Introduction

Special Relativity
General Relativity
Curriculum

The Geometry of Relativity

Tevian Dray

Department of Mathematics
Oregon State University
http://www.math.oregonstate.edu/~tevian

Tevian Dray The Geometry of Relativity

 Introduction
Special Relativity
General Relativity
Curriculum

Books

The Geometry of Special Relativity

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 General Relativity
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

Tevian Dray The Geometry of Relativity

 Introduction
Special Relativity Hyperbolic Trigonometry
General Relativity Applications
Curriculum

Trigonometry

t t’
ρ
A ρ sinh β

β β
•
B
x’ ρ cosh β
β
x

Tevian Dray The Geometry of Relativity

 Introduction
Special Relativity Hyperbolic Trigonometry
General Relativity Applications
Curriculum

Length Contraction

t t’ t t’

x’ x’

x x

ℓ
ℓ′ = cosh β

ℓ′ ℓ •

β • β

ℓ ℓ′
Tevian Dray The Geometry of Relativity
 Introduction
Special Relativity Hyperbolic Trigonometry
General Relativity Applications
Curriculum

Paradoxes
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 -20

barn frame pole frame

Tevian Dray The Geometry of Relativity

 Introduction
Special Relativity Hyperbolic Trigonometry
General Relativity Applications
Curriculum

Relativistic Mechanics
A pion of (rest) mass m and (relativistic) momentum p = 43 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. [E1 = mc 2 , E2 = 41 mc 2 ]
Β
p0 c sinh Β

p 0c
p0 c Β

E0 c cosh Β

sin
hΒ
p0 c Β
pc p2 c E0 p2 c
E2 Β
0

E2

E0
0 E0

cc
osh
mc2
Β

Β
E0 c cosh Β

Β
E mc2 0
E0
E0

E0
cc
p0 c
0

osh
Β

Β
E1

p 0c
p0 c sinh Β

E1
sin
hΒ
Β p0 c p1 c
Β
p1 c
Tevian Dray The Geometry of Relativity
 Introduction The Metric
Special Relativity Differential Forms
General Relativity Geodesics
Curriculum Einstein’s Equation

Line Elements

dr 2 + r 2 dφ2 dθ2 + sin2 θ dφ2 dβ 2 + sinh2 β dφ2

Tevian Dray The Geometry of Relativity

 Introduction The Metric
Special Relativity Differential Forms
General Relativity Geodesics
Curriculum Einstein’s Equation

Vector Calculus

ds 2 = d~r · d~r

dy |^
d~r d~r
dr r^
r d ^
dx ^{

d~r = dx ı̂ + dy ̂ = dr r̂ + r dφ φ̂

Tevian Dray The Geometry of Relativity

 Introduction The Metric
Special Relativity Differential Forms
General Relativity Geodesics
Curriculum Einstein’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: (d 2 = 0)
~ · d~r
df = ∇f
~ × ~F · d A
dF = ∇ ~
~ · ~F dV
d∗F = ∇
d∗f = 0

Tevian Dray The Geometry of Relativity

 Introduction The Metric
Special Relativity Differential Forms
General Relativity Geodesics
Curriculum Einstein’s Equation

The Geometry of Differential Forms

dx + dy r dr = x dx + y dy

dx

Tevian Dray The Geometry of Relativity

 Introduction The Metric
Special Relativity Differential Forms
General Relativity Geodesics
Curriculum Einstein’s Equation

Geodesic Equation

Orthonormal basis: d~r = σ i êi

Connection: ωij = êi · d êj

i i
dσ + ω j ∧ σ j = 0
ωij + ωji = 0

Geodesics: ~v dλ = d~r
~v˙ = 0

Symmetry: ~ · d~r = 0
dX
~ · ~v = const
=⇒ X

Tevian Dray The Geometry of Relativity

 Introduction The Metric
Special Relativity Differential Forms
General Relativity Geodesics
Curriculum Einstein’s Equation

Einstein’s Equation

Curvature:
Ωi j = dω i j + ω i k ∧ ω k j

Einstein tensor: 1
γ i = − Ωjk ∧ ∗(σ i ∧ σ j ∧ σ k )
2
G = ∗γ i = G i j σ j
i

~ = G i êi = G i j σ j êi
G
~ =0
=⇒d∗G

Field equation: ~ + Λ d~r = 8π T

G ~
(vector valued 1-forms, not tensors)

Tevian Dray The Geometry of Relativity

 Introduction
Special Relativity
General Relativity
Curriculum

Does it work?

I am a mathematician...
There is no GR course in physics department.
(I developed the SR course.)
Core audience is undergraduate math and physics majors.
(Many double majors.)
Hartle’s book:
Perfect for physics students, but tough for math majors.
My course: 10 weeks differential forms, then 10 weeks GR.
(Some physics students take only GR, after “crash course”.)

In this context: YES!

Tevian Dray The Geometry of Relativity

 Introduction
Special Relativity
General Relativity

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 metrics!
General relativity can be described without tensors!
BUT: Need vector-valued differential forms...

THE END

Tevian Dray The Geometry of Relativity