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, ħ,

the speed of light in a vacuum, c,

the Coulomb constant, (4πε0)−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, ħ with quantum mechanics, ε0 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.

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

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 T −1

2.99792458×108 m s−1

(exact by definition of metre)

Gravitational

constant

G L3 M−1 T −2 6.67384(80)×10−11 m3 kg−1 s−2[7]

Reduced Planck

constant

ħ = h/2π

where h is Planck

constant

L2 M T −1 1.054571726(47)×10−34 J s[8]

Coulomb constant

(4πε0)−1

where ε0 is

the permittivity of free

space

L3 M

T −2 Q−2

8.9875517873681764×109 kg

m3 s−2 C−2

(exact by definitions of ampere and

metre)

Boltzmann

constant

kB L2 M

T −2 Θ−1 1.3806488(13)×10−23 J/K[9]

Key: L = length, M = mass, T = time, Q = electric charge, Θ = 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[6] (SI units)

Planck length Length (L)

1.616 199(97) ×

10−35 m[10]

Planck mass Mass (M)

2.176 51(13) × 10−8 kg[11]

Planck time Time (T)

5.391 06(32) × 10−44 s[12]

Planck charge

Electric

charge (Q)

1.875 545 956(41) ×

10−18 C[13][14][15]

Planck

temperature

Temperature (Θ)

1.416 833(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

ent

Planck

area Area (L2)

2.61223 × 10−70 m2[citation

needed]

Planck

volume Volume (L3)

4.22419 × 10−105 m3[citation

needed]

Planck

momentu

m

Momentum (LMT−1)

6.52485 kg m/s

Planck

energy

Energy (L2MT−2)

1.9561 × 109 J

Planck

force

Force (LMT−2)

1.21027 × 1044 N

Planck

power Power (L2MT−3)

3.62831 × 1052 W

Planck

density Density (L−3M)

5.15500 × 1096 kg/m3

Planck

energy

density

Energy

density (L−1MT-2)

4.63298 × 10113 J/m3

Planck

intensity Intensity (MT−3)

1.38893 × 10122 W/m2

Planck

angular

frequency

Frequency (T−1)

1.85487 × 1043 s−1

Planck

pressure Pressure (L−1MT−2)

4.63309 × 10113 Pa

Planck

current

Electric

current (QT−1)

3.4789 × 1025 A

Planck

voltage Voltage (L2MT−2Q−1)

1.04295 × 1027 V

Planck

impedanc

e

Resistance (L2MT−1Q−2)

29.9792458 Ω

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, need not be written if their use is understood.

Table 4: How Planck units simplify the key equations of physics

SI form Nondimensionalized form

Newton's law of

universal gravitation

Einstein field

equations in general

relativity

Mass–energy

equivalence in speci

al relativity

Energy–momentum

relation

Thermal energy per

particle per degree

of freedom

Boltzmann's entropy

formula

Planck's relation for

energy and angular

frequency

Planck's

law (surface intensit

y per unit solid

angle per

unit angular

frequency) for black

body at temperature

T.

Stefan–Boltzmann

constant σ defined

Bekenstein–

Hawking black hole

entropy[17]

Schrödinger's

equation

Hamiltonian form

of Schrödinger's

equation

Covariant form of

the Dirac equation

Coulomb's law

Maxwell's equations

Other possible normalizations[edit]

As already stated above, Planck units are derived by "normalizing" the numerical

values of certain fundamental constants to 1. These normalizations are neither the

only ones possible nor necessarily the best. Moreover, the choice of what factors to

normalize, among the factors appearing in the fundamental equations of physics, is

not evident, and the values of the Planck units are sensitive to this choice.

There are several possible alternative normalizations.

Gravity[edit]

In 1899, Newton's law of universal gravitation was still seen as exact,[citation

needed] rather than as a convenient approximation holding for "small" velocities and

distances (the non-fundamental nature of Newton's law was shown to be true

following the development of general relativity in 1915). Hence Planck normalized to

1 the gravitational constant G in Newton's law. In theories emerging after

1899, G nearly always appears multiplied by 4π or a small integer multiple thereof.

Hence a fundamental choice that has to be made when designing a system of natural

units is which, if any, instances of 4π appearing in the equations of physics are to be

eliminated via the normalization.

Normalizing 4πG to 1:

Gauss's law for gravity becomes Φg = −M (rather than Φg = −4πM in

Planck units).

The Bekenstein–Hawking formula for the entropy of a black hole in

terms of its mass mBH and the area of its event horizon ABH simplifies

to SBH = πABH = (mBH)2, where ABH and mBH are both measured in a

slight modification of reduced Planck units, described below.

The characteristic impedance Z0 of gravitational radiation in free space

becomes equal to 1. (It is equal to 4πG/c in any system of units.)[18][19]

No factors of 4π appear in the gravitoelectromagnetic (GEM) equations,

which hold in weak gravitational fields or locally flat space-time. These

equations have the same form as Maxwell's equations (and the Lorentz

forceequation) of electromagnetism, with mass density replacing charge

density, and with 1/(4πG) replacing ε0.

Setting 8πG = 1. This would eliminate 8πG from the Einstein field

equations, Einstein–Hilbert action, Friedmann equations, and the Poisson

equation for gravitation. Planck units modified so that 8πG = 1 are known

as reduced Planck units, because the Planck mass is divided by √8π. Also, the

Bekenstein–Hawking formula for the entropy of a black hole simplifies

to SBH = 2(mBH)2 = 2πABH.

Setting 16πG = 1. This would eliminate the constant c4/(16πG) from the

Einstein–Hilbert action. The form of the Einstein field equations

with cosmological constant Λ becomes Rμν − Λgμν = (Rgμν − Tμν)/2.

Electromagnetism[edit]

Planck normalized to 1 the Coulomb force constant 1/(4πε0) (as does the cgs system

of units). This sets the Planck impedance, ZP equal to Z0/4π, where Z0 is

the characteristic impedance of free space.

Normalizing the permittivity of free space ε0 to 1:

Sets the permeability of free space µ0 = 1, (because c = 1).

Sets the unit impedance, ZP = Z0.

Eliminates 4π from the nondimensionalized form of Maxwell's

equations.

Eliminates ε0 from the nondimensionalized form of Coulomb's law,

leaving 1/4π

Temperature[edit]

Planck normalized to 1 the Boltzmann constant kB.

Normalizing 1/2 kB to 1:

o Removes the factor of 1/2 in the nondimensionalized equation for

the thermal energy per particle per degree of freedom.

o Introduces a factor of 2 into the nondimensionalized form of

Boltzmann's entropy formula.

o Does not affect the value of any base or derived Planck unit other than

the Planck temperature, which it doubles.

The factor 4π is ubiquitous in theoretical physics because the surface area of

a sphere is 4πr2. This, along with the concept of flux is the basis for the inverse-square

law. For example, gravitational and electrostatic fields produced by point charges

have spherical symmetry (Barrow 2002: 214–15). The 4πr2 appearing in the

denominator of Coulomb's law, for example, follows from the flux of an electrostatic

field being distributed uniformly on the surface of a sphere. If space had more than

three spacial dimensions, the factor 4π would have to be changed according to the

geometry of the sphere in higher dimensions. Likewise for Newton's law of universal

gravitation.

Hence a substantial body of physical theory discovered since Planck (1899) suggests

normalizing to 1 not G but 4nπG, for one of n = 1, 2, or 4. Doing so would introduce a

factor of 1/(4nπ) into the nondimensionalized form of the law of universal gravitation,

consistent with the modern formulation of Coulomb's law in terms of the vacuum

permittivity. In fact, alternative normalizations frequently preserve the factor of 1/(4π)

in the nondimensionalized form of Coulomb's law as well, so that the

nondimensionalized Maxwell's equations for electromagnetism and gravitomagnetism

both take the same form as those for electromagnetism in SI, which does not have any

factors of 4π.

Uncertainties in measured values[edit]

Table 2 clearly defines Planck units in terms of the fundamental constants. Yet

relative to other units of measurement such as SI, the values of the Planck units are

only known approximately. This is mostly due to uncertainty in the value of the

gravitational constant G.

Today the value of the speed of light c in SI units is not subject to measurement error,

because the SI base unit of length, the metre, is now defined as the length of the path

travelled by light in vacuum during a time interval of1⁄299792458 of a second. Hence the

value of c is now exact by definition, and contributes no uncertainty to the SI

equivalents of the Planck units. The same is true of the value of the vacuum

permittivity ε0, due to the definition ofampere which sets the vacuum

permeability μ0 to 4π × 10−7 H/m and the fact that μ0ε0 = 1/c2. The numerical value of

the reduced Planck constant ℏ has been determined experimentally to 44 parts per

billion, while that of G has been determined experimentally to no better than 1 part in

8300 (or 120000 parts per billion).[6] G appears in the definition of almost every

Planck unit in Tables 2 and 3. Hence the uncertainty in the values of the Table 2 and 3

SI equivalents of the Planck units derives almost entirely from uncertainty in the value

of G. (The propagation of the error in G is a function of the exponent of G in the

algebraic expression for a unit. Since that exponent is ±1⁄2 for every base unit other

than Planck charge, the relative uncertainty of each base unit is about one half that

of G. This is indeed the case; according to CODATA, the experimental values of the

SI equivalents of the base Planck units are known to about 1 part in 16600, or 60000

parts per billion.)

Discussion[edit]

Some Planck units are suitable for measuring quantities that are familiar from daily

experience. For example:

1 Planck mass is about 22 micrograms;

1 Planck momentum is about 6.5 kg⋅m/s;

1 Planck energy is about 500 kW⋅h;

1 Planck charge is slightly more than 11.7 elementary charges;

1 Planck impedance is very nearly 30 ohms.

However, most Planck units are many orders of magnitude too large or too small to be

of any practical use, so that Planck units as a system are really only relevant to

theoretical physics. In fact, 1 Planck unit is often the largest or smallest value of a

physical quantity that makes sense according to our current understanding. For

example:

A speed of 1 Planck length per Planck time is the speed of light in a vacuum,

the maximum possible speed in special relativity;[20]

Our understanding of the Big Bang begins with the Planck epoch, when the

universe was 1 Planck time old and 1 Planck length in diameter, and had a

Planck temperature of 1. At that moment, quantum theory as presently

understood becomes applicable. Understanding the universe when it was less

than 1 Planck time old requires a theory of quantum gravity that would

incorporate quantum effects into general relativity. Such a theory does not yet

exist;

At a Planck temperature of 1, all symmetries broken since the early Big Bang

would be restored, and the four fundamental forces of contemporary physical

theory would become one force.[citation needed]

Relative to the Planck Epoch, the universe today looks extreme when expressed in

Planck units, as in this set of approximations:[21][22]

Table 5: Today's universe in Planck units.

Property of

present-

day Universe

Approximate

number

of Planck units

Equivalents

Age 8.08 × 1060 tP 4.35 × 1017 s, or 13.8 × 109 years

Diameter 5.4 × 1061 lP 8.7 × 1026 m or 9.2 × 1010 light-years

Mass approx. 1060 mP

3 × 1052 kg or 1.5 × 1022 solar masses (only

counting stars)

1080 protons (sometimes known as the Eddington

number)

Temperature 1.9 × 10−32 TP

2.725 K

temperature of the cosmic microwave background

radiation

Cosmological

constant

5.6 × 10−122 tP−2 1.9 × 10−35 s−2

Hubble constant 1.24 × 10−61 tP−1 67.8 (km/s)/Mpc

The recurrence of large numbers close or related to 1060 in the above table is a

coincidence that intrigues some theorists. It is an example of the kind of large

numbers coincidence that led theorists such as Eddington and Dirac to develop

alternative physical theories. Theories derived from such coincidences have

sometimes been dismissed by mainstream physicists as "numerology".[citation needed]

History[edit]

Natural units began in 1881, when George Johnstone Stoney, noting that electric

charge is quantized, derived units of length, time, and mass, now named Stoney

units in his honor, by normalizing G, c, and the electron charge, e, to 1.[23] In

1898, Max Planck discovered that action is quantized, and published the result in a

paper presented to the Prussian Academy of Sciences in May 1899.[24][25] At the end

of the paper, Planck introduced, as a consequence of his discovery, the base units later

named in his honor. The Planck units are based on the quantum of action, now usually

known as Planck's constant. Planck called the constant b in his paper, though h is now

common. Planck underlined the universality of the new unit system, writing:

...ihre Bedeutung für alle Zeiten und für alle, auch außerirdische und

außermenschliche Kulturen notwendig behalten und welche daher als »natürliche

Maßeinheiten« bezeichnet werden können...

...These necessarily retain their meaning for all times and for all civilizations, even

extraterrestrial and non-human ones, and can therefore be designated as "natural

units"...

Planck considered only the units based on the universal constants G, ħ, c, and kB to

arrive at natural units for length, time, mass, and temperature.[25] Planck did not adopt

any electromagnetic units. However, since the non-rationalizedgravitational

constant, G, is set to 1, a natural generalization of Planck units to a unit electric

charge is to also set the non-rationalized Coulomb constant, ke, to 1 as

well. [26] Planck's paper also gave numerical values for the base units that were close

to modern values.

Planck units and the invariant scaling of nature[edit]

Some theorists (such as Dirac and Milne) have proposed cosmologies that conjecture

that physical "constants" might actually change over time (e.g. Variable speed of

light or Dirac varying-G theory). Such cosmologies have not gained mainstream

acceptance and yet there is still considerable scientific interest in the possibility that

physical "constants" might change, although such propositions introduce difficult

questions. Perhaps the first question to address is: How would such a change make a

noticeable operational difference in physical measurement or, more fundamentally,

our perception of reality? If some particular physical constant had changed, how

would we notice it, how would physical reality be different? Which changed constants

result in a meaningful and measurable difference in physical reality? If a physical

constant that is not dimensionless, such as the speed of light, did in fact change, would

we be able to notice it or measure it unambiguously? – a question examined

by Michael Duff in his paper "Comment on time-variation of fundamental

constants".[27]

George Gamow argued in his book Mr Tompkins in Wonderland that a sufficient

change in a dimensionful physical constant, such as the speed of light in a vacuum,

would result in obvious perceptible changes. But this idea is challenged:

[An] important lesson we learn from the way that pure numbers like α define the

world is what it really means for worlds to be different. The pure number we call the

fine structure constant and denote by α is a combination of the electron charge, e, the

speed of light, c, and Planck's constant, h. At first we might be tempted to think that a

world in which the speed of light was slower would be a different world. But this

would be a mistake. If c, h, and e were all changed so that the values they have in

metric (or any other) units were different when we looked them up in our tables of

physical constants, but the value of α remained the same, this new world would

be observationally indistinguishable from our world. The only thing that counts in the

definition of worlds are the values of the dimensionless constants of Nature. If all

masses were doubled in value [including the Planck mass mP ] you cannot tell because

all the pure numbers defined by the ratios of any pair of masses are unchanged.

— Barrow 2002[21]

Referring to Duff's "Comment on time-variation of fundamental constants"[27] and

Duff, Okun, and Veneziano's paper "Trialogue on the number of fundamental

constants",[28] particularly the section entitled "The operationally indistinguishable

world of Mr. Tompkins", if all physical quantities (masses and other properties of

particles) were expressed in terms of Planck units, those quantities would be

dimensionless numbers (mass divided by the Planck mass, length divided by the

Planck length, etc.) and the only quantities that we ultimately measure in physical

experiments or in our perception of reality are dimensionless numbers. When one

commonly measures a length with a ruler or tape-measure, that person is actually

counting tick marks on a given standard or is measuring the length relative to that

given standard, which is a dimensionless value. It is no different for physical

experiments, as all physical quantities are measured relative to some other like-

dimensioned quantity.

We can notice a difference if some dimensionless physical quantity such as fine-

structure constant, α, changes or the proton-to-electron mass ratio, mp/me, changes

(atomic structures would change) but if all dimensionless physical quantities remained

unchanged (this includes all possible ratios of identically dimensioned physical

quantity), we can not tell if a dimensionful quantity, such as the speed of light, c, has

changed. And, indeed, the Tompkins concept becomes meaningless in our perception

of reality if a dimensional quantity such as c has changed, even drastically.

If the speed of light c, were somehow suddenly cut in half and changed to c/2, (but

with the axiom that all dimensionless physical quantities continuing to remain the

same), then the Planck Length would increase by a factor of from the point of

view of some unaffected "god-like" observer on the outside.[citation needed] Measured by

"mortal" observers in terms of Planck units, the new speed of light would be remain as

1 new Planck length per 1 new Planck time – which is no different from the old

measurement. But, since by axiom, the size of atoms (approximately the Bohr radius)

are related to the Planck length by an unchanging dimensionless constant of

proportionality:

Then atoms would be bigger (in one dimension) by , each of us would be taller

by , and so would our metre sticks be taller (and wider and thicker) by a factor

of . Our perception of distance and lengths relative to the Planck length is, by

axiom, an unchanging dimensionless constant.

Our clocks would tick slower by a factor of (from the point of view of this

unaffected "god-like" observer) because the Planck time has increased by but we

would not know the difference (our perception of durations of time relative to the

Planck time is, by axiom, an unchanging dimensionless constant). This hypothetical

god-like observer on the outside might observe that light now propagates at half the

speed that it previously did (as well as all other observed velocities) but it would still

travel 299792458 of our new metres in the time elapsed by one of our new seconds (

continues to equal 299792458 m/s). We would not notice any difference.

This contradicts what George Gamow writes in his book Mr. Tompkins; there, Gamow

suggests that if a dimension-dependent universal constant such as c changed,

we would easily notice the difference. The disagreement is better thought of as the

ambiguity in the phrase "changing a physical constant"; what would happen depends

on whether (1) all other dimensionless constants were kept the same, or whether

(2) all other dimension-dependent constants are kept the same. The second choice is a

somewhat confusing possibility, since most of our units of measurement are defined

in relation to the outcomes of physical experiments, and the experimental results

depend on the constants. (The only exception is the kilogram.) Gamow does not

address this subtlety; the thought experiments he conducts in his popular works

assume the second choice for "changing a physical constant". And Duff or

Barrow[citation needed] would point out that ascribing a change in measurable reality,

i.e. α, to a specific dimensional component quantity, such as c, is unjustified. The very

same operational difference in measurement or perceived reality could just as well be

caused by a change in h or e.

This unvarying aspect of the Planck-relative scale, or that of any other system of

natural units, leads many theorists to conclude that a hypothetical change in

dimensionful physical constants can only be manifest as a change indimensionless

physical constants. One such dimensionless physical constant is the fine-structure

constant. There are some experimental physicists who assert they have in fact

measured a change in the fine structure constant[29] and this has intensified the debate

about the measurement of physical constants. According to some theorists[30] there are

some very special circumstances in which changes in the fine-structure

constant can be measured as a change indimensionful physical constants. Others

however reject the possibility of measuring a change in dimensionful physical

constants under any circumstance.[27] The difficulty or even the impossibility of

measuring changes in dimensionful physical constants has led some theorists to

debate with each other whether or not a dimensionful physical constant has any

practical significance at all and that in turn leads to questions about which

dimensionful physical constants are meaningful.[28]