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To represent the perturbations we must make a gauge choice.
A gauge transformation is a change in the correspondence between the
perturbation and the background represented by the coordinate shifts
*T* corresponds to a choice in time slicing and *L* a choice of
spatial coordinates. Since scalar and vector quantities
cannot be formed from tensor modes (),
no gauge freedom remains there. Under the condition that
metric distances be invariant,
they transform the metric as [17]
The stress-energy perturbations in different gauges are similarly
related by the gauge transformations
Note that the anisotropic stress is gauge-invariant.
Seed perturbations are also gauge-invariant to lowest order, whereas
a scalar field transforms as
| |
(37) |

The relation between the synchronous and Newtonian gauge equations follow
from these relations.

*Wayne Hu*

*9/9/1997*