**1.2. Relativistic viewpoint and gauge issues**

Many of the key aspects of the evolution of structure in the universe can be dealt with via a deceptively simple Newtonian approach, but honesty requires a brief overview of some of the difficult issues that will be evaded by taking this route.

Because relativistic physics equations are written in
a covariant form in which all quantities are independent
of coordinates, relativity does not distinguish between
*active* changes of coordinate (e.g. a Lorentz boost)
or *passive* changes (a mathematical change of variable,
normally termed a gauge transformation).
This generality is a problem, since it is not trivial
to know which coordinates should be used.
To see how the problems arise,
ask how tensors of different order change under a gauge
transformation *x*^{µ}
*x*' ^{µ} = *x*^{µ} +
^{µ}.
Consider first a scalar quantity *S* (which might be
density, temperature etc.).
A scalar quantity in relativity is normally taken to be
independent of coordinate frame, but this is only for the
case of Lorentz transformations, which do not involve a
change of the spacetime origin. A gauge transformation
therefore not only induces the usual transformation
coefficients *dx*' ^{µ} /
*dx*^{},
but also involves a translation that relabels spacetime points.
We therefore have to deal with *S* '(*x*^{µ} +
^{µ}) =
*S*(*x*^{µ}), so the rule
for the gauge transformation of scalars is

(2) |

Similar reasoning yields the gauge transformation laws for higher tensors, although we need to account not only for the translation of the origin, but also for the usual effect of the coordinate transformation on the tensor.

Consider applying this to the case of a uniform universe; here only depends on time, so that

(3) |

An effective density perturbation is thus produced by a local
alteration in the time coordinate: when we look at
a universe with a fluctuating density, should we really think
of a uniform model in which time is wrinkled?
This ambiguity may seem absurd, and in the laboratory it
could be resolved empirically by constructing
the coordinate system directly - in principle by using light
signals. This shows that the cosmological horizon plays an
important role in this topic: perturbations with wavelength
*ct* inhabit
a regime in which gauge ambiguities
can be resolved directly via common sense. The real difficulties
lie in the super-horizon modes with
*ct*. However,
at least within inflationary models, these difficulties can be overcome.
According to inflation, perturbations on scales greater than the
horizon were originally generated via quantum fluctuations on
small scales within the horizon of a nearly de Sitter exponential
expansion. There is thus no problem in
understanding how the initial density field is described, since
the simplest coordinate system can once again be constructed directly.

The most direct way of solving these difficulties is to construct
perturbation variables that are explicitly independent of gauge.
Comprehensive technical discussions of this method are given by
Bardeen (1980),
Kodama & Sasaki
(1984),
Mukhanov, Feldman &
Brandenberger (1992).
The starting point for a discussion of metric perturbations is
to devise a notation that will classify the possible
perturbations. Since the metric is symmetric, there are
10 independent degrees of freedom in
*g*^{µ}; a
convenient scheme that captures these possibilities is to write
the cosmological metric as

(4) |

In this equation,
is conformal
time, and
_{ij}
is the comoving spatial part of the Robertson-Walker metric.

The total number of degrees of freedom here is apparently 2 (scalar fields
and
)
+ 3 (3-vector field **w**) + 6 (symmetric 3-tensor *h*_{ij})
= 11. To get the right number, the tensor *h*_{ij} is
required to be traceless:
^{ij}
*h*_{ij} = 0.
The perturbations can be split into three classes: scalar perturbations,
which are described by scalar functions of spacetime coordinate, and which
correspond to the growing density perturbations studied above; vector
perturbations, which correspond to vorticity perturbations, and tensor
perturbations, which correspond to gravitational waves.
Here, we shall concentrate mainly on scalar perturbations.
Since vectors and tensors can be generated from derivatives
of a scalar function, the most general scalar perturbation
actually makes contributions to all the
*g*_{µ}
components in the above expansion:

(5) |

where four scalar functions
,
, *E* and
*B* are involved. It turns out that this situation can be
simplified by defining variables that are
unchanged by a gauge transformation:

(6) |

where primes denote derivatives with respect to conformal time.

These gauge-invariant `potentials' have a fairly direct
physical interpretation, since they are closely related
to the Newtonian potential. The easiest way to evaluate
the gauge-invariant fields is to make a specific gauge
choice and work with the longitudinal gauge in
which *E* and *B* vanish, so that
=
and
=
. A second key result is
that inserting the longitudinal metric into the Einstein equations shows
that and
are identical
in the case of fluid-like perturbations where off-diagonal elements of the
energy-momentum tensor vanish. In this case, the
longitudinal gauge becomes identical to the Newtonian
gauge, in which perturbations are described by a single scalar
field, which is the gravitational potential.
The conclusion is thus that the gravitational potential
can for many purposes give an effectively gauge-invariant measure of
cosmological perturbations, and this provides a sounder
justification for the Newtonian approach that we now adopt.