**2.2. Einstein's "greatest blunder"**

General relativity combined with homogeneity and isotropy leads to a startling conclusion: spacetime is dynamic. The universe is not static, but is bound to be either expanding or contracting. In the early 1900's, Einstein applied general relativity to the homogeneous and isotropic case, and upon seeing the consequences, decided that the answer had to be wrong. Since the universe was obviously static, the equations had to be fixed. Einstein's method for fixing the equations involved the evolution of the density with expansion. Returning to our analogy between General Relativity and electromagnetism, we remember that Maxwell's equations (7) do not completely specify the behavior of a system of charges and fields. In order to close the system of equations, we need to add the conservation of charge,

(14) |

or, in vector notation,

(15) |

The general relativistic equivalent to charge conservation is stress-energy conservation,

(16) |

For a homogeneous fluid with the stress-energy given by Eq. (9),
stress-energy conservation takes the form of the *continuity
equation*,

(17) |

where *H* = /
*a* is the Hubble parameter from Eq. (12). This equation relates
the evolution of the energy density to its equation of state
*p* = *w*
. Suppose we have
a box whose dimensions are expanding along with the universe, so that
the volume of the box is proportional to the cube of the scale factor,
*V*
*a*^{3}, and we fill it with
some kind of matter or radiation. For example, ordinary matter in a box
of volume *V* has
an energy density inversely proportional to the volume of the box,
*V*^{-1}
*a*^{-3}.
It is straightforward to show using the continuity equation that this
corresponds to zero
pressure, *p* = 0. Relativistic particles such as photons have
energy density that goes as
*V*^{-4/3}
*a*^{-4}, which corresponds to equation of state
*p* = / 3.

Einstein noticed that if we take the stress-energy
*T*_{µ} and
add a constant ,
the conservation equation (16) is unchanged:

(18) |

In our analogy with electromagnetism, this is like adding a constant to the
electromagnetic potential, *V'*(*x*) = *V*(*x*) +
. The constant
does not
affect local dynamics in any way, but it does affect the
cosmology. Since adding this
constant adds a constant energy density to the universe, the continuity
equation tells
us that this is equivalent to a fluid with *negative* pressure,
*p*_{} =
-_{}. Einstein chose
to give a closed,
static universe as follows
[2].
Take the energy density to consist of matter

(19) |

and cosmological constant

(20) |

It is a simple matter to use the Friedmann equation to show that this
combination of matter
and cosmological constant leads to a static universe
=
= 0. In order for
the energy densities to be positive, the universe must be closed,
*k* = + 1. Einstein was
able to add a kludge to get the answer he wanted.

Things sometimes happen in science with uncanny timing. In the 1920's,
an astronomer named
Edwin Hubble undertook a project to measure the distances to the spiral
"nebulae" as
they had been known, using the 100-inch Mount Wilson telescope. Hubble's
method involved
using Cepheid variables, named after the star Delta Cephei, the best
known member of the
class. ^{(1)}
Cepheid variables have the useful property that the period of their
variation, usually 10-100 days, is correlated to their absolute
brightness. Therefore, by
measuring the apparent brightness and the period of a distant Cepheid,
one can determine
its absolute brightness and therefore its distance. Hubble applied this
method to a number
of nearby galaxies, and determined that almost all of them were receding
from the earth.
Moreover, the more distant the galaxy was, the faster it was receding,
according to a roughly linear relation:

(21) |

This is the famous Hubble Law, and the constant *H*_{0} is
known as Hubble's constant. Hubble's
original value for the constant was something like
500 km/sec/Mpc, where one megaparsec
(*Mpc*) is a bit over 3 million light years.
^{(2)}
This implied an age for the universe of about a
billion years, and contradicted known geological estimates for the age
of the earth. Cosmology had its first "age problem": the universe can't
be younger than
the things in it! Later it was realized that Hubble had failed to
account for two distinct
types of Cepheids, and once this discrepancy was taken into account, the
Hubble constant fell to well under
100 km/s/Mpc. The current best estimate, determined using the Hubble space
telescope to resolve Cepheids in galaxies at unprecedented distances, is
*H*_{0} = 71 ± 6 km/s/Mpc
[5].
In any case, the Hubble law is exactly
what one would expect from the Friedmann equation. The expansion of the
universe predicted (and rejected) by Einstein had been observationally
detected, only a few years after the development of General
Relativity. Einstein later referred to the introduction
of the cosmological constant as his "greatest blunder".

The expansion of the universe leads to a number of interesting
things. One is the
cosmological redshift of photons. The usual way to see this is that from
the Hubble law,
distant objects appear to be receding at a velocity *v* =
*H*_{0}*d*, which means that photons
emitted from the body are redshifted due to the recession velocity of
the source. There is
another way to look at the same effect: because of the expansion of
space, the wavelength of a photon increases with the scale factor:

(22) |

so that as the universe expands, a photon propagating in the space gets
shifted to longer
and longer wavelengths. The redshift *z* of a photon is then given
by the ratio of the scale
factor today to the scale factor when the photon was emitted:

(23) |

Here we have introduced commonly used the convention that a subscript 0
(e.g., *t*_{0} or *H*_{0})
indicates the value of a quantity *today*. This redshifting due
to expansion applies
to particles other than photons as well. For some massive body moving
relative to the
expansion with some momentum *p*, the momentum also "redshifts":

(24) |

We then have the remarkable result that freely moving bodies in an
expanding universe eventually
come to rest relative to the expanding coordinate system, the so-called
*comoving* frame.
The expansion of the universe creates a kind of dynamical friction for
everything moving
in it. For this reason, it will often be convenient to define comoving
variables, which
have the effect of expansion factored out. For example, the physical
distance between two
points in the expanding space is proportional to *a*(*t*). We
define the comoving distance between two points to be a constant in time:

(25) |

Similarly, we define the comoving wavelength of a photon as

(26) |

and comoving momenta are defined as:

(27) |

This energy loss with expansion has a predictable effect on systems in
thermal equilibrium.
If we take some bunch of particles (say, photons with a black-body
distribution) in thermal
equilibrium with temperature *T*, the momenta of all these
particles will decrease linearly with expansion, and the system will cool.
^{(3)}
For a gas in thermal equilibrium, the temperature is in fact inversely
proportional to the scale factor:

(28) |

The current temperature of the universe is about
3 K. Since it has been cooling with
expansion, we reach the conclusion that the early universe must have
been at a much higher
temperature. This is the "Hot Big Bang" picture: a hot, thermal
equilibrium universe
expanding and cooling with time. One thing to note is that, although the
universe goes to
infinite density and infinite temperature at the Big Bang singularity,
it does *not*
necessarily go to zero size. A flat universe, for example is infinite in
spatial extent
an infinitesimal amount of time after the Big Bang, which happens
*everywhere* in the
infinite space simultaneously! The observable universe, as measured by
the horizon size,
goes to zero size at *t* = 0, but the observable universe
represents only a tiny patch of the total space.

^{1} Delta Cephei is not, however the
nearest Cepheid. That honor goes to Polaris, the north star
[3].
Back.

^{2} The *parsec* is an archaic
astronomical unit corresponding to one second of arc of parallax
measured from opposite sides of the earth's orbit: 1 pc = 3.26 ly.
Back.

^{3} It is not hard to convince yourself that
a system that starts out as a blackbody stays a blackbody with expansion.
Back.