Scalar-Tensor Theories of Gravity: Some Personal History Carl H

Scalar-tensor Theories of Gravity: Some personal history Carl H. Brans

Loyola University Tel.: +1504-865-3643 email:[email protected]

Abstract. From a perspective of some 50 years or more, this paper reviews my recall of the early days of scalar-tensor alternatives to standard Einstein general relativistic theory of gravity. Of course, the story begins long before my involvement, going back to the proposals of Nordström in 1914, and that of Kaluza, Klein, et al., a few years later, so I include reviews of these seminal ideas and those that followed in the 1920’s through the 1940’s. This early work concerned the search for a Unified Field Theory, unifying gravity and Electromagnetism, using five dimensional manifolds. This formalism included not only the electromagnetic spacetime vector potential within the five- metric, but also a spacetime scalar as the five-five metric component. Although this was at first regarded more as a nuisance, to be set to a constant, it turned out later that Fierz, Jordan, Einstein and Bergmann noticed that this scalar could be a field, possibly related to the Newtonian gravitational constant. Relatively little theoretical and experimental attention was given to these ideas until after the second world war when Bob Dicke, motivated by the ideas of Mach, Dirac, and others, suggested that this additional scalar, coupled only to the metric and matter, could provide a reasonable and viable alternative to standard Einstein theory. This is the point of my direct involvement with these topics. However, it was Dicke’s prominence and expertise in experimental work, together with the blossoming of NASA’s experimental tools, that caused the explosion of interest, experimental and theoretical, in this possible alternative to standard Einstein theory. This interest has waxed and waned over the last 50 years, and we summarize some of this work. Keywords: Scalar-Tensor, General Relativity, History PACS: 04.20.-q, 04.50.Kd,04.50.-h SOME HISTORY

The paradoxical phrase, “Varying...constant” was first used by me in the title of my Princeton PhD thesis of 1961. There I developed a formalism making explicit modifications of Einstein’s theory introducing a scalar field variable to determine the Newtonian universal gravitational “constant,” G. These ideas were first suggested to me by R. H. Dicke. By the 1950’s Bob had established himself as a leading experimentalist in many fields (ball point pen, washing machines, wartime radar, etc.) but most especially, motivated by his undergraduate contacts with Wheeler, in the study of gravity. In the late 1950’s Bob started looking in some detail into the historical data of Eötvös. Bob suggested that the famous claims about the equivalence principle based on this experiment were greatly exaggerated. So, Bob and his group set out to do this experiment with great care. His results did re-affirm, but with much more confidence, the fact that the ratio of gravitational to inertial mass is the same for a variety of atoms. But Bob was careful to point out that this experiment only confirmed the • Weak Equivalence Principle, WEP: The acceleration effects of gravity are the same on all (or at least a wide variety of) small objects (neglecting tidal and 2nd order effects). However, he was adamant in also affirming that this type of experiment did not of itself confirm a • Strong Equivalence Principle, SEP: The only effect of gravity is to accelerate particles. This SEP is, of course, a tacit assumption of Einstein’s formulation of his general relativistic theory of gravity, SET, (Standard Einstein Theory). In 1957 I was a first year graduate student at Princeton who was very interested in the applications of topology and bundle theory to the differential geometry of general relativity, motivated largely by lectures of Charlie Misner. Looking for a thesis topic I approached Charlie with the idea of incorporating and extending bundle theory further into general relativity and even quantum theory. Perhaps this might have developed into an early form of gauge theory (I’ll never know now). Misner thought this was too abstract, and besides, he knew that Bob Dicke was looking for a theorist to provide some formalism for his ideas, motivated by proposals of Mach, Dirac, et al. So, he suggested I approach Bob for some thesis ideas. Somewhat reluctantly, being very experimentally challenged myself, I nevertheless worked up the courage to approach Bob. To my great relief, Bob was very kind and tolerant of my experimental incompetence. He even invited me to the regular meetings of his experimental group. They were very kind and forbearing of a dumb theorist who did not know a vacuum tube (yes, they existed at this time!) from a screwdriver. More significantly though, Bob took me aside in private meetings. One notable comment he made to me was that some theorists should be given swift kicks in the pants to make them realize the reality of inertial reactions. Of course, this was all in good humor, but the point was serious. As we will explore later, Bob thought that there would be some relationship of inertial forces to Dirac’s ideas of causally relating G−1 to M/R. He pointed me to the works of Dirac, Sciama et al. (but we were unaware of Jordan’s work at this time), and suggested I try to formulate a set of equations generalizing SET to allow for the almost quaintly paradoxical idea of a “varying gravitational constant.” This resulted in my PhD thesis of 1961. During the writing I discovered the work of Jordan et al. on this topic and almost quit writing. However, I was encouraged to continue, giving what I hope was sufficient and appropriate credit to Jordan and his group. From this thesis, Bob and I published the first paper on our version of this subject in 1961. Of course, these theories ought properly be called “Jordan-Brans-Dicke,” (JBD), but unfortunately many papers disregard Jordan’s ground breaking work and refer to it simply as BD. Remo Ruffini has showed me a letter he received from Jordan decrying the lack of credit he (Jordan) was receiving during the explosion of interest in “scalar-tensor,” ST (a more neutral term) theories. I really believe that much of the reluctance to credit or investigate Jordan’s work was due to his apparently strong, and perhaps even enthusiastic, membership in the Nazi party. But, back to the origins of our version. Among other things, Dicke pointed out to me that Einstein himself was strongly motivated by the ideas of Mach. In re-reading some early passages of Einstein’s book, The Meaning of Relativity, I noticed that Einstein claimed that his standard theory did indeed confirm that inertial mass would be influenced by other masses. However, it turns out that this was only a coordinate effect. I soon wrote a few papers on this and related ST topics. Wheeler, who introduced Dicke to Mach’s principle, was nevertheless not at all enthusiastic about changing Einstein’s beautifully simple, purely geometric, theory of gravity. But Johnny, as one of my examiners, was his usual, very kind and gentle self during my oral defense of my thesis, and apparently let it slip through to generate a Princeton PhD degree! I should point out that as time went by many other theorists seemed also to be offended to have Einstein’s theory contaminated by an additional field. As a young, easily intimidated attendee at meetings, I often heard this disapproval. However, experimentalists, especially those at NASA, were effusively happy to have an excuse to challenge Einstein theory, long thought to be beyond further experimentation. Although happy to have a viable alternative to standard Einstein theory, they were even more happy to shoot it down! More on this later. Personally by the mid 1960’s I began to drift inexorably toward my first love: mathe- matical physics, and dropped my interest in ST’s.

SPECIAL RELATIVITY AND NEWTONIAN (SCALAR) GRAVITY

As a background for scalar ﬁelds in general relativity we review a very brief, familiar, and not necessarily totally historically accurate, account of one way to get from Newto- nian, spacetime scalar, gravity through special relativity to Einstein’s formulation. Many more complete discussions have been available for many years, especially since the cel- ebration of the Einstein centennial. A much more complete and accurate one is that of Norton [1]. Recall classical, Newtonian, gravitational theory deﬁned by scalar U, Newton: ∇2U = 4πGρ, F = −m∇U What is the natural special relativistic extension? Compare to the transition: electro- statics → electrodynamics, try

∂ 2 κ 2U ≡ (∇2 − )U = ρ, ∂t2 2 where κ ≡ 8πG, and c = 1 as usual. But, the resulting force ﬁeld would be simply the gradient,

α Fα = −∂U/∂x .

This naturally leads to trivial force since in fact,

d2xα m = mF α , dτ2 implies that U must be constant along world line of particle,

dxα dU F = = 0, α dτ dτ That is, the gravitational ﬁeld cannot vary over the path of a freely falling object. Thus such a mode is trivial, there is no “simple” scalar theory of gravity consistent with SR. Abraham, Nordström, Einstein, et al. attacked this problem in a manner similar to the following: Consider particle and ﬁeld actions. For a non-relativistic, free particle, we have

Z v2 Sp0 = m( )dt. path, x(t) 2

Introduce special relativity with the Minkowski metric, and proper time,

3 2 2 2 2 2 µ ν dτ = dt − dx − dy − dz = − ∑ gµν dx dx . µ,ν=0

The special relativistic formalism parameterized by proper time, withz ˙µ ≡ dzµ /dτ, results in Z p µ ν Sp = − m −ηµν z˙ z˙ dτ, path, z(τ) which can be put into a volume integral (anticipating ﬁeld interactions), Z 4 Sp = − mIpath(x)d x, where Z ρ p µ ν 4 ρ ρ Ipath(x ) ≡ m −ηµν z˙ z˙ δ (x − z (τ))dτ. path, z(τ) To carry something like this over to gravity, we need to couple the mass to the ﬁeld, noting that mass appears both as inertial and as gravitational coupling. Z Z 4 4 Sp + SI + SU = − miJpathd x − mgU(x)Jpathd x

Z 1 ,µ 4 − U, U d x. κ µ The WEP, verified by Eötvös and Dicke implies mi = mg = m. Clearly the field variation results in the correct source equation, with ρ(xµ ) = mR δ 4(xµ − zµ (τ))dτ, κ 2U = ρ. 2 However, the variation over the particle’s variables, zµ (τ),z˙µ (τ) results in something quite surprising, d m(1 +U)z˙µ (τ) = −mU,µ . dτ So here is the hint: The coupling of inertial mass directly to gravity implies that the inertial mass may “change” in the presence of a gravitational field! In fact, recall that “energy=mass”, so gravitational energy, mU should itself be acted on by a gravitational force. Thus, from action-reaction, gravitational mass would seem to be affected by gravity. Another, and ultimately the most productive, interpretation is that the metric has been influenced by gravity. Thus, the sum of the particle’s action plus the particle-gravity interaction action, is Z Z p µ ν 4 µ µ 4 Sp + SI = − m −ηµν z˙ z˙ δ (x − z (τ))dτ d x

Z Z p µ ν 4 µ µ 4 − U m −ηµν z˙ z˙ δ (x − z (τ))dτ d x or, assuming U << 1, Z Z p µ ν 4 µ µ 4 Sp + SI = − m −gµν z˙ z˙ δ (x − z (τ))dτ d x, where, gµν = (1 + 2U)ηµν , so • First hint of general relativity: we can absorb the gravitational field into the metric. Explicitly, for a point mass, M at the origin, 2GM g ≈ −(1 − ) 00 r or proper time depends on location in the gravitational field: r 2GM dτ ≈ 1 − dt. r This leads to the suggestion that a gravitational field would give rise to frequency shifts, and ultimately to the deflection of light (although not the precise, observed, value at this early stage). Of course, these were only the first steps to what ultimately became known as Ein- stein’s equations. Einstein himself was aided in this search by many colleagues, especially mathematicians. Not only are the equations non-linear, but so was the path to them. After much soul-searching (at some points he apparently thought the differential geometry was too complicated) Einstein finally came to the mathematically “simplest” equations, now known as the Einstein equations:

Rαβ − (1/2)gαβ R = κTαβ , (1) derived from action, Z √ 4 δ −g(R + κLmatter)d x = 0. (2)

There is much good, interesting, and instructive history associated with the path of Einstein, and others, to this ﬁnal form as well described by Norton[1]. We will now refer to the theory described by (1) and (2) as “Standard Einstein theory”, SET. These equations can be regarded as second order partial differential equations for the metric components, with mass (encoded in Tαβ ). However, it can also be said that “gravity gravitates”, roughly

2 “ ”gαβ = κ(Tαβ (matter) + Tαβ (metric)). (3) SET works great: it satisfies correspondence principle limit to Newtonian gravity, but beats it in famous solar system tests. It is truly an extremely beautiful and successful theory. Does this mean the end to scalar fields in gravity? Not necessarily as we shall now review. Einstein’s theory left unresolved a few important questions. Among these are: 1. (Classical) Mach’s principle, Dirac’s Large number coincidences. 2. (Classical) Cosmological coincidences: Universe must have been fantastically well- tuned at its birth so that we can exist to see it now. (anthropic principle) 3. (Quantum) The fine structures on spacetime needed to do relativity are as unob- servable because of quantum theory as were the properties of the immutable ether a century ago. (is spacetime the new ether?). 4. (Experimental) Few experiments had been done, as was emphatically pointed out by Dicke. Might some of these issues be related? MACH’S PRINCIPLE (MP), INERTIAL INDUCTION.

Another issue leading Einstein from special to general relativity is somewhat loosely referred to as “Mach’s principle,” suggesting that the equality of gravitational and inertial masses (WEP) is not accidental, but, in fact, inertial forces are some sort of gravitational forces. In fact Einstein believed that general relativity would indeed predict this, as described on pages 99-108 of his book[2], However, this turns out to be a purely “coordinate effect,” and thus of no (proper) measurable effect[3]. However, Sciama did come up with a very clever “toy”1 model, not to be taken as a full gravity theory, in which inertial forces felt in reference frames would explain “why” otherwise free particles in local inertial reference frames accelerating relative to the average mass density of universe experience forces proportional to their gravitational mass. Or, the centrifugal forces in Newton’s pail are no accident, rather due to interaction of the local water with the distant stars. Of course, we earlier pointed out the difficulties with a pure scalar theory of gravity, so first consider a vector model (the scalar will return through another path later). Hund, 1948, noticing the similarities between the centrifugal and Coriolis inertial forces and electromagnetic forces, suggested that a vector field might be more consistent with a Mach’s principle than standard Einstein theory. Later Sciama[4] also formulated this in a suggestion we now describe.

SCIAMA’S TOY MODEL OF INERTIAL FORCES.

Sciama proposed a clever model theory in which inertial forces felt in an accelerating frame are “real” gravitational forces. Sciama’s idea was to relate the “inertial force” experienced in a reference frame accelerating relative the the average mass of the universe (the “fixed stars”) to gravitational forces (using a toy, vector, model of gravity) between the local test mass and the fixed stars. Thus, the fact that local inertial reference frames happen to see the fixed stars as unaccelerated has a causal basis in some appropriate gravitational theory. This is one form of the generally amorphous Mach’s principle. To this end we follow Sciama and consider a four vector model for gravity, with potential (φS,AS),and, as usual, c = 1. Approximate “all of the mass of the universe, M” by a large massive shell of mass M, radius R ≈ Hubble radius and our test particle, m at the center of the shell. This vector model is formally identical to the standard Maxwell formulation of magnetism, but with µ0 → −G and q → m we would have φS = −GM/R = const,AS = −GMvS/R. In this model the particle’s rest RF has velocity v and acceleration a relative to the shell. Thus, the shell appears to have velocity vS = −v and accelerationv ˙S = −a relative to the RF. Applying the results obtained from the electromagnetic analog we get a gravitomagnetic field, ∂A GM E = − S = − a, S ∂t R

1 I refer to this as a “toy” model since a pure vector theory of gravity is not consistent with observations. and a force according to Sciama, GM F = mE = −m a. S S R Note that this force is in the direction of the acceleration that the RF sees for the shell. In order to keep the particle at rest in this RF, the observer invents a “fictitious, inertial” force in the opposite direction,2 GM GM F = −F = m a = −m a . i S R R S This is in fact exactly the “fictitious” inertial force experienced when a particle is accelerated relative to the fixed stars if and only if GM = 1, R or, in the form that led to scalar-tensor theories, 1 = M/R, G 1 suggesting that G is a scalar field with mass as a source.

DICKE’S EARLY IDEAS.

Dicke in the 1950’s became interested in the radical idea of doing some gravitational experiments! Not many significant ones had followed on the amazing success of the three standard tests. But new tools were now available. First, Dicke carefully examined the reports of the Eötvös experiment, and found that it needed re-doing. He also was adamant in pointing out that it really only validated the Weak Equivalence Principle, WEP, as described above, since it only concerned itself with the acceleration effects of a gravitational field. But, of itself, WEP does not exclude the possibility that there might be some other gravitational effect than simply the geometric gαβ , which cannot be “transformed away” by going to a freely falling elevator. But what cannot be “transformed away?” A scalar field, of course. Another, perhaps initially unrelated, idea that influenced Jordan and Dicke is the following: • Dirac’s3 Large Numbers Hypothesis (LNH): The grouping of fundamental dimensionless numbers in sets of order 1040n, where n = 0,1,2 is no accident. There should be physical explanation for the “coincidences.”

2 Recall the inertial force called “centrifugal” which an object feels in an RF rotating relative to the ﬁxed stars (Newton’s bucket). 3 Actually Eddington had come up with some of these ideas earlier, but the standard reference is now to Dirac in this regard. Among these large number coincidences is GM/R ∼ 1, for cosmological mass and radius. Using “atomic units,” this can be dramatically expressed

GM 10−401080 ∼ ∼ 1. (4) R 1040 Putting MP and WEP together Dicke suggested considering a ﬁeld theory for which

φ ∼ 1/G ∼ M/R. (5)

Thus, there would be a new field, φ, with mass as source, and thus perhaps another aspect of gravity not described by the metric (which can always be locally transformed away). This additional field could then be consistent with the early verifications of WEP.4 However, such a field would not violate any known experiments on the SEP. Recall this is the statement that the only effects of gravity are geometric. Jordan[5] and his colleagues in prewar and wartime Germany, and independently, Ein- stein and Bergmann in the USA, had been considering similar ideas. Communications between groups in Germany and the US in the ’30’s and ’40’s were obviously not ideal. Both groups were primarily driven by the unified field efforts, Kaluza-Klein, expanding space-time to five dimensions, getting a five-metric decomposing into four-part, plus a four-vector, A and a scalar, g44, gαβ g4β (= Aβ ) (gAB) = gα4(= Aα ) g44(= φ), where α,β,··· = 0,...,3, and A,B,··· = 0...,4. This was originally considered in the context of a unified field theory for electromagnetism (Aα are the electromagnetic potential components) and gravity. In fact, the five dimensional Einstein reduce to Maxwell-Einstein if g44 is constant. However, if g44 is not constant, a whole new set of extended gravitational theories is opened up. This was the direction taken by Jordan and his group. Einstein and Bergmann also briefly explored these ideas. However, Einstein himself spent most of the rest of his life exploring alternative Unified Field Theory’s using asymmetric metrics, with the electromagnetic tensor as the anti-symmetric part. There has been no further extensive exploration of asymmetric metric theories. Dicke and Brans took a straightforward approach: “minimal coupling”. Look at original Einstein action, Z √ 4 δ −g(R + κLmatter)d x = 0. (6)

What if we just let κ be the ﬁeld variable? This would lead to non-conservation of mass (the mass term in the matter Lagrangian would be multiplied by the variable κ.) Also, it

4 However, to higher accuracy, this ﬁeld itself gravitates leading to very small discrepancies with WEP. These were not directly looked for until much later when solar system measurements of the Moon-Earth orbit did not agree with it. is 1/κ which seems to have matter for source. So, simply divide Einstein action by κ, let it be ﬁeld variable φ, Z √ 4 δ −g(φR + Lmatter + Lφ )d x = 0 (7)

This (apparently) keeps same equation for matter, i.e., geodesics for point particles (WEP). Minimal form for Lφ with dimensionless coupling ω turns out to be

αβ Lφ = −ωg φ,α φ,β /φ. (8) Resulting ﬁeld equations are like Einstein’s,

Rαβ − (1/2)gαβ R = (1/φ)Tαβ (matter) + Tαβ (φ) (9) together with T ρ 2φ = matter ≈ − . (10) g (2ω + 3) 2ω + 3 Call this the standard scalar-tensor theory, STT. 1/φ plays role of κ, but is a field variable, with matter as source, so 1 φ ∼ M/R, (11) (2ω + 3) as in Dirac’s LNH. • The varying gravitational “constant:” Equations (9), (10), and (11) provide the basis for the claim that STT results in a gravitational constant which is a function of the mass distribution in the universe. The predictions are similar to Einstein’s and reduce to them as ω → ∞, providing we impose some boundary condition on the solution to (10), requiring that it go to a constant 1 as 2ω+3 goes to zero. Of course, one perhaps unfortunate, but apparently unavoidable, feature is that the theory contains ω as a “fudge factor”, so it cannot be disproven, as long as ω can be chosen to be sufficiently large. Consequently, no observation with non-zero error can “disprove” STT. However, if the value of ω is sufficiently large, there is reason to say that STT is irrelevant. This last statement is, of course, highly context dependent, so what is irrelevant in solar system effects may be significant in cosmological or even quantum contexts. Thus, briefly, • Scalar-Tensor Summary: Standard Einstein theory is a theory in which gravitational effects are completely described by Lorentzian geometry, i.e., the metric alone. The Scalar-Tensor modifications of this theory involve the addition of a scalar field, typically coupled directly to the metric, and thus indirectly to all matter. Roughly, STT approaches SET as ω → ∞. For some of the contributions of Dicke and me, see [6],[7],[8],[9]. For a careful and extensive recent review of STT’s see the book by Fujii and Maeda[10]. EVOLUTION OF SCALAR-TENSOR THEORIES.

Over 30 years of accumulated observational data have now put some very high lower bounds for ω. Since scalar-tensor theories in the JBD form approach standard Einstein in the limit of large ω this means that STT is equivalent to the simpler Einstein formulation in solar system and similar contexts to within current measurements. Thus, as solar system observations, many of them supported by NASA related work, continued to refine the limits on ω, STT’s apparently became irrelevant in solar system work in the 1960’s and early 1970’s. Nevertheless, in an interesting historically paradoxical effort, Bob Dicke made an attempt to re-activate solar models with sufficient oblateness to cancel the scalar effects and lead to a situation in which observations would agree with scalar-tensor, rather than standard Einstein, theory. Recall that at the end of the 19th century, hints of deviations of planetary orbits from the Kepler-Newton form led to similar proposals of an oblate sun. These ideas, however, were quickly forgotten when Einstein’s general relativity predicted the famous rotation of the perihelion of Mercury. But it turns out that Bob’s suggested solar oblateness was not observed in careful solar studies. Thus, for a good while, the STT modifications of standard Einstein theory fell into disrepute because of experimental bounds. However, the proposal of a viable alternative to standard Einstein theory, as well as Dicke’s enthusiastic promotion of it, motivated a significant and valuable surge of experimental gravity, a field which had lain relatively dormant for 30 years or more. If nothing else, STT contributed this to physics. However, SET itself was, and is, not without problems. From the 1970’s on, observational cosmology began to show discrepancies with standard Einstein. In fact, more recently, several developments have re-activated interest in STT or a modified version of it. First consider the so-called “flatness” problem. In standard Einstein theory with Robertson-Walker metric and pressureless dust, Λ = 0, a quantity, Ω is defined by

2 Ω = κρ/H = ρ/ρc.

Here H is the present value of Hubble’s “constant” and ρc, the critical density, is the value of ρ for which Ω = 1. Observations lead to a value of Ω approaching one which ensures that on the cosmological scale space-time is approximately flat, but not too sparsely occupied. However, this is a variable. To achieve this present value, the “initial” value near the big bang had to be very finely tuned. Change of one part in 1050 would have disastrous consequences from the anthropic viewpoint. A tiny bit smaller and the universe would by now be essentially empty on even the solar system scale. A tiny bit larger and we would have long since vanished in the big crunch. In either of these cases, we could not be here-now to worry about it (the anthropic principle). Thus, • Flatness problem: How did Ω get to its present value which lets us be here to observe it? Another problem is the homogeneity or connectedness one. In the standard cosmology of standard Einstein theory, horizons develop, so that distant galaxies could not possibly have ever been connected by light rays. But we now see them and they seem to be quite similar. • Homogeneity problem: How did distant matter communicate? One way out of these is suggested by deSitter (Λ 6= 0) models, or equivalently those with negative pressure5 equal to density, p = −ρ. Such models, originally suggested by (Guth, Linde, et al) contain some mechanism associated with inflationary effects not consistent with SET, although this mechanism may have long since become negligible. A large class of such models is provided by a class of theories, STTV , for which

Lφ = −ωφ,α φ,β /φ +V(φ), for some V(φ). Sometimes such φ particles are called “inﬂatons.” Such models may be modiﬁcations of the original ST theories, reappearing in entirely different contexts. The experimental determinations of ω making the theories irrelevant and undetectable in solar system environment might not be a hindrance when considering cosmological numbers.

CONFORMAL INVARIANCE, DILATONS, HIGGS, STRINGS AND ALL THAT.

We close with an very cursory review of related scalar field ideas. A very natural extension of Lorentz invariance is to include general conformal transformations of the metric, gαβ → Φgαβ as suggested in Weyl’s electromagnetic theory. Electromagnetic gauge transformations are associated with conformal transformations of the metric, rather than U(1) for quantum fields. Thus Weyl’s gauge theory produced extra connection terms for the spacetime geometry, rather than internal U(1) space, making the spacetime connection non-metric. Scalars also are naturally associated with conformal metric changes, with φ of weight -2. This process can transform STT to standard Einstein theory, with an additional scalar field. However, which metric is observed? This can lead to some confusion in the terminology and interpretation in literature. The conformally related geometry is often referred to as the “Einstein frame” while the original one is the “Jordan frame.” Of course, neither geometry is a frame. If the original action is

αβ √ φ,α φ,β g¯ −g¯ φR¯ − ω + Lm(g¯) φ then the conformally related one is

,α √ 3 φ,α φ 1 −g R − (ω + ) + Lm(g/φ) 2 φ 2 φ 2

5 “Dark energy” is now the current terminology. The differences between geometries, especially the matter interaction, must be carefully considered and is still a matter of some dispute. An interesting special case which is of some interest is when ω = −3/2 making STT singular. This is the precise case in which the theory is conformally Einstein. Moving now to the quantum realm, the famous Higgs particle provides another scalar, a complex spatially scalar ﬁeld. This model contains kinetic terms,

√ αβ ∗ −gg φ,α φ,β . This term is manifestly not conformally invariant, even though all other terms in a Higgs action are. To make it conformally invariant, must add

−6φ ∗φR.

If we replace φ ∗φ → φ, these terms reduce to those in the “conformally Einstein” anomalous STT case above! Again at the quantum level, Isham, Salam, Strathdee in early 1970’s seem to be the first to use the “dilaton” term in the context of re-thinking Weyl’s gauge theory The dilaton field, χ, associated to the conformal group, is regarded as a quantum gauge group. The extra connection term is proportional to dχ/χ, and is thought to be an “internal” term in the quantum symmetry sense, originally formulated on background flat spacetime. Later, the term “ dilaton” occurs in string theory. At the basic bosonic string level, the arguments summarized in the Green-Schwarz-Witten book[11] discuss how the first non-tachyonic state is described by a tensor field of rank two, which they decompose into “graviton, dilaton, and axion.” The identification of these terms with previous usage is not always clear. Sticking only with the dilaton, the scalar field also appears as needed in the first order renormalization of the conformal anomaly, leading to a set of field equations that can be derived from an effective action,

p −2Φ ,µ |g|e (R − 4Φ,µ Φ + axion part).

But this was only lowest order perturbation result. Higher orders result in more general forms for scalar (and metric) actions, so the ﬁeld seems pretty wide open for choice of theory. Finally, an excellent general interest review of these scalar topics has been given by Kaiser[12]. Although not directly observed, scalar ﬁelds seem to refuse to go away.

REFERENCES

1. Norton, John, “How Einstein found his field equations,” Historical studies in the Physical Sciences, 14, 253 (1984). 2. Einstein, A., The Meaning of Relativity, pp 99-108, Princeton (1955) 3. Brans. Carl, Phys. Rev. 125, 388 (1962). 4. Sciama, D., Monthly Notices of the Royal Astronomical Society, 133, (1953). 5. Jordan, P., Schwerkraft und Weltall, Braunschweig (1955). 6. Brans, C., “Mach’s Principle and a Varying Gravitational Constant,” PhD thesis, Princeton University, 1961 (unpublished). 7. Brans, C., Dicke, R. Phys. Rev. 124, 925 (1961) 8. Brans, C. Phys. Rev. 125, 2194 (1962). 9. Dicke, R., The Theoretical Significance of Experimental Relativity, Gordon and Breach (1964). 10. The Scalar-Tensor Theory of Gravitation, Y. Fujii, K-I Maeda, Cambridge(2003). 11. Green, M., Schwarz, J., Witten, E. Superstring Theory, Cambridge (1987). 12. Kaiser, D., Scientific American p62, June 2007.