Course notes for General Relativity
Instructor: Bernard Whiting
Sept. 11 2008
Notes TEXd by: B.T. Hall, T.N. Kypeos
1
Tangent Vectors / spaces
In Euclidian space, in cartesian cordinates, all vectors, derivatives etc. lie in the
space of points Rn / This does not hold for more general spaces.
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The set {p ∈ M : Tp } is called the tangent bundle T (M ); if dim(M ) = n,
dim(T (M )) = 2n (n to label p, n to give components of ~v ).
2
Manifolds
Definition: A mapping (or set of mappings) between a set of point (and their
local open regions corresponding to these points and Rn / This mapping must
be continuous in the open ball surrounding each point.
In general, there will not be a single map that spans the manifold. To check
whether something is a manifold, check differential nature of the (local) map.
A set of maps that span the manifold is called an atlas; to each (unique) atlas
there corresponds a manifold provided the maps are smoothly connected at the
overlaps. The presence of a manifold allows us to use behavior in cartesian Rn
and infer (for tensors) the behaivor in the corresponding part of M.
2.1
Informal Def of Curvature
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φ
δ
Sum of Interior Angles
π
excess
0
π
2
3π
2
π
2
π
2π
π
3π
2
5π
2
3π
2
3π
2π
2π − δ
Excess scales linearly with φ, and with the area of the triangle. To investiate
curvature in M , draw triangles, take A → 0 and watch lima→0 (Σφi − π)
3
Symmetries (killing vectors)
0
For a vector k µ to be a symmetry of gab , we require that gab
(xc ) = gab (xc );
c
doing the transformation for arbitrary (∈ k ) gives
1
0
gab
(xc ) = gab (xc ) − [gcb ∂a k c + gac ∂b k c + (∂c gab ) k c ]
(1)
thus, for k c to be a symmetry, the term in brackets must vanish.
3.1
Examples
ds2 = −dt2 + dx2 + dy 2 + dz 2
(2)
3.1.1
kc =
∂ c
x = (a, b, c, d) (a,b,c,d constant is obvious by inspection))
∂x
(3)
3.1.2
kc =
∂
= Lz = (0, −y, x, 0)
∂φ
(4)
3.1.3
others:
∂
∂θ
∂
sin θ
∂θ
cos θ
−
cot θsin φ∂φ
(5)
+
cot θcos φ∂φ
(6)
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If k c corresponds to a symmetry, it is called a killing vector.
2