Background Lorentz Transformation
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The Lorentz transformation matrix for a boost in the +x direction with speed v (b = v/c) is
In the weak
field limit a background Lorentz transformation is a transformation
of the form
The
inverse transformation is
where
The
components of a contravariant vector A transform as
The
components of a covariant vector B transform as
The
components of a contravariant tensor T of rank 2 transforms as
The
components of a contravariant tensor G of rank 2 transforms as
Similar
relations hold for all other tensors. For simplicity of notation define
To
transform the components of the gravitational field to a new frame of reference
use
Substitute
the weak field relation
Therefore
Since
Eq. (12) reduces to
Thus hmn transforms like a tensor under a background Lorentz transformation. We may therefore think of a weak gravitational field as a flat spacetime with a tensor field hmn defined on it. It is not a true tensor field since it is only a piece of gmn.