WOLFRAM|DEMONSTRATIONS PROJECT

The Planck Scale

​
exponents
n
1
​,
n
2
​,
n
3
c
-
5
2
-
3
2
-1
-
1
2
0
1
2
1
ℏ
-1
-
1
2
0
1
2
1
G
-1
-
1
2
0
1
2
1
-
3
2
c
1
2
ℏ
1
2
G
1.61625×
-35
10
m
Planck length
L
P
It is speculated that the most natural units of mass, length, and time can be related to the fundamental physical constants associated with the theories of relativity, quantum mechanics, and gravitation. This originated from a proposal (ca. 1900) of Max Planck and is consequently designated as the Planck scale. The speed of light,
c=2.99792458×
8
10
m
-1
s
is inextricably associated with electromagnetism and the special theory of relativity. Likewise, Planck's constant, most conveniently expressed as
ℏ=h/2π=1.05457148×
-34
10
2
m
kg
-1
s
, is the essential parameter in quantum theory, while the gravitational constant
G=6.67300×
-11
10
3
m
-1
kg
-2
s
is fundamental in both Newtonian gravity and the general theory of relativity.
A product containing the three fundamental constants of the form
n
1
c
n
2
ℏ
n
3
G
will have the dimensions
n
2
-
n
3
kg
n
1
+2
n
2
+3
n
3
m
-
n
1
-
n
2
-2
n
3
s
. Your task is to find appropriate choices of the exponents
n
1
,
n
2
,
n
3
to give natural units of mass, length, and time in terms of
c
,
ℏ
, and
G
. The so-called Planck mass
M
P
, Planck length
L
p
, and Planck time
T
P
differ by many orders of magnitude from any currently accessible experimental realization. They are objects on the scale of the proposed fundamental entities of superstring theories, M-theory, loop quantum gravity, and other speculative attempts to create a "theory of everything".