A First-Principles Implementation of Scale Invariance Using Best Matching
aa r X i v : . [ g r- q c ] O c t A First-Principles Implementation of Scale InvarianceUsing Best Matching
Hans Westman ∗ Physics Building A28, University of Sydney, NSW 2006Sydney, New South Wales, Australia
November 3, 2018
Abstract
We present a first-principles implementation of spatial scale invariance as a local gaugesymmetry in geometry dynamics using the method of best matching [1]. In addition tothe 3-metric, the proposed scale invariant theory also contains a 3-vector potential A k as adynamical variable. Although some of the mathematics is similar to Weyl’s ingenious butphysically questionable theory [14, 15, 16], the equations of motion of this new theory aresecond order in time-derivatives. Thereby we avoid the problems associated with fourthorder time derivatives that plague Weyl’s original theory. It is tempting to try to interpretthe vector potential A k as the electromagnetic field. We exhibit four independent reasonsfor not giving into this temptation. A more likely possibility is that it can play the roleof “dark matter”. Indeed, as noted in [20, 21] scale invariance seems to play a role in theMOND phenomenology. Spatial boundary conditions are derived from the free-endpointvariation method and a preliminary analysis of the constraints and their propagation inthe Hamiltonian formulation is presented. In practice we are never able to measure an objects “intrinsic” length. Rather, a measure-ment of the length of some object is a demonstration that x reference rulers can be fittedalongside it. But if both the reference ruler and the physical object are simultaneouslyrescaled, the measurement outcome would remain unaltered. Thus, we cannot measurean objects “intrinsic” length but only the ratio of lengths. Therefore, on epistemologicalgrounds, there seems to be good reasons to expect a fundamental theory to be scale in-variant so that the fundamental equations of motion make no reference to intrinsic sizes.Furthermore, since length measurements are ultimately done locally in space, it also seemsreasonable to expect a fundamental theory to be also invariant with respect to local scaletransformations.This heuristic argument, based on basic epistemology, is very similar to the well-knownarguments about absolute space. In practice we are never able to measure a body’s “ab-solute” position. Instead we can only measure a body’s position relative to other bodies.Therefore, on epistemological grounds we expect that a fundamental theory should notmake reference to absolute positions. This heuristic reasoning, dating back at least to thefamous Newton-Leibniz debates over absolute space, was undoubtedly important for thediscovery of general relativity in which absolute positions play no role.Given that this type of heuristic reasoning has proven very successful, it is only naturalto try to continue this process of “epistemological refinement” of physical theories. On thisview, the fact that general relativity is not scale invariant can be considered an “epistemo-logical defect”. In fact, a number of scale invariant theories has indeed been put forth inthe literature. An important early attempt in 1918 is by Hermann Weyl [14, 15, 17] and ∗ Email: [email protected] more recently by Mannheim [19]. Furthermore, Barbour has constructed scale invariantparticle model whose predictions seem to be approximately that of Newtonian gravity [3].In [4, 5] Anderson et. al. presented an intriguing first-principles derivation of York’sconformal technique [6, 7, 8] for solving the Hamiltonian constraint in general relativity byenforcing volume preserving spatial conformal transformations as a gauge symmetry. Theystart from a special form of the Einstein-Hilbert action called the Baierlein-Sharp-Wheeleraction [10]. Then they perform an arbitrary volume preserving conformal transformation.Since the Beierlein-Sharp-Wheeler action is not gauge invariant with respect to conformaltransformations both the scale parameter φ and its velocity ˙ φ will appear in the so obtainedconformalized action. From the free-endpoint variation method they argue that ˙ φ and φ should be varied independently. As a intriguing consequence, the constant mean curvaturefoliation time gauge as well as the Lichnerowicz-York equation falls out.Although this work sheds light on the importance of conformal structures in geometrydynamics for solving the Hamiltonian constraint, this trick of introducing conformal in-variance by conformalizing the Beierlein-Sharp-Wheeler action is analogous to the way onecan turn the Klein-Gordon equation in flat spacetime into a 4-diffeomorphism invarianttheory by parametrizing the Minkowski coordinates [11, 12]. One of the purposes of thisthis paper is to investigate whether conformal invariance can be implemented from firstprinciples and not by starting from the conformalized Baierlein-Sharp-Wheeler action. Ofcourse, this is a more risky project and there is no guarantee that the theory presentedhere is empirically adequate. We shall now start building in scale invariance in our theory from first principles. Considerthen a ruler. If we move the ruler from a point A to B , how can we be sure that the rulerhas the same length after it has been moved? How do we know that there is not a newkind of force that rescales the ruler when we move it from point to point is space? If therescaling effect is a universal so that all objects, irrespective of internal composition, arerescale in the same way, then it is pointless to introduce another reference ruler to check tosee if the length of the first ruler has changed. This is so since by assumption the new fieldwill have rescaled both rulers in the same way. Thus, the rescaling effect will be locallyunobservable. Note however that a measurement of an angle reduces to a measurement of ratios oflengths of the sides of a triangle. Angles are therefore already fully relational quantities (asis also indicated by their lack of dimension) and can therefore be unambiguously measuredlocally. Thus, angles have a different epistemological status than lengths.The key mathematical idea is therefore the following: allow for a connection Γ kij thatpreserves angles between parallely transported vectors (representing ideal rulers) but notnecessarily their individual lengths. The Levi-Civita connection is too restrictive for thispurpose since it preserves both lengths and angles. Instead we should only require that
DDs „ U i V j g ij | U || V | « = X k ∇ k „ U i V j g ij | U || V | « = 0 (1)for all X k , and all parallely transported U i and V i , i.e. DDs U i = DDs V i = 0, where | U | = p U i U j g ij and | V | = p V i V j g ij . By expanding this expression, rearranging terms,and making use of that the vectors are parallely transported we get ∇ k g ij » U i V j U · V − U i U j | U | − V i V j | V | – = 0 (2)The only way to satisfy this equation is if ∇ k g ij = A k g ij . (3)Thus, the connection will in general be non-metrical, i.e. ∇ k g ij = 0. If we assume vanishingtorsion Γ kij = Γ kji and make use of that ∇ k g ij = ∂ k g ij − Γ lki g lj − Γ lkj g il this implies thatthe connection takes the formΓ kij = kij ff −
12 ( δ ki A j + δ kj A i − g ij g kl A l ) (4) Of course, there might be interesting global effects. We will shortly discuss this in section 3.3. where kij ff is the Levi-Civita connection. Under a conformal transformation g ij → e θ g ij the Levi-Civita connection transforms in the following way: kij ff → kij ff + 12 ( δ ki ∂ j θ + δ kj ∂ i θ − g ij g kl ∂ l θ ) (5)Thus, if the vector potential transforms as A i → A i + ∂ i θ , then the full non-metricalconnection Γ kij = kij ff − ( δ ki A j + δ kj A i − g ij g kl A l ) is not only conformally covariantbut also conformally invariant since all the θ ’s cancel out. Our connection is thereforeunchanged by a conformal transformation. This, in turn, means that the correspondingRiemann and Ricci curvatures R klij = ∂ i Γ kjl − ∂ j Γ kil + Γ kin Γ njl − Γ kjn Γ nil (6) R lj = R klkj (7)are also conformally invariant. However, R = g ij R ij is not invariant since it involvescontraction with the metric which comes with conformal weight of −
1, i.e. R → e − θ R .Note also that R ij = R ji in general. This is so since we are not dealing with a metricalconnection.The reader familiar with Weyl’s 1918 theory [14, 15] will see that, up till now, themathematics is identical to Weyl’s theory. One important exception is that we are imple-menting spatial scale invariance g ij → e θ g ij while Weyl implemented spatiotemporal scaleinvariance g µν → e θ g µν . As we shall see, this mathematical difference is going to be crucialwhen we try to construct a scale invariant dynamics of the theory and allows us to obtaina theory which has field equations of second order in time. Einstein’s reaction to Weyl’s 1918 unified theory was mixed [17]. On the one hand sideEinstein was genuinely impressed by its mathematical beauty and ingenuity calling it “astroke of genius of first rank”. But on the other hand Einstein expressed doubts aboutthe theory as a physical theory. In this section we are going to review Einstein’s mainmisgivings about Weyl’s 1918 theory.
First Einstein remarked that the “gravitational field equations will be of fourth order,against which speaks all experience until now”. More specifically, fourth order theoriesin general imply a Hamiltonian that does not have a lower bound [18]. This gives rise to“instability” problems which is not desirable for a physical theory. In contrast to Weyl weshall only implement spatial scale invariance and in the theory we shall present below, thefield equations are guaranteed to be second order in time-derivatives thereby avoiding theproblems associated with higher-order derivatives.
Einstein also raised an objection concerning proper time. His objection runs as follows.Assume that the line element dτ = g µν dx µ dx ν corresponds to the proper time of a clock.Since proper time (as read off from a ideal clock) is something observable it should have aunique predictable value in a theory. Secondly, assume with Weyl that conformally relatedmetrics are physically equivalent, i.e. g µν and e θ g µν describe the same physical situation.However, the line element also gets transformed dτ → e θ dτ and as a consequence thetheory cannot make a unique prediction regarding what proper time a clock will show whenmoved along some path in spacetime. Therefore, there is a flaw in Weyl’s theory.One obvious way to avoid such a conclusion is to challenge the assumption that propertime of a physical clock can be obtained from the line element of the metric. Instead Weylproposed that the proper time should be dτ = Rg µν dx µ dx ν , where R is the scalar curva-ture in Weyl’s theory. The object Rg µν dx µ dx ν has conformal weight zero and consequentlydoes not transform under conformal transformations and in this way a unique proper time is obtained. However, this choice is hard to motivate on physical grounds. What if R ≈ within the theory. However, to the best of our knowledge, Weyl never developed such atheory of clocks and Einstein’s objection was left unresolved.Nevertheless, Einstein’s objection concerning proper time might not be as serious asfirst thought. We are not aware of any later attempts to develop a theory of clocks withinWeyl’s theory so here we shall briefly sketch such a theory. The simplest theoretical clockone can imagine is the lightclock. Consider then any spacetime trajectory γ . Closelyaround it we put mirrors that would make a light-beam bounce back and forth crossingthe trajectory many times. The proper time measured by this light clock would be thenumber of times the light-ray crosses over the trajectory γ .However, we need to make sure that the mirrors are positioned so that they remainat the same proper distance throughout the trajectory. This creates an immediate prob-lem: how do we define an notion of equidistance in a conformally invariant theory inwhich distance is but a gauge degree of freedom? This problem is only apparent though.Weyl’s theory to contains the necessary mathematical machinery: At the beginning of thetrajectory we choose an arbitrary (small) vector representing a reference ruler. Then weFermi-Walker transport this reference ruler along the curve using the Weyl non-metricalconnection. In this way we can define a notion of equidistance within Weyl’s theory andthe notion of proper time becomes well-defined and presumably gauge-independent.For our theory to be viable it also needs to circumvent Einstein’s objection regardingproper time. However, since a similar analysis of lightclocks would require a universallightcone structure, something which has yet to be demonstrated for the our theory, wepostpone a full analysis to a future paper. The connection A µ in Weyl’s theory might be non-trivial F µν = ∂ µ A ν − ∂ ν A µ = 0. Thishas the following implication [17]: If we move two a hydrogen atoms from a point A to Bin spacetime, but along different paths, then the final size of the hydrogen atoms mightdiffer. As a concequence the spectral lines would be shifted. However, experiments providevery tight bounds on deviations from the standard predictions. We are not aware of anyattempts of a numerical quantification of this effect but it seems plausible that, if the vectorpotential A µ is taken to represent the electromagnetic field, then current experiments wouldprobably rule out the theory.However, as is argued in section 7, the vector potential A µ cannot represent the elec-tromagnetic field but could instead be thought of as a candidate for “dark matter”. But if A µ is not the electromagnetic field, its typical strength is not known and could very wellbe very small. It would therefore be worth speculating about whether a Weyl-type theorycould explain the (controversial) claim of a tiny increase of the fine structure constantover the last 10 billion years. The fine structure constant is estimated by observations ofspectral lines from distant astronomical objects and a varying fine structure constant couldperhaps instead be reinterpreted as a Weyl-type rescaling effect. For Weyl the task to find a scale and diffeomorphism invariant action for his theory wasrather straightforward: you need a spacetime scalar density with conformal weight zero.This would ensure 4-diffeomorphism invariance as well as conformal invariance as gaugesymmetries. However, our approach is explicitly 3-dimensional in nature and thereforewe have to resort to a different strategy to construct an action. Such a strategy, called“best matching”, has been developed by Julian Barbour and collaborators (see e.g. [1,3]). The technique of “best matching” yields specific Lagrangians and bosonic gauge field I am grateful to Lucian Hardy for suggesting this as a possible resolution to Einstein’s “proper time”objection. A parallel transport of the vector would not be appropriate since in general the vector will not remain onthe spatial slice orthogonal to the worldline of the lightclock. theories (e.g. Maxwell’s, Yang-Mill’s theories, and general relativity) are examples of “bestmatched” theories [9]. When carrying out the best-matching procedure in general relativity (and most impor-tantly requiring a “local square-root” [2]) we end up with the following Lagrangian L = Z d x √ g √ T V = Z d x √ g r G ijkl “ ˙ g ij − L−→ N g ij ” “ ˙ g kl − L−→ N g kl ” R (8)where T = G ijkl “ ˙ g ij − L−→ N g ij ” “ ˙ g kl − L−→ N g kl ” , G ijkl = g ik g jl − g ij g kl , and V = R .This Lagrangian, first discovered by Baierlein, Sharp, and Wheeler [10], and later signif-icantly illuminated by Barbour et. al. [2], yields the same equations of motion as theEinstein-Hilbert action. It has a peculiar local square root structure, i.e. the square root istaken before the integration over space. This is something unique to general relativity [2]and is shared neither by the Barbour-Bertotti model [1], nor Yang-Mill’s theories such asMaxwell’s theory even when written in the mathematically equivalent best-matched form[1]. However, this peculiar local square root leads to foliation invariance as a unexpectedadditional symmetry for a narrow set of potentials [2]. In this way 4-diffeomorphism in-variance is retained. The emergence of a universal lightcone structure is also cruciallydependent on having a local square root form [2].As we shall see in section 4.4, although the the local square root structure in generalrelativity is somewhat mysterious, it seems to be very natural when local spatial scaleinvariance is demanded. We shall assume that the Lagrangian density for our theory has a similar square root form L = √ g √ T V . We can now immediately write down the kinetic terms T g = G ijkl “ ˙ g ij − L−→ N g ij − φg ij ” “ ˙ g kl − L−→ N g kl − φg kl ” (9) T A = g ij “ ˙ A i − L−→ N A i − ∂ i φ ” “ ˙ A j − L−→ N A j − ∂ j φ ” (10)A new auxiliary field φ has been added in order to impose the best-matching condition(obtained by varying the Lagrangian with respect to φ ) corresponding to the new confor-mal gauge symmetry. Note that these are basically the only natural choices available forquadratic kinetic terms.These kinetic terms are conformally covariant only if φ → φ + ˙ θ − N k ∂ k θ and N k → N k under a conformal transformation g ij → e θ g ij and A i → A i + ∂ i θ . Given this transforma-tion rule of the auxiliary field φ the kinetic terms transforms according to T g → T g (11) T A → e − θ T A . (12)Thus, T g and T A has conformal weight 0 and − R . Thus, the most natural kinetic termwhould have the form T = T g R + aT A (13)where a is an arbitrary constant. The kinetic term will then have conformal weight − As of date, it has not been shown that spinors fields are compatible with the principle of best-matching.The main difficulty is the following: Actions for four-component Dirac spinor fields are not quadratic but linearin the field velocities while best-matched actions are always quadratic. A possible way to proceed could be toconsider van der Waerden’s reformulation of the Dirac field in which the equations for the two-component spinoris second order in spatio-temporal derivatives.
In order to ensure conformal invariance we need to make sure that the action has conformalweight zero. √ g has weight +3 / √ T V . Sincethe kinetic term T has conformal weight − V must haveconformal weight − R R ij R ij F ij F ij R ij F ij R klij R lijk (14)where F ij = ∂ i A j − ∂ j A i . We will keep the discussion general here and take the potential V to be a linear combination of all those terms. V = bR + cR ij R ij + dF ij F ij + eR ij F ij + fR klij R lijk (15)Note that R ij is not necessarily a symmetric tensor and therefore R ij F ij is not identicallyzero. This is because we are not dealing anymore with the Levi-Civita metrical connection.The action for the theory is then L = √ g p ( T g R + aT A ) V . (16)It is immediately recognized that, since only first order time-derivatives enter in the La-grangian density, the field equations will be second order in time. Thus, we have avoidedthe “instability” problems connected to higher-order time-derivatives.
One way to try to determine the constants a, b, c, d, e, f would be to require foliation in-variance (i.e. that the Hamiltonian constraint will propagate). It was noted in [2] that therequirement of foliation invariance (i.e. the propagation of the Hamiltonian constraint)quite uniquely picks out the Baierlein-Sharp-Wheeler action. We hope that somethingsimilar will happen for our conformally invariant theory. At this point it is not clear if it ispossible to make the Hamiltonian constraint propagate. We will postpone a full analysisfor a later paper.
The best-matched Lagrangian for the Maxwell field looks like [1] L = sZ d xδ ij ( ˙ A i − ∂ i φ )( ˙ A j − ∂ j φ ) „ E − Z d xδ ij B i B j « (17)where A i is the electromagnetic vector potential, B i = ǫ ijk ∂ j A k the magnetic field, E isthe total energy, and φ = A . Notice that the square-root is taken after we have integratedover space. The local square-root structure in the Beierlein-Sharp-Wheeler action, wherethe square-root is taken before the integration, is therefore quite surprising. It accounts forthe emergence of spacetime diffeomorphism invariance and a universal lightcone structurebut at the same time it also signifies an important structural difference from other best-matched theories, e.g. the Maxwell field [1] which has a global square root-structure asjust mentioned. It would therefore be interesting to see if there is some fundamental reasonform the local square-root in general relativity.Indeed, it seems that local scale invariance can shed some light on this issue: A globalsquare-root structure is rather unnatural in the scale invariant framework. A confrmallyinvariant Lagrangian with a global square-root could look like L = sZ d x √ g ( T g R + aT A ) V „ E − Z d x √ gV « (18)where V and V are potentials. They must have conformal weight − / − / p | R | and ( F ) / . These Lagrangians, although they cannot be excluded `a priori ,appear rather uggly and unnatural. Therefore it seems that if local spatial scale invarianceis imposed, then a local square-root structure is natural. We do not consider higher order curvature scalars which involves higher order spatial derivatives.
Let us now consider a variation of the action S = R d x L with respect to the auxiliaryfields N k and φ . Making use of Gauss theorem yields δ φ S = Z V d x „ ∂ L ∂φ − ∂ µ ∂ L ∂∂ µ φ « δφ + Z ∂V dAn µ ∂ L ∂ µ φ δφ (19) δ N k S = Z V d x „ ∂ L ∂N k − ∂ µ ∂ L ∂∂ µ N k « δN k + Z ∂V dAn µ ∂ L ∂ µ N k δN k (20)where V stands for a four-dimensional region in spacetime, ∂V its three-dimensional bound-ary, and n µ the corresponding unit normal vector. The variational principle requires thatthe variation δS is zero for all for all variations δφ and δN k . Normally one makes therestriction that δφ = δN k = 0 on the boundary. However, since we are dealing with un-physical auxiliary fields N k and φ whose only purpose is to enforce the “best matching”condition, this restriction is not compelling. Indeed, to enforce an arbitrary restriction onthe variation of the auxiliary fields on the boundary amounts to introducing an arbitraryabsolute element in the theory, something which should be avoided in a relational theory.Instead one should allow the variation δφ to be arbitrary even on the boundary [3]. Thisis called the free-endpoint variation method and yields the following equations of motion ∂ L ∂φ − ∂ µ ∂ L ∂∂ µ φ = 0 n µ ∂ L ∂ µ φ ˛˛˛˛ x ∈ ∂V = 0 (21) ∂ L ∂N k − ∂ µ ∂ L ∂∂ µ N k = 0 n µ ∂ L ∂ µ N k ˛˛˛˛ x ∈ ∂V = 0 . (22)If, as is usually done, the boundary ∂V is chosen so that n µ = δ µ for λ = λ , n µ = − δ µ for λ and n µ = (0 , n i ) for λ < λ < λ then these equations become ∂ L ∂φ − ∂ i ∂ L ∂∂ i φ = 0 n k ∂ L ∂ k φ ˛˛˛˛ x ∈ ∂Vλ ∈ [ λ λ ] = 0 (23) ∂ L ∂N k − ∂ i ∂ L ∂∂ i N k = 0 n l ∂ L ∂ l N k ˛˛˛˛ x ∈ ∂Vλ ∈ [ λ λ ] = 0 (24)where we have made use of the fact that the Lagrangian does not depend on the time-derivatives of the auxiliary fields so that ∂ L ∂ ˙ φ ≡ ≡ ∂ L ∂ ˙ N k . Using the definition (see eqs.(27)-(28) of section 6) of the canonical momenta we can rewrite the spatial boundaryconditions as π k n k ˛˛˛ x ∈ ∂Vλ ∈ [ λ λ ] = 0 (25) π ij n i ˛˛˛ x ∈ ∂Vλ ∈ [ λ λ ] = 0 (26)Thus, the method of free endpoint variation yields important spatial boundary condi-tions for the fields g ij and A k . The first constraint can be interpreted as there beingzero “charge” in the universe (see equation (35)). The second constraint implies that theassymptotic ADM-type momentum and angular momentum must be zero for the wholeuniverse, something which is expected in a Machian theory. That spatial boundary conditions follow from the free endpoint variation method seemsnot to have been noted in the literature before. However, it should be noted that it is Free endpoint variations are encountered in numerous problems in analytical mechanics where the endpointscannot be fixed `a priori . A standard example is a flexible hanging beam whose position and orientation are fixedat one end but free in the other. The action is the total energy which consists of potential energy and “bending”energy. The variation at the non-fixed end is unknown `a priori and must be allowed to be completely free. Theshape of the beam is found by minimizing the total energy. This implies non-trivial boundary conditions for thenon-fixed end. For explicit expressions of the ADM momenta in general relativity see [22] and [23]. normally argued that only a spatially closed universe is compatible with Mach’s principle.In such a case we must impose periodic spatial boundary conditions and therefore the issueof boundary conditions does not arise.
The momenta conjugate to g ij and A i are π ij ≡ ∂ L ∂ ˙ g ij = √ gRG ijkl ( ˙ g kl − L−→ N g kl − φg kl ) s VT g R + aT A (27) π i ≡ ∂ L ∂ ˙ A i = √ gag ij ( ˙ A j − L−→ N A j − ∂ j φ ) s VT g R + aT A (28)Since the Lagrangian is independent of ˙ N k and ˙ φ their canonical momenta vanishes iden-tically π k −→ N ≡ π φ ≡ ddλ π k −→ N = 0 = ddλ π φ . By rewriting the Lagrangian equations of motion0 = ∂ L ∂φ − ∂ k ∂ L ∂∂ k φ − ddλ ∂ L ∂ ˙ φ = ∂ L ∂φ − ∂ k ∂ L ∂∂ k φ − dπ φ dλ (31)0 = ∂ L ∂N l − ∂ k ∂ L ∂∂ k N l − ddλ ∂ L ∂ ˙ N l = ∂ L ∂N l − ∂ k ∂ L ∂∂ k N l − dπ −→ Ndλ (32)we see that the propagation of the constraints π k −→ N and π φ can be ensured if dπ φ dλ = ∂ L ∂φ − ∂ k ∂ L ∂∂ k φ = 0 (33) dπ −→ Ndλ = ∂ L ∂N l − ∂ k ∂ L ∂∂ k N l = 0 . (34)After some calculation making use of the definitions for the canonical momenta and recall-ing that these are tensor densities rather than tensors it can be shown that these constraintstake the form π ij g ij − ∇ k π k = 0 (35) ∇ i π ij + 12 π i F ij = 0 (36)These are, in Dirac’s terminology, secondary constraints. Since the symmetries correspod-ing to these constraints (3-diffeomorphism invariance and spatial conformal invariance) ismanifest in the Lagrangian it is clear that these constraints will propagate. Accordingto the first constraint, which resembles Gauss law, the trace of π ij acts as a source forthe “electric” field π i . The second constraint is similar to the one in standard geometrydynamics but is now modified with an extra term which is familiar from quantum gravityin the Ashtekar variables (see e.g. [11]).Since we are dealing with a reparametrization invariant Lagrangian we can immediatelyread off the following quadratic primary constraint from the definitions of the canonicalmomenta: 1 √ g aG ijkl π ij π kl + 1 √ g Rg ij π i π j − a √ gRV = 0 (37)where G ijkl G klmn = δ mi δ nj . This is the Hamiltonian constraint which arises from the globalreparametrization invariance of the action . The total Hamiltonian thus looks like H = Z d xN H + N k H k + u C (38) Whether or not the action is also locally reparametrization invariant, so that we would have a many fingeredtime as in general relativity, depends on whether the Hamiltonian constraint propagates or not. where H = 1 √ g aG ijkl π ij π kl + 1 √ g Rg ij π i π j − a √ gRV ≈ H j = ∇ i π ij + 12 π i F ij ≈ C = π ij g ij − ∂ k π k ≈ N , N k ,and u are Lagrangian multipliers.It would be interesting to see whether the Hamiltonian constraint, with suitable numer-ical values for the constants a, b, c, d, e and f , can be made to propagate. If the Hamiltonianconstraint does not propagate and more secondary constraints must be introduced, the the-ory could turn out to be inconsistent (e.g. we end up with more constraints than unknownsor that 1 = 0). Or perhaps the Hamiltonian constraint will perhaps not be first class. Orperhaps the constraints do close but not according to the Dirac-Teitelboim algebra so thata solution can not be embedded in a spacetime [24]. These are very important issues buta proper analysis will unfortunately involve lengthy calculations and will be postponed toa forthcoming paper. Our scale invariant theory contains more degrees of freedom than conventional geometrydynamics. In addition to the gravitational field g ij it also contains the vector field A k . Forthis theory to be empirically adequate it is imperative that the new degrees of freedom areactually represented in nature in some form or another. Given the mathematical similarityof the vector potential A k and the electromagnetic field it is tempting to try to identify A k with the electromagnetic field. However, there are good reasons for not giving in to thistemptation. First of all we have to note that this field couples to the gravitational fieldin a non-standard way. The vector potential in this theory is thoroughly intertwined withthe gravitational field through its presence in the connection and the curvature tensor.Secondly, since any coupling to matter fields has to be done so that conformal invariance ispreserved, the vector potential A k will couple universally to other fields (with the exceptionof fields which are already conformally covariant, e.g. the electromagnetic field). Thirdly,the electromagnetic field arise normally from gauging a global U (1) symmetry. However,our connection is not a U (1) connection. The Lie group manifold of U (1) has the topologyof a circle S where θ and θ + 2 π are identified. However, such an identification cannotbe maintained within the present theory. Indeed, e θ g ij = e θ +2 π g ij since the complex unit i is absent in the exponential. Thus the Lie group manifold is not S but R . These keymathematical differences indicate that we are not dealing with the electromagnetic fieldhere but rather a new type of universal field.A fourth reason, for believing that the vector potential must play a different role thanthe electromagnetic field, comes from cosmology. According to observations the universe iscontinuously expanding. At first this seems to contradict the very idea of scale invariance:if scale is nothing but an unphysical and unobservable gauge degree of freedom how is itpossible that we can determine from observations that the size of the universe is changing?This difficulty is only apparent. There is a fully scale invariant and relational way tounderstand the expansion of the universe. An apparent expansion of the whole universecan also be understood as the galaxies shrinking in size relative to the Hubble radius. Thisratio of galaxy sizes and the Hubble radius is a scale invariant quantity and does makesense within a scale invariant theory.Thus, if our theory can explain the expansion of the universe at all, it must be the casethat this vector potential A k acts effectively as a short-range universal “shrinking” force.By “short-range” we mean that the shrinking effect should be more pronounced on galacticscales (i.e. near mass concentrations) as compared to the Hubble scale. By “universal”we mean that the force should act on all material objects in the same way irrespective ofinternal composition. Only a universal shrinking force would create the appearance of anexpanding universe. As noted above, the vector field has indeed this universal character.However, the electromagnetic field is not a universal field in this sense since it would act Indeed, Weyl considered his theory as a unification of the gravitational and electromagnetic field [15]. on a positively charge body in a different way than on a negatively charged one, and notat all in the case of a neutral body. Thus, it seems clear that the vector potential shouldnot be identified with the electromagnetic field. A more likely scenario is for the vector potential to play the role of “dark matter”.Indeed, there are some indications that scale invariance is important for explaining thegalaxy rotation curves [20, 21]. It is also of key interest that the spectrum for themicrowave background is scale invariant for large wavelengths, i.e. at scales where themass density is roughly homogenous. Normally, this is explained by an inflationary typecosmological model but in the scale invariant theory presented here it seems natural toexpect the spectrum to be scale invariant at scales at which the mass density of the universeis approximately homogenuous. These issues should be explored further.Furthermore, if the connection A k is curved so that ∂ i A j − ∂ j A i = F ij = 0 then non-trivial global effects would be present. For example, consider two identical rulers at thesame point in space. Then move one of them around a closed spatial loop. If we have “scalecurvature” F ij = 0, then the two rulers might be of different size after this operation. Inaddition, one would also expect a shift in the spectral lines of hydrogen as discussed insection 3.3. From experience we know that these effects must be very small on humanscales but it remains a possibility that they would be observable on cosmological scales.As mentioned in section 3.3, one could speculate that the the tiny increase in the finestructure constant (a controversial claim) could be reinterpreted as a Weyl-type rescalingeffect.Finally we stress again that it is necessary that the notion of proper time can be recov-ered in some way from the theory. This would presumably happen is if the Hamiltonianconstraint could be made to propagate and that the constraints close according to theDirac-Teitelboim algebra. This would mean that the theory is foliation invariant wouldprobably ensure the emergence of the universal lightcone structure. We could then pre-sumably define a notion of proper time by aid of lightclocks (as sketched in section 3.2)and in this way we could potentially provide a satisfactory answer to Einstein’s objectionregarding proper time. Acknowledgements
This work was supported by FQXI and Perimeter Institute. I have benefitted greatly fromdiscussions with Julian Barbour, Sean Gryb, Lucien Hardy, Tim Koslowksi, SebastianoSonego, Rafael Sorkin, and Tom Zlosnik.
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