Chapter 3: Relativity
15
2. The
principle of equivalence
: inertial mass
≡
gravitational mass
⇒
gravitation is equivalent with a
curved spacetime were particles move along geodesics.
3. By a proper choice of the coordinate system it is possible to make the metric locally Fat in each point
x
i
:
g
αβ
(
x
i
)=
η
αβ
:=
diag
(

1
,
1
,
1
,
1)
.
The
Riemann tensor
is de±ned as:
R
μ
ναβ
T
ν
:=
∇
α
∇
β
T
μ
∇
β
∇
α
T
μ
, where the covariant derivative is given
by
∇
j
a
i
=
∂
j
a
i
+
Γ
i
jk
a
k
and
∇
j
a
i
=
∂
j
a
i

Γ
k
ij
a
k
. Here,
Γ
i
jk
=
g
il
2
±
∂
g
lj
∂
x
k
+
∂
g
lk
∂
x
j

∂
g
j
k
∂
x
l
²
,
for Euclidean spaces this reduces to:
Γ
i
jk
=
∂
2
¯
x
l
∂
x
j
∂
x
k
∂
x
i
∂
¯
x
l
,
are the
Christoffel symbols
. ²or a secondorder tensor holds:
[
∇
α
,
∇
β
]
T
μ
ν
=
R
μ
σαβ
T
σ
ν
+
R
σ
ναβ
T
μ
σ
,
∇
k
a
i
j
=
∂
k
a
i
j

Γ
l
kj
a
i
l
+
Γ
i
kl
a
l
j
,
∇
k
a
ij
=
∂
k
a
ij

Γ
l
ki
a
lj

Γ
l
kj
a
jl
and
∇
k
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This note was uploaded on 11/16/2010 for the course ENGR 201 taught by Professor Elder during the Spring '10 term at Blinn College.
 Spring '10
 elder

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