scalar-tensor theories of gravity: some personal history

history-st-cuba-slides 8:31 May 27, 2008 1

SCALAR-TENSOR THEORIES OF GRAVITY:

SOME PERSONAL HISTORY

Cuba meeting in Mexico 2008

Remembering Johnny Wheeler (JAW), 1911-2008

and Bob Dicke, 1916-1997

We all recall Johnny as one of the most prominent and productive leaders of relativ-ity research in the United States starting from the late 1950’s. I recall him as the manin relativity when I entered Princeton as a graduate student in 1957. One of his mostrecent students on the staff there at that time was Charlie Misner, who taught a reallynew, interesting, and, to me, exciting type of relativity class based on then “modernmathematical techniques involving topology, bundle theory, differential forms, etc. Cer-tainly Misner had been influenced by Wheeler to pursue these then cutting-edge topicsand begin the re-write of relativity texts that ultimately led to the massive Gravitationor simply “Misner, Thorne, and Wheeler. As I recall (from long! ago) Wheeler was thetype to encourage and mentor young people to investigate new ideas, especially mathe-matical, to develop and probe new insights into the physical universe. Wheeler himselfremained, I believe, more of a “generalist, relying on experts to provide details andrigorous arguments. Certainly some of the world’s leading mathematical work was beingdone only a brief hallway walk away from Palmer Physical Laboratory to Fine Hall.

Perhaps many are unaware that Bob Dicke had been an undergraduate student ofWheeler’s a few years earlier. It was Wheeler himself who apparently got Bob thinkingabout the foundations of Einstein’s gravitational theory, in particular, the ever presentmysteries of inertia. However, while open to all new ideas I believe that Wheeler himselfwas never convinced of the need for the additional scalar field that Bob proposed. How-ever, while not a proponent of such ideas, Wheeler certainly welcomed and encouragedtheir public investigation.

Of course I am certainly coming from a highly prejudiced position, but I do believethat Wheeler and Dicke must be included in any list of leading theoretical and experi-mental relativists of the 20th century.


IntroductionThe paradoxical phrase, “Varying...constant in the title was first used by me in my

Princeton PhD thesis of 1961. There I developed a formalism making explicit modifica-tions of Einstein’s theory introducing a scalar field variable to determine the Newtonianuniversal 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(ballpoint pen, washing machines, wartime radar, etc.) but most especially, motivatedby his undergraduate contacts with Wheeler, in the study of gravity. In the late 1950’sBob started looking in some detail into the historical data of Eotvos. Bob suggested thatthe famous claims about the equivalence principle based on this experiment were greatlyexaggerated. For example, a person walking within a hundred meters would disturb theoriginal experiment significantly. So, Bob and his group set out to do this experimentwith great care. His results did re-affirm, but with much more confidence, the fact thatthe ratio of gravitational to inertial mass is the same for a variety of atoms. But Bobwas careful to point out that this experiment only confirmed a Weak EquivalencePrinciple, WEP, stating that the acceleration effects of gravity are the same onall (or at least a wide variety of) small objects (neglecting tidal and 2nd order effects).However, he was adamant in also affirming that this experiment did not confirm aStrong Equivalence Principle, SEP, which asserts that the only effect of gravity is toaccelerate particles. This SEP is, of course, a tacit assumption of Einstein’s formulationof his general relativistic theory of gravity, SET.

In 1957 I was a first year graduate student at Princeton who was very interested inthe applications of topology and bundle theory to the differential geometry of generalrelativity, motivated largely by lectures of Charlie Misner. Looking for a thesis topic Iapproached Charlie with the idea of incorporating and extending bundle theory furtherinto general relativity and even quantum theory. Perhaps this might have developedinto an early form of gauge theory (I’ll never know now). Misner thought this was tooabstract, and besides, he knew that Bob Dicke was looking for a theorist to provide someformalism for his ideas, motivated by proposals of Mach, Dirac, et al. So, he suggestedI approach Bob for some thesis ideas. Somewhat reluctantly, being very experimentallychallenged myself, I nevertheless worked up the courage to approach Bob. To my greatrelief, Bob was very kind and tolerant of my experimental incompetence. He even invitedme to the regular meetings of his experimental group. They were very kind and forbearingof a dumb theorist who did not know a vacuum tube (yes, they existed at this time!) froma screwdriver. More significantly though, Bob took me aside in private meetings. Onenotable comment he made to me was that some theorists should be given swift kicks in thepants to make them realize the reality of inertial reactions. Of course, this was all in goodhumor, but the point was serious. As we will explore later, Bob thought that there wouldbe 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’swork at this time), and suggested I try to formulate a set of equations generalizing SETto 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 workof Jordan et al. on this topic and almost quit writing. However, I was encouragedto continue, giving what I hope was sufficient and appropriate credit to Jordan and hisgroup. From this thesis, Bob and I published the first paper on our version of this subjectin 1961. Of course, these theories ought properly be called “Jordan-Brans-Dicke, (JBD),but unfortunately many papers disregard Jordan’s groundbreaking work and refer to itsimply as BD. Remo Ruffini has showed me a letter he received from Jordan decryingthe 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 reluctanceto credit or investigate Jordan’s work was due to his apparently strong, and perhapseven enthusiastic, membership in the Nazi party.

But, back to the origins of our version. Among other things, Dicke pointed out tome that Einstein himself was strongly motivated by the ideas of Mach. In re-readingsome early passages of Einstein’s book, The Meaning of Relativity, I noticed that Ein-stein claimed that his standard theory did indeed confirm that inertial mass would beinfluenced 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 notat all enthusiastic about changing Einstein’s beautifully simple, purely ge-ometric, 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 slipthrough to generate a Princeton PhD degree! I should point out that as time went bymany other theorists seemed also to be offended to have Einstein’s theory contaminatedby an additional field. As a young, easily intimidated attendee at meetings, I often heardthis disapproval. However, experimentalists, especially those at NASA, were effusivelyhappy to have an excuse to challenge Einstein theory, long thought to be beyond fur-ther experimentation. Although happy to have a viable alternative to standard Einsteintheory, 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: math-ematical physics, and dropped my interest in ST’s. Although it is not properly on thetopic, I feel that I can now pause for a “commercial message: Be the first in your de-partment to buy and read that wonderful new book, Exotic Smoothness and Physics,by Aßelmeyer and Brans, WSP 2007. My parents are both dead, so I (and my wife) arealways looking for new customers...


Non-political AdvertisementThe organizing committee of this meeting assumes

no responsibility for the reading of this work,nor for its accuracy or relevance.


Special Relativity and Gravity NOSCALARGRAVITY?

Recall classical, Newtonian, gravitational theory defined by scalar U ,

Newton: ∇2U = 4πGρ, F = −m∇UWhat is the natural special relativistic extension? Compare to the transition: elec-

trostatics → electrodynamics, try

2U ≡ (∇2 − ∂2

∂t2)U =

κ

2ρ,

where κ ≡ 8πG, and c = 1 as usual.But!!, the resulting force field would be simply the gradient,

Fα = −∂U/∂xα.

This naturally leads to trivial force!! In fact,

md2xα

dτ 2= mFα,

implies that U must be constant along world line of particle,

Fαdxα

dτ=dU

dτ= 0,

So ... there is no gravity: the theory is trivial, there is no “simple scalartheory of gravity consistent with SR.

Is this surprising? No, say Abraham, Nordstrom, Einstein, et al. It is interesting tolook at their early steps in the development of standard GR gravity.

Consider particle and field actions. For a non-relativistic, free particle, we have

Sp0 =

∫

path, x(t)

m(v2

2)dt.

The relativistic generalization of this is

Sp = −∫

m√1− v2dt ≈

∫

m(v2

2)dt−m

∫

dt.

Introduce metric notation, and proper time,

dτ 2 = dt2 − dx2 − dy2 − dz2 = −3∑

µ,ν=0

gµνdxµdxν ,

where here the metric components, gµν , reduce to those of Minkowski, ηµν , and we uset = x0. Extend this to a special relativistic formalism parameterized by proper time,with zµ ≡ dzµ/dτ,

Sp = −∫

path, z(τ)m√

−ηµν zµ zνdτ,


which can be put into a volume integral (anticipating field interactions),

Sp = −∫

mIpath(x)d4x,

where

Ipath(xρ) ≡ m

∫

path, z(τ)

√

−ηµν zµ zνδ4(xρ − zρ(τ))dτ.

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.

Sp + SI + SU = −∫

miJpathd4x−

∫

mgU(x)Jpathd4x

−1

κ

∫

U,µU,µd4x.

But, Eotvos/Dicke impliesmi = mg = m. Clearly the field variation results in the correctsource equation, with ρ(xµ) = m

∫

δ4(xµ − zµ(τ))dτ ,

2U =κ

2ρ.

However, the variation over the particle’s variables, zµ(τ), zµ(τ) results in somethingquite surprising,

d

dτ(m(1 + U) zµ(τ)

)

= −mU ,µ.

METRIC-SCALARGRAVITY?

So here is the hint: The coupling of inertial mass directly to gravity impliesthat 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 actedon by a gravitational force! Thus, from action-reaction, gravitational mass would seemto be affected by gravity.

Another interpretation is that the metric has been influenced by gravity. Thus, thesum of the particle’s action plus the particle-gravity interaction action, is

Sp + SI = −∫

(

∫

m√

−ηµν zµ zνδ4(xµ − zµ(τ))dτ)

d4x

−∫

U(

∫

m√

−ηµν zµ zνδ4(xµ − zµ(τ))dτ)

d4x

or, assuming U << 1,

Sp + SI = −∫

(

∫

m√

−gµν zµ zνδ4(xµ − zµ(τ))dτ)

d4x,

where,gµν = (1 + 2U)ηµν ,


so we have now absorbed the gravitational field into the metric!! METRIC-SCALARGRAVITY?

Explicitly, for a point mass, M at the origin,

g00 ≈ −(1−2GM

r)

or proper time depends on location in the gravitational field:

dτ ≈√

1− 2GM

rdt.

This leads to redshift and deflection of light, etc., by gravity.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, especiallymathematicians. Not only are the equations non-linear, but so was the path to them. Af-ter much soul-searching (at some points he apparently thought the differential geometrywas too complicated) Einstein finally came to the mathematically “simplest equations,Einstein equations, are EINSTEIN

“SETRαβ − (1/2)gαβR = κTαβ,

derived from action,

δ

∫ √−g(R + κLmatter)d

4x = 0.

There is much good, interesting, and instructive history associated with the path ofEinstein, and others, to this final form.

Refer to this as “Standard Einstein theory, SET. These equations are simple, non-linear, interpretable via “gravity gravitates, roughly can be stated

“2gαβ = κ(Tαβ(matter) + Tαβ(metric)).

SET works great: it satisfies correspondence principle limit to Newtonian gravity, butbeats it in famous solar system tests. Really an extremely beautiful theory.


BUT...Does this mean the end to scalar fields in gravity????

probably not, since there are more slides left ...

Still questions/problems...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)4) (Quantum) The fine structures on spacetime needed to do relativity are as unob-

servable because of quantum theory as were the properties of the ether a century ago!(is spacetime the new ether?).

4) (Experimental) Few experiments had been done!!(Dicke) SETPROBLEMSMight some of these issues be related?

A few issues stand out:Mach’s principle (MP), inertial induction.

Einstein was motivated by Mach’s idea that inertia should be some sort of gravita-tional force, which would explain “why local inertial rf’s see average mass density ofuniverse non-accelerating. (Newton’s pail, no accident).

Newton sees flat water surface, and fixed stars (universe) at rest. INERTIALFORCES


Newton now sees curved water surface and fixed stars rotating.Is there a causal connection, G−1 ≈ M

R.??


But SET does not fully satisfy MP in the sense of predicting an explicit quantita-tive dependence of inertial reaction (mass) on universe structure, even though Einsteinthought it did.

Dicke (1950’s) got interested in the radical idea of doing some gravitational exper-iments! Looked back at Eotvos experiment, found it needed re-doing. He also wasadamant in pointing out that it really only validated

Weak Equivalence Principle (WEP): The acceleration effects of a gravita-tional field are equivalent to an acceleration of the rf.

In other words, WEP allows for the possibility that there might be some other grav-itational effect than simply the geometric gαβ, which cannot be “transformed away bygoing to a freely falling elevator.

What can not be “transformed away?

A Scalar!!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 andelectromagnetic forces, suggested that a vector field might be more consistent with aMach’s principle than standard Einstein theory. Later Sciama, 1953, also formulatedthis as a “toy model in which inertial forces felt in an accelerating frame are “realgravitational forces.

Sciama’s toy model of inertial forces.Sciama, 1953, came up with a clever “toy1 model 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 massof the universe (the “fixed stars) to gravitational forces (using a toy, vector, model ofgravity) between the local test mass and the fixed stars. Thus, the fact that local inertialreference frames happen to see the fixed stars as unaccelerated has a causal basis in someappropriate gravitational theory. This is one form of the generally amorphous Mach’sprinciple.

To this end we follow Sciama and consider a four vector model for gravity, withpotential (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 standardMaxwell formulation of magnetism, but with µ0 → −G and q → m we would haveφS = −GM/R = const,AS = −GMvS/R.

1I refer to this as a “toy” model since a pure vector theory of gravity is not consistent with observa-tions.


In this diagram the particle’s rest RF has velocity v and acceleration a relative tothe shell. Thus, the shell appears to have velocity vS = −v and acceleration vS = −arelative to the RF. Applying the results obtained from the electromagnetic analog weget a gravitomagnetic field,

ES = −∂AS

∂t= −GM

Ra,

and a force according to Sciama,

FS = mES = −mGM

Ra.

Note that this force is in the direction of the acceleration that the RF sees for theshell. In order to keep the particle at rest in this RF, the observer invents a “fictitious,inertial force in the opposite direction,2

Fi = −FS = mGM

Ra = −mGM

RaS.

This is in fact exactly the “fictitious inertial force experienced when a particle isaccelerated relative to the fixed stars if and only if

GM

R= 1,

2Recall the inertial force called “centrifugal” which an object feels in an RF rotating relative to thefixed stars (Newton’s bucket).


or, in the form that led to scalar-tensor theories,

1

G=M/R,

suggesting that 1Gis a scalar field with mass as a source.

Einstein thought he had something similar, with inertial mass directly dependenton universe structure, in General Relativity, but it was only “coordinate effect, not anabsolute one. To get some absolute effect need a scalar!

A related idea motivating Dicke (and earlier P. Jordan) wasDirac’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.Among these LNC’s is GM/R ∼ 1, for cosmological mass and radius. Using “atomic

units, this can be dramatically expressed

GM

R∼ 10−401080

1040∼ 1.

Putting MP and WEP together Dicke suggested considering a field theory for which

φ ∼ 1/G ∼M/R.

SCALARAGAINThus, there would be a new field, φ, with mass as source, an additional “gravitational

field, consistent with WEP, but not the experimentally un-verified SEP. Question is howto hook up to matter and gravity and formulate the field equations.

3Actually Eddington had come up with some of these ideas earlier, but the standard reference is nowto Dirac in this regard.


Jordan and his colleagues in prewar and wartime Germany, also Einstein and Bergmannin USA had been considering similar ideas. Communications between groups in Germanyand the US in the ’30’s and ’40’s were obviously not ideal! Both groups were primarilydriven by the unified field efforts, Kaluza-Klein, expanding space-time to five dimen-sions, getting a five-metric decomposing into four-part, plus a four-vector,A and a scalar,g44,

(gAB) =

(

gαβ g4β(= Aβ)gα4(= Aα) g44(= φ)

)

,

where α, β, . . . = 0, . . . , 3, and A,B, . . . = 0 . . . , 4. This was originally considered inthe context of a unified field theory for electromagnetism (Aα are the EM potentialcomponents) 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 extendedgravitational theories is opened up. This was the direction takenby Jordan and his group.

Einstein and Bergmann also briefly explored these ideas. However, Einstein himselfspent most of the rest of his life exploring alternative UFT’s using asymmetric metrics,with the electromagnetic tensor as the anti-symmetric part. There has been no furtherextensive exploration of asymmetric metric theories.4

4Although I did use them for my BS thesis at Loyola in 1957.


Dicke and Brans took straightforward approach, “minimal coupling. Look at originalEinstein action,

δ

∫ √−g(R + κLmatter)d

4x = 0.

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 is 1/κ which seems to have matter for source. So, simply divide Einstein action by κ,let it be field variable φ,

δ

∫ √−g(φR + Lmatter + Lφ)d

4x = 0

JBDSCALARTENSOR,“STT

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

Lφ = −ωgαβφ,αφ,β/φ.

Resulting field equations are like Einstein’s,

Rαβ − (1/2)gαβR = (1/φ)Tαβ(matter) + Tαβ(φ)

together with

2gφ =

Tmatter

(2ω + 3)≈ − ρ

2ω + 3.

Call this the standard scalar-tensor theory (zeroth form), STT0. 1/φ plays role of κ, butis a field variable, with matter as source, so

φ ∼ 1

(2ω + 3)M/R,

as in Dirac’s LNH.Varying gravitational “constant

Predictions are similar to Einstein’s and reduce to them as ω → ∞. Analysis showsthat 1/φ does indeed play role of a gravitational “constant, which is now a variable.

The theory contains ω as a “fudge factor, so it cannot be disproven, aslong as ω can be chosen to be sufficiently large. Many say this is not fair,something of a theoretical way of cheating.


Scalar-Tensor SummaryStandard Einstein theory is a theory in which gravitational effects are completely

described by Lorentzian geometry, i.e., the metric alone. The Scalar-Tensor modifica-tions of this theory involve the addition of a scalar field, typically coupled directly to themetric, and thus indirectly to all matter. Roughly, STT approaches SET as ω →∞.

For an excellent recent review, see The Scalar-Tensor Theory of Gravitation, Y. Fujii,K-I Maeda, Cambridge(2003).

Some good... STT leads to a “varying gravitational constant, and a MP in thissense. Presents an alternative to SET, so motivates much needed gravitational experi-mentation, and theoretical work on experimentation.

Some not so good... STT is disappointing in fully deriving the value of κ, needingboundary conditions, etc. Has same sort of singularity behavior as Einstein... Introducesa new dimensionless constant, ω in place of an old dimensional one, κ.... Cannot bedisproven, but one remarkable difference would be the prediction ofmonopole radiationin STT, whereas lowest order for SET is quadrupole.

End of my personal involvement,but much more theoretical and experimental work goes on...


STT Death...Over 30 years of accumulated observational data put some pretty high value on

ω, so in these observational contexts, STT is equivalent to simpler Einstein’s. Otherconsiderations, cosmological, collapse, black hole formation etc., produce results similarto Einstein. So, for a good while, the STT modifications of standard Einstein theoryfell into disrepute because of experimental bounds. Thus, experimental data seems toindicate that ω must be so big that the theory is irrelevant in solar system contexts.Others, especially Nordtvedt continued to work around in it. Also, on the plus side, itcertainly motivated a lot of beneficial experimental interest in gravity.

...and Resurrection(?)But recently, several developments re-activated interest in it or a modified version

(STTx) of it.Some Problems with Einstein Cosmology

One standard example is the so-called “flatness problem. In SET, for a standardRW model (pressureless dust, Λ = 0), there is a variable,

Ω = κρ/H2 = ρ/ρc,

where H is the present value of Hubble’s “constant. The present value of Ω is about1, which ensures that on the cosmological scale space-time is approximately flat, but nottoo 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 smallerand the universe would by now be essentially empty on even the solar system scale. Atiny bit larger and we would have long since vanished in the big crunch. Thus,

Flatness problem: How did Ω get to its present value which lets us be hereto observe it? Also known as the “anthropic principle

Another problem is the homogeneity or connectedness one. In the standard cosmologyof SET, horizons develop, so that distant galaxies could not possibly have ever beenconnected 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 pressure equal to density, p = −ρ. Such models, called “inflationarysince the radius of the universe expands exponentially, resolve the two problems above,and others, but are not consistent with what we see of the universe today. So, suggestionsdating to the 1980’s, (Guth, Linde, et al) is that some mechanism produced inflationaryeffects for a while, but have long since become negligible. A large class of such modelsis provided by a class of theories, STTV , for which

Lφ = −ωφ,αφ,β/φ+ V (φ),

for some V (φ). Sometimes such φ particles are called “inflatons. Such models maybe 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 cosmologicalnumbers.

Conformal invariance, dilaton, etc.A very natural extension of Lorentz invariance is to include general conformal trans-

formations of the metric,gαβ → Φgαβ

Interesting beginning in Weyl’s electromagnetic theory. Associated electromagneticgauge transformations to conformal transformations of the metric, rather than U(1)for quantum fields. Thus Weyl’s gauge theory produced extra connection terms for thespacetime geometry, rather than internal U(1) space, making the spacetime connectionnon-metric.

Conformal STT’s conformal metric changes, with φ of weight -2, transform SSTto standard Einstein theory, with an additional scalar field! Much abuse of terminologyand interpretation in literature. The conformally related geometry is often referred toas the “Einstein frame while the original one is the “Jordan frame. Of course, neithergeometry is a frame! If the original action is

√−g(

φR− ωφ,αφ,β gαβ

φ+ Lm(g)

)

then the conformally related one is

√−g(

R− (ω +3

2)φ,αφ

,α

φ2+

1

φ2Lm(g/φ)

)

The differences between geometries, especially the matter interaction must be carefullyconsidered. Still a matter of some dispute.

Also, consider the special case ω = −3/2 which is anomalous in SST. This is theprecise case in which the theory is conformally Einstein.

Higgs: A complex scalar Higgs model contains Higgs kinetic terms,

√−ggαβφ∗,αφ,β.

This term is manifestly not conformally invariant, even though all other terms in a HiggsAction 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 anoma-lous STT case above!

Dilaton, StringsIsham, Salam, Strathdee in early 1970’s seem to be the first to use this term in the

context of re-thinking Weyl’s gauge theory, coining “dilaton to be the field, χ, associ-ated to the conformal group regarded as a quantum gauge group. The extra connectionterm is proportional to dχ/χ, and is thought to be an “internal term in the quantumsymmetry sense, originally formulated on background flat spacetime.


Later, the term dilaton occurs in string theory (can we still talk about it, evenif it is “not even wrong??). At the basic bosonic string level, the arguments sum-marized in the Green-Schwarz-Witten book discuss how the first non-tachyonic state isdescribed 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 firstorder renormalization of the conformal anomaly, leading to a set of field equations thatcan be derived from an effective action,

√

|g|e−2Φ(R− 4Φ,µΦ,µ + axion part).

But this was only lowest order perturbation result. Higher orders result in more generalforms for scalar (and metric) actions, so the field seems pretty wide open for choice oftheory!

Conjectures:JBD Scalar↔Dilaton↔Inflaton

???


Some background references:

• P. Jordan, Schwerkraft und Weltall, Braunschweig (1955),

• C. H. Brans, The roots of scalar-tensor theory:an approximate history, gr-qc/0506063,

• R. H. Dicke The Theoretical Significance of Experimental Relativity, Gordon andBreach (1964)

• Fujii and Maeda, The Scalar-Tensor Theory of Gravity, Cambridge (2003)

• Kaiser Scientific American June 2007, p62.

• Also some old documents on http://loyno.edu/ brans/ST-history.

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