Breaking the Conformal Gauge by Fixing Time Protocols
BBreaking the Conformal Gaugeby Fixing Time Protocols
L. Fatibene , , S. Garruto , , M. Polistina Dipartimento di Matematica, Universit`a di Torino, Italy INFN Sezione Torino- Iniz. Spec. QGSKY Dipartimento di Fisica, Universit`a di Torino, Italy
October 8, 2018
Abstract
We review the definition by Perlick of standard clocks in a Weylgeometry and show how a congruence of clocks can be used to fix theconformal gauge in the EPS framework. Examples are discussed indetails.
In early ’70s Ehlers-Pirani-Schild (EPS) introduced an axiomatic approachto geometry of spacetime; see [1]. Instead of assuming a geometry on space-time, they assumed as primitive notion the families of the worldlines ofparticles and of light rays. They showed that requiring physically reason-able properties for these families one can define a conformal structure C anda projective structure P on the spacetime M .A conformal structure is an equivalence class of Lorentzian metrics; C =[ g ] is made of all the Lorentzian metrics ˜ g conformal to g , i.e. such thatthere exists a positive conformal factor Φ( x ) and ˜ g = Φ · g . By using aconformal structure C one can define light cones and timelike (spacelike orlightlike) directions. On the contrary, distances cannot be defined beingrather associated to a specific representative of g ∈ C .A projective structure is a class of (torsionless) connections which sharethe same autoparallel trajectories; P = [Γ] is made of all connections in theform ˜Γ αβµ = Γ αβµ − δ α ( β V µ ) for some covector V µ . A projective structure isassociated to a free fall; see [2].A projective structure P and a conformal structure C are EPS-compatible if lightlike geodesics of C are a subset of autoparallel trajectories of theprojective structure P . In this case one can always choose of representative1 a r X i v : . [ g r- q c ] O c t Γ for P in the form˜Γ αβµ = { g } αβµ − (cid:16) g α(cid:15) g βµ − δ α ( β δ (cid:15)µ ) (cid:17) A (cid:15) (1)for some covector A , a representative g of the conformal structure and where { g } denotes the Levi-Civita connection of the metric g . Then one also has ˜Γ ∇ λ g µν = − A λ g µν (2)The triple ( M, C , ˜Γ) is called a Weyl geometry where M is a manifoldof dimension m (hereafter m = 4), C is a conformal structure on M and ˜Γis a connection given by (1) which represents a EPS-compatible projectivestructure and free fall. As a matter of fact, the EPS analysis shows that aWeyl geometry is a more natural description of spacetime geometry whichcontain as a particular case the standard Lorentzian geometry (for A = 0)though it is more general.Recently Perlick defined standard clocks in Weyl geometries; see [3]. Heproved that on any trajectory one con always choose a parametrization(modulo affine transformations) which makes the clock standard. In section2 we shall briefly review Perlick’s proposal for standard clocks and reviewthe main properties for them.Then, in section 3, we shall show that any clock is standard for a suitablerepresentative of the conformal structure. Finally, in section 4, we shallexplore some consequences of different conformal fixings induced by differentclocks. In this section we will briefly review the definition of standard clock given byPerlick (see [3]) which is suitable for a Weyl geometry. A new definition isneeded because in a Weyl geometry all definitions are supposed to be local,hence the one usually given (see [4]) is not suitable in this context.
Definition 1: A clock is a timelike curve in M , i.e. a C ∞ map γ : I → M : s (cid:55)→ γ µ ( s ), for a real interval I ⊂ R , such that its tangent vector ˙ γ µ ( s ) istimelike for all s ∈ I .Two clocks γ µ and ˜ γ µ may share the same worldline (Im( γ ) ≡ Im(˜ γ ) ⊂ M ) and differ by the parametrization : this happens if there exists a (orien-tation preserving) diffeomorphism φ : R → R that map one into the other,i.e. if γ µ = ˜ γ µ ◦ φ . Two such clocks are said to be equivalent . We will alsoeventually refer to the whole class of equivalent curves [ γ µ ] as a particle andto the submanifold assicated in M by γ µ as worldline of that particle.2hen a clock γ µ is given is natural to define its 4-velocity and its 4-acceleration by: v µ ( s ) = ˙ γ µ ( s ) a µ (Γ) ( s ) = ¨ γ µ ( s ) + Γ µαβ ( γ ( s )) ˙ γ α ( s ) ˙ γ β ( s ) (3)Let us stress that while the 4-velocity depends on the curve only, the 4-acceleration depends on a connection as well. Hereafter, we shall often usethe 4-acceleration associated to the free fall connection ˜Γ, as well as the oneassociated to a representative g ∈ C , namely a µ ( g ) ( s ). Definition 2
A clock γ µ : I → M is said to be a standard clock (withrespect to a Connection Γ and a Lorentzian metric g ) if its 4-acceleration a µ (Γ) ( s ) is g -orthogonal to the 4-velocity v µ ( s ), so that: g ( a (Γ) , v ) = 0 (4)The parametrization of a standard clock is usually called a proper time .Perlick proved that given a Weyl geometry ( C , ˜Γ) and a conformal repre-sentative g ∈ C , then one can always find a parametrization for any particlewhich corresponds to proper time. Consider in fact two equivalent clocks γ µ ( s ) = ˜ γ µ ◦ φ ( s ) (cid:0) s (cid:48) = φ ( s ) (cid:1) (5)Their velocities and accelerations (with respect to any connection Γ) arerelated by v λ ( s ) = ˙ φ ( s ) ˜ v λ ( φ ( s )) a λ (Γ) ( s ) = ˙ φ ( s )˜ a λ (Γ) ( φ ( s )) + ¨ φ ( s )˜ v λ ( φ ( s )) (6)where the dots denote derivatives with respect to s (or s (cid:48) ). From here onwe shall understand evaluation at s or s (cid:48) when it is clear from context.Then ˜ γ µ is a standard clock iff we have:0 = g (˜ a (Γ) , ˜ v ) ⇒ ¨ φ = g ( a (Γ) , v ) g ( v, v ) ˙ φ (7)which has to be regarded as a differential equation for the unknown function φ ( s ). The solution is unique up to an affine transformation φ (cid:55)→ aφ + b withconstants a , b ∈ R .One can also show that this definition of standard clock is consistentwith standard General Relativity (GR). Let us consider a free falling clockin a general Weyl geometry ( M, [ g ] , ˜Γ); one has dds ( g ( v, v )) = ( ∂ λ g µν ) ˙ γ µ ˙ γ ν ˙ γ λ + 2¨ γ µ ˙ γ ν g µν ==( ˜Γ ∇ λ g µν + 2 g µρ ˜Γ ρνλ ) ˙ γ µ ˙ γ ν ˙ γ λ + 2¨ γ µ ˙ γ ν g µν ==2 (cid:16) g ( a (˜Γ) , v ) − A ( v ) g ( v, v ) (cid:17) (8)3ith A ( v ) = A µ v µ .In the Riemannian case, i.e. when ˜Γ = { g } , one has: dds ( g ( v, v )) = 2 g ( a ( g ) , v ) (9)which vanishes for standard clocks. Hence the length of the 4-velocity of astandard clock is constant, which is the definition of proper time in standardGR. Then one can use the affine freedom to fix | v | = − γ : s (cid:55)→ ( s, r , θ , ϕ ) is standard in aSchwarzschild metric though it is not free falling. Also in this case of coursethe tangent vector ˙ γ has constant length and the clock is parametrized byits proper time.We already showed in [5] that is we fix a representative g of a confor-mal structure C , then any congruence of timelike curves induces an EPS-compatible connection ˜Γ for which the timelike curves are autoparallel. Hereafter we shall show that given a congruence of clocks, there is a represen-tative of the conformal structure for which the clocks are standard. In otherwords, given a particle there are three relevant objects: a parametrizationto make the particle a clock, a representative g of the conformal structure,and a Weyl connection describing free fall. Given two of this objects thethird can determined so that the clock is standard with respect to g and ˜Γ.Let us consider first a clock γ ( s ) and a representative ˆ g ∈ C . In viewof equation (9), the clock γ is standard with respect to the metric ˆ g iff andonly if ˆ g ( v, v ) is constant along the curve γ , which in general is not the case.However, one can always choose a conformal factorΦ( γ ( s )) = − α ˆ g ( v, v ) ⇒ g = Φ ˆ g (10)so that g ( v, v ) = − α along the curve γ . Then for such a representative g ∈ C the clock γ is standard with respect to g . Let us stress that this resultis achieved without changing the clock parametrization.Of course, this procedure does not fix the representative g of the confor-mal factor since we have no information about the value of the conformalfactor outside the worldline of γ . However, we we have a congruence of clockswhich fills (an open set of) the spacetime M , then the conformal factor andthe metric g will be (locally) fixed uniquely (up to affine transformations)by the requirement of the clocks to be standard.Once again Weyl geometries show to be less strict than Lorentzian ge-ometries as shown in [5]. We have just shown that considered a representa-tive of the conformal structure g , a connection for free fall ˜Γ and a clock γ single clock in factdoes the trick of fixing the conformal factor, just because a single clock doesin fact allow to define a congruence of clocks. This is already contained inEPS and Perlick framework; see [1], [3].One of the EPS axioms prescribes that if one fixes a clock γ , then forany event x ∈ Im( γ ) there exists two neibourhoods U x ⊂ V x ⊂ M such thatfor any point e ∈ U x there exists two light rays r ± through e which intersectthe worldline of the clock γ at two points p ± ( e ) ∈ V x . Since p ± ∈ Im( γ )they correspond to two values of the parameter along the clock, say s ± ( e ).Following [3], one can define two (local) functions τ = ( s + + s − ) and ρ = ( s + − s − ) (which are smooth outside Im( γ )) which provide an operationaldefinition of time and distance (which of course depend on the clock). Inparticular one can define the hypersurfaces Σ t = { e ∈ U x : τ ( e ) = t } whichare isochronous hypersurfaces. If one has enough clocks γ i (one clock isenough when dim( M ) = 2) the functions ρ i singles out a point in each Σ t ,defining a worldline P which is timelike. Therefore a point p ∈ P belongs tosome Σ t and one can use t as a parameter along P to define a clock. In thisway one has a congruence of clocks conventionally defined out of the originalclock. Then by requiring that all these clocks are standard one can singleout a conventional breaking of the conformal gauge defining a representative g ∈ C which in turn defines distances and time lapses.This construction defines a local chart on spacetime and it is then associ-ated to a special class of observers. In the next section we shall discuss howone can compute (for the sake of simplicity in 2d special relativity) transitionfunctions between coordinates associated to this kind of observers. For simplicity let us hereafter restrict to 2-dimensional spacetimes. Let usconsider M = R ; let ( x, t ) be coordinates on M and let particles and lightrays be straight lines γ u : R → M : s (cid:55)→ ( us + x , s + t ), with − < u < u = ± x, t ) chosen M have no special meaning, they are just a mean to write parametrizationsof curves. Let us first fix a clock γ P : s (cid:55)→ ( x , t + s ) through the point P = ( x , t )which corresponds to the worldline x = x . For any event P = ( x , t ) thereare always exactly two light rays through P which intersect the worldline5f the clock at s ± ( P ) = − ( t − t ) ± | x − x | ⇒ (cid:40) τ = t − t ρ = | x − x | (11)Accordingly, if the clocks at rest are set to zero on an isochronous line τ = const , then they are synchronized. The two functions ( ρ, τ ) define acoordinate grid and thus defined Cartesian coordinates ( x, t ) as | x | = ρ and t = τ . These coordinates, which incidentally coincide with the originalcoordinates we chose on M , are centered at ( x , t ), and their origin can bemoved anywhere by a translation.Following EPS prescription an observer comoving with the clocks definesa function G P on M as G P ( P ) = − s + s − = − ( t − t ) + ( x − x ) (12)and defines the metric at P as the coefficient of the second order expansionof G P , namely G P ( P ) (cid:39) g µν ( P )( x µ − x µ )( x ν − x ν ) which corresponds tothe Minkowski metric g = η µν dx µ ⊗ dx ν = − dt + dx at the point P .Accordingly, choosing the congruence of clocks γ P as above correspondsto an inertial observer which sees the clocks at rest, which establishes stan-dard coordinates ( x, t ) and defines the metric as Minkowski anywhere. Allclocks γ P are standard with respect to η . For v = ˙ γ P one has g ( v, v ) = − Let us now consider another congruence of clocks γ (cid:48) P : s (cid:55)→ ( x + βαs, t + αs )for some velocity − < β < α ∈ R + . For any other event P one hastwo light rays through P hitting the clock at the parameter s ± = − ( t − t ) + β ( x − x ) ± | ( x − x ) − β ( t − t ) | α (1 − β ) (13)so that it defines the functions τ (cid:48) = ( t − t ) − β ( x − x ) α (1 − β ) ρ (cid:48) = | ( x − x ) − β ( t − t ) | α (1 − β ) (14)Accordingly, if the clocks are set to zero on an isochronous line τ (cid:48) = const ,then they are synchronized. The functions ( ρ (cid:48) , τ (cid:48) ) define a coordinate gridwhich is associated to Cartesian coordinates ( x (cid:48) , t (cid:48) ) and they are centeredon the clock starting event ( x , t ).Following EPS prescription the observer comoving with the clocks definesa function G (cid:48) P on M as G (cid:48) P ( P ) = − s + s − = − (( t − t ) − β ( x − x )) + (( x − x ) − β ( t − t )) α (1 − β ) (15)6nd defines the metric at P as the coefficient of the second order expansionof G (cid:48) P , namely G (cid:48) P ( P ) (cid:39) g (cid:48) µν ( P )( x µ − x µ )( x ν − x ν ) which corresponds tothe metric g (cid:48) = g (cid:48) µν dx µ ⊗ dx ν = − dt + dx α (1 − β ) (16)at the point P . The velocities of the new clocks are v (cid:48) = α ( ∂ t + β∂ x ) andone has g (cid:48) ( v (cid:48) , v (cid:48) ) = −
1, as for the rest clocks.By this choice one has coordinates in the form (cid:40) t (cid:48) = αα (1 − β ) ( t − βx ) x (cid:48) = αα (1 − β ) ( − βt + x ) (17)One can easily check that g (cid:48) = 1 α (1 − β ) ( − ( dt ) + ( dx ) ) = − ( dt (cid:48) ) + ( dx (cid:48) ) (18)so that also the new observer defines, in its coordinates ( x (cid:48) , t (cid:48) ), the Minkowskimetric.The phace α of the clock must be fixed so that g ( ˙ γ (cid:48) P , ˙ γ (cid:48) P ) = − α (1 − β ) = −
1, thus α = (1 − β ) − / . Then the transformation between the twoobservers is (cid:40) t (cid:48) = α ( t − βx ) x (cid:48) = α ( − βt + x ) α = 1 (cid:112) − β (19)which are known as Lorentz boosts. Accordingly, one can completely recoverspecial relativity from these assumptions. Let us now consider an apparently less trivial example; let us consider aclock at rest which is parametrized in a non-standard fashion (so that it isnot standard for Minkowski metric g ). Let us fix a congruence of clocks γ (cid:48)(cid:48) Q : ( − , + ∞ ) → M : σ (cid:55)→ ( x , log(1 + σ )) (20)and fix a value σ of the parameter so that 1 + σ = e t . One can use anaffine reparametrization to define a clock starting at ( x , t ), namely γ (cid:48)(cid:48) P : s (cid:55)→ ( x , log(1 + σ + s )) (21)Being this a reparametrization of the clock at rest we know in generalthat the observer associated to this clock will define a metric g which isconformal to the metric associated to the clock at rest (i.e. the Minkowskimetric g = − dt + dx ). In other words we expect g (cid:48)(cid:48) = Φ · g . By consideringlight rays through an event P one has intersections with the clock at s ± = e t ±| x − x | − e t (22)7nd define the two functions τ (cid:48)(cid:48) = e t ch( x − x ) − e t ρ (cid:48)(cid:48) = e t sh | x − x | (23)Notice that this time the coordinate grid is curvilinear (i.e. the lines of thegrid are not geodesics of g ).For the metric at P we obtain the function G (cid:48)(cid:48) P ( P ) = − (cid:16) e t + | x − x | − e t (cid:17) (cid:16) e t −| x − x | − e t (cid:17) (24)Expanding it to second order one gets the metric g (cid:48)(cid:48) = e t (cid:0) − dt + dx (cid:1) (25)which is as expected conformal to the Minkowski metric g . The clocks arestandard with respect to the metric g (cid:48)(cid:48) . Thus as a first result one seesexplicitly that the choice of the congruence of clocks does in fact fixes theconformal gauge.The conformal factor is clearly non-constant. However, one can find re-gions in which the conformal factor is approximately constant. For example,around an event at t = 10, in the region t ∈ [9 . , . (cid:40) t (cid:48) = e t ch( x ) − x (cid:48) = e t sh( x ) (26)In these new coordinates the metric reads as g (cid:48)(cid:48) = − dt (cid:48) + dx (cid:48) (27)In fact one can easily check that the metric (25) was itself flat, beside beingconformal to (another) flat metric g . These coordinates do not cover thewhole Minkowski, due to the singularities of the clocks. In fact what wehave is two different Minkowski spaces and a region in the first Minkowski isconformally mapped into a region of the other. The map does not preservethe metric, but being the image metric flat it can be written as Minkowskiin another coordinate system.An observer living in the new copy of Minkowski can use coordinates (26)and see a spacetime which is exacly Minkowski in a Cartesian coordinatesystem. Such an observer is entitled as the original one to define specialrelativity. However, both observer are looking at the same spacetime and inthat spacetime free falling particles either falls along geodesics of g or alonggeodesics of g (cid:48)(cid:48) . 8igure 1: Grid for coordinates ( x (cid:48) , t (cid:48) ) with a free falling particles passing through P = ( x = 0 . , t = 0 . with respect to original coordinates ( x, t ) . If we decide they fall along geodesics of gP : s (cid:55)→ ( x + us, t + s ) (28)for some − < u < x, t ) then such worldlines will becurved (i.e. non-geodesics) for g (cid:48)(cid:48) and they read in the observer coordinates( x (cid:48) , t (cid:48) ) as P : s (cid:55)→ ( e t + s sh( x + us ) , e t + s ch( x + us ) −
1) (29)This of course is not a geodesics of g (cid:48)(cid:48) and the observer would see itaccelerating as soon as it exit the safe region in which the conformal factoris approximately constant.Figure 2: Free falling particles passing through P = ( x = 0 . , t = 0 . (cid:39) ( x (cid:48) =0 . , t (cid:48) = 0 . with respect to coordinates ( x (cid:48) , t (cid:48) ) The observer would judge that forces are acting on the particle to de-flecting it. It would see the force acting universally on any particle in thesame way, so it would probably judge the force to be gravitational. How-ever, in its spacetime there is no gravitational source (since we started offfrom Minkowski spacetime). Accordingly, the observer would judge thatits spacetime is filled with some mysterious dark source which affects themotion of objects at non-local scales while it compensates at local scales.9his is a toy model, and of course we are not claiming that it is arealistic explanation of dark sources. Still it is based on a somehow simpleassumption (atomic clocks are not standard on long times) which we thinkshould not be dismissed on an a priori basis but should be tested and provenfalse.
In the previous examples we chose non-standard clocks which eventuallyturned out to be standard for another Minkowski metric. However, thatwas due to the specific form of clocks we considered. Essentially, any otherchoice gives a new metric which is not flat anymore.Let us now try and fix a congruence of non-standard clocks: γ (cid:48)(cid:48)(cid:48) Q : [0 , + ∞ ) → M : σ (cid:55)→ ( x , σ ) (30)and fix a value σ of the parameter so that σ = √ t . One can use an affinereparametrization to define a clock starting at ( x , t ), namely γ (cid:48)(cid:48)(cid:48) P : s (cid:55)→ ( x , ( σ + s ) ) (31)Being this a reparametrization of the clock at rest we know in generalthat the observer associated to this clock will define a metric g (cid:48)(cid:48)(cid:48) which isconformal to the metric associated to the clock at rest (i.e. the Minkowskimetric g = − dt + dx ). In other words we expect g (cid:48)(cid:48)(cid:48) = Φ · g . By consideringlight rays through an event P one has intersections with the clock at s ± = (cid:112) t ± | x − x | − √ t (32)and define the two functions τ (cid:48)(cid:48)(cid:48) = (cid:16)(cid:112) t + | x − x | + (cid:112) t − | x − x | (cid:17) − √ t ρ (cid:48)(cid:48)(cid:48) = (cid:16)(cid:112) t + | x − x | − (cid:112) t − | x − x | (cid:17) (33)Notice that again the coordinate grid is curvilinear.For the metric at P we obtain the function G (cid:48)(cid:48)(cid:48) P ( P ) = − (cid:16)(cid:112) t + | x − x | − √ t (cid:17) (cid:16)(cid:112) t − | x − x | − √ t (cid:17) (34)Expanding it to second order one gets the metric g (cid:48)(cid:48)(cid:48) = 14 t (cid:0) − dt + dx (cid:1) (35)which is as expected conformal to the Minkowski metric, though this time g (cid:48)(cid:48)(cid:48) is not flat. The clocks are standard with respect to the metric g (cid:48)(cid:48)(cid:48) . Againthe choice of the congruence of clocks does in fact fixes the conformal gauge.10he conformal factor is clearly non-constant. However, one can find re-gions in which the conformal factor is approximately constant. For example,around an event at t = 1, in the region t ∈ [0 . , .
1] the conformal factordoes change by less than 1%.An observer living in that region with instruments with an precision of1% does not appreciate any curvature of spacetime. Let us also stress thatthe absolute value of the conformal factor would be irrelevant. The observerin fact would define its unit so that its metric is exactly Minkowskian attime t = 1 and sees no deviation within the region where the conformalfactor is approximately constant.Only the change of the conformal factor is relevant and one does notneed to justify why conformal factor today is approximately 1. We definedunit 3 century ago and we are fine until our instruments cannot appreciatea change in the conformal factor at the scales of centuries.An observer which is comoving with the congruence of clocks does alsodefine new coordinates (cid:40) t (cid:48) = (cid:0) √ t + x + √ t − x (cid:1) x (cid:48) = (cid:0) √ t + x − √ t − x (cid:1) (36)In these new coordinates the metric reads as g (cid:48)(cid:48)(cid:48) = 2 t (cid:48) − x (cid:48) t (cid:48) + x (cid:48) (cid:0) − dt (cid:48) + dx (cid:48) (cid:1) (37)These coordinates do not cover the whole Minkowski, due to the singu-larities of the clocks.Figure 3: Grid for coordinates ( x (cid:48) , t (cid:48) ) with a free falling particles passing through P = ( x = 0 . , t = 0 . with respect to original coordinates ( x, t ) . However, an observer living in the covered region, quite far away from theboundary, can use them and if it lives in a small region where the conformalfactor is approximately constant consider them as approximately Cartesian.However, if the observer believes that its clock is standard with respect toMinkowski metric it would regard its coordinates as being exactly Cartesianand free falling particles to follow straight lines in spacetime. In fact freefalling particle would fall the line P : s (cid:55)→ ( x + us, t + s ) (38)11or some − < u < x, t ) which read in the observercoordinates ( x (cid:48) , t (cid:48) ) as P : s (cid:55)→ (cid:32) (cid:0) √ t + s + x + us − √ t + s − x − us (cid:1) (cid:0) √ t + s + x + us + √ t + s − x − us (cid:1)(cid:33) (39)This of course is not a geodesics of g (cid:48)(cid:48)(cid:48) and the observer would see it accel-erating as soon as it exit the safe region in which the conformal factor isapproximately constant.Figure 4: Free falling particles passing through P = ( x = 0 . , t = 0 . (cid:39) ( x (cid:48) =0 . , t (cid:48) = 0 . with respect to coordinates ( x (cid:48) , t (cid:48) ) Here we showed how one can fix the conformal gauge by considering a clock(or a congruence of clocks). We also investigate the effects of erroneouslyregarding non-standard clocks for being standard. Here we consider theeffect at a kinematical level as already back to EPS it was noted that (see[1]):
This last step [imposing the Riemannian axiom] seems unavoidable onempirical grounds if equality of gravitational time (as given by Weylarc length and measurable, for example, by the method of Kundt andHoffmann) and atomic time is assumed. This is because the latter istransported in an integrable fashion, which was pointed out by Einsteinin his criticism of Weyl’s theory and supported by (among other things)the consistency of the interpretations of observed red-shifts. (But howcompelling is time-equality postulate?)
We agree that there is no fundamental or real reason to accept a priori that atomic clocks are standard and that this is something to be experimen-tally tested.We also have to notice that there are dynamics in which such effectsnaturally arise and are dynamically controlled. Let us consider a Palatini12 ( R )-theory of gravitation (see [6]) in which one has two metrics, an originalmetric g and a conformal metric ˜ g by a conformal factor f (cid:48) ( R ) which on-shell can be seen to be a function of the trace of matter stress tensor. Forexample in cosmology the conformal factor is expected to be a function ofcosmological time.When we provide an operational definition for atomic clocks we of coursecannot assume to be standard with respect to g or ˜ g , especially to theprecision needed to guarantee it to be standard up to cosmological scales,i.e. for time scales up to billion years, nor we are able to test it at these scales.Thus we can speculate about what would happen if the atomic clocks were not standard with respect to g or ˜ g .If atomic clocks were standard with respect to ˜ g then we could performa Legendre transformation and obtain standard GR (as already noticed inEPS analysis; see also [9]). However, if atomic clocks were not standardwith respect to ˜ g then we showed that it would be standard with respectto a suitable metric g conformal to ˜ g . This would allow to extend EPSinterpretation to the metric sector, assuming that atomic clocks (and thenrulers and scales; see [7]) are standard with respect to the original g . Thiswill eventually lead (at least) to a wrong estimation of cosmological distanceswith respect to standard GR cosmology. This effect being in addition to theeffective sources which are typical of extended theories of gravitation whichare used to model dark sources; see [8]. References [1] J. Ehlers, F.A.E. Pirani and A. Schild, in:
General Relativity, Papersin Honor of J.L. Synge , edited by ORaifeartaigh, Oxford UniversityPress, Oxford, (1972).[2] L. Fatibene, M. Francaviglia, G. Magnano,
On a Characterization ofGeodesic Trajectories and Gravitational Motions , Int. J. Geom. Meth.Mod. Phys.; arXiv:1106.2221v2 [gr-qc][3] V.Perlick,
Characterization of Standard Clocks by Means of Light Raysand Freely Falling Particles , Gen. Rel. Grav. 19(11), (1987) 1059-1073[4] R. F. Marzke and J. A. Wheeler.
Gravitation as geometry. I: The geom-etry of space-time and the geometro-dynamical standard meter , Gravi-tation and Relativity, Eds. H.-Y. Chiu and W.F. Hoffmann, W.A. Ben-jamin, New York, page 40, 1964.[5] L. Fatibene, M. Francaviglia,
Weyl Geometries and TimelikeGeodesics , Int. J. Geom. Methods Mod. Phys. 9(5) (2012) 1220006;arXiv:1106.1961v1 [gr-qc][6] L. Fatibene, M. Ferraris, M. Francaviglia, G. Magnano,
Extended The-ories of Gravitation Structure of Spacetime and Fundamental Princi- les of Physics, following Ehlers-Pirani-Schild Framework , EPJ Webof Conferences Volume 58, 2013 TM 2012 The Time Machine Factory[unspeakable, speakable] on Time Travel in Turin ;L. Fatibene, M. Ferraris, M. Francaviglia, G. Magnano, Extended Theo-ries of Gravitation Observation Protocols and Experimental Tests
EPJWeb of Conferences Volume 58, 2013 TM 2012 The Time MachineFactory [unspeakable, speakable] on Time Travel in Turin[7] L. Fatibene, M.Francaviglia,
Mathematical Equivalence versus PhysicalEquivalence between Extended Theories of Gravitation . Int. J. Geom.Meth. Mod. Phys., 11(01) (2013), 1450008; arXiv:1302.2938 [gr-qc][8] S.Capozziello, M.F.De Laurentis, M.Francaviglia, S.Mercadante,
FromDark Energy and Dark Matter to Dark Metric , Foundations of Physics39 (2009) 1161-1176 gr-qc/0805.3642v4;S. Capozziello, M. De Laurentis, M. Francaviglia, S. Mercadante,
FirstOrder Extended Gravity and the Dark Side of the Universe II: MatchingObservational Data , Proceedings of the Conference Univers Invisibile,Paris June 29 July 3, 2009 to appear in 2012[9] L.Fatibene, S.Garruto,