The VSL Discussion: What Does Variable Speed of Light Mean and Should we be Allowed to Think About ? Alexander Unzicker Pestalozzi-Gymnasium M¨unchen February 1, 2008 Abstract In the past years, variable speed of light (VSL) theories have been of growing interest but also a sub- ject of controversial discussion. They have been accused both for tautologies and for violating special relativity, and concerns have been expressed about the validity of such approaches in general (e.g. Ellis, astro-ph/0703751). Without trying completeness on the issue, the example of Einstein’s VSL attempts (1911) and Dicke’s ‘electromagnetic’ theory (1957) are urges to give some comments on the above criticism. Introduction should be proven by that statement. All that follows indeed from the definition of SI units, but in my Exotic theories. Of course, ‘variability’ can en- humble opinion something can either be measured compass a lot of aspects. One may introduce dis- or defined, not both. Thus c = 1 is not a physical ne- persion, considering a dependency on λ, or on v, vi- cessity but at best a mathematical convention; one olating Lorentz-invariance. Most of these proposals may further ask if it is a possible, reasonable or even do not have sufficient experimental support at the the only practical one. At the very end, this is not a moment, though many of them are interesting and scientific question; to illuminate the practical value seem as good as inflation for resolving the flatness of c = 1, we investigate the following toy theory: and horizon problems in cosmology; this however is not the focus of interest here, since comments on [1] with respect to modern VSL theories [2] have Meteorology at constant temperature already be given [3]. There, appropriate reference Fortunately, glass and most fluids have different and a clear discussion of older attempts are how- thermal expansion coefficients (TEC), and for that ever missing. These so-called [2] reason we easily construct thermometers based on the expansion of fluids with respect to the contain- Conservative theories suffered an even harsher ing tube. But imagine all substances had the same arXiv:0708.2927v3 [physics.gen-ph] 9 Sep 2007 ‘Not even wrong’- criticism of being tautological. TEC, things wouldn’t be that easy! All thermome- The argument is the following: ‘One assumes a good ters made in that fashion would show the same tem- clock can be constructed, and then uses the timing of perature. Well, one still could measure temperature reflected electromagnetic radiation to determine the by means of the mean quadratic velocities of parti- distance. But then the (physical) speed of light of cles in a gas. Determining the velocities with clocks necessity has to be unity, precisely because all elec- and rods and deriving the temperature would still tromagnetic radiation travels at the speed of light, be possible. But what if the same velocity is used for and distances are being determined by use of such the definition of time and length scales ? A gas ther- radiation.’ ([1], sec. 2). One may wonder what fact mometer in a cold location would then just mimic a 1 slower running time and/or a contraction of length the place, we may easily infer, by means scales. One realizes that in such a world it is not of Huygens’s principle, that light-rays easy to detect temperature differences, but there is propagated across a gravitational field an effect: the velocity of sound waves, depending on undergo deflexion’. temperature, would be a function of place and time, and hence, differences in temperature would cause As a consequence of a variable speed of light, he a deflection and focussing of sound waves. considered variable time scales only and postulated Mind now the following mathematical insight: dc df there cannot be any doubt that the numerical value = , (1) c f of temperature depends on arbitrary chosen units, and since it is a dimensionful quantity, it can be set which, as it is well-known, led to he (wrong) half to unity in every point (see argument [1], sec. 2). value for the classical light deflection.2. It was then Therefore, mathematicians should feel free in for- Dicke3 with his ‘electromagnetic’ theory of gravita- mulating meteorology (or, appreciating generaliza- tion [8] who discovered that the classical tests could tions, thermodynamics) with T = 1, but the de- be described by mand that any weather forecast should be expressed dc dλ df in this manner will be of limited usefulness. People = + , (2) who do not shy elementary material should have a c λ f look at the textbook example in Feynman’s lectures II, chap. 42 [4]1. considering variable length scales, too. We shall not go into further details and refer the reader to the ‘polarisable vacuum representation’ of GR by [9], Differential geometry. I shall like to draw see also [10]. It is however at least an open question attention to the fact that such a convention if GR can be formulated by a scalar VSL theory, (T=1, c=1) leads to a curved space, which equiv- instead of a 10-component metric! I shall not enter alently can be described by a metric. However it is the fruitless question whether this is ‘simple’ or not - quite a difference if one can choose an -arbitrary- it’s up to you whether you consider this an approach unit globally or if you have to do this locally in ev- worth thinking about or share the above criticism: ery point. In the later case such a choice T = 1 can what a pity that Einstein in 1911 could not make turn out to be complicated. The ‘proof’ instead that use of check-lists like [1] - maybe he had stopped to ‘physically’ c is always a constant is like the proof develop weird theories about a curved spacetime... of a differential geometer that physically no moun- tains exist, since in every point of a differentiable Lorentz invariance. manifold one can attach a flat tangent space (the Such useful methodic necessity to change direction when walking uphill is guidelines would also have prevented Einstein nothing physical, just a ‘connection’.). What a nice from being in conflict with special relativity and revival of the earth as a plane! We proceed a lit- remaining so blatantly unfamiliar with underlying tle further in history and listen to those who first principles of his own work - I wonder if this is the considered a variable speed of light: message the reader should learn from [1], sec. 4. In 1911, Einstein wrote: Einstein and a VSL. The constancy of the velocity of light can be maintained only insofar as one The first who realized that a variable speed of light restricts ... to ... regions with constant may cause astronomical light bending was Einstein gravitational potential... in 1911 [6]: 2It should be noted that though c being a scalar field here, ‘From the proposition which has just this theory is not a ‘scalar’ theory coupled to matter to which been proved, that the velocity of light Einstein later expressed caveats. See also [7] for clarifying that point. in the gravitational field is a function of 3 It is not quite clear why Dicke gave up this interesting 1Though being a toy theory, there are very interesting approach and followed up the quite different Brans-Dicke the- comments regarding the topic given by Landau [5], par. 8. ory. 2 Indeed, such a VSL theory would require consider- The need and the fear to change equations. able reformulation on a technical level, but there is Paradoxically, postulating exotic new fields does little doubt that this can be done as a matter of prin- usually little harm to the standard models, while ciple, as long as the local c is the limiting velocity. the speed of light has a dominant role in various In continuum mechanics it is well-known that spe- fields of theoretical physics [16]. VSL has obviously cial relativistic effects show up [11], and a variation to consider the influence on other fields since chang- of the (corresponding) speed of transversal sound ing c in one context only would be a rather weird and arises naturally. Further clarifying explanations are fruitless trial. On the other hand, despite all tech- given by Dicke [8] and Ranada [12]. nical difficulties that may arise, the ultimate test remains the agreement with experiment, and due to The Conditions a good physical theory the usual minute deviations a VSL causes we can- has to satisfy not expect that the corresponding observations be- come visible in all facets simultaneously. Of course, The necessity of compatibility with special relativ- as [1] states, if one changes one equation, one has ity or Lorentz-invariance of a theory stressed by [1] to change many ones, but this elucidates also the is one of the most basic requirements new propos- psychological problem that may arise: for somebody als have to satisfy in order to be taken seriously. who has written a book full of formulas containing Other requirements would be the possibility of a c, any VSL proposal becomes a nightmare. Lagrangian formulation, satisfying the equivalence principle, the agreement with general relativity, with Physicists in the ptolemaic period may have quantum electrodynamics, to be renormalizable etc. felt similarly when hearing about the earth being All these are nice properties of successful physical in motion. For somebody living in the 17th cen- theories. Setting up guidelines for the development tury, surely it wasn’t easy to get familiar with such of possible new ideas however does not provide any a counterintuitive fact.