Inertialframe Minkowski
space
Newton's
first law holds line element
freeparticle at rest or moves
with a constant velocity ds2 Colt 01 2 dy2 d22
6 taxi taxy
Manifold
topological spacethat locally resembles Euclidean space
Summation convention Dimension thenumber independent parameters
DX'd
of
dxb
8
implies summation over to
Required to specify anypoint in the
b setuniquelly
2bX d
curve
Tangentspace a a
Parameter
Tc c
B
degree of freedom
0
aa
no
mI.gumaiiE
iiiiith
N
dimensional
distance A B
doublesumimplied
massiveparticles
withEinstein's convention
dg2 dyad b time Ee masslessparticles
gab spacelike
metric tensor
afunction that tells how to compute distance between two points
Mathematicsa
functionthattakes apair tangent
of gabXC _gbaX NzInn
vectors andw as input and produces a real independent
numberscalar
guw coordinatesdefinedonsome issymmetric members
themanifold
pathof
way
Physics inlocalcoordinates in the metric
canbewrittenas dx dN Xa x'a Lenghof the curve
g gov transformation
gig 319,7 Lars
gas
doff I du
,Vector calculus on
manifolds
Vector on a manifold
field
vector field Xb ve eafxb vb xYeb
f
f
ariant component
contravariant
component
basisvectors that
span To
if 0 atb
e eb ff L if 1 a b
q.m gig
tafngentavector infinitiesatem between 2points
coordinate basis vectors a
geo 8K c coordinatecurve
dsealxtdxagrab
ea.es jgab ea.es
eachother's inverse
gbgbc
ffgaiswb
wo.eag.be
wagabwb
e gabe
coordinate transformations
3 e b e'a
ftp.eebea
ea
similarly V'be Va etc
zxfebe.a gxifeb
orthonormalbasisvectors Length of a vector
eaeb Baby
unitlength gab diagh e
AM GµuAtta
orthogonaltooneanother
, Derivative in the manifold
Parallel transport
t
def dduvoddufvaeatflueaeai.uaGalea
t
1 9 94
vaeatvage.ie
jaea ve acebxc
Jef baceb
A f ineconnection Christoffel symbol 4 Intrinsic
apreservingparallelrelationships
a degrivative
baeEbJoe Jcea baeeb day f bcub Yea DDVudea O
w f
ZEE Abc doesNOT
transform forparalleltransport
as thecomponentsofa dog beubdofu
tensor
Torsiontensor bae l bae l Baa InGR manifolds are torsionless 1 bae I baa
The metric connection be Lgad 2bJac degacddgbc
tells howto compute theconnection
atanypoint in a manifold
abc Gad
dbc Abc 2 2bJac JCOJba Jag
be Gad dbc Iabet bae 2cgab
Jcgab acgdb begad
Differentiation comparingparalleltransportedfield vector at P with the vector at a
Derivative of vector field fu t
IcVb ea
Covariant derivative bit 2bWVa
wa
Directional Dvd Qb tangentvector
b e
derivative Da Dbt l2bVatTabcV4xb It aboveEb
derivativeinthedirectionofi
space
Newton's
first law holds line element
freeparticle at rest or moves
with a constant velocity ds2 Colt 01 2 dy2 d22
6 taxi taxy
Manifold
topological spacethat locally resembles Euclidean space
Summation convention Dimension thenumber independent parameters
DX'd
of
dxb
8
implies summation over to
Required to specify anypoint in the
b setuniquelly
2bX d
curve
Tangentspace a a
Parameter
Tc c
B
degree of freedom
0
aa
no
mI.gumaiiE
iiiiith
N
dimensional
distance A B
doublesumimplied
massiveparticles
withEinstein's convention
dg2 dyad b time Ee masslessparticles
gab spacelike
metric tensor
afunction that tells how to compute distance between two points
Mathematicsa
functionthattakes apair tangent
of gabXC _gbaX NzInn
vectors andw as input and produces a real independent
numberscalar
guw coordinatesdefinedonsome issymmetric members
themanifold
pathof
way
Physics inlocalcoordinates in the metric
canbewrittenas dx dN Xa x'a Lenghof the curve
g gov transformation
gig 319,7 Lars
gas
doff I du
,Vector calculus on
manifolds
Vector on a manifold
field
vector field Xb ve eafxb vb xYeb
f
f
ariant component
contravariant
component
basisvectors that
span To
if 0 atb
e eb ff L if 1 a b
q.m gig
tafngentavector infinitiesatem between 2points
coordinate basis vectors a
geo 8K c coordinatecurve
dsealxtdxagrab
ea.es jgab ea.es
eachother's inverse
gbgbc
ffgaiswb
wo.eag.be
wagabwb
e gabe
coordinate transformations
3 e b e'a
ftp.eebea
ea
similarly V'be Va etc
zxfebe.a gxifeb
orthonormalbasisvectors Length of a vector
eaeb Baby
unitlength gab diagh e
AM GµuAtta
orthogonaltooneanother
, Derivative in the manifold
Parallel transport
t
def dduvoddufvaeatflueaeai.uaGalea
t
1 9 94
vaeatvage.ie
jaea ve acebxc
Jef baceb
A f ineconnection Christoffel symbol 4 Intrinsic
apreservingparallelrelationships
a degrivative
baeEbJoe Jcea baeeb day f bcub Yea DDVudea O
w f
ZEE Abc doesNOT
transform forparalleltransport
as thecomponentsofa dog beubdofu
tensor
Torsiontensor bae l bae l Baa InGR manifolds are torsionless 1 bae I baa
The metric connection be Lgad 2bJac degacddgbc
tells howto compute theconnection
atanypoint in a manifold
abc Gad
dbc Abc 2 2bJac JCOJba Jag
be Gad dbc Iabet bae 2cgab
Jcgab acgdb begad
Differentiation comparingparalleltransportedfield vector at P with the vector at a
Derivative of vector field fu t
IcVb ea
Covariant derivative bit 2bWVa
wa
Directional Dvd Qb tangentvector
b e
derivative Da Dbt l2bVatTabcV4xb It aboveEb
derivativeinthedirectionofi
