SOME COSMOLOGICAL MODELS T T S A S M

Transcription

1 Mogens True Wegener SOME COSMOLOGICAL MODELS T T S A S M HEIR IME CALES ND PACE ETRICS Revised version (2011) of a paper to PIRT 8, Imperial College, London, 2002 SUMMARY Accepting clock retardation as an empirical fact, e provisionally adopt Whitro's derivation of the Robertson-Walker Metric ( RWM) of Cosmology from the gamma-factor of SR. Recalling 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. Taking 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". It is then easy to sho that the ideas of "big bang" and "steady state" are not mutually exclusive. 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 Taking clock retardation as an empirical fact e provisionally adopt Whitro's derivation of the standard Robertson-Walker metric ( RWM) from the -factor of special relativity ( SR). According to Milne & Walker, to kinds of observers must be distinguished: 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". The difference, naturally, is statistical and a matter of degree. Thus it is possible to make sense of a graduation of clocks according to their approximation to the ideal of a universal time: e classify particles by estimating the departure of their distribution from universal isotropy. Recalling the fact 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 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, it is easy to sho that the ideas of "big bang" and "steady state" are not mutually exclusive after all: a universe starting ith a "big bang" at the dan of creation may very ell approximate to a "steady state" in the course of infinite time. In agreement ith our provisional analysis 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: + ) that hich keeps atomic sizes constant hile light is being stretched, and,) that 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. Being derived directly from. œ.>.=, it is even simpler than the Bondi-Gold model. 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= With - >+82=, the LT are expressible in temporal coordinates: > œ > -9=2= B =382=. B œ B -9=2= > =382= It is interesting that the LT are derivable from the 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^Ñ in O, Ð>ßBßCßDÑin W and Ð>ßBßCßD Ñin 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=! Ñ X =382 = Î-9=2! (2,) B œ ÖB -9=2Ð=! Ñ X =382 = Î-9=2! Finally, introducing non-standard frame-times - & - for frames W& W defined by means of : (3) X =2! ß -9=2! and using A >+82=, e find the Tangherlini transformations ( TT) as generalized by Selleri: -9=2Ð=! Ñ (4 + ) B œ B - B Ð" A@Ñ A =382= œ - È -9=2! " A -9=2Ð=! Ñ B Ð" Ñ A -9=2! = - È" A (4,) B œ B - =382 œ 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. Notice that TT reduce to GT if all observations are referred 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 the substitution:.< Ä fðxñ. 5. With this move, X is no longer the private frame time >, but more like the public proper time of all fundamental observers, i.e. all observers at rest in the universe, e.g. relative to the cosmic background radiation ( CBR). Putting X, this transforms the standard invariant of SR into the standard RWM metric:. g œ.>.< œ. f ÐÑ. 5 f is the expansion or scale factor for the universe, and 5 a fixed "comoving" coordinate. No, for fundamental observers,. 5 œ!, i.e.. g œ., shoing that all fundamental observers participate in the same common cosmic time g. By implication, any devitation of proper time from g is restricted to non-fundamental or accidental observers distinguished by a variable 5. Considering g Á Á >, one can ask if all this amounts to more than mere 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. Everything depends on convention in the sense that it follos from a preferred point of vie. Please, notice that this does not involve us in any conflict ith the results of experiment. The only conflict at stake is one relating to the standard interpretation of SR. Mogens True Wegener

5 -5- Hence, if the clocks of fundamental observers sho the true time g, then the clock of an accidental particle ill be more or less astray. 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. The natural ay of interpreting this retardation of moving clocks is as an effect of gravitation. In this ay e have found a natural 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 Ñ. 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 emerge from 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: here Mach claimed that inertia should be reduced to gravitation, Milne instead held that gravitation should be reduced to inertia - and demonstrated ho to do it! But all this is a repetition of ideas presented earlier. With polar coordinates the RWM becomes: (). g œ. f ÐÑ {. 3 - Ð. ) =38 ). 9 Ñ} (8) e fðñ3 ß T '. ÎfÐÑ -98=>Þ ß VÐTÑ dtî. "ÎfÐÑ 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 ÐÑÖ. 3 3 Ð. ) =38 ). 9 Ñ,=!. g œ. f ÐÑÖ. 3 =38 3 Ð. ) =38 ). 9 Ñ,=". g œ. f ÐÑÖ. 3 =382 3 Ð. ) =38 ). 9 Ñ,= ". 4 É, œ!. 3.- Î È",-. 4 ÎÐ",4 Î%Ñ œ. +<->+8Ð4 ÎÑ É, œ ". +<>+82Ð4 ÎÑ É, œ ". 4 4 Ð. ) =38 ). 9 Ñ. 0. (. ' ",4 Î% ",4 Î%. 3 - Ð. ) =38 ). 9 Ñ The folloing versions comprise all possible values of the constant of curvature,,: (9 + ). g œ. f ÐÑ {.- ÎÐ",- Ñ - Ð. ) =38 ). 9 Ñ} (9,) œ. ÐÑÖ. 4 4 Ð. ) =38 ). 9 4 f Ñ ÎÐ", Ñ (9-) œ V ÐTÑÒ dt Ö. 0.(.' ÎÐ", ÑÓ In the T-scale, the expansion of cosmos is explained by a shrinking of its atoms! 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 ÑÎ " 9 9 (10) fðñ. Î dt / V ÐTÑ Let us assume that radar signals are being exchanged beteen a pair of observers, T & U, : in Ð" x "Ñ 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 =ÐÑ / 5. Generalizing, and introducing Einsteinian standard coordinates > & < for T (priming those of U), e at once get: " " > Ð$ " Ñ Ì< Ð$ " Ñ 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 9ÑÎ. g.>.< œ.. 5 œ/ 9 ÐdT. 5 Ñ This invariant is easily expanded into a hyperbolic time-space of Ð" x $Ñ dimensions if e put: =382 3 Ð. ) =38 ). 9 Ñ Ö. 0.(.' ÎÐ" Ñ The standard SR invariant is thus transmuted into the hyperbolic metric of an expanding universe ith expansion function fðñ, hich can be transformed into another hyperbolic metric, viz. that of a stationary universe hose atoms all contract in accordance ith the Hubble " ÐT 9ÑÎ9 function V ÐTÑ /, here T œ {1 691ÐÎ Ñ}, being a constant of calibration: (12 + ). g œ.>.< < Ð. ) =38 ). 9 Ñ (12,) œ. Ö. 3 =382 3 Ð. ) =38 ). 9 Ñ T (12-) œ / Ò dt Ö. 0.(.' ÎÐ" ÑÓ 9 " Ð "Ñ 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 expansion function f Ð Ñ is changed from to /, to hich all "steady state" models must approximate. So: (13) fðñ /. Î dt V ÐTÑ e / 3 >+82< is a candidate to the proper distance beteen fundamental particles, > just as / "Î È " 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: " " T (14). œ.> >+82<.< Ì /. 3 œ.< >+82<.>. g œ. /. 3 œ ".>.< dt. 3-9=2 < Ð" TÑ 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 fðñ œ / and the third encompasses the public flat 3-space of atoms shrinking in step ith VÐTÑ œ " T: " -9=2 < (15 + ). g œ Ò.> Ö.< =382 <Ð. ) =38 ). 9 Ñ Ó (15,) œ. / Ö. 3 3 Ð. ) =38 ). 9 Ñ œ (15-) œ ÒdT Ö. 3 3 Ð. ) =38 ). 9 Ñ Ó 1T. g œ Ò.> Ö.< =382 <Ð. ) =38 ). 9 Ñ ÓÎ-9=2 < does not compete ell ith the standard invariant of SR hich is the much simpler one. g œ.>.< <Ð. ) =38 ). 9 Ñ. 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 ether 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 that 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: Ð" Ñ (16 + ). g œ.> Ö.< =382 <Ð. ) =38 ). 9 Ñ " " / 3 (16,) œ Ò. / Ö. 3 3 Ð. ) =38 ). 9 Ñ Ó (16-) œ ÒdT Ö. 3 3 Ð. ) =38 ). 9 Ñ Ó T 3 " " (" Ñ 1 /(" TÑ 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 model seems flaed: " is not a genuine cosmic time, since the external contraction factor " / also applies to 3. Þ This clearly shos that the model does not conform to the standard RWM..relativity.me

8 -8-6. A NEW STEADY STATE MODEL Let us start from scratch by adopting. g œ.>.< of SR. Hoever, as e need not accept that standard frames are flat, 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: 1 (1). g.> Ö.< =382 < Ð. ) =38 ). 9 Ñ ( st metric) No the shortcoming alluded to in 5 can be remedied by adopting the folloing definitions: > t < (18) 3 =382</ >+82</ >+82 / ÒÓ V / < < Please, notice that e are free to choose V >+82 Ä " or V >+82 Ä From the above definitions e immediately derive the folloing relationships: > " (19) /. 3 œ.<-9=2<.>=382<œð.<. t >+82<Ñ-9=2 <œ.<. =382< < < From these e further obtain. œ.>.< >+82 œ dt Ð" >+82 <Ñ.< >+82 < >+82 ; then, for. 3 œ 0 ( fundamental observers) e >+82<, A.<Î. =382<, hence: È " " >+82 < (20). œ.>.< >+82 < œ.>î-9=2 < œ.>î < > t Applying 3 =382</ >+82</ to our 1st metric, e obtain the steady-state like metric: t " " / t 3 2 (21). g œ Òdt / Ö. 3 3 Ð. ) =38 ). 9 Ñ Ó ( nd metric) t " " T 2 " (" T Ñ 3 3 Applying / dt Î. T to the nd, e get a metric for a static universe of shrinking atoms: (22). g œ ÒdT Ö. 3 3 Ð. ) =38 ). 9 Ñ Ó ( rd metric) The SR -like metric. œ.>.< =382 <Ð.. Ñ 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 these metrics is conformal to RWM. t < " Applying / œ " >+82 œ " / 3 to our 2nd metric, e derive this non- RWM: g % " ". 3 " %. " Ð" / 3 Ñ (23). g œ Ò. Ö" / 3 / 3 / Ö. 3 3 Ð. ) =38 ). 9 Ñ Ó ( thm) 4 % This metric holds for fundamental particles only. Noticing that eq. (23), for.) œ. 9 œ!, is reducible to the identity. g œ., both in the case of. 3 œ!, or.vºv, folloing a single fundamental particle on its course outards in the line of sight, and in the case of. 3 œ 3., or.v œ!, folloing a series of fundamental particles passing an imaginary border V œ const., e have tested our basic assumption, viz. that the master clocks of fundamental particles alays keep the same cosmic rhythm,. g œ invar. Postulating. g œ. in general, eq. (23) becomes: " " ). 9 " (24) 1 œ ÒÖ" / 3 / 3 / ÖÐ Ñ 3 ÐÐ Ñ =38 ) Ð Ñ Ñ Ó " %.... Ð" / 3 Ñ % hich is clearly the equation, expressed ith, 3, ), 9, for an invariant cosmic hyperbola. So is a genuine cosmic time, although eq. (23) does not conform to the standard RWM. In line ith Selleri [199f.], e finally suggest that the proper transformations to be used locally for the inertial motion of an accidental particle relative to a fundamental observer are: (25) < œ Ñ œ œ œ. < œ - g Ñ Mogens True Wegener

9 -9-. CONCLUSION Any orld model ith an RWM type of metric can be described in three different ays: The 1st description is based on the idea that the average light speed is a universal constant. We claim the metric common to all observers, fundamental or accidental, to be reducible to:. g œ.>.< for.) œ. 9 œ! Hoever, this is not the case for that of the first "steady state" model; its metric is not SR-like. The safest ay to ensure that the claim is fulfilled is, of course, to start ith the SR metric. 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: (26). g œ.> œ.>.< (2). g œ. f ÐÑ. 5 œ V ÐTÑÖdT. 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 presentation of the "big bang" model of Milne & Prokhovnik, of the "steady state" model of Bondi & Gold, and of our on alternative to the latter, offers precisely such rules. The 2nd and 3rd ays of description treat the universe as a spatial totality unfolding in a common orld time, thereby invoking the idea of public timespace: +) According to the 2nd ay, the spatial extension of the substratum is assumed to expand relative to the extension of its material content hich is determined by the structure of its atoms, i.e., distances beteen fundamental observers are increasing relative to their internal structure. So the proper distance beteen to fundamental observers is given by e fðg Ñ5, here f is the scale function, g is cosmic time, and 5 is a fixed coordinate of a fundamental observer.,) According to the 3rd ay, the spatial extension of the substratum is taken to be stationary hereas the dimensions of its contents contract secularly in pace ith the reduction of atoms, i.e., the inner structure of fundamental observers is shrinking relative to their proper distances. This shrinking takes place in step ith V " ÐT Ñ here V is a contraction function, the inverse of the expansion function f, T being an auxiliary time scale defined by: T '. ÎfÐÑ -98=>Þ But common to all possible descriptions is that our invariant.g of time-space becomes the universal element of a cosmic super-time, as suggested by my mentor André Mercier. REFERENCES 1. Selleri, F., 199, in: Found.Phys.Lett.10,1. 2. Whitro, G.J.: cf. ch.v, 6, of (Edinb. 1961), or ch.6, 6.4, of (Oxf. 1980). 3. Milne, E.A., 1948/1951: Kinematic Relativity, Oxford Univ.Pr. 4. Prokhovnik, S.J., 196: The Logic of Special Relativity, Cambr.Univ.Pr. 5. Mercier, A., 2000, in: Duffy & Wegener: Recent Advances in Relativity Theory I, Hadronic Pr..relativity.me