Summary of General Theory of Relativity (3ERX0) course. The summary includes the contents of the lectures (by Kamp, Heefer, and Fuster) and lecture notes by Leon Kamp as well as information from the General Relativity book by Hobson.
Looks fun and is complete but the lay-out and handwriting does not make it very readable. Also the summery might be a bit too consise.
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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
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
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