Adapted from P. Coles, 1999, *The Routledge Critical
Dictionary of the New Cosmology*, Routledge Inc., New York. Reprinted
with the author's permission. To order this book click here:
http://www.routledge-ny.com/books.cfm?isbn=0415923549

The theory of **general relativity** has to be
modified if it is to be applied to situations where the matter density
is extremely high, in order to take account of the effects of to
**quantum physics** (see also **quantum gravity**). In cosmology, this
basically means that form when quantum effects manifest themselves on
the scale of the **horizon**.

When do we expect quantum corrections to become significant? Of
course, in the absence of a complete theory (or even *any* theory at
all) of quantum gravity, it is not possible to give a precise answer
to this question. On the other hand, we can make fairly convincing
general arguments that yield estimates of the timescales and energy
scales where we expect quantum gravitational effects to be large, and
for which therefore we should distrust calculations based only on the
classical (non-quantum) theory of general relativity. It turns out
that the limit of validity of general relativity in the **Friedmann
models** is fixed by the *Planck time*, which is of the order of
10^{-43} seconds after the Big Bang.

The Planck time *t*_{p}, is the time for which quantum
fluctuations
governed by the Heisenberg uncertainty principle exist on the scale of
the *Planck length*, *l*_{p} = *ct*_{p},
where *c* is the speed of light. From these
two scales we can construct other Planck quantities such as the *Planck
mass*, *m*_{p}, the *Planck energy*,
*E*_{p}, and so on. Starting from the
Heisenberg uncertainty principle in the form

*E*
*t* *h* /
2

and ignoring any factors of 2 from
now on, we can see that on
dimensional grounds alone we can identify the energy term with some
mass *m*_{p} through the relation *E* =
*m*_{p}*c*^{2}. Assuming that
*t* can be
represented as the Planck time *t*_{p}, we have

*m*_{p}*c*^{2}*t*_{p} *h*

We can express the Planck mass as a Planck density
_{p} times a
*Planck
volume* (or rather the cube of a Planek length). We want to bring
**gravity** into these considerations, in the shape of the Newtonian
gravitational constant *G*. We can do this by noting that the free-fall
collapse time for a self-gravitating body of density
_{p} is given
by *t*^{2}
= 1 / G_{p}. Replacing *m* by
_{p}(*ct*_{p})^{3} and then
_{p} by
1 / *Gt*_{p}^{2} in the above
expression leads to

*c*^{5}*t*_{p}^{2} / *G* *h*

which finally leads us to an expression for the Planck time in terms of fundamental constants only:

*t*_{p} (*hG / c*^{5})

which is around 10^{-43} seconds. The Planck length is simply this
multiplied by the speed of light, *c*, and is consequently around
10^{-33}
cm. This, for example, is about the size of the cosmological **horizon**
at the Planck time (assuming that the concept of a horizon is
meaningful at such an early time). The *Planck density* is phenomenally
high: about 10^{96} grams per cubic centimetre. Interestingly, however,
the Planck mass itself is not an outrageous number: *m*_{p}
= (*hc / G*) = 10^{-5}
g. We can carry on with this approach to calculate the *Planck energy*
(about 10^{19} GeV) and the *Planck temperature* (the
Planck energy divided
by the Boltzmann constant, which gives about 10^{32} K).

In order to understand the physical significance of the Planck time
and all the quantities derived from it, it is useful to think of it in
the following manner, which ultimately coincides with the derivation
given above. We can define the *Compton time* for a particle of
mass *m*
to be *t*_{C} = *h / mc*^{2}; this represents
the time for which it is permissible
to violate the conservation of energy by an amount equal to the mass
of the particle, as deduced from the uncertainty principle. For
example, a pair of virtual particles of mass *m* can exist for a time of
about *t*_{C}. We can also defined the *Compton
radius* of a body of mass *m*
to be equal to the Compton time times the velocity of light:
*l*_{C} = *ct*_{C}
= *h / mc*. Obviously both these quantities decrease as *m*
increases. These scales indicate when phenomena which are associated
with quantum physics are important for an object of a given mass.

Now, the *Schwarzschild radius* of a body of mass *m* is given
by *l*_{S} =
2*Gm / c*^{2}. This represents, to within an order of
magnitude, the radius
that a body of mass in must have for its rest-mass energy
*mc*^{2} to equal
to its internal gravitational potential energy *U*
*Gm*^{2} / *l*_{S}. General
relativity leads us to the conclusion that no particle (not even a
photon) can escape from a region of radius *l*_{S} around a
body of mass *m*;
in other words, speaking purely in terms of classical mechanics, the
escape velocity from a body of mass *m* and radius
*l*_{S} is equal to the
velocity of light. We can similarly define a *Schwarzschild time* to be
the quantity *t*_{S} = *l*_{S} / *c* =
2*Gm/c*^{3}; this is simply the time taken by
light to travel a **proper distance** *l*_{S}. A body of
mass *m* and radius *l*_{S}
has a free-fall collapse of the order of *t*_{S}. Note that
both *t*_{S} and *l*_{S}
increase as *m* increases.

We can easily verify that, for a mass equal to the Planck mass, the Compton and Schwarzschild times are equal to each other and to the Planck time.

Likewise, the relevant length scales are all equal. For a mass greater
than the Planck mass, that is to say for a *macroscopic* body,
*t*_{C} < *t*_{S}
and *l*_{C} < *l*_{S}, and quantum corrections
are expected to be negligible in
the description of the gravitational interactions between different
parts of the body. Here we can describe the self-gravity of the body
using general relativity or even, to a good approximation, Newtonian
theory. On the other hand, for bodies of the order of the Planck
mass, that is to say for *microscopic* entities such as elementary
particles, *t*_{C} > *t*_{S} and
*l*_{C} > *l*_{S}, and quantum corrections will be
important in a description of their self-gravity. In the latter case
we must use a theory of quantum gravity in place of general relativity
or Newtonian gravity.

At the cosmological level, the Planck time represents the moment
before which the characteristic timescale of the expansion is such
that the cosmological horizon, given roughly by *l*_{P},
contains only one
particle (with mass equal to the Planck mass) for which
*l*_{C}
*l*_{S}. On
the same grounds as above, we therefore have to take into account
quantum effects on the scale of the cosmological horizon.

It is interesting to note the relationship between the Planck
quantities and the properties of **black holes**. According to theory, a
black hole of mass *M*, because of quantum effects, emits **Hawking
radiation** like a **black body**. The typical energy of photons
emitted by
the black hole is *kT*, where the temperature *T* is given by

*T*
*hc*^{3} / 4* k G M*

The time needed for such a black hole to evaporate completely (i.e. to
lose all its rest-mass energy *Mc*^{2} via Hawking
radiation) is given by

*t*
*G*^{2}*M*^{3} / *hc*^{4}

By taking these two equations and inserting *M* =
*m*_{p}, we arrive at the
interesting conclusion that a Planck-mass black hole evaporates on a
timescale of the order of the Planck time.

These considerations show that quantum gravitational effects are
expected to be important not only at a cosmological level at the
Planck time, but also continuously on a microscopic scale for
processes operating over distances of about *l*_{p} and
times of about
*t*_{p}. In particular, the components of the **metric**
describing **space-time**
geometry will suffer fluctuations of the order of
*l*_{p} / *l* on a length
scale l and of the order of *t*_{p} / *t* on a timescale
*t*. At the Planck time,
the fluctuations are 100% on the spatial scale *l*_{p} of
the horizon and
on the timescale *t*_{p} of the expansion. We might imagine
the Universe at
very early times as behaving like a collection of Planck-mass black
holes, continually evaporating and recollapsing in a Planck time. This
picture is very different from the idealised, perfect-fluid universe
described by the Friedmann models, and it would not be surprising if
deductions from these equations, such as the existence of a
**singularity**, were found to be invalid in a full quantum description.

FURTHER READING:

Kolb, E.W. and Turner, M.S., *The Early Universe* (Addison-Wesley,
Redwood City, CA, 1990).