Books
The Geometry of Special Relativity
Tevian DrayA K Peters/CRC Press 2012ISBN: 978-1-4665-1047-0http://physics.oregonstate.edu/coursewikis/GSR
Differential Forms and the Geometry of General Relativity
Tevian DrayA K Peters/CRC Press 2014http://physics.oregonstate.edu/coursewikis/GDF
http://physics.oregonstate.edu/coursewikis/GGR
Tevian Dray The Geometry of Relativity
IntroductionSpecial RelativityGeneral Relativity
Books
The Geometry of Special Relativity
Tevian DrayA K Peters/CRC Press 2012ISBN: 978-1-4665-1047-0http://physics.oregonstate.edu/coursewikis/GSR
Differential Forms and the Geometry of General Relativity
Tevian DrayA K Peters/CRC Press 2014http://physics.oregonstate.edu/coursewikis/GDF
http://physics.oregonstate.edu/coursewikis/GGR
The Geometry of Vector Calculus
Tevian Dray & Corinne A. Manogue
online only: http://physics.oregonstate.edu/coursewikis/GVC
Tevian Dray The Geometry of Relativity
IntroductionSpecial RelativityGeneral Relativity
Hyperbolic TrigonometryApplications
Trigonometry
β
βB
t’
A
x’
t
x
•
β coshρ
β sinhρρ
β
•
Tevian Dray The Geometry of Relativity
IntroductionSpecial RelativityGeneral Relativity
Hyperbolic TrigonometryApplications
Length Contraction
x’
t’t
x
x’
t’t
x
ℓ ′ = ℓcoshβ
β β
ℓ
ℓ ′
•
ℓ
ℓ ′
•
Tevian Dray The Geometry of Relativity
IntroductionSpecial RelativityGeneral Relativity
Hyperbolic TrigonometryApplications
Paradoxes
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
Tevian Dray The Geometry of Relativity
IntroductionSpecial RelativityGeneral Relativity
Hyperbolic TrigonometryApplications
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
Tevian Dray The Geometry of Relativity
IntroductionSpecial RelativityGeneral Relativity
The MetricDifferential FormsGeodesicsEinstein’s Equation
Line Elements
a
a
dr2 + r2 dφ2 dθ2 + sin2 θ dφ2 dβ2 + sinh2 β dφ2
Tevian Dray The Geometry of Relativity
IntroductionSpecial RelativityGeneral Relativity
The MetricDifferential FormsGeodesicsEinstein’s Equation
Line Elements
a
a
dr2 + r2 dφ2 dθ2 + sin2 θ dφ2 dβ2 + sinh2 β dφ2
Tevian Dray The Geometry of Relativity
IntroductionSpecial RelativityGeneral Relativity
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
Tevian Dray The Geometry of Relativity
IntroductionSpecial RelativityGeneral Relativity
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)
Tevian Dray The Geometry of Relativity
IntroductionSpecial RelativityGeneral Relativity
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)
Exterior derivative: (d2 = 0)
df = ~∇f · d~r
dF = ~∇× ~F · d~A
d∗F = ~∇ · ~F dV
d∗f = 0
Tevian Dray The Geometry of Relativity
IntroductionSpecial RelativityGeneral Relativity
The MetricDifferential FormsGeodesicsEinstein’s Equation
The Geometry of Differential Forms
dx
Tevian Dray The Geometry of Relativity
IntroductionSpecial RelativityGeneral Relativity
The MetricDifferential FormsGeodesicsEinstein’s Equation
The Geometry of Differential Forms
v×
vx
dx
Tevian Dray The Geometry of Relativity
IntroductionSpecial RelativityGeneral Relativity
The MetricDifferential FormsGeodesicsEinstein’s Equation
The Geometry of Differential Forms
v×
vx
dx
dx + dy r dr = x dx + y dy
Tevian Dray The Geometry of Relativity
IntroductionSpecial RelativityGeneral Relativity
The MetricDifferential FormsGeodesicsEinstein’s Equation
Geodesic Equation
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
Tevian Dray The Geometry of Relativity
IntroductionSpecial RelativityGeneral Relativity
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
(vector valued 1-forms, not tensors)
Tevian Dray The Geometry of Relativity
IntroductionSpecial RelativityGeneral Relativity
SUMMARY
http://physics.oregonstate.edu/coursewikis/GSR
http://physics.oregonstate.edu/coursewikis/GDF
http://physics.oregonstate.edu/coursewikis/GGR
http://physics.oregonstate.edu/coursewikis/GVC
Tevian Dray The Geometry of Relativity
IntroductionSpecial RelativityGeneral Relativity
SUMMARY
http://physics.oregonstate.edu/coursewikis/GSR
http://physics.oregonstate.edu/coursewikis/GDF
http://physics.oregonstate.edu/coursewikis/GGR
http://physics.oregonstate.edu/coursewikis/GVC
Special relativity is hyperbolic trigonometry!
Tevian Dray The Geometry of Relativity
IntroductionSpecial RelativityGeneral Relativity
SUMMARY
http://physics.oregonstate.edu/coursewikis/GSR
http://physics.oregonstate.edu/coursewikis/GDF
http://physics.oregonstate.edu/coursewikis/GGR
http://physics.oregonstate.edu/coursewikis/GVC
Special relativity is hyperbolic trigonometry!
Spacetimes are described by metrics!
Tevian Dray The Geometry of Relativity
IntroductionSpecial RelativityGeneral Relativity
SUMMARY
http://physics.oregonstate.edu/coursewikis/GSR
http://physics.oregonstate.edu/coursewikis/GDF
http://physics.oregonstate.edu/coursewikis/GGR
http://physics.oregonstate.edu/coursewikis/GVC
Special relativity is hyperbolic trigonometry!
Spacetimes are described by metrics!
General relativity can be described without tensors!
Tevian Dray The Geometry of Relativity
IntroductionSpecial RelativityGeneral Relativity
SUMMARY
http://physics.oregonstate.edu/coursewikis/GSR
http://physics.oregonstate.edu/coursewikis/GDF
http://physics.oregonstate.edu/coursewikis/GGR
http://physics.oregonstate.edu/coursewikis/GVC
Special relativity is hyperbolic trigonometry!
Spacetimes are described by metrics!
General relativity can be described without tensors!
BUT: Need vector-valued differential forms...
Tevian Dray The Geometry of Relativity
IntroductionSpecial RelativityGeneral Relativity
SUMMARY
http://physics.oregonstate.edu/coursewikis/GSR
http://physics.oregonstate.edu/coursewikis/GDF
http://physics.oregonstate.edu/coursewikis/GGR
http://physics.oregonstate.edu/coursewikis/GVC
Special relativity is hyperbolic trigonometry!
Spacetimes are described by metrics!
General relativity can be described without tensors!
BUT: Need vector-valued differential forms...
THE END
Tevian Dray The Geometry of Relativity
IntroductionSpecial RelativityGeneral Relativity
Tevian Dray The Geometry of Relativity
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