186
11 Linearized General Relativity and Gravitational Waves
˜
A ν = A ν + ϕ ,ν , ˜
A
,ν
,ν = A
ν ,ν + ϕ
,ν ,ν = A
ν ,ν = 0.
(11.93)
Thus if we begin in a Lorentz gauge and make a transformation with a solution of
the wave equation we remain in the Lorentz gauge; this is convenient and elegant.
Most importantly we can choose ϕ so that in the tilde gauge ˜
A
0
= ˜
A
3
= 0. For
simplicity we align the vector k ν along the time and z axis, k ν = (1, 0, 0, −1), so
that U = ct − z. Then the Lorentz gauge condition is
A
ν ,U k ν = A
0 ,U − A
3 ,U = 0, A
0
= A
3
, A 0 = −A 3 .
(11.94)
Here the last step follows from integration, since all components are functions of
only U , and any constant would be irrelevant. Exactly the same Formula holds in the
tilde gauge since the Lorentz condition holds there also. Finally, in the tilde gauge we
may then force the 0 and 3 components to be zero according to (11.93) by choosing
˜
A 0 = A 0 + ϕ ,0 = 0, ϕ ,0 = −A 0 , ϕ ,U = −A 0
˜
A 3 = A 3 + ϕ ,3 = 0, ϕ ,3 = −A 3 , ϕ ,U = A 3 .
(11.95)
The two expressions for the U derivative of ϕ are same according to (11.94). Thus
we can integrate to give ϕ as a function of U , with an irrelevant constant. Thereby
the vector potential has only 1,2 components in the tilde system. Note also that the
1,2 components of the vector potential are not changed by the gauge transformation.
From the discussion of gravitational waves in the text and the above comments
on electromagnetic waves the following mathematical analogies are apparent:
A μ ↔ h αβ vector potential, metric perturbation
(11.96)
μ ↔ αβ polarization vector, metric polarization
A
ν = A ν + ϕ ,ν ↔ h
αβ = h αβ + ( f α,β + f β,α )
gauge transformation, small coordinate transformation
F μν ↔ R αβγ δ physical electromagnetic, gravitational tidal fields
These analogies are rather elegant and simple. However this does not mean that
electromagnetism and gravity are in any sense the same thing with a few indices
altered. Einstein spent many of his later years trying to establish a deep physical
connection between gravity and electromagnetism and did not succeed.
11 Linearized General Relativity and Gravitational Waves
˜
A ν = A ν + ϕ ,ν , ˜
A
,ν
,ν = A
ν ,ν + ϕ
,ν ,ν = A
ν ,ν = 0.
(11.93)
Thus if we begin in a Lorentz gauge and make a transformation with a solution of
the wave equation we remain in the Lorentz gauge; this is convenient and elegant.
Most importantly we can choose ϕ so that in the tilde gauge ˜
A
0
= ˜
A
3
= 0. For
simplicity we align the vector k ν along the time and z axis, k ν = (1, 0, 0, −1), so
that U = ct − z. Then the Lorentz gauge condition is
A
ν ,U k ν = A
0 ,U − A
3 ,U = 0, A
0
= A
3
, A 0 = −A 3 .
(11.94)
Here the last step follows from integration, since all components are functions of
only U , and any constant would be irrelevant. Exactly the same Formula holds in the
tilde gauge since the Lorentz condition holds there also. Finally, in the tilde gauge we
may then force the 0 and 3 components to be zero according to (11.93) by choosing
˜
A 0 = A 0 + ϕ ,0 = 0, ϕ ,0 = −A 0 , ϕ ,U = −A 0
˜
A 3 = A 3 + ϕ ,3 = 0, ϕ ,3 = −A 3 , ϕ ,U = A 3 .
(11.95)
The two expressions for the U derivative of ϕ are same according to (11.94). Thus
we can integrate to give ϕ as a function of U , with an irrelevant constant. Thereby
the vector potential has only 1,2 components in the tilde system. Note also that the
1,2 components of the vector potential are not changed by the gauge transformation.
From the discussion of gravitational waves in the text and the above comments
on electromagnetic waves the following mathematical analogies are apparent:
A μ ↔ h αβ vector potential, metric perturbation
(11.96)
μ ↔ αβ polarization vector, metric polarization
A
ν = A ν + ϕ ,ν ↔ h
αβ = h αβ + ( f α,β + f β,α )
gauge transformation, small coordinate transformation
F μν ↔ R αβγ δ physical electromagnetic, gravitational tidal fields
These analogies are rather elegant and simple. However this does not mean that
electromagnetism and gravity are in any sense the same thing with a few indices
altered. Einstein spent many of his later years trying to establish a deep physical
connection between gravity and electromagnetism and did not succeed.