First ingredient: quantum mechanics
In the early 20th century, it was realized that the
stability of atomic matter could not be explained using the Maxwell
equations of classical electrodynamics. This triumph belonged to quantum
mechanics. The hydrogen atom was stable because the
possible energy states of the electron in the atom are quantized
by the rule
where n is an integer, and m
is (approximately) the electron mass.
So when the electron changes energy for some reason,
say by absorbing or emitting electromagnetic radiation, it can only
absorb or emit light of a wavelength corresponding to the difference
in quantized energy states of the electron. The collection of wavelengths
of light emitted by hydrogen gas is called the emission spectrum of
hydrogen, and there is a corresponding spectrum for absorption. One
of the great successes of quantum mechanics was the calculation of the
wavelengths in the observed hydrogen spectrum.
Second ingredient: relativity
The other great revolution that started the 20th
century was the spacetime revolution of special and general relativity.
In special relativity, when a source of light of wavelength l_{em}
is moving away from an observer at some velocity v, the observer sees
the light at some other wavelength l_{obs},
determined by the principle that the speed of light is the same for
all observers. The fractional difference between l_{em}
and l_{obs} is called
the red shift, denoted by the letter
z, and is computed from the relative velocity v between the source and
observer by
where c is the speed of light. If the source and observer are moving
towards one another, the red shift becomes a blue shift and is given
is given by taking v > v in above.
Conclusion: the Universe is expanding
Stars are made mostly out of hydrogen and helium,
and the emission spectrum of the hydrogen atoms in a star in a far away
galaxy ought to be the same as that of hydrogen atoms in a tube of gas
in a laboratory on Earth. But that's not what Edwin Hubble found when
he compared the emission spectra of different stars and galaxies. Hubble
found that the emission wavelengths of the hydrogen gas were red shifted
by an amount proportional to their distance from our solar system. Hubble's
Law relates the red shift z to the distance D through
where the empirical constant H_{0} is called Hubble's
constant.
Hubble's observation suggested that the stars and
galaxies in the Universe are hurtling away from one another with a velocity
that increases with distance, as if the whole Universe was expanding,
like in a big explosion. When physicists extrapolated that motion backwards
in time, it suggested that the Universe started out very hot and dense
and somehow exploded into the huge cold place that we see today. Hubble's
Law was an empirical observation that demanded, and received, very intense
attention from modern theoretical physics after it was first proposed
in 1924.
The equation of motion
When physicists want to study a given system, they
turn to the equations of motion for that system. According to the theory
of general relativity, the correct equation of motion for describing
a Universe is the Einstein equation
relating the curvature of the spacetime in a given Universe to the
distribution of energy and momentum in that Universe. The energymomentum
tensor T_{mn} includes
all of the energy from all nongravitational sources such as matter,
electromagnetism or even quantum vacuum energy as we shall see later.
The standard cosmological solution to the Einstein
equation is written in the form of the FriedmanRobertsonWalker metric
The function a(t) is called the scale factor,
because it tells us the size of the Universe. The scale factor a(t)
and the constant k are both determined by the particular type of matter
and/or radiation present in the Universe. This will be described in
the next section.
For any value of a(t) or k, the gravitational red
shift z of light due to the changing size of the Universe satisfies
where t_{obs} is the time in the Universe that the light is
being observed and t_{em} is the time when the light
was first emitted.
The Hubble parameter H(t) gives the relative rate
of change in the scale factor a(t) by
The observed Hubble constant is just the current value of the dynamically
evolving Hubble parameter. The uncertainties of the currently observed
value of the Hubble constant have been lumped into the parameter h_{0}.
How old?
A quick approximation for the age of the Universe
can be approximated by the inverse of the Hubble constant. The calculated
age turns out to be
Current best estimates of h_{0} are
so the Universe is most likely somewhere between 12
and 16 billion years old, at least according to this method of
estimation.
But recall that according to relativity, time is relative.
We can guess the amount of time likely to have elapsed since the time
when time was a meaningful quantity that could be measured. But we can't
say anything about any processes that might have occurred before the
notion of time made sense. In some sense, quantum gravity could be an
eternal stage of the Universe, and the Big Bang could be regarded as
the end of eternity and the beginning of time itself.
