THE PLANCK TIME
The Planck time has a value of 5.39121 × 10-44 second (current uncertainty 40 × 10-44). It
is the smallest time that can be operationally defined, that is, measured even in principle.
The Planck length has a value of 1.61624 × 10-35 (current uncertainty 12 × 10-35) meter
and represents the smallest length that can be operationally defined.
By international agreement (see also Time, Units of and Time, Operational
Definition of), the distance or length, L, between two points in space is defined as the
time, t, it takes for light to travel between the points in a vacuum, multiplied by a
constant, c
L = ct
(1)
where c = 299,792,458 meters per second is the speed of light in a vacuum. (This number
is exact by definition). In order to measure t, and thus L, we need a clock with an
uncertainty Δt no larger than t. The time-energy uncertainty principle says that the
product of Δt and the uncertainty in a measurement of energy in that time interval, ΔE,
can be no less than /2, where  = h/2π and h = 6.6260693 × 10-34 Joule-sec (current
uncertainty 11 × 10-34) is Planck's constant. That is,
ΔEt ≥ ΔEΔt ≥

2
(2)
Thus,
ΔE ≥
 c
≥
2t 2L
This energy is equivalent to the rest energy of a body of mass m,
(3)
ΔE = mc 2
(4)
Equation (3) implies that within a spherical region of space of radius L we cannot
determine, by any measurement, that it contains a mass less than
m=

2cL
(5)
Now, a spherical body of mass M will be a black hole if its radius R is less than or
equal to
R=
2GM
c2
(6)
where G = 6.6742 × 10-11 cubic meters per kilogram per square second (current
uncertainty 10 × 10-11) is Newton's gravitational constant. This is called the
Schwarzschild radius.
Consider a body of mass m given in (5). Its Schwarzschild radius will be
1
! G $ 2
LPL = # 3 &
" c %
(7)
which is called the Planck length. Notice it is simply the length formed from the three
basic constants in physics, , c, and G. It represents the smallest length that can be
operationally defined, that is, defined in terms of measurements that can be made by any
instrument. If we tried to measure a smaller distance, the time interval would be smaller,
the uncertainty in rest energy larger, the uncertainty in mass larger, and the region of
space would be experimentally indistinguishable from a black hole. Since nothing inside
a black hole can climb outside its gravitational field, we cannot see inside and thus cannot
make smaller measurement of distance.
Similarly, we can make no smaller measurement of time than the Planck time,
1
t PL
LPL
! G $ 2
=
=# 5 &
c
" c %
(8)
The Planck time and Planck length are the most basic units of time and space.
Although distance and time are assumed continuous variables, they are fundamentally
discrete. However, since physics experiments have not yet even come close to probing
space and time on the Planck scale, treating them as continuous remains a good
approximation.
General relativity, they theory of gravity introduced by Albert Einstein in 1915, has
so far passed every empirical test to high precision. However, not being a quantum theory
it can be expected to break down at the Planck scale, where it will have to be replaced by
a quantum theory of gravity, still not developed.
The Planck time represents the earliest time that can be operationally defined for our
Universe on the positive side of the time axis. However, this does not mean that "time
began" at that moment. Nothing forbids and time symmetry (see Time, symmetry of)
implies another universe at earlier times, on the negative side of our time axis.
FURTHER READING
Stenger, Victor J. (2006). The Comprehensible Cosmos: Where Do the Laws of Physics
Come from? Amherst, NY: Prometheus Books.