the geometry of relativity - natural sciences &...

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

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


Books

The Geometry of Special Relativity

Tevian DrayA K Peters/CRC Press 2012ISBN: 978-1-4665-1047-0http://physics.oregonstate.edu/coursewikis/GSR


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


Books



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



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



Books



Differential Forms and the Geometry of General Relativity

Tevian DrayA K Peters/CRC Press 2014http://physics.oregonstate.edu/coursewikis/GDF


The Geometry of Vector Calculus

Tevian Dray & Corinne A. Manogue

online only: http://physics.oregonstate.edu/coursewikis/GVC





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


Hyperbolic TrigonometryApplications

Trigonometry

β

βB

t’

A

x’

t

x

•

β coshρ

β sinhρρ

β

•




Length Contraction

x’

t’t

x

x’

t’t

x

ℓ ′ = ℓcoshβ

β β

ℓ

ℓ ′

•

ℓ

ℓ ′

•




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




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



The MetricDifferential FormsGeodesicsEinstein’s Equation

Line Elements

a

a

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




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




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)




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




The Geometry of Differential Forms

dx





v×

vx

dx





v×

vx

dx

dx + dy r dr = x dx + y dy




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




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)



SUMMARY











SUMMARY





Special relativity is hyperbolic trigonometry!







SUMMARY






Spacetimes are described by metrics!







SUMMARY







General relativity can be described without tensors!







SUMMARY








BUT: Need vector-valued differential forms...







SUMMARY








BUT: Need vector-valued differential forms...

THE END






the geometry of relativity - natural sciences &...

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