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  • Planck units

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    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",[1][2] 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.[3] 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)].[4]

    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.


    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

    7 See also

    8 Notes

    9 References

    10 External links

    Base units[edit]

    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."[5]

    Table 1: Dimensional universal physical constants normalized with Planck units

    Constant Symbol Dimension Value in SI units

    with uncertainties[6]

    Speed of light in

    vacuum c L T1

    2.99792458108 m s1

    (exact by definition of metre)


    constant G L3 M1 T2 6.67384(80)1011 m3 kg1 s2[7]

    Reduced Planck


    = h/2 where h is Planck


    L2 M T1 1.054571726(47)1034 J s[8]

  • Coulomb constant

    (40)1 where 0 is the permittivity of free


    L3 M

    T2 Q2

    8.9875517873681764109 kg

    m3 s2 C2

    (exact by definitions of ampere and



    constant kB

    L2 M

    T2 1 1.3806488(13)1023 J/K[9]

    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




    Name Dimension Expression Value[6] (SI units)

    Planck length Length (L)

    1.616199(97) 1035 m[10]

  • Planck mass Mass (M)

    2.17651(13) 108 kg[11]

    Planck time Time (T)

    5.39106(32) 1044 s[12]

    Planck charge Electric

    charge (Q) 1.875545956(41) 1018 C[13][14][15]


    temperature Temperature ()

    1.416833(85) 1032 K[16]

    Derived units[edit]

    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



    area Area (L2)

    2.61223 1070 m2[citation



    volume Volume (L3)

    4.22419 10105 m3[citation





    Momentum (LMT1)

    6.52485 kg m/s

  • Planck

    energy Energy (L2MT2)

    1.9561 109 J


    force Force (LMT2)

    1.21027 1044 N


    power Power (L2MT3)

    3.62831 1052 W


    density Density (L3M)

    5.15500 1096 kg/m3





    density (L1MT-2)

    4.63298 10113 J/m3


    intensity Intensity (MT3)

    1.38893 10122 W/m2




    Frequency (T1)

    1.85487 1043 s1


    pressure Pressure (L1MT2)

    4.63309 10113 Pa




    current (QT1)

    3.4789 1025 A

  • Planck

    voltage Voltage (L2MT2Q1)

    1.04295 1027 V




    Resistance (L2MT1Q2)


    Simplification of physical equations[edit]

    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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