Naber G.L. The Geometry of Minkowski Spacetime. An Introduction to the Mathematics of the Special Theory of Relativity
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314 Symbols
¯
ß=ß×{1} “conjugate” of ß, 157
¯
φ (φ, 1) ∈
¯
ßforφ ∈ ß, 157
{¯s
˙
X
}, {
¯
ˆs
˙
X
},... conjugate spin frames, 158
¯
φ
˙
X
,
¯
ˆ
φ
˙
X
,... components of
¯
φ, 158
¯
G =
,
¯
G
˙
X
˙
Y
-
conjugate of G =
G
A
B
, 158
¯
ß
∗
dual of
¯
ß, 158
{¯s
˙
X
}, {
¯
ˆs
˙
X
},... dual conjugate spin frames, 158
¯
φ
∗
element of
¯
ß
∗
conjugate to φ
∗
∈ ß
∗
, 159
¯
φ
˙
X
,
¯
ˆ
φ
˙
X
,... components of
¯
φ
∗
, 159
rs
mn
valence of a spinor, 159
ξ
A
1
···A
r
˙
X
1
···
˙
X
s
B
1
···B
m
˙
Y
1
···
˙
Y
n
spinor components, 159
=
*
0 −1
10
+
=
*
11
10
01
00
+
=[
AB
]=[¯
˙
X
˙
Y
]=¯
˙
X
˙
Y
, 161
ß
rs
mn
space of spinors of valence
rs
mn
, 164
C
kl
(ξ), C
˙
k
˙
l
(ξ),... contractions of ξ, 166, 167
¯
ξ spinor conjugate of ξ, 167
ξ
(AB)
symmetrization of ξ
AB
, 168
ξ
[AB ]
skew-symmetrization of ξ
AB
, 169
α
(A
β
B)
symmetrization of α
A
β
B
, 169
σ
a
A
˙
X
,σ
a
A
˙
X
Infeld-van der Waerden symbols, 169
V
A
˙
X
spinor equivalent of v ∈M, 170
V
A
˙
X
spinor equivalent of v
∗
∈M
∗
, 175
F
A
˙
XB
˙
Y
spinor equivalent of the bilinear form F , 179
φ
AB
symmetric spinor determined by F
A
˙
XB
˙
Y
, 180
ξ
A
→ e
iθ
ξ
A
phase change, 185
∇
A
˙
X
spinor differential operator, 197
R
n
real n-space, 202
p, q Euclidean inner product on R
n
, 202
p Euclidean norm on R
n
, 202
U
ε
(p) open ball of radius ε about p in R
n
, 203
C
∞
smooth, 203
S
n
n-sphere, 203
(U, ϕ) chart, 206
χ : U → M coordinate patch, 211
χ
i
coordinate velocity vector, 218
T
p
(M) tangent space to M at p, 218
(p, q) Minkowski inner product on R
5
, 221
M
5
5-dimensional Minkowski space, 221
Γ
r
ij
Christoffel symbols, 230
Symbols 315
H
3
(r) hyperbolic 3-space, 240
R Riemann curvature tensor, 246
R
ij
Ricci tensor, 249
R scalar curvature, 249
G
ij
Einstein tensor, 249
T
ij
energy-momentum tensor, 250
F
∗p
derivative of F at p, 252
E Einstein static universe, 255
F
∗
g pullback metric, 260
I
−
past null infinity, 266
I
+
future null infinity, 266
i
−
past timelike infinity, 273
i
+
future timelike infinity, 273
i
0
spacelike infinity, 274
d
E
E-distance, 279
N
E
ε
(x
0
) open Euclidean ε-ball about x
0
, 280
M
E
M with the Euclidean topology, 280
Cl
E
A, bdy
E
A,... Euclidean closure, boundary,... of A ⊆M
C(x) C
−
T
(x) ∪C
+
T
(x) ∪{x}, 284
N
P
ε
(x
0
) P -open ε-ball about x
0
, 284
M
P
M with the path topology, 285
Cl
P
A, bdy
P
A,... path closure, boundary, ... of A ⊆M
H(X) homeomorphism group of X, 292
[α] homotopy class of the path α, 297
α
−1
inverse of the path α, 297
βα product of the paths α and β, 297
π
1
(X, x
0
) fundamental group of X at x
0
, 297
π
1
(X) fundamental group of X, 297
(
˜
X, p) universal covering manifold of X, 299
RP
2
real projective plane, 300
Z
2
group of integers mod 2
RP
3
real projective 3-space, 301
RP
n
real projective n-space, 301
SO(3) rotation group, 302
Index
A
aberration formula, 86
accelerations, 38, 48, 49, 82, 201, 228, 230
active transformations, 73, 77, 80
addition of velocities formula, 26
admissible basis, 19
admissible frame of reference, 19
local, 201
nonexistence of, 6
advanced null coordinates, 270
affine parametrization, 262
α-emission, 87
anti-isomorphism, 158
B
barn paradox, 41
bilinear form, 7
components of, 120
matrix of, 120
nondegenerate, 7
skew-symmetric, 119
spinor equivalent of, 192
symmetric, 7
binding energy, 88
binomial expansion, 82, 85, 91
Biot-Savart Law, 127
bivector, 120
spinor equivalent of, 180
boost, 26
C
canonical basis, 107
canonical forms, 107, 109, 195
carriers of a representation, 138
Cartan, E., 135
catenary, 117
causal automorphism, 59, 60, 223, 292
causal precedence, 58
causality assumption, 4
causality relations, 58, 222, 227
Cayley-Hamilton Theorem, 103
change of basis formula, 119
characteristic equation, 99
charge, 93
charge-to-mass ratio, 116
charged particle, 93
Christoffel symbols, 230
chronological precedence, 58
Clock Hypothesis, 48
clocks, 2, 36, 48
atomic, 2
synchronization of, 3
closed
in R
n
, 203
in X, 203
commutation relations, 169
Compton effect, 88
Compton wavelength, 90
cone, 4
null, 11, 221, 268
time, 16, 49, 221
configuration space, 302
conformal diffeomorphism, 259
conformal embedding, 260
conformally related metrics, 260
317
318 Index
conjugate isomorphism, 158
conjugate spin covector, 158
conjugate spin vector, 158
conjugate spinor, 158, 168
conjugate transpose, 69
conjugation representation, 146
connectible by a light ray, 14
conservation of energy, 87, 90
conservation of momentum, 81, 87
conservation of 4-momentum, 87
conservation laws, 81, 88
constant curvature, 248, 253
constant electromagnetic fields, 123
constant loop, 297
contact interaction, 87
continuous, 203
contraction, 166
contravariant rank, 141
contravariant vector, 141
coordinate curve, 218
coordinate functions, 50, 202, 203
coordinate patch, 211
coordinate transformation, 219
coordinates
spatial, 2, 10, 29, 253
time, 3, 10, 253
cosmic rays, 23
cosmological constant, 250
cosmological model, 252
Coulomb field, 123
total energy of, 125
covariant rank, 141
covariant vector, 141
covector, 141
covering map, 299
covering transformation, 299
curve, 47, 217
component functions, 47
E-continuous future- (past-) timelike,
280
null, 47, 226
reparametrization of, 47
smooth, 47, 217
spacelike, 47, 226
timelike, 47, 226
cylinder, 242
locally flat, 242
D
decay, 87
derivative of a smooth map, 252
de Sitter spacetime, 224
Christoffel symbols for, 231
conformal coordinates, 215
constant curvature, 249
Einstein tensor, 249
geodesics of, 240
global coordinates, 214
hyperbolic coordinates, 217
line element for, 226
Lorentz metric for, 224
null infinity, 266
planar coordinates, 215
Ricci tensor, 249
Riemann curvature tensor, 247
and Robertson-Walker metrics, 254
scalar curvature, 249
diffeomorphism, 203
dilation, 60
time, 22
Dirac
equation, 135, 136
Scissors Problem, 295
direct sum, 43
direction 3-vector, 22
of a photon, 84
of a reference frame, 22
disintegration, 87
displacement vector, 10
null, 10
spacelike, 12
timelike, 12
distance, 57
measured with clocks, 57
divergence, 118
dominant energy condition, 113, 123, 195
Doppler effect, 85
transverse, 85
double cover, 304
dual
basis, 139, 157
of a bivector, 122
duration, 43
E
E-continuous curve, 280
future-timelike, 281
future-timelike at t
0
, 281
past-timelike, 281
past timelike at t
0
, 281
Index 319
timelike, 281
timelike vs smooth timelike, 281
E-continuous map, 280
E-distance, 279
E-open ball, 280
E-open set, 280
E-topology, 280
eigenspace, 98
eigenspinor, 190
eigenvalue
of a linear transformation, 98
of a spinor, 190
eigenvector, 98
Einstein field equations, 250
empty space solutions, 250
Einstein static universe, 255
geodesics, 258
line element, 256
Einstein summation convention, xi
electric 3-vector, 95
electromagnetic field, 94, 120
constant, 113, 123
null, 99
regular, 99
spinor, 186
electromagnetic spinor, 186
electromagnetic wave
simple plane, 129
spherical, 3, 17, 35
electron, 88
spin, 135, 294
elevator experiment, 200
energy, 82
Coulomb field, 125
density, 111
of a photon, 84
total relativistic, 82
energy-momentum tensor, 250
energy-momentum transformation, 109
spinor form of, 195
equation of motion, 93
Euler angles, 72
event horizon, 268
events, 1, 9
expanding universe, 255
extended complex plane, 73, 76, 79
exterior derivative, 120
F
Feynman path, 288
Feynman track of an electron, 288
field equations, 198, 250, 255, 277
field of vision, 79
Fizeau procedure, 3
flag pole of a spin vector, 179
4-acceleration, 51
4-momentum
of a material particle, 81
of a photon, 84
4-tensor, 141
4-vector, 169
contravariant, 141
covariant, 141
4-velocity, 50
fractional linear transformation, 73, 77
frame of reference, 3, 10
admissible, 4, 19
free charged particle, 93
free particle, 87
frequency of a photon, 84
fundamental group, 297
of a product, 298
of real projective space, 300, 301
of the rotation group, 303
future-directed
null curve, 47
null vector, 17
timelike curve, 47
timelike vector, 16
future null cone, 17
future null direction, 74
future null infinity, 266, 274
future time cone, 16
future timelike infinity, 273
G
Gaussian curvature, 243
of cylinder, 243
of S
2
, 243
general linear group, 138
general theory of relativity, 202
geodesic, 230
affine parametrization, 262
causal character, 234
constant speed, 233
degenerate, 230
existence and uniqueness, 232
of de Sitter spacetime, 235
of Minkowski spacetime, 234
of S
2
, 234
of S
3
, 235
reparametrization of, 232
320 Index
geodesic hypothesis, 250
gravitation, 6, 199, 249
group representation, 138
H
Hermitian matrix, 170
homeomorphism, 203, 292
homeomorphism group, 292
homomorphism, 14, 138
homotopic paths, 297
homotopy, 297
hyperbolic form of special Lorentz
transformations, 27
hyperbolic motion, 55
I
identity representation, 138, 141, 146
index of an inner product, 9
inelastic collision, 90
inertial mass, 81
Infeld-van der Waerden symbols, 173
initial point on a worldline, 87
inner product, 7
indefinite, 7
index of, 9
Lorentz, 9
negative definite, 7
positive definite, 7
instantaneous rest frame, 53
invariant subspace
of a linear transformation, 99
of a representation, 143
Inverse Function Theorem, 208
isometry, 252
K
Kennedy-Thorndike experiment, 3
kernel, 68, 97
L
length, 39
contraction, 40
Levi-Civita symbols, 121
spinor equivalent of, 196
light cone, 11
future and past, 17
light ray, 11
as an intersection of null cones, 11
light signals, 3, 17
light travel time, 3
lightlike vector, 10
line element, 225
loop, 297
constant (trivial), 297
Lorentz contraction, 40
invisibility of, 79
Lorentz 4-Force Law, 93
Lorentz group, 19
general homogeneous, 14
inhomogeneous, 19
2-valued representations of, 142, 144
Lorentz inner product, 9
spinor equivalent of, 192
Lorentz invariant, 99, 136, 141
Lorentz transformations, 14, 19
boost, 26
decomposition of, 28
determined by three past null directions,
79
effect on past null directions, 79
general homogeneous, 14
hyperbolic form, 27
improper, 19
invariant null directions, 79
nonorthochronous, 15
orthochronous, 15
proper, 19
special, 25
vs fractional linear transformations, 77
Lorentz World Force Law, 93
lowering indices, 162
M
magnetic 3-vector, 95
manifold, 206
complete, 238
n-dimensional, 206
smooth, 206
Index 321
mass
inertial, 81
proper, 81
relativistic, 82
mass-energy equivalence, 83
massless free-field equations, 198, 277
material particle, 81
worldline of, 47
free, 81
matrix group, 137
order of, 137
representation of, 138
matrix of a bilinear form, 120
Maxwell’s equations, 118, 122
solutions of, 123, 124, 129, 130
spinor form of, 197
measuring rods, 39, 40
metric, 219
component functions, 219
Lorentzian, 220
Riemannian, 219
Michelson-Morley experiment, 3
Minkowski diagram, 32, 266
Minkowski spacetime, 9
multilinear functional, 140, 159
N
neutrino equation, 198
null basis, 10
null cone, 11
future, 17
past, 17
null direction, 61
future, 74
past, 74
null electromagnetic field, 99
null flag, 185
null vector, 10
future-directed, 17
parallel, 10
past-directed, 17
orthogonal, 10
null worldline, 11
O
observer, 1
admissible, 1
open
in Minkowski spacetime, 117
in R
n
, 203
in X, 203
orientation-entanglement relation, 304
orthochronous, 15
orthogonal complement, 7
orthogonal transformation, 12, 222
and causal automorphisms, 60, 227
and fractional linear transformations, 77
and homeomorphisms, 292
associated matrices, 13
invariant null directions, 79
orthochronous, 60
orthogonality, 7
of null vectors, 10
of spacelike and null vectors, 58
of spacelike vectors, 57
with timelike vectors, 15
orthonormal basis, 8
outer product, 165
P
P-continuous curve, 289
characterized as Feynman paths, 289
vs E-continuous curves, 288
vs timelike curves, 288
P-open set, 284
not E-open, 284
paradox
barn, 41
twin, 36
parallel postulate, 235
parity nonconservation, 198
particle
charged, 93
free, 87
free charged, 93
horizon, 268
material, 81
passive transformation, 73
past-directed
null curve, 47
null vector, 17
timelike curve, 47
timelike vector, 16
past null cone, 17
past null direction, 74
past null infinity, 266, 274
past time cone, 16
past timelike infinity, 273
322 Index
path, 296
inverse of, 297
products of, 297
path connected, 296
path topology, 284
basis for, 285
homeomorphisms of, 292
topological properties, 286
Pauli spin matrices, 69
Penrose diagram, 266
phase factor, 182
photon, 84
direction 3-vector of, 84
energy, 84
4-momentum of, 84
frequency, 84
propagation of, 2
wavelength, 84
worldline, 4, 11
world momentum of, 84
pions, 36
Planck’s constant, 84
Poincar´e group, 19
point event, 1
Poynting 3-vector, 111
principal null directions, 108, 109
product path, 297
proper mass, 81
proper spatial separation, 56
proper time function, 50
proper time parameter, 227
proper time separation, 44
pullback, 260
Pythagorean Theorem, 58
Q
quadratic form, 7
quantum mechanics, 88, 142, 198, 294
R
raising indices, 162
range, 97
rank
contravariant, 141
covariant, 141
real projective n-space, 301
fundamental group of, 301
regular electromagnetic field, 99
relative 3-momentum, 81
relativistic electron, 137
relativistic energy, 82
relativistic mass, 82
relativity
of simultaneity, 23
principle, 5
representations, 138
carriers of, 138
equivalent, 143
irreducible, 143
reducible, 143
spinor, 145
two-valued, 144
retarded null coordinates, 270
reversed Schwartz inequality, 44
reversed triangle inequality, 44
Ricci flat, 250
Ricci tensor, 249
Riemann curvature tensor, 246
Riemann sphere, 73
rigidity, 40
Robb’s theorem, 57
Robertson-Walker metrics, 254
rocket twin, 37, 55
rotation group, 302
rotation in the Lorentz group, 20
rotation subgroup, 20
S
Schur’s lemma, 143
Schwartz inequality
for R
3
,10
reversed, 44
signals, 3, 33, 57, 268
simply connected, 298
simultaneity, 23
relativity of, 23
simultaneous, 23
skew-symmetric bilinear form, 119
skew-symmetric linear transformation, 94
null, 99
regular, 99
skew-symmetrization, 169
smooth
assignment, 118
map, 203
real-valued function, 203
vector field, 118
Index 323
spacelike displacement, 57
spacelike infinity, 274
spacelike vector, 12
spacetime, 220
spatially homogeneous and isotropic, 252
spatial coordinates, 29, 254
special linear group, 69
special Lorentz transformation, 25
hyperbolic form, 27
speed
of light, 3
of material particles, 82
spin, 88, 135, 137
spin covector, 155
spin frame, 153
spin space, 153
spin transformation, 69
spin vector, 153
spinor, 135
components of, 159
conjugate, 158, 168
contravariant indices, 159
covariant indices, 159
dotted indices, 159
electromagnetic field, 186
equivalents (See spinor equivalent)
essential two- valuedness of, 185, 293
form of Maxwell’s equations, 198
Hermitian, 168
lower indices, 159
skew-symmetric, 168
symmetric, 168
undotted indices, 159
upper indices, 159
valence of, 159
spinor-covector, 172
spinor equivalent
of a bilinear form, 192
of a bivector, 180
of a covector, 176
of differential operators, 197
of the dual of a bivector, 196
of a 4-vector, 172
of Levi-Civita symbols, 196
of the Lorentz inner product, 192
of a vector, 172
of a world vector, 172
spinor map, 71
spinor representation, 145
type of, 145
spinorial object, 142, 293, 305
standard configuration, 25
stereographic projection, 75
stress tensor, 111
subgroup
of the Lorentz group, 19, 20, 24
of transformations, 72, 138
summation convention, xi
symmetric linear transformation, 110
symmetrization, 168
synchronization, 2
lack of, 36
T
tangent space, 218, 219
tangent vector, 47, 217
temporal order, 2, 4, 34, 56
tensor product, 140
tensorial objects, 303
terminal point of a worldline, 87
3-vector
direction, 22
relative momentum, 81
velocity, 21, 52, 85
time
axis, 43
cone, 16
coordinates, 3
dilation, 22, 36
orientation, 16
in units of distance, 3
timelike curve
E-continuous, 281
smooth, 47
timelike straight line, 43
timelike vector, 12
future-directed, 16
past-directed, 16
timelike worldline, 47
topological manifold, 296
products of, 298
topology
Euclidean (or E-), 280
fine, 279
path (or P-), 284
total 4-momentum, 87
total relativistic energy, 82
total world momentum, 87
trace, 109
trace free, 110
transformation equations, 40, 206, 246
transformation matrix, 13, 137
translation, 4, 60