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Chapter 3: Relativity
17
3.2.4
The trajectory of a photon
For the trajectory of a photon (and for each particle with zero restmass) holds
ds
2
=0
. Substituting the
external Schwarzschild metric results in the following orbital equation:
du
d
ϕ
±
d
2
u
d
ϕ
2
+
u

3
mu
²
=0
3.2.5
Gravitational waves
Starting with the approximation
g
μ
ν
=
η
μ
ν
+
h
μ
ν
for weak gravitational ±elds and the de±nition
h
±
μ
ν
=
h
μ
ν

1
2
η
μ
ν
h
α
α
it follows that
±
h
±
μ
ν
=0
if the gauge condition
∂
h
±
μ
ν
/
∂
x
ν
=0
is satis±ed. From this, it
follows that the loss of energy of a mechanical system, if the occurring velocities are
±
c
and for wavelengths
²
the size of the system, is given by:
dE
dt
=

G
5
c
5
³
i,j
±
d
3
Q
ij
dt
3
²
2
with
Q
ij
=
´
±
(
x
i
x
j

1
3
δ
ij
r
2
)
d
3
x
the mass quadrupole moment.
3.2.6
Cosmology
If for the universe as a whole is assumed:
1. There exists a global time coordinate which acts as
x
0
of a Gaussian coordinate system,
2. The 3dimensional spaces are isotrope for a certain value of
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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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