Re: Cosmology

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding Philip Zell's questions:

> ...        My question: The expansion is of space-time, is it not?

No, it is not.  The cosmolgical expansion of the universe is the
expansion *in time* of *space* not *spacetime*.  In *spacetime* the
phenomenon is manifest as a *curvature* of spacetime that causes
geometric distortions in how the various spatial sections of spacetime
are laminated and stacked together relative to each other as the
(temporal) time-like coordinate is increased.  In particular, the way the
spatial sections are stacked is so that those which are labeled by a
greater value of the time-like coordinate are stretched relative to those
which are labeled by a lesser value of the time-like coordinate.

> In GR, do you not explain the orbits of planets in terms of geodesics?

Yes.

> Are these geodesics not curved as a result of the mass distribution in
> our solar system?

This is tricky.  The very fact that they *are* geodesics means that they
are the analogs of *straight* lines which happen to inhabit a curved
spacetime.  In fact, they they are defined by a process of 'parallel
transport' which *locally* defines paths which do not curve in any
direction w.r.t. the local "fabric" of the geometric space, i.e. the
manifold.  But because the geodesics are embedded in the curved manifold
they must follow that manifold's intrinsic curves as they traverse it.
For instance, the great circle routes on the Earth's surface are the
geodesics on the curved surface of the Earth.  They are locally straight
in that if a person follows one locally, that person neither veers off to
the left nor to the right and goes as straight as possible while staying
on the Earth's surface.  But the one who follows a great circle *must*
stay confined to the surface of the Earth (no boring through the Earth's
interior is allowed) and will follow its intrinsic curvature.  But the
*only* reason why such a geodesic has any curvature when it is observed
from the vantage point of a higher-dimensional flat space that embeds
the manifold of interest, is because it must follow the curves of that
curved manifold.  A geodesic has no contribution to any possible
extrinsic curvature from any tendency of it to locally veer off in
another direction in the curved manifold.  Geodesics are locally straight
*in* the manifold of interest.

>  Yet, is not the space time in our solar system
> flat, because there isn't that much mass contained within it?

It is very *nearly* flat, but it is curved.  It it this tiny curvature
which manifests itself as teh gravitational effects that we see.

> Is flatness defined strictly in terms of the geodesic followed by light?

Null geodesics are the paths that light follows in spacetime.  Time-like
geodesics are those followed by freely falling massive test particles.
Space-like geodesics would be the paths followed by tachyons, *if* they
existed.

David Bowman
David_Bowman@georgetowncollege.edu