# The scaling of metric coefficients by changing units

1. Aug 24, 2009

### oldman

The Robertson-Walker (RW) metric is used for gauging the accepted spatially flat and dynamic model of the universe. This metric may, along any straight line, be written:

ds^2 =c^2 dt^2 - a(t)^2 dr^2.

The coordinates t and r are gauged with Systeme Internationale (SI) units of seconds and metres. Here the invariant speed of light c may be regarded as a unit conversion factor that in the SI system changes light-seconds to metres. The dimensionless metric coefficients are: for time, static and equal to unity and: for space, dynamic and known as the scale factor, a(t). The accepted (FLRW) model of the universe is highly symmetric. It describes the universe as a homogeneous, isotropic and continuous fluid, while ignoring its lumpiness of atoms, stars and galaxies. In space sections local clocks are synchronized by setting them to an agreed time (say zero -- or "now") when the density of the local fluid or the temperature or of the local cosmic microwave background is everywhere the same (arbitrarily chosen value).

The units for the space coordinate r of the metric can be dynamically rescaled from a SI unit to the non-SI unit r’ = a(t) r. Then r’ is called a comoving coordinate. When a(t) increases with time the proper separation of fluid elements increases with time and the universe is described as expanding. This rescaling makes the RW metric look like a space-rescaled version of the Minkowskian metric of Special Relativity (SR).

The Schwarzchild metric is used for gauging the spherically symmetric static spacetime of a point mass. This metric may, along any radial line, may be written:

ds^2 = alpha c^2 dt^2 - (1/alpha) dr^2.

Alpha and its reciprocal are (the square roots of) dimensionless static metric coefficients that vary with position along the line.

It seems to me that in this case both time and space units could also be rescaled from SI units so as to make the metric look like the space-and-time-rescaled Minkowskian metric of SR or, for that matter, the RW metric at any one chosen instant of so-called universal time.

This similarity between the RW and Schwarzchild metrics is, I think, why the interior of massive star collapsing into a black hole has been described as behaving:

The Schwarzchild metric as written above has a (to me) satisfying reciprocal symmetry between its coefficients. Time and space are here treated as equals, as it were. Not so the Robertson-Walker metric. As written above it is asymmetric in that only the space coefficient is chosen to be (or has to be, for physical reasons I don't understand) differentfrom that of the Minkowskian metric.

Now come a few questions:

Could one choose to write the RW in a similarly symmetric way, for example as:

ds^2 =c^2 1/a(t) dt^2 - a(t) dr^2)

by rescaling the SI units of t and r?

Would such a change be observable?

If it is not observable, would we still speak of an “expanding” universe with such certainty?

Could such rescaling of units in the metric be regarded as a global symmetry of the metric?

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Last edited: Aug 24, 2009