[Relativity FAQ] - [Copyright]

original by Michael Weiss 1994

A detailed explanation requires looking at Friedmann-Robertson-Walker (FRW) models of spacetime. The famous "expanding balloon speckled with galaxies" provides a visual analogy for one of these; like any analogy, it will mislead you if taken too literally, but handled with caution it can furnish some insight.

Draw a latitude/longitude grid on the balloon. These define
*co-moving coordinates*. Imagine a couple of speckles ("galaxies")
enbedded in the rubber surface. The co-moving coordinates of the speckles
don't change as the balloon expands, but the distance between the speckles
steadily increases. In co-moving coordinates, we say that the speckles
don't move, but "space itself" stretches between them.

A bug starts crawling from one speckle to the other. A second after the first bug leaves, his brother follows him. (Think of the bugs as two light-pulses, or successive wave-crests in a beam of light.) Clearly the separation between the bugs will increase during their journey. In co-moving coordinates, light is "stretched" during its journey.

Now we switch to a different coordinate system, this one valid only
in a neighborhood (but one large enough to cover both speckles). Imagine a
clear, flexible, non-stretching patch, attached to the balloon at one
speckle. The patch clings to the surface of the balloon, which slides
beneath it as the balloon inflates. (The bugs crawl along *under* the
patch.) We draw a coordinate grid on the patch. In the patch coordinates,
the second speckle recedes from the first speckle. And so in patch
coordinates, we can regard the redshift as a Doppler shift.

Is this visually appealing? I think so. However, this explanation glosses over one crucial point: the time coordinate. FRW spacetimes come fully-equipped with a specially distinguished time coordinate (called the co-moving or cosmological time). For example, a co-moving observer could set her clock by the average density of surrounding speckles, or by the temperature of the Cosmic Background Radiation. (From a purely mathematical standpoint, the co-moving time coordinate is singled out by a certain symmetry property.)

We have many choices of time-coordinate to go with the
space-coordinates drawn on our patch. Let's use cosmological time. Notice
that this is *not* the choice usually made in Special Relativity: though
the two speckles separate rapidly, their cosmological clocks remain
synchronized. Bugs embarking on their journey from the "moving" speckle
appear to crawl "upstream" against flowing space as they head towards the
"home" speckle. The current diminishes as they approach home. (In other
words, bug-speed is anisotropic in these coordinates.) These differences
from the usual SR picture are symptoms of a deeper fact: besides the
obvious "spatial" curvature of the balloon's surface, FRW spacetimes have
"temporal" curvature as well. Indeed, not all FRW spacetimes exhibit
spatial curvature, but (with one exception) all have temporal curvature.

You can work out the magnitude of the redshift using patch
coordinates. I leave this as an exercise, with a couple of hints. (1)
Since bug-speed is anisotropic far from the home speckle, consider also a
patch attached to the "moving" speckle. Compute the initial distance
between the bugs (the "wavelength") in both patch coordinate systems, using
the standard *non-relativistic* Doppler formula for a stationary source,
moving receiver. (2) Now think about how the bug-distance changes as the
bugs journey to the home speckle (this time sticking with home patch
coordinates). The bug-distance does *not* propagate unchanged. Consider
instead the analog of the period of a lightwave: the time between
bug-crossings of a grid line on the patch. This *does* propagate almost
unchanged, *provided* the rate of balloon expansion stays pretty much the
same throughout the bugs' perilous trek. The final result: the magnitude
of the redshift, computed using Doppler's formula, agrees to first-order
with magnitude computed using the "stretched-light" explanation. (To the
cognoscenti: the assumptions are that Hx<<1 and (dH/dt)x<<1, where
H(t)=dR(t)/dt, R(t) is the scale factor, t is cosmological time, and x is
the average distance between the "speckles" (co-moving geodesics) during
the course of the journey.)

(This long-winded "proof of equivalence" between the Doppler and "stretched-light" explanations substitutes a paragraph of imagery for a half-page of calculus.)

Let me close by emphasizing the word "approximation" from the first paragraph of this entry. The Doppler explanation fails for very large redshifts, for then we must consider how Hubble's "constant" changes over the course of the journey.

M.V.Berry, "Principles of Cosmology and Gravitation", chapter 6.

Steven Weinberg, "The First Three Minutes", chapter 2, especially pp. 13 and 30.