GR tarea3

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Tarea 3
Se entrega el Jueves 22 de Marzo
1. a) Given a frame O whose coordinates are {xα}, show that
∂xα
∂xβ
= δαβ (1)
b) For any two frame, we know that
∂xβ
∂xα¯
= Λβα¯ (2)
Show that a) and the chain rule imply
Λβα¯Λ
α¯
µ = δ
β
µ (3)
2. In Euclidean three space in Cartesian coordinates, we don’t normally
distinguish between vectors and one-forms, because their components
transform identically. Prove this in two steps.
a) Show that
Aα¯ = Λα¯βA
β (4)
and
Pβ¯ = Λ
α
β¯Pα (5)
are the same transformation if the matrix {Λα¯β} equals the transpose
of its inverse. Such matrix is said to be orthogonal.
b) The metric of such a space has components {δij, i, j = 1, . . . , 3}.
Prove that a transformation from one cartesian coordinate system to
another must obey
δi¯j = Λ
k
i¯ Λ
l
j¯δkl (6)
and that this implies {Λki¯ } is an orthogonal matrix.
3. Show that if A is a
(
2
0
)
tensor and B a
(
0
2
)
tensor, then
AαβBαβ (7)
is frame invariant, i.e. a scalar.
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4. From the definition fαβ = f(~eα, ~eβ) for the components of a
(
0
2
)
tensor,
prove that the transformation law is
fα¯β¯ = Λ
µ
α¯Λ
ν
β¯fµν (8)
and that the matrix version of this is
(f¯) = (Λ)T (f)(Λ) (9)
where (Λ) is the matrix with components Λµα¯.
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