The possibility that fundamental constants can change in time is predicted
by some unified field theories (see, e.g. [1]).
The detection of such a variation
would be an important confirmation of these theories. The analysis of the
spectra of distant quasars [2] does indicate that the fine structure
constant *alpha* (the constant which measures the intensity of the
electromagnetic interaction) might be changing in time.
Fine structure constant *alpha* is in fact a dimensionless combination
of three other fundamental constants: *alpha = e²/hc*
(*e* - electron charge, *h* - Planck constant, *c* -
speed of light).
A recent publication in Nature [3] suggests that this
variation of *alpha* should be interpreted in terms of a changing speed
of light. The claim that speed of light might be changing received huge
publicity in mass media.
However, it is well-known in scientific circles
dealing with the problem of variation of the fundamental constants
that only dimensionless constants (like *alpha*) should be considered
in this context (see, e.g. [4,5]). Speed of light, in contrast,
is a dimensionful constant.
Recent works by Duff [6] and Flambaum [7] explain
why arguments presented in Ref. [2] are wrong and cannot
lead to any conclusion about a changing speed of light.
However, changing speed of light is meaningless just from consideration
of the problem of measurements, regardless of how people try to get around
it.

The problem of measurements is discussed in scientific literature in context of varying fundamental constants (see, e.g. [4]). However, big public interest to the changing speed of light shows that the problem of measurements deserves consideration on a more elementary level.

First, the term *fundamental constants* needs to be explained.
Fundamental constants can be considered as natural standards against
which everything else can be measured. If something is changing it
can be detected by consecutive measurements against natural standards.
But if the standards themselves are changing, any detection of this
seems to be questionable. In fact, the only chance to see any change
comes when a change in the natural standards (fundamental constants)
happens in a disproportionate way, so that some dimensionless ratio of
constants changes. For example, if absolutely everything in the
Universe suddenly increases in size, it cannot be noticed.
If, in contrast, the Earth becomes larger but the Sun remains the same,
it can be detected by comparing their sizes. This comparison comes in a
form of (dimensionless) ratio *size of Earth/size of Sun*.
It tells us that the relative sizes of Earth and
Sun have changed. However, there is no way to say whether the Earth
has become larger or the Sun smaller.

Now suppose that the speed of light is changing. It can be meaningful only if it can be detected by measurements. To perform measurements we need units. Units do not exist in nature and are invented by people to express quantitative relations in nature which exist. Units to measure length, time, speed, etc. are always expressed in terms of some combination of fundamental constants. Performing measurements means comparing the measured value to a particular combination of fundamental constants. Measuring a fundamental constant means comparing fundamental constants between themselves. Physical laws must not depend on the particular choice of units. Below we illustrate that whatever units are used to measure the speed of light, the claim that the speed of light is changing leads to nonsense.

The speed of light *c* is most commonly expressed in metres
(*m*) per second (*s*).
Its value is *c*=299792458 *m/s*. However, the metre is defined
as the distance which light travels in 1/299792458 *s* [8].
If the speed of light is changing, its value in *m/s* will still be
the same.
One may argue that this definition of the metre is not good in a situation
where the speed of light is changing. What if we use the old definition
instead, 1*m* = 1/10000000 of the distance from the North Pole to the
equator? This just moves the problem into another area: there is no way
to distinguish between a change in the speed of light and a change in the
size of the Earth (and there is no way to say that one is more likely
than the other!).

Let's now take a stick 1*m* long and say that this is going to be
our standard unit for length. And let's measure the speed of light via the
time needed for light to travel along the stick. Since we always use the
same stick we probably should expect that its length remains the same.
One would argue that if consecutive measurements produce different results,
the speed of light is changing. However, the measurement of the speed of light
using a stick as a standard can be interpreted as the measurement of the
length of the stick using the speed of light as a standard. The speed of
light is not less fundamental than the length of the stick [9].
Here again we cannot say what is changing, the speed of light or the length
of the stick.

The best unit to use to get to a paradox the fastest possible way is the
speed of light itself. This is the only single fundamental constant
(not a combination of constants) which has the dimension of speed and
can be used as a unit to measure any speed. Then *c*=1 by definition and
cannot change!

We see that depending on the units used, the speed of light either remains the same or its change cannot be distinguished from a change in other fundamental constants. Recalling that physical laws must not depend on units, we come to the conclusion that a changing speed of light is nonsense.

The question remains, why then changing speed of light is so often mentioned in the literature? This is mostly due to two reasons. One is just lack of understanding. Other reason is much more interesting. In fact, theories can operate with abstract quantities which have no direct connection to observations. They are usually introduced into the theory for convenience and play an intermediate role between the basic principles of the theory and observable effects. One of the most common examples of this sort is probably the vector potential used in electrodynamics. Electric and magnetic fields in electrodynamics are often expressed in terms of the vector potential. The fields can be observed and measured while the vector potential cannot [10]. Similarly, a theory dealing with variation of the fine structure constant can be formulated in terms of a changing speed of light. This only means that there must be an equivalent theory formulated in terms of a changing electron charge. Neither theory claims that a changing speed of light or electron charge can be observed. However, when it comes to observable effects both theories give exactly the same results (see, e.g. [11]).

We can say in conclusion that changing speed of light (and other dimensionful constants) is at most a pure mathematical abstraction which cannot be observed or measured. In contrast, the change of a dimensionless combination of fundamental constants is meaningful and is the subject of the study in Ref. [2].

I am grateful to A. I. Milstein, V. V. Flambaum and O. P. Sushkov for valuable discussions.

[1] W. Marciano, *Phys. Rev. Lett.* **52**, 489 (1984);
J. D. Barrow, *Phys. Rev. D* **35** 1805 (1987);
T. Damour and A. M. Polyakov, *Nucl. Phys. B * **423**, 596 (1994).

[2] J. K. Webb, V. V. Flambaum, C. W. Churchill, M. J. Drinkwater,
and J. D. Barrow, *Phys. Rev. Lett.* **82**, 884 (1999);
J. K. Webb, M. T. Murphy, V. V. Flambaum, V. A. Dzuba, J. D. Barrow,
C. W. Churchill, J. X. Prochaska, and A. M. Wolfe,
*Phys. Rev. Lett.* **87**, 091301 (2001).

[3] P. C. W. Davies, T. M. Davis, C. H. Lineweaver,
*Nature* **418**, 602 (2002).

[4] J.-P. Uzan, e-print hep-ph/0205340.

[5] M. J. Duff, L.B. Okun, G. Veneziano, e-print physics/0110060.

[6] M. J. Duff, e-print hep-th/0208093.

[7] V. V. Flambaum, e-print astro-ph/0208384.

[8] *Particle Physics Booklet*, Particle Data Group, Springer (2000).

[9] The length of a stick depends on size of atoms, which is roughly
*h²/me²* (Bohr radius). This is another combination of
fundamental constants with the dimension of length (*h* - Planck
constant,*m* - electron mass, *e* - electron charge).

[10] Some integral characteristics of vector potential can in fact be measured. However, in contrast to electric and magnetic fields, the value of vector potential at given point is not defined and cannot be measured.

[11] H. B. Sandvik, J. D. Barrow, J. Magueijo,
*Phys. Rev. Let.* **88**, 031302 (2002).

Last modified: September 7, 2002