On time-variation of the fundamental constants Tobias Nilsson
[email protected]
January 31, 2003
Abstract Recently, evidence indicating that the fine structure constant, α, might vary in time was observed. In this thesis it is investigated what consequences such a variation might have and how the variation may be explained. It is sometimes stated that time-variation of dimensionful constants, like the velocity of light c, can be detected. It is argued here that it is meaningless to talk about variation of dimensionful constants unless we specify what units we use.
Contents 1 Introduction 1.1 Fundamental constants . . . . . 1.2 The fine structure constant . . 1.3 Constants that not are constant 1.4 Notations . . . . . . . . . . . .
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3 3 3 4 5
2 Experimental evidence 2.1 Oklo . . . . . . . . . 2.2 QSO constraints . . 2.3 Atom clocks . . . . . 2.4 Variations of G . . . 2.5 Weak interaction . .
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3 Is it meaningful to talk about time-variation of dimensionful constants? 3.1 The number of fundamental constants . . . . . . . . . . . . . . . 3.2 Making measurements . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Other ways to identify variation of dimensionful constants . . . . 3.3.1 Black holes constraints . . . . . . . . . . . . . . . . . . . . 3.3.2 VSL vs. varying e theories . . . . . . . . . . . . . . . . . .
15 15 20 22 22 24
4 Some early theories 26 4.1 Mach’s Principle and Milne’s law of gravitation . . . . . . . . . . 26 4.2 The Large Number hypothesis . . . . . . . . . . . . . . . . . . . . 27 5 Varying constants from higher dimensions 5.1 Kaluza-Klein theories . . . . . . . . . . . . 5.1.1 Charge quantisation . . . . . . . . . 5.1.2 What about time varying constants 5.2 Superstring theories . . . . . . . . . . . . . 6 VSL theories 6.1 Formulation of a VSL theory . 6.2 VSL cosmology . . . . . . . . . 6.2.1 A ψ dominated universe 6.3 VSL as solution to cosmological 6.3.1 The horizon problem . . 6.3.2 The flatness problem . . 1
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30 30 32 34 35
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37 37 39 40 40 40 41
On time-variation of fundamental constants
6.3.3
The cosmological constant problem . . . . . . . . . . . . .
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7 A varying-e theory 44 7.1 A varying ~ dual . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 7.2 A varying c dual . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 8 Varying constants and the equivalence principle 8.1 WEP-violations form α-gradients . . . . . . . . . 8.2 WEP-violations from a “fifth force” . . . . . . . 8.3 The case of no WEP violations . . . . . . . . . . 8.4 WEP violations in different varying-α models . .
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51 52 53 55 57
9 Discussion 59 9.1 Time-variation of dimensionful constants . . . . . . . . . . . . . . 59 9.2 Variation of α and other dimensionless constants . . . . . . . . . 60 A Black hole thermodynamics
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B Einstein’s equations from the action principle
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Bibliography
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2
Chapter 1
Introduction 1.1
Fundamental constants
In the theories of physics certain important fundamental quantities appear. Their values are usually considered to be constant, so they are called fundamental constants. Examples are the speed of light c, which occur in electromagnetism and relativity, Planck’s constant ~, which occur in quantum mechanics, the charge of the electron e, which occurs in electromagnetism, the mass of the electron me , which occurs for example in atomic physics, and the gravitational constant G, which occurs in theories concerning gravity. The above mentioned constants all have dimensions, i.e. we need units to measure them in. For example, the speed of light c = 299792458 m/s are measured in some unit of velocity, m/s in SI units. The values of these constants will be different in different units. Some constants however are dimensionless, which means that their values are independent of the choice of units. Examples are the fine structure constant, α, and the ratio between the mass of the electron me . These constants are the same in all units. and the mass of the proton, µ = m p
1.2
The fine structure constant
The fine structure constant, α, is a dimensionless constant,which is (in SI units): α=
e2 4π0 ~c
(1.1)
Arnold Sommerfeld introduced it in 1917 as a constant to determine the fine structure of the atomic energy levels. It is however a more important constant than that, in fact α is a constant determining the strength of the electromagnetic field. It is also possible to find similar constants determining the strengths of the weak and strong interactions (αw and αs ) and one determining the strength 2 of gravity (αG = Gm ~c , where m is the mass of some elementary particle). The value of the inverse of α turns out to be close to the integer 137, α−1 = 137.03599976(50). This fact caused some speculation about if the inverse of α had some deeper meaning (see for example [1]). For example, Arthur Eddington argued that there was 137 varieties of the electron. According to him the electron could be assigned 16 different labels. This was, argued Eddington, because the 3
On time-variation of fundamental constants
Dirac matrices have 16 elements, or that space-time have four dimensions and 42 = 16. Counting the number of states two electrons could have (only counting combinations like (1, 2) and (2, 1) once) Eddington arrived at the number 136, which missed 137 by one. According to Eddington this was close enough, and he thought that the 137:th state soon would turn up. However, it was later discovered that α is not constant, but depends on the energy. Due to Heisenberg’s uncertainty relation, ∆E∆t & ~, particle-antiparticle pairs are continuously created (and then quickly annihilates again) in vacuum. If the (virtual) particles in the pair have electrical charge (for example an electronpositron pair) they will be affected by an electromagnetic field, for example an electric field from a (real) charged particle. This will lead to that the vacuum around a charged particle gets polarised, leading to that the charge seen by an observer at a large distance will be smaller than the real charge of the particle. The effect is similar to how an electrical field from a charge becomes weaker in a dielectricum. At high energy (i.e. small distances) this effect becomes less important; hence the size of the charge as seen by the observer becomes larger. Also the coupling constants of strong and weak interaction, αs and αw , will have an energy dependence due to similar effects, although the dependence will be a little different. The fact that the coupling constants are running (i.e. they are energy dependent) makes us suspect that at some high energy the coupling constants of the electromagnetic, weak and strong (and perhaps even gravity) will be equal, hence all forces will be unified.
1.3
Constants that not are constant
The constants of nature are generally considered to be constant, that is why we call them constants. However, there are some ideas that suggests that they are not, i.e. that they maybe are time and/or space dependent. The possibility that the constants may not be constant was first (at least seriously) considered in the 1930:s by Milne[2][3] and Dirac [4][5], who both considered the possibility that the gravitational constant G might vary with time. Although the reputable origin (Dirac) the concept of varying constant did not receive too much attention, however some work was done in the area (mostly concerning a varying G). In attempts to unify the interactions (gravity, electromagnetism, weak and strong interaction) higher dimensional theories, like Kaluza-Klein and superstring theories, have been developed. In these theories some extra dimensions are introduced. It turns out that in the couplings, (G, the fine structure constant α, αw and αs ), can be calculated. It turns out that these constants will depend on the size of the extra dimensions and some new scalar fields introduced by the theory, like the dilaton in superstring theory. This means that if the size of the extra dimensions or the scalar fields changes, then the constants will also change. In the higher dimensional theories there is no reason why the extra dimensions or the new scalar fields should be constants, in fact it might even be hard to explain why they should be constant, so it is possible that the constants vary. Also, even if the higher dimensional theories do not necessarily require the constants to vary, they give a framework for describing the variations if they do vary. Moffat [6] showed that some cosmological problems, like the horizon and 4
Tobias Nilsson
flatness problems, can be solved if the speed of light was much larger that its present value in the very early universe. The same conclusions was later, independent of the Moffats results, drawn by Albrecht and Maugeijo [7]. This was followed by a lot of other work on the subject. However, whatever theoretical reasons for the constants to be constant or not to be constant there are, the only way to really find out if the constants are constant is by trying to measure the possible variation of the constants in experiments. For a long time all experimental evidence was consistent with constant constants. However, an experiment can only reach a finite precision, so all these experiments gave was just upper limits on the variability of the constants (see [8] for most current bounds on the variation of various constants). Recently, evidence was detected which suggest that the fine structure constant, α, was smaller in the past[9][10]. The possibility that α varies have raised the question of which of the factors that α consists of that is responsible for the variation: is it e, c or ~. It is argued in this thesis that since we are free to choose whichever units we want when making our measurement, and since the values and the possible variation of the (dimensionful) constants are dependent on the units we use to measure them, we can not tell (without specifying units) which dimensionful constant that is varying (or rather, we are free to choose any one of them (or any combination) to explain the variation of α with). However, there are people who do not agree with that, who believes that it is possible to determine which of the constants that varies (most of them seems to believe in the speed of light). The evidence of a varying α has also inspired the creation of theories trying to explain the variation. Some of these theories explain the variation by a varying e, others by a varying c. However, for some reason there are not many theories trying to explain the varying α by a varying ~.1 The fact that a theory is formulated modelling a varying α as a varying e (c) does only mean that the theory becomes simple when the variation of α is explained by the e (c). It is possible to reformulate the theory so that another constant is varying instead, for example reformulate a varying e theory to a varying ~ or varying c theory, however these theories might be more complicated than the original theory.
1.4
Notations
Throughout most of this work we use only three units: the units of length, time e2 and mass. To measure electric charge we let 4π 7→ e2 . Hence the fine structure 0 2
e constant will be α = ~c . The notations used in this thesis are, unless otherwise stated:
c The speed of light in vacuum. h The ordinary Planck’s constant. ~ Planck’s modified constant (~ = h/2π), from here on called Planck’s constant. e The charge of the electron. 1 Some theories with a varying c also have a simultaneous variation of ~ (ie ~ ∝ c), but very few theories consider a varying ~ alone.
5
On time-variation of fundamental constants
α The fine structure constant (α =
e2 ~c ).
G The gravitational constant (Newtons constant). gµν The metric. We will use metrics with the signature (− + ++). g The determinant of the metric. Unless otherwise stated, greek indices (µ, ν, . . . ) runs from 0 to 3 and latin indices (i, j, . . . ) from 1 to 3. Whenever an index are occurring twice in an expression, one time as a raised index and one time as a lowered index, a summationPover that index is understood, i.e. with an expression like xµ xµ we really mean µ xµ xµ .
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Chapter 2
Experimental evidence 2.1
Oklo
Oklo is an uranium mine in Gabon in West Africa, where evidence has been discovered suggesting that a natural fission reactor was in operation there about 1.8 · 109 years ago. At Oklo the abundance of 149 Sm is lower than what is found elsewhere, which is due to neutron capture during the time the reactor was in operation: n +149 Sm →150 Sm + γ (2.1) By measuring the abundance of 149 Sm at Oklo, it is possible to estimate its neutron capture cross-section, which depends on the values of α and other constants. Hence it is possible to estimate α at Oklo 1.8·109 years ago. It should be noted that the cross-section also will depend on other constants, like the strong coupling constant αs , and that the effect of a change in one constant may be cancelled be a change in another constant. Shlyakhter[11] did this and arrived at the upper bound for the variability of α1 : α˙ < 10−17 yr−1 (2.2) α
The bound was later reanalysed by Damour and Dyson[12]. Their results where (95% C.L.): ∆α < 1.2 · 10−7 α
(2.3)
α˙ < 5.0 · 10−17 yr−1 α
(2.4)
−0.9 · 10−7 < −6.7 · 10−17 yr−1
