SpacetimeTrigonometry:a Cayley-Klein geometry

approach to Special and General Relativity

SpacetimeTrigonometry:a Cayley-Klein geometry

approach to Special and General Relativity

General Relativity and Gravitation: A Centennial Perspective (Penn State U., June 2015)

Roberto B. Salgado (rsalgado@uwlax.edu)Physics Dept., U.Wisconsin-La Crosse

AbstractIn two dimensions, there are 9 Cayley-Klein geometries: Euclidean, Elliptic, and Hyperbolic, plus 6 others of physical interest: Minkowski, deSitter, and anti-deSitter, and their Galilean limits. Using familiar techniques from Euclidean geometry and trigonometry, we present a unifiedformalism for these nine geometries. We develop the corresponding analogues for Galilean and Minkowski spacetimes and immediately provide them with physical interpretations.We lay out the foundations of an idealized curriculum intended for physics students from high-school geometry to intro General-Relativity.

Cayley-Klein Geometries

1= −2ε 0=2ε 1= +2

ε

2 1η = −

2 1η = +

2 0η =

Measure of the angle between two lines

Measure

of

the l

en

gth

betw

een

tw

o p

oin

ts

elliptic parabolic hyperbolic

elliptic Elliptic anti-Newton-Hooke(co-Euclidean)

anti-DeSitter(co-Hyperbolic)

parabolic Euclidean Galilean Minkowski

hyperbolic Hyperbolic Newton-Hooke(co-Minkowski)

DeSitter

2 22

2 2 2

2 22 2

2

2

2

2(1 ) (1 ) 2(1 ( ))

y t tydt dydS

t yη η η

η+ − −=

− −

2 2dt dyε ε ε

ε

−Metric signature and curvature(1, )− 2

ε κ η= − 2

from the cross-ratio of projective geometry

aside: the algebra of hypercomplex numbers

( ) ( )( ) ( )2

1 0 0: 1 , 0 1 0

0: , 1 0

aa a

aa b b a

= =

= + =−=

real unit

complex unit 1 i- -b1

εε

( ) ( )( ) ( )

2

2

0: , 1 0 (but 0)

0: , 1 0 (but 1)

aa b b a

aa b b a

= + == ≠

= + == ≠

dual unit

double unit

0 00

1 b+1

ε εε ε

ε εε ε

Circles and MetricsSurveyors [observers] travel along all possible directions [velocities], stopping when their odometers read 1 mi [1 sec]. This defines a “circle”, with “perpendicular”defined as “being tangent to that circle”.

2dS = 2 2dt dy− 2ε

-1

1

y

-1 1 t

-1

1

y

1-1 t

-1

1

y

-1 1 t

-1

1

y

-1 1 t

-1

1

y

-1 1 t

-1

1

y

-1 1 t

With coinciding tangents, “absolute simultaneity”

EuclideanEuclidean GalileanGalilean MinkowskiMinkowski

1 = −2ε 0 =2ε 1 = +2

ε

2R = 2 2t y− 2ε

points withconstant t’for this surveyor

“relativity ofsimultaneity”

2( / )ligh mt axc c=2ε

Angle from Spacelike Arc-Length2 2 2 21 1

22 2

( ) ( ) for 0( ) for 0.dy dt dS

dLdy

− − ≠≡=

=

ε

ε

2 2ε ε

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

y

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 t

( )( )

2 2 111 11

1

sinh

y

R RyR

R R

dy dtdL

d y

−− =Θ ≡ = =

⌠⌡∫∫

2ε

εε

sin

sinh

cos

cosh

tan

tanh

( )

( )

( )

E

G

M

E

M

E

G

M

yt

RR

R

RR

Rt

R

R

y

v

θθ

θ

θ

θ

θθ

θ∆∆

= =

=

Θ

Θ

Θ

=

= = =

SINH

COSH

TANHΘ

1tanE vθ −=

G vθ =

1tanhM vθ −=

real-valued generalized

trig functions of real-values

real-valued generalized

trig functions of real-values

slope v

Spacetime Trigonometry

2 54 3

2! 4! 3! 5!

cos sincosg 1 singcosh sinh

1

EXP

COSH S

( ) exp( )

( ) ( )INH

E E

G G G

M M

θ θ θ θ

θ θθ θ θθ θ

θ

θ

≡ + ≡

+ + + + + +

≡

≡

= +

Θ

+ Θ

=

Θ

⋯ ⋯2 4 2 4ε ε ε ε

ε

ε

ε

ε I.M. Yaglom named the “Galilean trig functions”

real-valuedgeneralized

trig functionsof real-values

real-valuedgeneralized

trig functionsof real-values

2

EXP EXP

SINH COSH

COSH SINH

VERSINH SINH

ddd

dd

dd

d

Θ = ΘΘ

Θ = ΘΘ

Θ = ΘΘ

Θ = ΘΘ

ε

ε

2 2 2

1 21 2 2

1 2

2 2 1/2

22

1 COSH SINH

TANH TANHTANH( )

1 TANH TANH

COSH (1 TANH )

1 COSH 2SINVERSIN H2

H

−

= Θ − ΘΘ + ΘΘ + Θ =

+ Θ Θ

Θ = − Θ− Θ ΘΘ ≡ =

ε

ε

ε

ε

Differential identities Algebraic identities

Boosts and Eigenvectors

For a metric , the linear transformation R

( satisfying )

that preserves it is:

det 1 (0)( ) ( ) ( )

R R IR G RGR R R=

= =Θ Φ = Θ + Φ⊤

21 00

G = − ε

2COSH(SINH CO

SINH)SH

R Θ Θ =

Θ Θ Θ ε

Minkowski

Galilean

eigenvectoreigenvalue

( )01

( ) ( )1 1,1 1−1 tanh

exp(1 ta

)nh

MM

M

θθ θ±± =∓

1 “absolute length”

Doppler-Bondi

“absolute time”

“absolute speed

of light”

Components by Projection

0

1

1 t

sin Eθ

sing Gθ

sinh Mθ slope v=TANHΘ

cos Eθ cosg Gθ cosh Mθ

y

“time dilation”

Law of Cosines (“Clock Effect”)

A

C

Ba�

c�

b�

S

( )

2 2 2

2 2 2

2 COSH( )

2

AB AC CB AB CB

AC CB AC CB AC CB

BCt t t t t

t t t

S

t t t

= + +

+ + +≡

∡

Upon comparing…

“triangle inequality”

non-“clock-effect”

“clock-effect”

2

2

2

for for

1

fo0

1r

AB AC CB

AB AC CB

AB AC CB

t tt

t t

tttt

< += +>

= −=

++ =

ε

ε

ε

Doppler Effect

O TS

TR

SINHRT Θ

SINHRT Θ

COSHRT Θ

2

(1)(1COSH

COS) (1 )

(1 TANH ) 1H

SIN

1(1 )11

HR

R R

RR R

S RTT T

T T T

v vvvv

T T

v

Θ −− −

Θ − Θ −− +

Θ=

−

== = =

COSH

Gal (1 )1Mink ex

(1

p(

ANH

1

T )

)M

S

S

R

S S

f

f v

f f

f

vv

θ−−− +

Θ − Θ

= =

=

Receding receiver

Θ

ΘO TR

TS

COSHST Θ SINHST Θ

2

1 1(1)(1 ) (1 )

(1 TAN

COSH1

(COSH SINH )

1CO H )SH 1 11

(1 1

S NH

)

IR

R

R R

R

S

R R

S

S

TT

T

T TT

v vvvT T

T

v

T

v

Θ +

Θ +==

+ +Θ + Θ −− + +

=Θ

==

Θ

=

COSH1Gal

(1 )1 1Min

1

exp( ) 1

(1 TANH )

S

M

S

S S

R f

vv

f

f

fv

fθ

Θ + Θ

=

=

+

+= −

Receding source

SINHST Θ

Curve of Constant-Curvature “Uniform Acceleration”

The “curvature ρ of a plane curve y(t)”measures

how the angle φ of the tangent vector varies with arc-length s.

( )3/22 21 ( )

ytd d dds dt ds y

φ φρ = =≡−ɺɺ

ɺε

φs

1

2

1 2

( ) ( ){

2

212

2 2

2

2 2

1 1/ 2)

1

2 (

sin sinh, , 1 1

Euc

1 1cos cosh

) 1) 1

1(

( (

Gal ) ( ) 02

) ( 1)

Mink ( 1

M

MG

GE

E

SINHt SINHy VERSI

t y

t y

NH SINH

t y

ρ ρθθ θ

θ θθρ ρ ρ

ρ ρ

ρ ρ

ρ ρ

= =

= − −

ΘΘΘ Θ

− =

=

− + = −

+= −

By integration,

Lagrangian and Hamiltonian?

Kinetic Energy

2

0

2COSH 1

ˆ( )ˆ( ) ·

TANH ( SINH ) CO

Work

SH

SINH

VERSINH 2SINH2

f

fv

ff

d pyd y v dp

dt

d m m d

m d

m m mΘ −

=

= Θ Θ = ΤΑΝΗΘ Θ Θ= Θ Θ

Θ = Θ ≡ ≡

= ⌠⌡ ∫

∫ ∫∫

ℓ

ε

( )2

2

2 2 2 2

1cosh 1 11

1 1 1 2sing tang2

2sinh2

2 2 2G G

MM

G

m mv

m m m m

m

v

θθ θ θ

θ

= =

= − = − = −

=

( ) ( )COSHSINH

E mp = ΘΘ

Lagrangian? Hamiltonian?

Can the free-particle Lagrangians be unified? With Optics?

( ) ( )

( )

22 2 2 2 2 2

2

2

22 2

?( ) ( ) ( )

COSH

Mink ( )( sech

( )

Euc[Optics]

2SINH ( / 2)1 SECH VERSINHCOSH

)1 1Gal

( )(2 2

sec )

M

E

L mc mc mc

mc

mc

n

v mvc

θ

θ

= = =

−

=

Θ− Θ Θ− − −Θ Θ

=

ε ε εε

Visual Tensor AlgebraaV“pole [vector]”

aab bg V = V

“polar [hyperplane]”abgabg

“metric”Through the pole,

draw the tangents to the conic.This construction is due to W.Burke

Applied Differential Geometry

In the spirit of the visualization of tensors by Schouten, Misner-Thorne-Wheeler, and Burke… we can draw accurate representations of the metric tensors and the operations of index-lowering and raising.

Euclidean

Galilean Minkowskian

123

45

12

34

1

2

30

Electromagnetism?

· 0

·

BB E

dDD H jd

d

t

dt

α

ρ β

∇ = ∇× = −

∇ = ∇× = +

�� � � �

�� � � � �

LeBellac & Levy-Leblond’s Galilean-invariant electromagnetism as a stepping stone to Lorentz-invariant electomagnetism, as suggested by Jammer & Stachel?

2

SINH

SINH COSHx

x x

COSH j

j jβ

ρ ρβρ′

′ = Θ + ΘΘ += Θ

ε

2

2

COSH ( )

( )

COSH ( )

COSH

COSH

) (

E E E E v B

B B B B v E

D D D D v H

H H H H v D

α

β

α

β

′ ′⊥ ⊥

′ ′⊥ ⊥

′ ′⊥ ⊥

′ ′⊥ ⊥

= = Θ + ×

= = Θ − ×

= = Θ + ×

= = Θ − ×

� � � � ��

� � � � ��

� � � � ��

� � � � ��

‖ ‖

‖ ‖

‖ ‖

‖ ‖

ε

ε

, D E B Hκ µ= =� � � �

2

22

( ,1,0) "electric limit"(1, ,0) "magnetic limit"( , , )(1,1,0) "ether limit"(1,1,1) "Lorentz-invariant"

α β

ε

εε

Acknowledgments and Thanks…• to the Physics Department at UW-La Crosse for funding.• to students at UW-La Crosse and Mount Holyoke who took my relativity classes and tried out some material based on this work.

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Emch, G.G., Mathematical and conceptual foundations of 20th-century physics, North-Holland, 1984, Ch. 4.

Birman, G.S. & Nomizu, K., “Trigonometry in Lorentzian Geometry“, The American Mathematical Monthly, Vol. 91, No. 9 (1984), pp. 543-549

Herranz, F.J., Ortega, R., & Santander, M., “Trigonometry of spacetimes: a new self-dual approach to a curvature/signature (in) dependent trigonometry”, Journal of Physics A: Mathematical and General, Vol. 33, No. 24 (2000), p4525.

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Burke, W.L., Spacetime, Geometry, Cosmology, University Science Books, 1980.Burke, W.L., Applied Differential Geometry, Cambridge,1985.

Schouten, J.A., Tensor Analysis for Physicists, Clarendon Press, 1954

LeBellac, M. & Levy-Leblond, J.M., “Galilean electromagnetism”, Il Nuovo Cimento, Vol. 14B, No. 2 (1973), pp 217-234

Jammer, M. & Stachel, J., "If Maxwell had worked between Ampere and Faraday:An historical fable with a pedagogical moral“, Am. J. Phy, Vol. 48, No. 1 (1980), pp 5-7