spacetime trigonometry: a cayley-klein geometry approach...
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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 ([email protected])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.
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