SOME COSMOLOGICAL MODELS T T S A S M

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1 Mogens True Wegener SOME COSMOLOGICAL MODELS T T S A S M HEIR IME CALES ND PACE ETRICS Revised version (2015) of a paper to PIRT 8, Imperial College, London, 2002 SUMMARY Provisionally accepting the standard Robertson-Walker Metric ( RWM) of cosmology, e recall that the principle of cosmic isotropy can be used as an argument for the definability of an all-embracing universal time, at least statistically, e propose to reverse this procedure by postulating such time as a regulative idea in the sense of Kant. Using RWM as our formal point of departure, e then investigate the properties of to standard models of modern cosmology: 1) the uniform expansion model of Milne & Prokhovnik, the simplest model of a cosmic big bang, and 2) the exponential expansion model of Bondi & Gold, (supposed to be) the simplest model of a cosmic steady state. Finally e derive the metric of a ne steady state model hich does not conform to the RWM. 1. INTRODUCTION 2. A Classical Alternative to SR 3. The Robertson-Walker Metric 4. Milne's Simple Big Bang Model 5. The First Steady State Model 6. A Ne Steady State Model. CONCLUSION

2 -2-1. INTRODUCTION We provisionally accept the Robertson-Walker Metric ( RWM) of modern cosmology as defining an universal substratum of observer-particles, or monads, subject to cosmic isotropy. Recalling the fact that the principle of cosmic isotropy is here used as an argument for the definability of an all-embracing universal time, at least statistically, e propose to reverse this procedure by postulating such time as a regulative idea in the sense of Kant. Folloing Milne & Walker, to kinds of monads are distinguishable: fundamental ones, defining the geometrical structure of the specific cosmological model under consideration by constituting its substratum, and accidental ones hich are superposed on the substratum in a ay that refers the description of their motion to the substratum as a universal frame of rest. Their difference, naturally, is statistical and a matter of degree. So it is possible to make sense of a graduation of clocks according to their approximation to the ideal of an universal time. Using RWM as a formal point of departure, e investigate the properties of to standard models of modern cosmology:!) the uniform expansion model of Milne & Prokhovnik, hich is the simplest model of a cosmic big bang, and ) the exponential expansion model of Bondi & Gold, supposed to be the simplest model of a cosmic steady state. Rejecting the so-called perfect cosmological principle of the latter, a different approch is suggested. In agreement ith our tentative acceptance of RWM e consider the relationship beteen our choice of time scale for a certain model of the universe and its corresponding space metric. As it turns out, there are at least to important ays of mapping the expansion of the universe: + ) one hich keeps atomic sizes constant hile light is being stretched, and,) one hich keeps distances beteen fundamental observers constant hile their atoms are shrinking. Finally a ne steady state model of the universe is proposed hich deviates from RWM by alloing atoms to be contracted due to universal dispersion. In this model, spatial curvature is apparently increasing ith the distance at hich an object is seen by a fundamental observer. The model implies orld map and orld vie to be identical as regards their formal structure. Taken directly from. œ.>.= œ, it is even simpler than the model of Bondi-Gold. The basic properties of this model and related ones are examined and discussed. Mogens True Wegener

3 -3-2. A CLASSICAL ALTERNATIVE TO SR The hyperbolic formulae corresponding to the standard addition!! = are: -9=2! œ -9=2!-9=2= =382! =382= =382! œ =382! -9=2= -9=2! =382= Using - >+82=, the LT of SR ith temporal coordinates become: > œ > -9=2= B =382=. B œ B -9=2= > =382= It is interesting that the LT are derivable from these addition formulae if and only if in: LT W W >Î -9=2!. >Î -9=2!. BÎ =382!. BÎ =382! It is natural to identify ith the SR invariant: g > B C D œ > B C D. The standard (1x3) for three inertial frames, O, &, in relative motion are: (1 + ) \ B -9=2! > =382! œ B -9=2! > =382!. ] C C (1,) X > -9=2! B =382! œ > -9=2! B =382!. ^ D D Consider O to be a preferred frame ith the privileged observer S situated in its origo. Let the observers S& S be situated in the origos of W& W, resp., and let the frame times > & > of W& W be synchronized to the proper time X of S by choosing X > >! hen S& S both coincide ith S. Suppose that an event I occurs at particle T, as observed by S, S& S. Let the standard coordinates of I be ÐXß\ß]ß^ in O, Ð>ßBßCßDin W and Ð>ßBßCßD in W Then, by eliminating the irrelevant frame times > & > from the expressions for X &\, e get: \œbî-9=2! X>+82! œbî-9=2! X>+82! LT Further, using = œ!! œ = to eliminate! or!, e recover for the privileged time X: (2 + ) B œ ÖB -9=2Ð=! X =382 = Î-9=2! (2,) B œ ÖB -9=2Ð=! X =382 = Î-9=2! Finally, introducing non-standard frame-times - & - for frames W& W defined by means of : (3) - - XÎ. XÎ -9=2! ß -9=2! and using A >+82=, e find the Tangherlini transformations ( TT) as generalized by Selleri: -9=2Ð=! (4 + ) B œ B - B Ð A@ A -9=2! =382= œ - È A -9=2Ð=! (4,) B œ B - B Ð A -9=2! =382= œ - È A In standard SR, it is alays the proper time of a single moving clock hich is said to be retarded relative to the slave clocks distributed as a netork over the rest frame of the observer; but if e refer the inertial motion of particles to a privileged frame e should use TT instead. Please notice that TT reduce to GT if all observations refer to the frame of the miday particle: = (5)! œ Ê B B œ - = =382= œ X =382 Applying - - = = > B>+82, > B>+82 directly to LT, e get the same result, viz. GT : (6) - - = = - œ ß œ ß B œb =382= ß C œcß D œd.relativity.me

4 -4-3. THE ROBERTSON-WALKER METRIC In his monumental Natural Philosophy of Time (1961/1980), G.J. Whitro sketched a method to derive the RWM of relativistic standard cosmology from the -factor of SR. Let: - 9 Þ -> 9 9 < 9 and let the origo of the comoving standard rest frame W of an observer T be T himself. No suppose an event I, taking place at some object S, to be triggered by a light signal hich instantaneously released a visible flash. Suppose further that this light signal as sent off by T at the instant, and that the flash as perceived by T at the instant $, both & $ being instants of proper time of T as read off his on standard atomic clock G. We then recover the Einstein coordinates of the Cartesian frame W of T by means of the usual definitions: $ > <. $ > < > <. > < From the standard SR invariant. $. e get the -factor for the retardation of moving clocks:. g. $. œ.>.<.> Whitro no suggested:.< Ä fðx. 5. The parameter X of his function fðx is then no longer the private frame time >, but rather the public proper time of fundamental observers, i.e. all observers at rest in the universe, e.g. relative to the cosmic background radiation ( CBR). With X, the standard invariant of SR is finally transformed into the standard RWM metric:. g œ.>.< œ. f Ð. 5 Here f is taken to be the universal scale factor and 5 a fixed (comoving) coordinate. No, for fundamental observers,. 5 œ!, i.e.. g œ., shoing that all fundamental observers are in pace ith the same common cosmic time g. By implication, any deviation of proper time from g is restricted to non-fundamental or accidental observers distinguished by a variable 5. Considering g Á Á >, one may ask if all this amounts to more than a vague analogy. According to the standard vie, it is alays the proper time of a moving particle hich is claimed to be slo relative to the frame time of a stationary observer. So coordinate time, or frame time, is thereby tacitly assumed to represent the true extended time of any observer. The cosmic time g implied by RWM is seldom taken seriously, but mostly ignored or explained aay as being of statistical origin and thus ill defined. Nevertheless, it is the firm stance of the present riter that a fundamental importance should be ascribed to the cosmic time g. If e define true time by the readings of the master clocks of our fundamental observers hen they have been properly synchronized - e.g. by letting a definite non-local cosmic event, such as the beginning of everything in a socalled big bang, represent a common time zero - then it is no longer true to say that the master clock of a fundamental observer is slo relative to the frame clocks of another observer, fundamental or not. Much rather it is true to say that it is the clocks of accidental particles that are slo relative to the clocks of fundamental observers. But the only conflict at stake here is one relating to the standard interpretation of SR. Hence, if the clocks of fundamental observers sho the true time g, then the clock of an accidental particle ill be more or less slo. In fact, the greater its distance to that fundamental particle relative to hich it is momentarily at rest, and hich thus constitutes the natural origo of its on rest frame, the sloer its clock ill run and the more it ill deviate from true time. Mogens True Wegener

5 -5- The natural ay of interpreting this retardation of moving clocks is as an effect of gravitation. In this ay e have found a simple coupling beteen the rates of non-fundamental clocks and hat seems to be a gravitational potential due to the substratum of fundamental particles. The reason for this dependence is that the deviation of the clock of an accidental particle from true time g is found by direct comparison ith the clock of that fundamental particle ith hich it momentarily coincides; and the greater the distance of an accidental particle is to the origo of that rest frame to hich it belongs, the faster its speed relative to that fundamental particle ith hich it coincides ill appear; this follos from the expansion function f Ð. What e have disclosed is the possibility of an influence of the substratum on particles hich do not belong to the substratum and hich represent deviations from cosmic symmetry. This supports a conjecture of Whitro's former master, E.A. Milne. The essential point of his Kinematic Relativity, devised as an alternative to Einstein's theories, SR & GR, is precisely that hat e call gravitational effects may be due to local deviations from cosmic symmetry. Indeed, if elevated to a universal principle, Milne's conjecture amounts to nothing less than an inversion of Mach's principle: hile Mach meant that inertia should be reduced to gravitation, Milne insisted that gravitation should be reduced to inertia - and demonstrated ho to do it! All this is but a repetition of ideas presented earlier. With polar coordinates the RWM becomes: (). g œ. f Ð {. 3 - Ð. ) =38 ). 9 } (8) e fð3 ß T '. ÎfÐ -98=>Þ ß VÐT dtî. ÎfÐ Here e is proper distance, V is an inverse scale function, and T is an auxiliary time parameter..- É, œ!. 3.- Î È,- œ.+<-=38- É, œ. +<=382- É, œ. g œ. f ÐÖ. 3 3 Ð. ) =38 ). 9,=!. g œ. f ÐÖ. 3 =38 3 Ð. ) =38 ). 9,=. g œ. f ÐÖ. 3 =382 3 Ð. ) =38 ). 9,=. 4 É, œ!. 3.- Î È,-. 4 ÎÐ,4 Î% œ. +<->+8Ð4 Î É, œ. +<>+82Ð4 Î É, œ. 4 4 Ð. ) =38 ) (. ',4 Î%,4 Î%. 3 - Ð. ) =38 ). 9 The folloing versions comprise all possible values of the constant of curvature,,: (9 + ). g œ. f Ð {.- ÎÐ,- - Ð. ) =38 ). 9 } (9,) œ. ÐÖ. 4 4 Ð. ) =38 ). 9 4 f ÎÐ, (9-) œ V ÐTÒ dt Ö. 0.(.' ÎÐ, Ó Applying the T-scale, the expansion of cosmos is explained aay as a shrinking of atoms! In standard presentations, considerable relevance is often ascribed to the Hubble function [:. [ Ð fð/ fð 4 % %.relativity.me

6 -6-4. MILNE'S SIMPLE BIG BANG MODEL In hat follos e thro light on the RWM by discussing some simple orld models. One of the simplest is Milne's model of uniform expansion, adopted by Prokhovnik and others: % ÐT Î 9 9 (10) fð Î 9. Î dt / V ÐT. [ Ð fð/ fð º Î Let us assume that radar signals are being exchanged beteen a pair of observers, T & U, : in Ð x time-space. Suppose that a photon 9 is emitted from T at œ, reflected by U at ; : œ, and received by T at œ $. Then, according to the relativity principle as interpreted : ; ; : by Milne, $ is the same function of as is of - call it =Ð / 5. Generalizing, and introducing Einsteinian standard coordinates > & < for T (priming those of U), e at once get: > Ð$ Ì< Ð$ 5-5 $ œ / œ > <. œ / œ > < (11) > œ -9=25. < œ =3825 Let us next assume that 5 is not a constant, but a variable; then, by differentiation:.> œ. -9=25. 5=3825.< œ. = =25 ÐT 9Î. g.>.< œ.. 5 œ/ 9 ÐdT. 5 This invariant is easily expanded into a hyperbolic time-space of Ð x $ dimensions if e put: =382 3 Ð. ) =38 ). 9 Ö. 0.(.' ÎÐ The standard SR invariant is thus transmuted into the hyperbolic metric of an expanding universe ith expansion function fð, hich can be transformed into another hyperbolic metric, viz. that of a stationary universe hose atoms all contract in accordance ith the Hubble ÐT 9Î9 function V ÐT /, here T œ {1 691ÐÎ }, being a constant of calibration: (12 + ). g œ.>.< < Ð. ) =38 ). 9 (12,) œ. Ö. 3 =382 3 Ð. ) =38 ). 9 Ð T (12-) œ / Ò dt Ö. 0.(.' ÎÐ Ó 9 % The 1st of these metrics, incorporating the universal constancy of the velocity of light, yields an infinity of private time-spaces, comprising the flat 3-spaces of fundamental observers. The folloing to metrics both yield a public time-space, each containing a curved 3-space: that of the 2nd metric being associated ith - time, relative to hich atoms keep a constant size hile the distances beteen fundamental observers steadily expand in proportion to f, ith the consequence that light is stretched, as suggested by Prokhovnik, and that of the 3rd metric being associated ith T- time, relative to hich distances beteen fundamental observers remain invariant hereas the sizes of their atomic constituents are contracting in proportion to T V œ /, due to a secular reduction of the velocity of light, as explained by Whitro. 9 = 4 4 % Mogens True Wegener

7 -- 5. THE FIRST STEADY STATE MODEL Passing from Milne's orld model to that of Gold & Bondi - and of Hoyle - the scale / factor f Ð is changed from Î to / / 9, characterizing all steady state models. So: 9 (13) fð /. Î dt T V ÐT. [ Ð fð/ fð º const. e / 3 >+82< is a candidate to the proper distance beteen fundamental particles, > just as / ÎÈ e œ -9=2< is a plausible relationship of frame time > to proper time. Hence, if Bondi & Gold, and Hoyle, ant to retain. /. 3 as a fundamental invariant of > their model, in face of the definitions / -9=2< & / 3 >+82<, they have to accept: (14). œ.> >+82<.< Ì /. 3 œ.< >+82<.>. g -9=2 < œ. /. 3 œ Ð T Generalizing these to (1x3) dimensional time-space e find the folloing three metrics, of hich the first one is closest to represent the private 3-spaces of the standard frames of SR, hereas the second comprises the public flat 3-space of a universe expanding ith fð œ / and the third encompasses the public flat 3-space of atoms shrinking in step ith VÐT œ T:.>.< dt. 3-9=2 < (15 + ). g œ Ò.> Ö.< =382 <Ð. ) =38 ). 9 Ó (15,) œ. / Ö. 3 3 Ð. ) =38 ). 9 œ 1 Ð T (15-) œ ÒdT Ö. 3 3 Ð. ) =38 ). 9 Ó. g œ Ò.> Ö.< =382 <Ð. ) =38 ). 9 ÓÎ-9=2 < does not compete ell ith the standard invariant of SR hich is the much simpler one. g œ.>.< <Ð. ) =38 ). 9. This rather serious problem is caused by the external factor -9=2 < hich is much less akin to the LT of SR than to the Voigt transformations ( VT) of some competing aether theory. Maybe, hen evidence accumulates, e shall have to recur to the ether hypothesis again. Hoever, since neither LT nor SR have yet been finally disproved, and since the steady state assumption does not necessarily exclude the conjecture that the universe is of finite age and originated in a big bang, it is orth hile to search for alternative steady state models hich are not at variance ith. g œ.>.<. As a step on the ay I shall propose the model belo hich follos from the first line by means of the definitions stated first in this section: (16 + ). g œ.> Ö.< =382 <Ð. ) =38 ). 9 / 3 (16,) œ Ò. / Ö. 3 3 Ð. ) =38 ). 9 Ó (16-) œ ( ÒdT Ö. Ð. =38. Ó T 3 3 ) ) /( T This, at the very least, is compatible ith the standard invariant. g œ.>.< of SR. > Hoever, hen interpreted by means of / -9=2<, / 3 >+82<, the eq.(16 b) is flaed: is not a genuine cosmic time, since the external contraction factor / also applies to 3. Þ This clearly shos that the metric (16 b) does not conform to the general RWM..relativity.me

8 -8-6. A NEW STEADY STATE MODEL Let us make a fresh start by adopting. g œ.>.< of SR. But as e need not accept that standard frames are flat, it seems e are free to assume that the orld is better described, hen referring to frame time >, by taking the geometry of frames, or 3-spaces, to be hyperbolic ( 2015: I have lately realized that my assumption of hyperbolic 3-space should be retracted): (1). g.> Ö.< =382 < Ð. ) =38 ). 9 No the shortcomings alluded to in 5 can be remedied by adopting the folloing definitions: > t < (18) 3 =382</ >+82</ >+82 / ÒÓ V / Interpreting V as tice the distance to the miday particle beteen to fundamental observers, < < e shall choose V >+82 Ä rather than V >+82 Ä, deleting the factor ÒÓ above. From the above definitions e immediately derive the folloing relationships: > (19) /. 3 œ.<-9=2<.>=382<œð.<. t >+82<-9=2 <œ.<. =382< < < From these e further obtain. œ.>.< >+82 œ dt Ð >+82 <.< >+82 < >+82 ; then, for fundamental observers,. 3 œ 0, e >+82<, A.<Î. =382<, hence: È >+82 < (20). œ.>.< >+82 < œ.>î-9=2 < œ.>î < > t Using 3 =382</ >+82</ ith our metric (1), e obtain the steady-state like metric: t (21). g œ Òdt / Ö. 3 3 Ð. ) =38 ). 9 Ó / t 3 This metric is identical to (16,) of the preceding section, only ith the symbol replaced by t; the reason for this shift is that e ant to retain for the proper time of fundamental particles. t Using / dt Î. T ith our metric Ð, e get one for a static orld of shrinking atoms: T g 3 3 ) ) 9 ( T 3 (22). œ ÒdT Ö. Ð. =38. Ó This metric is identical to (16-) of the preceding section, ithout any further proviso. The SR -like metric. g œ.>.< =382 <Ð.. is thus changed into to other metrics: the 2nd depicting a universe in exponential expansion, and the 3rd depicting the same universe as stationary, but ith shrinking atoms. Hoever, none of the metrics is conformal to RWM, since neither t in (21), nor T in (22), yields a cosmic time. So they are irrelevant to our purpose. t < No, using / œ >+82 œ V œ / 3 - cf. (18) - ith our metric (21), e get: % %. 3 Ö %/ 3 / 3. Ð / 3 Ð / 3 (23). g œ. / Ö. 3 3 Ð. ) =38 ). 9 % % Applying this metric to fundamental particles e notice that, for.) œ. 9 œ!, eq. (23) is reduced to. g œ., both a) in the case. 3 œ!. or.v/. º V, folloing a single fundamental particle on its course outards in the line of sight, and b ) in the case.v œ!, or. 3 œ 3., folloing a series of fundamental particles passing an imaginary border at distance Vœconst. Thus e have tested our basic assumption, viz. that the master clocks of fundamental particles alays keep the same cosmic rhythm. g œ. œ invar. Although eq. (23) does not conform to the standard RWM, e shall hence claim that g in (23) is a genuine cosmic time. Folloing Selleri [199f.], e finally suggest that the approximate inertial motion of an accidental particle S, as observed by to fundamental observers J & J, should be described in accordance ith the generalized inertial transformations sketched above, cf. 2, p.5. Mogens True Wegener

9 -9-. CONCLUSION We claim the metric for all observers, fundamental as accidental, to be reducible to:. g œ.>.< for.) œ. 9 œ! The safest ay to ensure that our claim is fulfilled is, of course, to start ith the SR invariant. As pointed out, this timespace is private as a consequence of the retardation of clocks and the contraction of rods or - ith a single expression - the relativization of our metrical units. The SR-like metric in itself says nothing about cosmic expansion or atomic contraction. In spite of Whitro's claim to have derived RWM from the relativistic -factor hich bears an affinity to an SR-like metric, it is hard to see more than a mere analogy beteen these to: (24). g œ.> œ.>.< (25). g œ. f Ð. 5 œ V ÐTÖdT. 5 Further, if the former applies to accidental and fundamental observers ithout distinction hile the latter represents the structure of an expanding substratum of fundamental observers, e shall obviously need some rules of interpretation hich can take us from the former to the latter by explicitly narroing the perspective, thus giving privilege to fundamental observers. Our ne Steady State model of Continued Creation describes the universe as a spatial totality unfolding in a common orld time g, thereby invoking the idea of a public timespace. What is ne in comparison ith the old model is that this ne one conforms to the no-horizon postulate of Milne; thus it survives the number-redshift statistics hich refuted the old one. A very simple alternative construction of our ne CC-model ignoring important details, since the metric is only valid for fundmental particles, might run as follos, taking c = unityß using > Ð> > $ & < Ð> > $ for frame-time & frame-distance, resp., and postulating: (2-28).>Î. g œ -9=2Ð<Î< Ì.<Î. g œ =382Ð<Î< 9 9 From these to simple differential equations all important consequences follo in due order: Ð 29-30). g œ.>.< œ>+82ð<î< 9 (3-32).>Î. g œ Î Þ.<Î. g (33) DÐ> Ð.>.<Î. g œ /B: Ö<Ð>Î< 9 œ. g ÎÐ.>.< (34) DÐ> œ /B: Ð<Î< 9 œ / Í < œ < 9 > 9?83>C < (35) e >+82Ð Ê i. eî. g º e.. (36) [ fð/ fð œ eð/ eð œ const. From the Hubble proportionality (36), velocity-space being hyperbolic (see Ungar 2008), it is natural to conclude that distance-space must be hyperbolic too; so e shall postulate:* (3). g œ.>.< < =382 < Ð. ) =38 ). 9 A basic feature of this CC -metric is that the element.g of time-space can be vieed as the universal element of a cosmic super-time, as suggested by my mentor André Mercier. * 2015: I am no longer convinced by this argument. On p.0, I have stated my present vie..relativity.me