c, ℏ, and G are

*dimensional*constants in the sense that they possess physical dimensions, and their values must be expressed relative to a choice of physical units. Note in this context that there are three fundamental physical dimensions: length [L], time [T], and mass [M]. Each physical quantity is represented to have dimensions given by some combination of powers of these fundamental dimensions, and each value of a physical quantity is expressed as a multiple of some chosen unit of those dimensions. The speed of light has dimensions [L][T]

^{-1}, and in CGS (Centimetre-Gramme-Second) units has the value c ≈ 3 x 10

^{10}cm/s; Planck's constant has the value ℏ ≈ 10

^{-27}g cm

^{2}s

^{-1}in CGS units; and Newton's gravitational constant has the value G ≈ 6.67 x 10

^{-8}cm

^{3}g

^{-1}s

^{-2}in CGS units.

The laws of physics define the necessary relationships between dimensional quantities. The values of these quantities are variable even within a fixed system of units, hence the lawlike equations can be said to define the relationships between dimensional

*variables*. Nevertheless, the laws of physics also contain dimensional

*constants*. In particular, the fundamental equations of relativity and quantum theory, such as the Einstein field equation, the Maxwell equation, the Schrodinger equation, and the Dirac equation, contain the fundamental dimensional constants, c, ℏ, and G.

Ultimately, dimensional constants are necessary in equations which express the possible relationships between physical variables, because the dimensional constants change the units on one side of the equation into the units on the other side. As an example, consider the most famous case in physics, E=mc

^{2}. This equation can be seen as expressing a necessary relationship between the energy-values and mass-values of a system. In CGS units the energy is in ergs, where an erg is defined to equal one g cm

^{2}/s

^{2}, and the mass is in grammes. To convert the units of the quantity on the right-hand-side of the equation into the same units as the quantity on the left-hand-side, the mass is multiplied by the square of the speed of light in vacuum, which has units of cm

^{2}/s

^{2}. One might argue that the reason why the (square root of the) conversion factor should be ≈ 3 x 10

^{10}in CGS units, rather than any other number, follows from the definition of the cm and the s. Like all dimensional quantities, the value of fundamental constants such as c changes under a change of physical units.

Intriguingly, the fundamental dimensional constants can also, heuristically at least, be used to express the limiting relationships between fundamental theories. Thus, classical physics is often said to be the limit of quantum physics in which Planck's constant ℏ → 0, and non-relativistic physics is often said to be the limit of relativistic physics in which the speed of light in vacuum c → ∞. The flip side of this coin is that ℏ is said to set the scale at which quantum effects become relevant, and c is said to set the speeds at which relativistic effects become relevant. ℏ sets the scale at which quantum effects become relevant, in the sense that a system with action A is a quantum system if the dimensionless ratio A/ℏ is small. If this ratio is large, then the system is classical. As ℏ → 0, A/ℏ becomes large even for very small systems, hence classical physics is said to be the limit of quantum physics in which ℏ → 0. Similarly, c sets the speeds at which relativistic effects become relevant in the sense that a system with speed ν is relativistic if the dimensionless ratio ν/c is close to 1. If the ratio is a small fraction, then the system is non-relativistic. As c → ∞, ν/c becomes a small fraction even for very fast systems, hence non-relativistic physics is said to be the limit of relativistic physics in which c → ∞. In a similar manner, G sets the scale of gravitational forces, and determines whether a system is gravitational or not.

Duff (2002), however, argues that no objective meaning can be attached to variation in the values of the dimensional constants. According to Duff, "the number and values of dimensional constants, such as ℏ, c, G, e, k etc, are quite arbitrary human conventions. Their job is merely to convert from one system of units to another...the statement that c = 3 x 10

^{8}m/s, has no more content than saying how we convert from one human construct (the meter) to another (the second)."

To understand Duff's point, consider 'geometrized' units, in which the speed of light is used to convert units of time into units of length. Thus, for example, c • s is a unit of length defined to equal the distance light travels in a second. If time is measured in units of length, then all velocities are converted from quantities with the dimensions [L][T]

^{-1}to dimensionless quantities, and in particular, the speed of light acquires the dimensionless value c = 1. In geometrized units, anything which has a speed ν less than the speed of light has a speed in the range 0 ≤ ν

_{geo}< 1:

ν

_{geo}= ν

_{cgs}/ c

_{cgs}.

Thus, the speed of light can be used to convert velocities expressed in CGS and geometric units as follows:

ν

_{cgs}= ν

_{geo}• c

_{cgs}.

Similarly, in geometrized units, the gravitational constant G converts units of mass to units of length. In fact, in geometrized units, all quantities have some power of length as their dimensions. In general, a quantity with dimensions L

^{n}T

^{m}M

^{p}in normal units acquires dimensions L

^{{n+m+p}}in geometrized units, after conversion via the factor c

^{m}(G/c

^{2})

^{p}, (Wald,

*General Relativity*, 1984, p470).

Whilst in geometrized units, c = G = 1, if one changes to so-called 'natural units' (such as Planck units), then ℏ = c = G = 1, and these constants disappear from the fundamental equations. Theories expressed in these natural units provide a non-dimensional formulation of the theory, and the dimensional variables becomes dimensionless variables in this formulation. Duff, for example, points out that "any theory may be cast into a form in which no dimensional quantities ever appear either in the equations themselves or in their solutions," (2002, p5). Whilst in geometrized units, all quantities have dimensions of some power of length [L]

^{n}, in Planck units all quantities are dimensionless, as a result of division by l

_{P}

^{n}, the n-th power of the Planck length l

_{P}= √(G ℏ/c

^{3}) ≈ 1.616 x 10

^{-33}cm. In particular, in natural units all lengths are dimensionless multiples of the Planck length.

The existence of theoretical formulations in which the dimensional constants disappear, is held to be one of the reasons why a postulated variation in the values of the dimensional constants cannot be well-defined. Different choices of units certainly result in different formulations of a theory, and the dimensional constants can indeed be eliminated by a judicious choice of units. Nevertheless, it should be noted that the most general formulation of a theory and its equations is the one which contains the symbols denoting the dimensional constants as well as the symbols denoting the dimensional variables.

Whilst the arguments recounted above are to the effect that variations in the fundamental dimensional constants cannot be well-defined, these arguments are also often conflated or conjoined with arguments that such changes are not

*operationally*meaningful. Here, it is argued that a change in a dimensional constant cannot be measured because there is no way of discriminating it from a change in the units of which that constant is a multiple. For example, if the length of a physical bar, stored at a metrological standards institute, is used to define the unit of length, one might try to measure a change in the speed of light from a change in the time taken for light to travel such a length. In such a scenario, it could be argued that it is the length of the bar which has changed, not the speed of light.

Whilst it is indeed true that a change in the value of a dimensional variable could be explained by a change in one's standard units, this is a truth which applies to the measurement of dimensional

*variables*, just as much as it applies to the measurement of dimensional

*constants*. The logical conclusion of this line of argument is that only dimensionless ratios of dimensional quantities can be determined by measurements; individual lengths, times and masses cannot be determined, only ratios of lengths, ratios of times, and ratios of masses. Duff duly follows this line of reasoning to its logical conclusion, asserting that "experiments measure only dimensionless quantities," (2002, p5).

However, unless the dimensional quantity being measured is itself used to define the units in which the quantity is expressed, the question of whether one can discriminate a change in a dimensional quantity from a change in the units of which that quantity is a multiple, is an empirical-epistemological question rather than an ontological question. Whilst the value of a dimensional constant does indeed change under a change of units, so does the value of a dimensional variable, and there is no reason to infer from this that a dimensional variable is merely a human construct. For example, the rest-mass energy of a system changes under a change from MeV to keV, but this is no reason to conclude that rest-mass energy is a human construct. Hence, the question of operational meaning may be something of a red-herring.

It is, however, certainly true that the units of time and length can themselves be defined as functions of the fundamental dimensional constants. Thus, the standard unit of time is defined in terms of the frequency ν of hyperfine transitions between ground state energy levels of caesium-133 atoms:

ν = m

_{e}

^{2}c

^{-2}e

^{8}/ h

^{5}m

_{N}≡ T

^{-1},

where e is the charge of the electron, m

_{N}is the mass of the neutron, and m

_{e}is the mass of the electron. The period of any cyclic phenomenon is the reciprocal of the frequency, 1/ν, and in 1967 the second was defined in the International System (SI) of units to consist of 9,192,631,770 such periods.

From 1960 until 1983, the SI metre was defined to be 1,650,763.73 wavelengths (in vacuum) of the orange-red emission line of krypton-86. This is determined by the Rydberg length R

_{∞}:

4 π R

_{∞}= m

_{e}e

^{4}/ c h

^{3}≡ L .

As Barrow and Tipler comment, "if we adopt L and T as our standards of length and time then they are

*defined*as constant. We could not measure any change in fundamental constants which are functions of L and T," (

*The Anthropic Cosmological Principle*, 1986, p242). Since 1983 the metre has been defined in terms of the unit of time, the second, so that a metre is defined to be the distance travelled by light, in a vacuum, during 1/299 792 458 of a second. Such considerations led cosmologist George Ellis to claim in 2003 that "it is...not possible for the speed of light to vary, because it is the very basis of measuring distance."

Magueijo and Moffat (2007) acknowledge that if the unit of length is defined in such a manner, then the constancy of the speed of light is indeed a tautology. However, they then provide the following riposte: "An historical analogy may be of use here. Consider the acceleration of gravity, little g. This was thought to be a constant in Galileo’s time. One can almost hear the Ellis of the day stating that g cannot vary, because 'it has units and can always be defined to be constant'. The analogy to the present day relativity postulate that c is an absolute constant is applicable, for the most common method for measuring time in use in those days did place the constancy of g on the same footing as c nowadays. If one insists on defining the unit of time from the tick of a given pendulum clock, then the acceleration of gravity is indeed a constant by definition. Just like the modern speed of light c. And yet the Newtonian picture is that the acceleration of gravity varies."

Whilst there is considerable disagreement that the values of fundamental dimensional constants have any theoretical significance, there is a consensus that each different value of a fundamental dimensionless constant, such as the fine structure constant α = e

^{2}/ℏ c, defines a different theory. The values of the dimensionless constants are, by definition, invariant under any change of units, they remain obstinately in the dimensionless formulation of a theory, and their values have to be set by observation and measurement. Dimensionless constants, however, are themselves merely functions f(c, ℏ, G) of dimensional constants, in which the dimensions of the units cancel. If the variation of dimensionless constants is meaningful, and if dimensionless constants are functions of the dimensional constants, then one might ask how variation in the former can be achieved without variation in the latter. As Magueijo (2003) comments: "If α is seen to vary one cannot say that all the dimensional parameters that make it up are constant. Something - e, ℏ, c, or a combination thereof - has to be varying. The choice amounts to fixing a system of units, but that choice has to be made...In the context of varying dimensionless constants, that choice translates into a statement on which dimensional constants are varying."