 Planck Impedance
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Planck Impedance

In particle physics and physical cosmology, Planck units are a set of units of measurement defined exclusively in terms of five universal physical constants, in such a manner that these five physical constants take on the numerical value of 1 when expressed in terms of these units.

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 nature and not from any human construct. Planck units are only one system of several systems of natural units, but Planck units are not based on properties of any prototype object or particle (that would be arbitrarily chosen), but rather on only the properties of free space. They are relevant in research on unified theories such as quantum gravity.

The term Planck scale refers to the magnitudes of space, time, energy and other units, below which (or beyond which) the predictions of the Standard Model, quantum field theory and general relativity are no longer reconcilable, and quantum effects of gravity are expected to dominate. This region may be characterized by energies around (the energy-equivalent of the Planck mass), time intervals around (the Planck time) and lengths around (the Planck length). At the Planck scale, current models are not expected to be a useful guide to the cosmos, and physicists have no scientific model to suggest how the physical universe behaves. The best known example is represented by the conditions in the first 10-43 seconds of our universe after the Big Bang, approximately 13.8 billion years ago.

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

Each of these constants can be associated with a fundamental physical theory or concept: c with special relativity, G with general relativity, ? with quantum mechanics, kB with thermodynamics, and ?0 with electromagnetism.

## Introduction

Any system of measurement may be assigned a mutually independent set of base quantities and associated base units, from which all other quantities and units may be derived. In the International System of Units, for example, the SI base quantities include length with the associated unit of the metre. In the system of Planck units, a similar set of base quantities may be selected, and the Planck base unit of length is then 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 selected equations of physical law when physical quantities are expressed in terms of Planck units. For example, Newton's law of universal gravitation,

{\begin{aligned}F&=G{\frac {m_{1}m_{2}}{r^{2}}}\\\\&=\left({\frac {F_{\text{P}}l_{\text{P}}^{2}}{m_{\text{P}}^{2}}}\right){\frac {m_{1}m_{2}}{r^{2}}}\\\end{aligned}} can be expressed as:

${\frac {F}{F_{\text{P}}}}={\frac {\left({\dfrac {m_{1}}{m_{\text{P}}}}\right)\left({\dfrac {m_{2}}{m_{\text{P}}}}\right)}{\left({\dfrac {r}{l_{\text{P}}}}\right)^{2}}}.$ 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 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 explicitly by their corresponding unit:

$F={\frac {m_{1}m_{2}}{r^{2}}}\ .$ This last equation (without G) is valid only if F, m1, m2, and r are 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 , 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."

## Definition

Table 1: Dimensional universal physical constants normalized with Planck units
Constant Symbol Dimension in SI Quantities Value (SI units)
c L T-1 
(exact by definition)
Gravitational constant G L3 M-1 T-2 
? =
where h is the Planck constant
L2 M T-1 
(defined as exactly)
Boltzmann constant kB L2 M T-2 ?-1 
(exact by definition)
Coulomb constant ke =
where ?0 is the permittivity of free space


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

A property of Planck units is that in order to obtain the value of any of the physical constants above it is enough to replace the dimensions of the constant with the corresponding Planck units. For example, the gravitational constant (G) has as dimensions L3 M-1 T-2. By replacing each dimension with the value of each corresponding Planck unit one obtains the value of (1 lP)3 × (1 mP)-1 × (1 tP)-2 = (1.616255×10-35m)3 × (2.176435×10-8kg)-1 × (5.391247×10-44s)-2 = 6.674...×10-11 m3 kg-1 s-2 (which is the value of G).

This is the consequence of the fact that the system is internally coherent. For example, the gravitational attractive force of two bodies of 1 Planck mass each, set apart by 1 Planck length is 1 coherent Planck unit of 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:

$l_{\text{P}}=c\ t_{\text{P}}$ $F_{\text{P}}={\frac {m_{\text{P}}l_{\text{P}}}{t_{\text{P}}^{2}}}=G\ {\frac {m_{\text{P}}^{2}}{l_{\text{P}}^{2}}}$ $E_{\text{P}}={\frac {m_{\text{P}}l_{\text{P}}^{2}}{t_{\text{P}}^{2}}}=\hbar \ {\frac {1}{t_{\text{P}}}}$ $E_{\text{P}}={\frac {m_{\text{P}}l_{\text{P}}^{2}}{t_{\text{P}}^{2}}}=k_{\text{B}}\ T_{\text{P}}.$ $F_{\text{P}}={\frac {m_{\text{P}}l_{\text{P}}}{t_{\text{P}}^{2}}}={\frac {1}{4\pi \varepsilon _{0}}}\ {\frac {q_{\text{P}}^{2}}{l_{\text{P}}^{2}}}$ 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
Name Dimension Expression Value (SI units)
Planck length Length (L) $l_{\text{P}}={\sqrt {\frac {\hbar G}{c^{3}}}}$ Planck mass Mass (M) $m_{\text{P}}={\sqrt {\frac {\hbar c}{G}}}$ Planck time Time (T) $t_{\text{P}}={\frac {l_{\text{P}}}{c}}={\frac {\hbar }{m_{\text{P}}c^{2}}}={\sqrt {\frac {\hbar G}{c^{5}}}}$ Planck temperature Temperature (?) $T_{\text{P}}={\frac {m_{\text{P}}c^{2}}{k_{\text{B}}}}={\sqrt {\frac {\hbar c^{5}}{Gk_{\text{B}}^{2}}}}$ Planck charge Electric charge (Q) $q_{\text{P}}={\sqrt {\frac {\hbar c}{k_{\text{e}}}}}={\sqrt {4\pi \varepsilon _{0}\hbar c}}={\frac {e}{\sqrt {\alpha }}}$ 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 due to uncertainty in the values of the gravitational constant G and ?0 in SI units.

The values of c, h, e and kB in SI units are exact due to the definition of the second, metre, kilogram and kelvin in terms of these constants, and contribute no uncertainty to the values of the Planck units expressed in terms of SI units. The vacuum permittivity ?0 has a relative uncertainty of . The numerical value of G has been determined experimentally to a relative uncertainty of .G appears in the definition of every Planck unit other than for charge 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 ± for every base unit other than Planck charge, the relative uncertainty of each base unit is about one half that of G.)

## 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.

Table 4: Coherent derived units of Planck units
Derived unit of Expression Approximate SI equivalent
area (L2) $l_{\text{P}}^{2}={\frac {\hbar G}{c^{3}}}$ volume (L3) $l_{\text{P}}^{3}=\left({\frac {\hbar G}{c^{3}}}\right)^{\frac {3}{2}}={\sqrt {\frac {(\hbar G)^{3}}{c^{9}}}}$ momentum (LMT-1) $m_{\text{P}}c={\frac {\hbar }{l_{\text{P}}}}={\sqrt {\frac {\hbar c^{3}}{G}}}$ energy (L2MT-2) $E_{\text{P}}=m_{\text{P}}c^{2}={\frac {\hbar }{t_{\text{P}}}}={\sqrt {\frac {\hbar c^{5}}{G}}}$ force (LMT-2) $F_{\text{P}}={\frac {E_{\text{P}}}{l_{\text{P}}}}={\frac {\hbar }{l_{\text{P}}t_{\text{P}}}}={\frac {c^{4}}{G}}$ density (L-3M) $\rho _{\text{P}}={\frac {m_{\text{P}}}{l_{\text{P}}^{3}}}={\frac {\hbar t_{\text{P}}}{l_{\text{P}}^{5}}}={\frac {c^{5}}{\hbar G^{2}}}$ acceleration (LT-2) $a_{\text{P}}={\frac {c}{t_{\text{P}}}}={\sqrt {\frac {c^{7}}{\hbar G}}}$ frequency (T-1) $f_{p}={\frac {c}{l_{\text{P}}}}={\sqrt {\frac {c^{5}}{\hbar G}}}$ Most Planck units are many orders of magnitude too large or too small to be of practical use, so that Planck units as a system are typically only relevant to theoretical physics. In fact, 1 Planck unit is often the largest or smallest value of a physical quantity that makes sense within present-day theories of physics. For example, 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. 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.

An exception to the general pattern of Planck units being "extreme" in magnitude is the Planck mass, which is about 22 micrograms: very large compared to subatomic particles, but well within the mass range of living things.

## Significance

Planck units have little anthropocentric arbitrariness, but do still involve some arbitrary choices in terms of the defining constants. Unlike the metre and second, which exist as base units in the SI system for historical reasons, the Planck length and Planck time are conceptually linked at a fundamental physical level. Consequently, 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)].

While 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, this is not about the relative strengths of the two fundamental forces. From the point of view of Planck units, this is comparing apples to oranges, because mass and electric charge are incommensurable quantities. Rather, the disparity of magnitude of force is a manifestation of the fact that the charge on the protons is approximately the unit charge but the mass of the protons is far less than the unit mass.

### Cosmology

In Big Bang cosmology, the Planck epoch or Planck era is the earliest stage of the Big Bang, before the time passed was equal to the Planck time, tP, or approximately 10-43 seconds. There is no currently available physical theory to describe such short times, and it is not clear in what sense the concept of time is meaningful for values smaller than the Planck time. It is generally assumed that quantum effects of gravity dominate physical interactions at this time scale. At this scale, the unified force of the Standard Model is assumed to be unified with gravitation. Immeasurably hot and dense, the state of the Planck epoch was succeeded by the grand unification epoch, where gravitation is separated from the unified force of the Standard Model, in turn followed by the inflationary epoch, which ended after about 10-32 seconds (or about 1010 tP).

Relative to the Planck epoch, the observable universe today looks extreme when expressed in Planck units, as in this set of approximations:

Table 5: Today's universe in Planck units.
Property of
present-day Observable Universe
Approximate number
of Planck units
Equivalents
Age 8.08 × 1060tP 4.35 × 1017 s, or 13.8 × 109 years
Diameter 5.4 × 1061lP 8.7 × 1026 m or 9.2 × 1010light-years
Mass approx. 1060mP 3 × 1052 kg or 1.5 × 1022solar masses (only counting stars)
1080 protons (sometimes known as the Eddington number)
Density 1.8 × 10-123?P 9.9 × 10-27 kg m-3
Temperature 1.9 × 10-32TP 2.725 K
temperature of the cosmic microwave background radiation
Cosmological constant 5.6 × 10-122t -2
P
1.9 × 10-35 s-2
Hubble constant 1.18 × 10-61t -1
P
2.2 × 10-18 s-1 or 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 (e.g. a variable speed of light or Dirac varying-G theory). After the measurement of the cosmological constant in 1998, estimated at 10-122 in Planck units, it was noted that this is suggestively close to the reciprocal of the age of the universe squared. Barrow and Shaw (2011) proposed a modified theory in which ? is a field evolving in such a way that its value remains ? ~ T-2 throughout the history of the universe.

## History

The concept of natural units was introduced 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.

In 1899 (one year before the advent of quantum theory), Max Planck introduced what became later known as the Planck constant. At the end of the paper, Planck proposed, 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 the Planck constant. Planck called the constant b in his paper, though h (or the closely related ?) is now common. However, at that time it was part of Wien's radiation law, which Planck thought to be correct. Planck underlined the universality of the new unit system, writing:

... die Möglichkeit gegebenist, Einheiten für Länge, Masse, Zeit und Temperatur aufzustellen, welche, unabhängig von speciellen Körpern oder Substanzen, ihre Bedeutung für alle Zeiten und für alle, auch außerirdische und außermenschliche Culturen notwendig behalten und welche daher als »natürliche Maßeinheiten« bezeichnet werden können.
... it is possible to set up units for length, mass, time and temperature, which are independent of special bodies or substances, necessarily retaining their meaning for all times and for all civilizations, including extraterrestrial and non-human ones, which can be called "natural units of measure".

Planck considered only the units based on the universal constants G, ?, c, and kB to arrive at natural units for length, time, mass, and temperature. Planck did not adopt any electromagnetic units. Planck's paper also gave numerical values for the base units that were close to modern values.

The original base units proposed by Planck in 1899 differed by a factor of ${\sqrt {2\pi }}$ from the Planck units in use today. This is due to the use of the reduced Planck constant ($\hbar$ ) in the modern units, which did not appear in the original proposal.

Original Planck units
Name Dimension Expression Value in SI units Value in modern Planck units
Original Planck length Length (L) ${\sqrt {\frac {hG}{c^{3}}}}$ ${\sqrt {2\pi }}\times l_{\text{P}}$ Original Planck mass Mass (M) ${\sqrt {\frac {hc}{G}}}$ ${\sqrt {2\pi }}\times m_{\text{P}}$ Original Planck time Time (T) ${\sqrt {\frac {hG}{c^{5}}}}$ ${\sqrt {2\pi }}\times t_{\text{P}}$ Original Planck temperature Temperature (?) ${\sqrt {\frac {hc^{5}}{Gk_{\text{B}}}}}$ ${\sqrt {2\pi }}\times T_{\text{P}}$ ## List 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 6 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 6: How Planck units simplify the key equations of physics
SI form Lorentz-Heaviside version Planck form Gaussian version form
Newton's law of universal gravitation $F=G{\frac {m_{1}m_{2}}{r^{2}}}$ $F={\frac {m_{1}m_{2}}{4\pi r^{2}}}$ $F={\frac {m_{1}m_{2}}{r^{2}}}$ Einstein field equations in general relativity ${G_{\mu \nu }=8\pi {G \over c^{4}}T_{\mu \nu }}\$ ${G_{\mu \nu }=2T_{\mu \nu }}\$ ${G_{\mu \nu }=8\pi T_{\mu \nu }}\$ Mass-energy equivalence in special relativity ${E=mc^{2}}\$ ${E=m}\$ Energy-momentum relation $E^{2}=m^{2}c^{4}+p^{2}c^{2}\;$ $E^{2}=m^{2}+p^{2}\;$ Thermal energy per particle per degree of freedom ${E={\tfrac {1}{2}}k_{\text{B}}T}\$ ${E={\tfrac {1}{2}}T}\$ Boltzmann's entropy formula ${S=k_{\text{B}}\ln \Omega }\$ ${S=\ln \Omega }\$ Planck-Einstein relation for energy and angular frequency ${E=\hbar \omega }\$ ${E=\omega }\$ Planck's law (surface intensity per unit solid angle per unit angular frequency) for black body at temperature T. $I(\omega ,T)={\frac {\hbar \omega ^{3}}{4\pi ^{3}c^{2}}}~{\frac {1}{e^{\frac {\hbar \omega }{k_{\text{B}}T}}-1}}$ $I(\omega ,T)={\frac {\omega ^{3}}{4\pi ^{3}}}~{\frac {1}{e^{\omega /T}-1}}$ Stefan-Boltzmann constant ? defined $\sigma ={\frac {\pi ^{2}k_{\text{B}}^{4}}{60\hbar ^{3}c^{2}}}$ $\sigma ={\frac {\pi ^{2}}{60}}$ Bekenstein-Hawking black hole entropy $S_{\text{BH}}={\frac {A_{\text{BH}}k_{\text{B}}c^{3}}{4G\hbar }}={\frac {4\pi Gk_{\text{B}}m_{\text{BH}}^{2}}{\hbar c}}$ $S_{\text{BH}}=\pi A_{\text{BH}}=m_{\text{BH}}^{2}$ $S_{\text{BH}}={\frac {A_{\text{BH}}}{4}}=4\pi m_{\text{BH}}^{2}$ Schrödinger's equation $-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} ,t)\psi (\mathbf {r} ,t)=i\hbar {\frac {\partial \psi (\mathbf {r} ,t)}{\partial t}}$ $-{\frac {1}{2m}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} ,t)\psi (\mathbf {r} ,t)=i{\frac {\partial \psi (\mathbf {r} ,t)}{\partial t}}$ Hamiltonian form of Schrödinger's equation $H\left|\psi _{t}\right\rangle =i\hbar {\frac {\partial }{\partial t}}\left|\psi _{t}\right\rangle$ $H\left|\psi _{t}\right\rangle =i{\frac {\partial }{\partial t}}\left|\psi _{t}\right\rangle$ Covariant form of the Dirac equation $\ (i\hbar \gamma ^{\mu }\partial _{\mu }-mc)\psi =0$ $\ (i\gamma ^{\mu }\partial _{\mu }-m)\psi =0$ Unruh temperature $T={\frac {\hbar a}{2\pi ck_{B}}}$ $T={\frac {a}{2\pi }}$ Coulomb's law $F={\frac {1}{4\pi \epsilon _{0}}}{\frac {q_{1}q_{2}}{r^{2}}}$ $F={\frac {q_{1}q_{2}}{4\pi r^{2}}}$ $F={\frac {q_{1}q_{2}}{r^{2}}}$ Maxwell's equations $\nabla \cdot \mathbf {E} ={\frac {1}{\epsilon _{0}}}\rho$ $\nabla \cdot \mathbf {B} =0\$ $\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$ $\nabla \times \mathbf {B} ={\frac {1}{c^{2}}}\left({\frac {1}{\epsilon _{0}}}\mathbf {J} +{\frac {\partial \mathbf {E} }{\partial t}}\right)$ $\nabla \cdot \mathbf {E} =\rho$ $\nabla \cdot \mathbf {B} =0\$ $\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$ $\nabla \times \mathbf {B} =\mathbf {J} +{\frac {\partial \mathbf {E} }{\partial t}}$ $\nabla \cdot \mathbf {E} =4\pi \rho \$ $\nabla \cdot \mathbf {B} =0\$ $\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$ $\nabla \times \mathbf {B} =4\pi \mathbf {J} +{\frac {\partial \mathbf {E} }{\partial t}}$ Ideal gas law $PV=nRT$ $PV=NT$ ## Alternative choices of normalization

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.

The factor 4? is ubiquitous in theoretical physics because the surface area of a sphere of radius r is 4?r2. This, along with the concept of flux, are the basis for the inverse-square law, Gauss's law, and the divergence operator applied to flux density. For example, gravitational and electrostatic fields produced by point charges have spherical symmetry (Barrow 2002: 214-15). The 4?r2 appearing in the denominator of Coulomb's law in rationalized form, for example, follows from the flux of an electrostatic field being distributed uniformly on the surface of a sphere. Likewise for Newton's law of universal gravitation. (If space had more than three spatial dimensions, the factor 4? would have to be changed according to the geometry of the sphere in higher dimensions.)

Hence a substantial body of physical theory developed since Planck (1899) suggests normalizing not G but either 4?G (or 8?G or 16?G) to 1. Doing so would introduce a factor of (or or ) into the nondimensionalized form of the law of universal gravitation, consistent with the modern rationalized formulation of Coulomb's law in terms of the vacuum permittivity. In fact, alternative normalizations frequently preserve the factor of in the nondimensionalized form of Coulomb's law as well, so that the nondimensionalized Maxwell's equations for electromagnetism and gravitoelectromagnetism both take the same form as those for electromagnetism in SI, which do not have any factors of 4?. When this is applied to electromagnetic constants, ?0, this unit system is called "rationalized" Lorentz-Heaviside units. When applied additionally to gravitation and Planck units, these are called rationalized Planck units and are seen in high-energy physics.

The rationalized Planck units are defined so that $c=4\pi G=\hbar =\varepsilon _{0}=k_{\text{B}}=1$ .

There are several possible alternative normalizations.

### Gravity

In 1899, Newton's law of universal gravitation was still seen as exact, rather than as a convenient approximation holding for "small" velocities and masses (the approximate nature of Newton's law was shown 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 in formulae multiplied by 4? or a small integer multiple thereof. Hence, a choice to be made when designing a system of natural units is which, if any, instances of 4? appearing in the equations of physics are to be eliminated via the normalization.

• Normalizing 4?G to 1:

### Electromagnetism

Planck units normalize to 1 the Coulomb force constant ke = (as does the cgs system of units). This sets the derived unit of impedance, ZP equal to , where Z0 is the characteristic impedance of free space.

$e={\sqrt {4\pi \alpha }}\cdot q_{{\text{P}}'}\approx 0.30282212\cdot q_{{\text{P}}'}$ where ${\alpha }$ is the fine-structure constant. This convention is seen in high-energy physics.

### Temperature

Planck normalized to 1 the Boltzmann constant kB.

• Normalizing kB to 1:
• Removes the factor of in the nondimensionalized equation for the thermal energy per particle per degree of freedom.
• Introduces a factor of 2 into the nondimensionalized form of Boltzmann's entropy formula.
• Does not affect the value of any of the base or derived Planck units listed in Tables 3 and 4.

## Planck units and the invariant scaling of nature

Some theorists (such as Dirac and Milne) have proposed cosmologies that conjecture that physical "constants" might actually change over time (e.g. a 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, or 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".

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 ? 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 ? 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 ? 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

Referring to Duff's "Comment on time-variation of fundamental constants" and Duff, Okun, and Veneziano's paper "Trialogue on the number of fundamental constants", 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, ?, changes or the proton-to-electron mass ratio, , changes (atomic structures would change) but if all dimensionless physical quantities remained unchanged (this includes all possible ratios of identically dimensioned physical quantity), we cannot 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 (but with the axiom that all dimensionless physical quantities remain the same), then the Planck length would increase by a factor of 2 from the point of view of some unaffected observer on the outside. Measured by "mortal" observers in terms of Planck units, the new speed of light would 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:

$a_{0}={\frac {4\pi \epsilon _{0}\hbar ^{2}}{m_{e}e^{2}}}={\frac {m_{\text{P}}}{m_{e}\alpha }}l_{\text{P}}.$ Then atoms would be bigger (in one dimension) by 2, each of us would be taller by 2, and so would our metre sticks be taller (and wider and thicker) by a factor of 2. 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 4 (from the point of view of this unaffected observer on the outside) because the Planck time has increased by 4 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 unaffected 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 of our new metres in the time elapsed by one of our new seconds (c × 4 ÷ 2 continues to equal ). 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 significantly, 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. 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 would point out that ascribing a change in measurable reality, i.e. ?, 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 if ? is changed and no other dimensionless constants are changed. It is only the dimensionless physical constants that ultimately matter in the definition of worlds.

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 in dimensionless 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 and this has intensified the debate about the measurement of physical constants. According to some theorists there are some very special circumstances in which changes in the fine-structure constant can be measured as a change in dimensionful physical constants. Others however reject the possibility of measuring a change in dimensionful physical constants under any circumstance. 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.