# planck units

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• Planck units

In physics, Planck units are physical units of measurement defined exclusively in

terms of five universal physical constants listed below, in such a manner that these

five physical constants take on the numerical value of 1 when expressed in terms of

these units. Planck units have profound significance for theoretical physics since they

elegantly simplify several recurring algebraic expressions of physical

law by nondimensionalization. They are particularly relevant in research on unified

theories such as quantum gravity.

Originally proposed in 1899 by German physicist Max Planck, these units are also

known as natural units because the origin of their definition comes only from

properties of the fundamental physical theories and not from interchangeable

experimental parameters. Planck units are only one system of natural units among

other systems, but are considered unique in that these units are not based on properties

of any prototype object or particle (that would be arbitrarily chosen), but rather on

properties of free space alone.

The universal constants that Planck units, by definition, normalize to 1 are:

the gravitational constant, G,

the reduced Planck constant, , the speed of light in a vacuum, c,

the Coulomb constant, (40)1 (sometimes ke or k), and the Boltzmann constant, kB (sometimes k).

Each of these constants can be associated with at least one fundamental physical

theory: c with electromagnetism and special relativity, G with general

relativity and Newtonian gravity, with quantum mechanics, 0 with electrostatics, and kB with statistical mechanics and thermodynamics.

Planck units are sometimes called "God's units", since Planck units are free

of anthropocentric arbitrariness. Some physicists argue that communication

with extraterrestrial intelligence would have to employ such a system of units in order

to be understood. Unlike the metre and second, which exist as fundamental units in

the SI system for historical reasons, the Planck length and Planck time are

conceptually linked at a fundamental physical level.

• Natural units help physicists to reframe questions. Frank Wilczek puts it succinctly:

We see that the question [posed] is not, "Why is gravity so feeble?" but rather, "Why

is the proton's mass so small?" For in natural (Planck) units, the strength of gravity

simply is what it is, a primary quantity, while the proton's mass is the tiny number

[1/(13 quintillion)].

It is true that the electrostatic repulsive force between two protons (alone in free

space) greatly exceeds the gravitational attractive force between the same two protons.

From the point of view of Planck units, however, this is not a statement about the

relative strengths of the two forces; rather, it is a manifestation of the fact that

the charge on the protons is approximately the Planck charge but the mass of the

protons is far less than the Planck mass.

Contents

1 Base units

2 Derived units

3 Simplification of physical equations

4 Other possible normalizations

o 4.1 Gravity

o 4.2 Electromagnetism

o 4.3 Temperature

5 Uncertainties in measured values

6 Discussion

o 6.1 History

o 6.2 Planck units and the invariant scaling of nature

8 Notes

9 References

Base units

All systems of measurement feature base units: in the International System of

Units (SI), for example, the base unit of length is the metre. In the system of Planck

units, the Planck base unit of length is known simply as the Planck length, the base

unit of time is the Planck time, and so on. These units are derived from the five

dimensional universal physical constants of Table 1, in such a manner that these

constants are eliminated from fundamental equations of physical law when physical

• quantities are expressed in terms of Planck units. For example, Newton's law of

universal gravitation,

can be expressed as

Both equations are dimensionally consistent and equally valid in any system of units,

but the second equation, with G missing, is relating only dimensionless

quantities since any ratio of two like-dimensioned quantities is a dimensionless

quantity. If, by a shorthand convention, it is axiomatically understood that all physical

quantities are expressed in terms of Planck units, the ratios above may be expressed

simply with the symbols of physical quantity, without being scaled by their

corresponding unit:

In order for this last equation to be valid (without G present), F, m1, m2, and r are

understood to be the dimensionless numerical values of these quantities measured in

terms of Planck units. This is why Planck units or any other use of natural units

should be employed with care; referring to G = c = 1, Paul S. Wesson wrote that,

"Mathematically it is an acceptable trick which saves labour. Physically it represents a

loss of information and can lead to confusion."

Table 1: Dimensional universal physical constants normalized with Planck units

Constant Symbol Dimension Value in SI units

with uncertainties

Speed of light in

vacuum c L T1

2.99792458108 m s1

(exact by definition of metre)

Gravitational

constant G L3 M1 T2 6.67384(80)1011 m3 kg1 s2

Reduced Planck

constant

= h/2 where h is Planck

constant

L2 M T1 1.054571726(47)1034 J s

• Coulomb constant

(40)1 where 0 is the permittivity of free

space

L3 M

T2 Q2

8.9875517873681764109 kg

m3 s2 C2

(exact by definitions of ampere and

metre)

Boltzmann

constant kB

L2 M

T2 1 1.3806488(13)1023 J/K

Key: L = length, M = mass, T = time, Q = electric charge, = temperature.

As can be seen above, the gravitational attractive force of two bodies of 1 Planck

mass each, set apart by 1 Planck length is 1 Planck force. Likewise, the distance

traveled by light during 1 Planck time is 1 Planck length. To determine, in terms of SI

or another existing system of units, the quantitative values of the five base Planck

units, those two equations and three others must be satisfied to determine the five

unknown quantities that define the base Planck units:

Solving the five equations above for the five unknowns results in a unique set of

values for the five base Planck units:

Table 2: Base Planck units

Base Planck units

v

t

e

Name Dimension Expression Value (SI units)

Planck length Length (L)

1.616199(97) 1035 m

• Planck mass Mass (M)

2.17651(13) 108 kg

Planck time Time (T)

5.39106(32) 1044 s

Planck charge Electric

charge (Q) 1.875545956(41) 1018 C

Planck

temperature Temperature ()

1.416833(85) 1032 K

Derived units

In any system of measurement, units for many physical quantities can be derived from

base units. Table 3 offers a sample of derived Planck units, some of which in fact are

seldom used. As with the base units, their use is mostly confined to theoretical physics

because most of them are too large or too small for empirical or practical use and

there are large uncertainties in their values (see Discussion and Uncertainties in

values below).

Table 3: Derived Planck units

Name Dimension Expression Approximate SI equival

ent

Planck

area Area (L2)

2.61223 1070 m2[citation

needed]

Planck

volume Volume (L3)

4.22419 10105 m3[citation

needed]

Planck

momentu

m

Momentum (LMT1)

6.52485 kg m/s

• Planck

energy Energy (L2MT2)

1.9561 109 J

Planck

force Force (LMT2)

1.21027 1044 N

Planck

power Power (L2MT3)

3.62831 1052 W

Planck

density Density (L3M)

5.15500 1096 kg/m3

Planck

energy

density

Energy

density (L1MT-2)

4.63298 10113 J/m3

Planck

intensity Intensity (MT3)

1.38893 10122 W/m2

Planck

angular

frequency

Frequency (T1)

1.85487 1043 s1

Planck

pressure Pressure (L1MT2)

4.63309 10113 Pa

Planck

current

Electric

current (QT1)

3.4789 1025 A

• Planck

voltage Voltage (L2MT2Q1)

1.04295 1027 V

Planck

impedanc

e

Resistance (L2MT1Q2)

29.9792458

Simplification of physical equations

Physical quantities that have different dimensions (such as time and length) cannot be

equated even if they are numerically equal (1 second is not the same as 1 metre). In

theoretical physics, however, this scruple can be set aside, by a process

called nondimensionalization. Table 4 shows how the use of Planck units simplifies

many fundamental equations of physics, because this gives each of the five

fundamental constants, and products of them, a simple numeric value of 1. In the SI

form, the units should be accounted for. In the nondimensionalized form, the units,

which are now Planck units

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