Starlight and Time is the Big BangSAMUEL R. CONNER AND DON N. PAGE

ABSTRACT

The physics of Dr. Russell Humphreys new cosmological modelpresented in Starlight and Time is profoundly flawed and the conclusionsdrawn from this model are seriously mistaken. An accurate treatment ofthe physics indicates that this model is actually a trivial variant of thestandard Big Bang model, with its attendant implications for the age of theUniverse and the Earth time required for light to travel from distant galaxiesto the Earth.

1. INTRODUCTION

Dr D. Russell Humphreys Starlight and Time:Solving the Puzzle of Distant Starlight in a YoungUniverse1 purports to solve the light-travel-time problemin a way which is compatible with astronomicalobservations of the Universe and the General Theory ofRelativity, the best present description of gravity and thestructure of the Universe. Humphreys alleges that hisalternative cosmology lifts from the neck of the young-Universe movement the doubly burdensome yoke of thelight-travel problem and the movements past denials ofthe validity of General Relativity, a very thoroughly testedand proven theory.

The arguments presented in Starlight and Time soundingenious and are attractively presented, but they are flawedin ways which are so serious as to totally invalidate them.These flaws centre around a series of misunderstandingsabout the meaning of General Relativity concepts and theinterpretation of cosmological models based on GeneralRelativity. When these misunderstandings are corrected itis found that the approach to cosmology advocated inStarlight and Time leads to the same time-scaleconclusions as standard Big Bang cosmology. The modelof Starlight and Time is in fact a trivial variant of the BigBang model.

The misunderstandings exhibited in Starlight andTime may be grouped into several clusters. The body ofthis paper is organised into sections correcting each of thesesets of misunderstandings. The first set centres around theCosmological Principle, or the Copernican Principle.Starlight and Time affirms, wrongly, that the long-time-scale implications of Big Bang cosmology theory arecrucially dependent on the global validity of this principle/assumption and that the relaxation of this assumption,through the introduction of a boundary to the matter of the

Universe, produces dramatic differences in the gravitationalproperties of the Universe. In fact, the gravitationalproperties of the visible part of the Universe are identicalfor unbounded and spherical bounded models regardlessof the validity of the Copernican Principle in more distantregions. A second cluster of misunderstandings centresaround the nature of time in cosmological models.Starlight and Time affirms, wrongly, that the physicalclock synchronisation properties which occur in thestandard Big Bang model are due to the boundaryconditions implied by the Copernican principle and thatmodification of these boundary conditions can change theway that physical clocks behave in the Universe. In fact,physical clocks located on Earth and on distant galaxiesbehave identically in bounded and unbounded Universeswhich have identical interior matter distributions. A thirdset of misunderstandings centres around the nature andeffects of event horizons in the Universe. Starlight andTime affirms, wrongly, that observers who pass throughevent horizons observe dramatic changes in the rate of timepassage in distant parts of the Universe. In fact, no suchchanges are observed; the distant-clock time phenomenaobserved in a bounded Universe are identical to thoseobserved in an unbounded Universe. A fourth cluster ofmisunderstandings centres around the observableconsequences of this model. Starlight and Time and otherof Humphreys writings affirm, wrongly, that this modelcan be adjusted to accurately predict observed phenomenawhile limiting the passage of time on Earth, since thebeginning of the Universe, to only a few thousand years.In fact, the short-time-scale expansion proposal hasprofound and easily testable consequences which are notobserved. Examination of a number of these phenomenademonstrates that the short-time-scale cosmic expansionand short-time light travel from distant galaxies proposedin Starlight and Time did not occur in the history of the

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real Universe.The physical model of the Universe presented in

Starlight and Time, a locally homogeneous and isotropicspherical matter distribution, is possible, and in fact wasproposed more than 20 years prior to the unveiling ofStarlight and Time in 1994.2 The young-Universeinterpretations which Humphreys derives from this modelare, however, mistaken. The time-scale implications ofthe bounded cosmology of Starlight and Time areidentical to the standard Big Bang model.

Before proceeding to the substance of our rebuttal ofStarlight and Time, it is important to state the physicaland mathematical assumptions on which this rebuttaldepends. We assume that:(1) General Relativity is an accurate description of gravity;(2) gravity is the most important force acting over

cosmologically large distances, so that the conventionalapplication of General Relativity to cosmology theoryis valid;

(3) the fundamental parameters of nature, such as thegravitational constant G and the speed of light c, areinvariant over the observable history of the Universe;

(4) the visible region of the Universe is approximatelyhomogeneous and isotropic on large distance scales;and

(5) the events which we witness by the light of distantgalaxies and quasi-stellar objects are real events andnot appearances impressed onto the Universe by theintention of the Creator.

Assumption (1) is an established fact,3 there is excellentevidence for assumptions (2) through (4), and assumption(5) is a metaphysical postulate which is necessary if one isto do cosmology at all. Humphreys agrees explicitly orimplicitly with all five of these assumptions.4 Theassumption which may attract the most scepticism fromreaders of this journal is (3), in particular the invariance ofc. The hypothesis that c was much larger in the past is acommon view in the young-Universe movement and is, infact, a competing solution to the light-travel-time problem.However, there is compelling evidence, overlooked by theoriginators and promoters of the declining-c hypothesis,that c has not varied significantly in the past on Earth, andpersuasive evidence that it has not varied in the past atdistant galaxies.5

Humphreys claims that it is possible to buildcosmological models which satisfy these five assumptionsand which are also compatible with the short time-scalesrequired by young-Universe interpretations of the Bible.Starlight and Time purports to show how to build suchmodels. As we shall show below, the approach tocosmology advocated in Starlight and Time fails and noplausible variation of it can succeed.

Readers not familiar with the concepts and themathematical apparatus of General Relativity may findparts of the following discussion obscure. We haveprepared a much more user-friendly version of this paper

titled The Big Bang Cosmology of Starlight and Time,but henceforth referred to simply as the Supplement.6 TheSupplement is much longer than the present article, is lessterse and is equipped with an extensive set of explanatoryand supplementary appendices which will be helpful toreaders not familiar with General Relativity and itsapplication to cosmology theory. It also discussesnumerous additional errors in Starlight and Time whichcannot be mentioned in this paper due to lack of space.Readers interested in the Supplement should write to theaddress given in reference 6.

2. MISUNDERSTANDINGS ABOUT THESIGNIFICANCE OF THE

COSMOLOGICAL PRINCIPLE

The Copernican Principle or CosmologicalPrinciple is an assumption which was adopted in the earlyyears of General Relativity-based cosmology theory inorder to render tractable the mathematical problem ofapplying General Relativity to the structure of theUniverse.7 The assumption is that the Universe is, on largescales, very smooth and that there are no preferreddirections in the Universe homogeneity and isotropy.One of the consequences of the Cosmological Principle isthat, if it is valid globally (throughout the Universe) asopposed to locally (only in the region of the Universe visiblefrom Earth), then the Universe is unbounded. The centralcontention of Starlight and Time is that the long-time-scale consequences of Big Bang cosmology are the faultof this unbounded feature of globally homogeneous andisotropic universes. The book lays considerable emphasison the widespread acceptance of this assumption amongcosmologists, and criticises professional cosmologytheorists and popular science writers for uncriticallyadopting this hypothesis. Much of the remainder of thebook is devoted to attempting to demonstrate that therelaxation of the global validity of the CosmologicalPrinciple through the introduction of a spherical boundaryto the matter of the Universe completely changes thegravitational properties of the Universe and overthrows thelong time-scale implications of standard cosmology theory.

This argument of Starlight and Time is mistaken. Theglobal validity of the Cosmological Principle is notnecessary for standard cosmology theory to be valid as adescription of the Universe visible from Earth. This hasbeen recognised for decades and is especially prominentin recent work on inflationary cosmology. The impositionof a spherical boundary to the matter of the Universe hasno effect on the gravitational and clock time-keepingproperties in the interior of such a boundary.

2.1 Misunderstanding About Cosmologists’Acceptance of the Cosmological PrincipleTo begin with, Starlight and Time misrepresents

professional cosmology theorists views of the

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Cosmological Principle. The fact of the matter is thattheorists do not, in general, believe that it must apply tothe entire Universe. Examples of this can be found in recentgraduate-level textbooks on General Relativity andcosmology, for example,8,9 and even in some popularwriting.10 It is true that not every text mentions this fact,and it is probably the case that some, perhaps many, of theauthors of older textbooks who do not mention this factbelieved that the Cosmological Principle was globally valid.The silence of these earlier works does not establishHumphreys claims, however. Present-day cosmologytheorists readily acknowledge the possibility that theUniverse may be very inhomogeneous at very largedistances or on very large distance scales. As a singleexample of this we cite an important recent paper on thevery large-scale structure of the Universe:

. . . one of the main principles of the big bang theoryis the homogeneity of the Universe. The assertion ofhomogeneity seemed to be so important that it wascalled the cosmological principle. Indeed, withoutusing this principle one could not prove that the wholeuniverse appeared at a single moment of time, whichwas associated with the big bang. So far, inflationremains the only theory which explains why theobservable part of the Universe is almosthomogeneous. However, almost all versions ofinflationary cosmology predict that on a much largerscale the Universe should be extremelyinhomogeneous, with energy density varying from thePlanck density [≈1094 g/cm3] to almost zero.11

(Emphasis in original)Despite the global inhomogeneity discussed by this author,he affirms that the local, observable part of the Universe iswell-described by standard Big Bang theory.12 This citationand the other references show that Starlight and Time ismistaken in its assertions about cosmology theoristsreliance on the global validity of the CosmologicalPrinciple. Modern cosmology theorists do not believe inthe model of the Universe which Humphreys critiques.Despite this, these theorists still affirm, with good reason,that general relativistic cosmology theory implies that theUniverse is very old.

The bounded cosmology proposed in Starlight andTime is not the innovation it has been represented to be.13

It is nearly identical in its most important features to themodel proposed in 1971 by Oskar Klein.2 Kleins model,like Humphreys, is spherical, bounded and starts in a stateof contraction. Also, like Humphreys model, Kleinsmodel is motivated by dissatisfaction with theCosmological Principle. The principal differences are thatKleins model does not have an optically thick boundary,employs a different method for reversing the initialcontraction, has a different explanation for the cosmicmicrowave background, and permits the Earth to be not atthe centre. Regarding the differences between his boundedmodel and the standard unbounded Big Bang model, Klein

writes:Although there is hardly any difference between acosmological Friedmann [that is, a standard unboundedBig Bang] solution and a bounded one so long asobservations are not approaching the border [whichwould reveal a deficit of galaxies beyond the edge ofthe matter] and the extrapolations backward in timedo not approach the state of the so-called fireball, thedifference becomes significant when these conditionsare not satisfied; in fact, it is enormous at the earlystage assumed for the fireball . . .14

The fireball state Klein mentions is the hot, dense statein which primordial nucleosynthesis occurs in the standardBig Bang model, when the Universe is about 1010 timessmaller than its present size, with densities of order 104 g/cm3 and temperatures of order 1010 K. Kleins model doesnot resemble the Big Bang at this early stage becauseradiation can escape from the Universe through theboundary, resulting in greater cooling at the edge than atthe centre, so that the Universe does not remain radiallyhomogeneous. At late times (Z < 1100), when radiationplays an insignificant role in the dynamics of the Universe,there is, in Kleins words, hardly any difference betweenhis bounded model and the standard unbounded model.Humphreys avoids the excessive early cooling which wouldoccur in Kleins model, if extrapolated back to the fireballstate, by postulating an optically thick boundary, whichabsorbs and re-emits into the interior the radiation whichescapes from Kleins model. In such a case, there is hardlyany difference between the observable behaviour andproperties of the bounded and unbounded cases at any time.

The problem of Starlight and Time and the debatesurrounding it can be summarised thus: either Humphreysis right that boundaries matter, in which case many pastand present-day cosmology theorists have committed acolossal intellectual blunder in admitting globalinhomogeneity but not recognising its implications for thetime-scale of the visible Universe, or else the cosmologytheorists are right and Humphreys is mistaken non-localinhomogeneity is irrelevant. As we shall see, the answerto this problem is the second case; the boundary conditionsdo not matter. For this reason, cosmology theorists do notpay attention to them when studying the part of the Universewhich we can observe from Earth. The visible Universe isvery nearly homogeneous and isotropic on large scales,6,15

and this is sufficient for the results of standard cosmologytheory, applied to the visible part of the Universe, to bevalid.

In sum, the Cosmological Principle is a mathematicalconvenience which does not have to be globally valid inorder for the results of standard cosmology theory to applyto the part of the Universe we can see. Humphreys hasmisinterpreted the historical fact that this principle wasinvoked to simplify the mathematics of cosmology to implythat, without this assumption, the standard model falls apart,even when applied only to the visible portion of the

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Universe. This is not so. Contemporarycosmology theorists do not implicitly believethat the Cosmological Principle is truethroughout the Universe, but this does notaffect their work since it is necessary only thatit be valid locally.

2.2 Misunderstanding About theGravitational Properties of Bounded

and Unbounded UniversesThe other side of the erroneous assertion

that the global validity of the CosmologicalPrinciple is essential to Big Bang models isthe erroneous assertion that relaxing thisprinciple produces profound changes in thegravitational properties of the Universe. Inparticular, Starlight and Time asserts,wrongly, that the supposition of a boundary tothe matter of the Universe, beyond which thereis only vacuum, completely changes thegravitational properties of the Universe andconsequently the time-keeping properties ofphysical clocks in the Universe.

Confining our attention to universes whichare spherically symmetric about Earth, thereare two classes of bounded universes: thosewith a boundary within the Earths particlehorizon, and those for which the boundary ismore distant than the particle horizon. Thepresence of a boundary is manifestly irrelevantin the latter case, since gravitational influencesfrom the matter distribution beyond theboundary, travelling at the speed of light,16

have not yet reached Earth. The observedinterior properties of such a universe cannotdepend in any way on the presence or absence of matterbeyond the horizon. The standard Big Bang model isconsequently an accurate description of what we are ableto observe in such bounded universes, and it is for thisreason that cosmology theorists do not worry about thevery large-scale inhomogeneity of the Universe.

Starlight and Time proposes that the boundary iswithin Earths particle horizon. This can be seen from thelow value of the comoving radius of the edge proposed inits example model17 (which is actually too low,corresponding to a redshift of Zedge = 1.375, well below thelargest redshifts observed for discrete objects, around Zmax≈ 5),6 and from its appeal to an optically thick boundary topreserve the thermal character of the observed cosmicmicrowave background radiation (this is not necessary ifthe boundary is beyond the particle horizon). Thus, inHumphreys model, there is sufficient time for gravitationalinfluences from the matter distribution (or lack thereof)beyond the boundary to produce detectible effects at Earth.What are these effects? It is easy to show that, in fact,there are none. The arguments of Starlight and Time on

this score are basically Newtonian, so we will first disprovethis thesis using Newtonian arguments and only then citethe corresponding results from General Relativity.

Starlight and Time claims that the interiorgravitational properties of a spherical bounded universeare profoundly different from those of an unboundeduniverse, both of which are assumed to be locallyhomogeneous and isotropic within the matter-filled region.The two cases are illustrated in panels (a) and (b) of Figure1. The two cases differ in that the unbounded case has aspherically symmetric matter distribution extending withoutlimit outside the boundary which marks the edge of thebounded case. Any difference in the interior properties(that is, the properties within the actual boundary of thebounded case and the corresponding region of theunbounded case) of the two universes must be due to theadditional exterior matter in the unbounded case.

Starlight and Time alleges that the bounded Universehas a large-scale pattern of gravitational potential andgravitational force which points toward the centre, but thatthe unbounded Universe does not.18-20 This situation is

Figure 1. (a) Bounded, and(b) Unbounded cosmos matter distributions.(c) Gravitational field configuration for the bounded cosmos.(d) Humphreys’ claim of no fields in the unbounded cosmos.

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illustrated in panels (c) and (d) of Figure 1. In the limit ofthe validity of Newtonian gravity, the gravitational fieldand potential at any point is a linear function of the matterdistribution, so that we can split the unbounded Universeinto two parts, the interior and exterior matter distributions,and calculate the field contribution from each of these parts.The sum of these contributions equals the interior field,which, according to Humphreys, vanishes. We mayexpress the situation in terms of the standard Newtonianexpression for the gravitational field g(x) at some positionx within the interior region:

g x d x Gx x

x x

g x g x

d x Gx x

x xd x G

x x

x x

according to Humpreys

all space

ext

allspace ext

( ) = −−

= ( ) + ( )

= −−

+ −−

=

∫

∫ ∫

33

33

33

0

''

'

''

''

'

'

int

ρ

ρ ρ

( )

In these expressions, g int(x) is thegravitational field at locations x (inside theboundary) which is produced by the matter withinthe boundary while gext(x) is the field at the samelocation produced by the matter external to theboundary. Similar expressions can be written forthe gravitational potential. The field contributionin the interior region of the unbounded matterdistribution which is due to the interior matterdistribution itself, gint(x), is the same for thebounded and unbounded cases, since the interiormatter distribution is identical for the boundedand unbounded cases (see Figure 2, panel (a)).In order for the total field (that produced by theinterior and exterior matter) to vanish in theinterior region, it necessarily follows that the fieldcontribution in the interior region due to theexternal spherical matter distribution mustexactly cancel out the field contribution due tothe interior matter distribution:

g x d x Gx x

x x

g x d x Gx x

x x

extext

( ) = −−

= − ( ) = − −−

∫

∫

33

33

''

'

''

'int

int

ρ

ρ

The claim that the gravitational field vanishes inthe interior region of the unbounded Universerequires that the field produced in the interiorregion by the external spherically symmetricmatter distribution be exactly the opposite of thefield of the bounded matter distribution, asillustrated in Figure 2, panel (b). Thisrequirement is in conflict, however, with a well-

known theorem of Newtonian gravity:The gravitational field in the empty interior of a hollowspherically symmetric matter distribution vanishes. Thegravitational potential in such a region is spatially non-varying.In fact, the field due to the exterior matter distribution

vanishes in the interior region, gext (x) = 0, so that the fieldin the interior region is due solely to the interior matterdistribution:

g x g x d x Gx x

x x( ) = ( ) = −

−∫intint

''

'3

3ρ

as illustrated in panels (c) and (d) of Figure 2. This field isidentical to the field of the bounded Universe. A similarargument leads to the conclusion that the pattern ofNewtonian gravitational potential in the interior of thebounded and unbounded Universes is identical, with thepossible exception of a spatially non-varying offset betweenthe two cases. The gravitational effects on clocks which

(1)

Figure 2. (a) Actual field configuration due to the interior matter in the unboundedcosmos.

(b) Field which the exterior matter must produce in order to cancel thefield produced by the interior matter.

(c) Actual field configuration produced by the exterior matter.(d) Actual field configuration in the unbounded cosmos.

(2)

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Humphreys alleges to occur in the bounded Universe are,in the Newtonian limit, the result of differences in potential,and the pattern of potential differences in the interior ofthe bounded and unbounded cases is identical.

The identity of the gravitational field in the interior ofthe bounded and unbounded cases may be easily seen byconsideration of the motion of galaxies in the interior regionof both cases. If we consider a galaxy located at the sameposition in the two models and follow its motion as bothmodels expand, we will find that the motion is identicalfor the two cases: radial recession (with identical speeds)from the centre, but with identical radial accelerationstoward the centre which slow the recession as time passes.In the Newtonian description, the deceleration which slowsthe expansion of the Universe, whether bounded orunbounded, is due to the gravitational field of the Universe.This deceleration is identical for the two cases because thegravitational field is identical. Indeed, if Humphreys claimof zero field in the unbounded Universe were correct, thenin the Newtonian description of such a Universe, theUniverse would expand without decelerating. In fact,however, an unbounded Universe decelerates in exactlythe same way as a bounded Universe with the same interiorproperties.

This Newtonian counter-argument to Humphreyserroneous Newtonian arguments may be readily extendedto a truly relativistic Universe. The General Relativityresult which corresponds to the Newtonian hollow spheretheorem is a corollary of a famous General Relativitytheorem called Birkhoffs theorem. This corollary has theconsequence that

the metric inside an empty spherical cavity at thecenter of a spherically symmetric system must beequivalent to the flat-space Minkowski metric ηαβ . . .This corollary is analogous to another famous resultof Newtonian theory, that the gravitational field of aspherical shell vanishes inside the shell . . . Itsimportance arises from the fact that the Birkhofftheorem is a local theorem, not depending on anyconditions on the metric for → ∞ (aside fromspherical symmetry), so that space must be flat in aspherical cavity at the center of a spherically symmetricsystem, even if the system is infinite even, in fact, ifthe system is the whole universe.21

The flat space-time described by the Minkowksi metrichas no gravitational fields. In the general relativisticdescription as well as the Newtonian approximation, theexternal spherically symmetric matter produces nogravitational effects on the interior region, so that theinterior gravitational properties are identical.

Another way of recognising the identity of the interiorgravitational properties of the bounded and unboundedcases is to note that the space-time metric, gµν, is identicalin the interior of both kinds of Universe. The identicalmetric means that the two classes of Universe have identicalgeometrical properties, such as space-time curvature, in

their interior. These identical properties result in identicalgravitational behaviours (interested readers are referred tothe Supplement for a discussion of the relation betweenthe metric and more familiar gravitational properties suchas the gravitational field and gravitational potential).Although Humphreys agrees that the metric is identical inthe interior of the matter boundary for both bounded andunbounded cases, he fails to recognise the implications ofthis fact. Responding to our 1995 rebuttal of Starlightand Time,22 he writes:

. . . What they are actually trying to say is true: withinthe matter sphere, the local curvature of space is thesame in both types of cosmos. But the local curvatureof space is not the property which makes the essentialdifference. The distinguishing properties are thegravitational field strength and the gravitationalpotential, both of which are in principle objectivelymeasurable. In a bounded-matter universe there existsa small but definite large-scale pattern of gravitationalforce pointing toward the center; in an unboundeduniverse there is none. In a bounded-matter universethere is a large difference in gravitational potentialenergy between the center and the edges; in anunbounded universe there is none. How could thechallengers overlook such obvious differences?20

We have shown above that there is no difference inthe pattern of gravitational field or the spatial pattern ofpotential differences in the interior regions of the boundedand unbounded cases. Humphreys fails to recognise thatit is the curved geometry of space-time which producesall of the familiar gravitational properties such as theNewtonian field and potential, and less familiar phenomenasuch as gravitational time dilation. Identical interior space-time metric implies identical interior space-time geometrywhich implies identical interior gravitational properties.The failure to recognise this fact lies at the heart of theerrors of Starlight and Time.

3. MISUNDERSTANDINGS ABOUT TIME INCOSMOLOGICAL MODELS

The metric in the interior of a locally homogeneousand isotropic space-time may be written, using coordinateswhich are at rest with respect to the matter of the Universe(comoving coordinates), in Robertson-Walker form:23

where τC is the cosmic time coordinate, η a dimensionlessradial coordinate, θ and ϕ the standard spherical angles,and a(τc) the time-dependent radius of curvature or scalefactor of the Universe. The parameter k takes on the values-1, 0 and 1 for spaces of negative, zero and positivecurvature, respectively. The example cosmological modelemployed in Starlight and Time uses k = +1. The metrictells how to compute the space or time distance between

ds c d a td

kd dc c

2 2 2 22

22 2 2 2

1= − ( )

−+ +[ ]

τ ηη

η θ θ ϕsin (3)

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events at different locations in space-time. It thus mustfigure prominently in any discussion of time passage inthe Universe. Starlight and Time and other ofHumphreys cosmological writings exhibit seriousmisunderstandings about what time measures are relevantin discussions of cosmology and how to use the metric tocompute them. We will illustrate these misunderstandingsby a number of quotes from Humphreys writings. Thefirst comes from the introductory part of the relativityappendix in Starlight and Time:

The cosmic time τ[τc in equation (3)] in theRobertson-Walker metric is same [sic] as the propertime or natural time in eq. (1). It is the timemeasured by a set of clocks throughout the universe,each one riding with a galaxy as it moves with theexpansion of space. These clocks can all besynchronized with one another. Later on I willintroduce a distinctly different time, t, often calledSchwarzschild time, or coordinate time. Thedifference between these two types of timemeasurements constitutes the essence of this paper, sobe alert for the distinctions.24

This statement, which is correct, clearly affirms that thetime measured by a physical clock located in a galaxy (forexample, a clock on Earth, in the Milky Way galaxy, or aclock located in some other galaxy) is the same as thecosmic time of standard cosmology theory. Thus, theamount of physical time elapsed on a physical clock locatedon Earth or in a distant galaxy is simply the elapsed cosmictime. This should settle the matter, since it tells us how tomeasure time intervals elapsing on real clocks in theUniverse. Instead, Humphreys introduces a new timecoordinate and prefers to make measurements in thatcoordinate. The reader should note that the differencebetween the real, physical, natural time measured by real,physical clocks and Humphreys preferred Schwarzschildtime coordinate is, by his own declaration, the essence ofhis model. We shall see that this essential difference isirrelevant to what really happens on real, physical clocksin the Universe.

The second quote comes from Humphreys reply toour 1995 rebuttal22 of Starlight and Time:

. . . Conner and Page again spend much effortasserting another point I had already made. It is onpage 114, the fact that the same comovingcoordinate system equation can be used to describethat part of the bounded biblical cosmos which hasmatter in it. In it, clocks can similarly be synchronized.Next they say (section 2, end of first subsection):As the universe expands from a certain size a1 to alarger size a2, exactly the same amount of [cosmic]time passes on every comoving clock.They imply that somehow this fact is damaging to mytheory, but nowhere in their article do they say exactlywhy that should be. Therefore, I will spell out theirimplication for them. They are implying that right

now, all parts of the cosmos are the same age. Ifthat implication were so, it would invalidate mytheory.25

In this statement, Humphreys agrees with us that, as theUniverse expands from small initial size a1 to its presentlarge size a2, all comoving clocks (these are the clockswhich ride along with each galaxy as Humphreys puts itin his first quote) experience the same passage of properor natural time. This clearly means that the same amountof time, as measured by physical clocks, has elapsed inevery part of the Universe as the Universe has expandedto its present size. Therefore, at the present moment in thehistory of the expansion of the Universe, that is, on space-like hypersurfaces for which the radius of curvature a isthe same at all locations in the interior of the matter sphere(a definition of simultaneity which is also adopted byHumphreys),26 it is inescapable that the same amount ofphysical time has elapsed in every part of the Universe.To borrow Humphreys words,

right now, [that is, on space-time surfaces ofconstant spatial curvature, which is clearly whatHumphreys intends in his definition of cosmicsimultaneity, demonstrated in his Figure 226] all partsof the cosmos are the same age.The third quote comes from Humphreys reply to Dr

John Byls rebuttal of Starlight and Time published inCreation Research Society Quarterly:27

Byls assertion, All galactic clocks tick at the samerate, contains the loose ends of its own unravelling.To specify the rate of one type of clock, we need tohave another type of clock in the same location forcomparison. If I tell you your watch is running slow,you have the right to ask what clock I am saying iscorrect. To what other type of clock is Byl comparinghis cosmic clocks? He does not say. That renders hisclaim of same rate meaningless. Furthermore,according to Schwarzschild clocks, his statement iswrong: in a bounded universe, Byls clocks tick atdifferent rates (relative to my clocks) in different places.Byls clocks really should not be called cosmic atall. The name comes from applying his metric to anunbounded Big Bang cosmos, where a similar typeof analysis would show that cosmic clocks andSchwarzschild clocks would indeed tick at the samerate everywhere [this claim is false, as we show below].For the bounded, finite-mass, non-Big-Bang cosmos Ipropose, perhaps Byls system should be called localclocks and local time.28

The reader should note that there is an evolution ofHumphreys ideas between the first (1994) and third (1997)quotes. In the first quote, Humphreys acknowledges thatthe cosmic time coordinate τc of the standard Robertson-Walker metric of cosmology is the time measured by aphysical clock on Earth or accompanying a distant galaxy.In the second quote, he again acknowledges this fact buttries to avoid its force. In the third quote, he rejects the

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legitimacy of the cosmic time coordinate in a boundedUniverse. Byl is saying that as the Universe expands agiven amount, all galactic clocks register the samepassage of physical time. Humphreys is wrong in sayingthat there is no time reference for Byls statement. Byl isusing the expansion of the Universe as his reference. Allgalactic clocks tick at the same rate with respect to theexpansion of the Universe, a fact which Humphreysaffirms in the second quote. Byls statement is true and isvirtually identical to the statement of ours whichHumphreys quoted in his quote # 2 above, and whichHumphreys agreed is a fact. Although Humphreysacknowledged in the first and second quote that cosmictime (or, if he prefers, local time) is indeed the time keptby physical clocks in real galaxies, he rejects this timemeasure because he prefers to use Schwarzschild time. Itis true that Schwarzschild clocks do not keep the sametime as galactic clocks (this difference is the essence ofthe model), but the question which must be answered is,Which clock actually tells how much time has elapsed ata particular galaxy or on the Earth. By Humphreys ownadmission (quote # 1), it is the cosmic, or local, ornatural, or galactic (these terms are all interchangeable)clocks which tell how much time has elapsed on Earth orat distant galaxies.

3.1 The Metric Tells How to Measure Distanceand Time in the Universe

From the third quote above, one might get theimpression that Humphreys and his critics are engaged ina childish shouting match: My time is better than yours!No! My time is better! Is there no objective time measureto which appeal can be made in order to resolve thisdisagreement? In fact, there is. General Relativity providesa very specific method for calculating how much timepasses on a physical clock travelling along a physicaltrajectory through space-time. By using this method, it iseasy to unambiguously answer the question, Which timemeasure gives the true age of the cosmos in the boundedUniverse model of Starlight and Time? This bookserrors stem in part from a failure to follow the mathematicalrules of General Relativity for measuring physical timepassage in the Universe.

Starlight and Time affirms, rightly, that the space-time metric tells us how to measure space-time intervalsbetween space-time events which are separated by someinfinitesimal coordinate differential dxµ.29 The square ofthe space-time interval, ds2, is given by

ds g dx dx g dx dx2

0

3

0

3

= ≡= =

∑ ∑µ

µνµ ν

µνµ ν

ν

In this expression, the dxα are coordinate differentialsbetween two events at slightly different coordinate locationsin space-time, and gαβ are the components of the space-time metric, which gives the space-time interval inaccordance with this equation. The Greek indices µ and ν

are, in this equation, dummy indices which are summedover their range of 0 to 3. The right-most expression givesthe definition of the Einstein summation convention(repeated upper and lower indices are summed over theirfull range), a convenient shorthand notation which we willemploy. The index 0 customarily refers to the timecoordinate and the indices 1, 2, 3 refer to the spacecoordinates. An important fact which must be kept in mindin thinking about this equation is that ds2 is the square of aphysical distance, either in space or time, along thedifferential trajectory element dxµ. With the signconvention adopted in Starlight and Time, ds2 > 0 meansthat ds2/c2 is the square of the physical time elapsed on aphysical clock traversing the coordinate trajectorydx ds c d proper

µ τ: 2 2 2= . This physical time interval is calledthe proper time interval, dτproper. Humphreys also refersto this as the natural time interval.30 If ds2 is negative,then ds2 is the square of the physical distance intervalbetween the two space-time events which would bemeasured by an observer for whom the two events aresimultaneous: − =ds dlsimult

2 2 . The intermediate case, ds2

= 0, corresponds to a trajectory dxµ which is traversibleonly by particles moving with the speed of light.Humphreys affirms these facts.30

The quantity ds2 is an indexless quantity; it has nosuperscripts or subscripts. The expression used to computeds2, ds2 = gµνdxµdxν, has two sets of super- and subscripts,but these are summed over and so do not persist in thefinal result. The quantity ds2 is known in tensor calculus,the mathematical language of relativity, as a scalarinvariant its value is independent of the coordinatesystem used to express or compute it. Thus, if one were todescribe the coordinate differentials dxµ in a new coordinatesystem, x′, then when one computed the space-time intervalin the new coordinate system, one would find

ds g dx dx g dx dx ds' ' '' '2 2= = =α β

α βµν

µ ν

This is an elementary fact of tensor calculus which occursbecause, in the new coordinate system, the metric takes onnew values gα′β′ which are not the same as the old valuesgµν in the old coordinate system. The changes in the valuesof the metric coefficients exactly counteract the changesin the values of the coordinate differentials in such a waythat the scalar (indexless) quantity ds2 is conserved. Thisfact is shown in any elementary textbook of tensor analysis;a simple demonstration is also given in the Supplement. Itis also eminently reasonable: the physical time elapsed ona clock cannot depend on the coordinate system used todescribe the clocks trajectory.

The crucial fact to keep in mind is that the physicalspace-time distance along a trajectory is independent ofthe coordinate system used to describe that trajectory. So,for example, if one uses a particular coordinate system tocalculate the physical time elapsed along a time-like (thatis, ds2 > 0 for each differential step dxµ) trajectory, one

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will obtain the identical physical elapsed time along thisphysical trajectory using any other coordinate system.The time elapsed on a physical clock is determined by theclocks trajectory through space-time, not by the coordinatesystem used to describe the trajectory. This means thatthere is something incorrect in the appeal, in Starlight andTime, to a different coordinate system to argue againstphysical results which are derived from the standardcomoving coordinate system of cosmology theory.

3.2 Time Measurements Using theRobertson-Walker and Klein Metrics

We can apply the principle that the physical timeelapsed on a physical clock is determined by the clockstrajectory and not by the coordinate system to the physicaltime elapsed on physical clocks in a bounded universe andcompare the results with the claims of Starlight and Time.We will start with the standard coordinate system andmetric of cosmology theory and derive the desired resultin that simple coordinate system. Then we shall derive thesame result in the coordinate system which Humphreysprefers.

The metric in the matter-filled portion of a locallyhomogeneous and isotropic space-time geometry is givenby equation (3).31 How can this equation be used todetermine the age of the Universe? All that is needed is tochoose an observer to measure the age of the Universe,determine what the coordinate trajectory of that observeris, and then turn the crank of equation (3) to compute thetime passage which this observer would measure at theUniverse ages. The question which is most relevant toStarlight and Time is, how old is the Universe asmeasured by observers on Earth and by observers on distantgalaxies? The trajectories of such observers are simplythe trajectories of their host galaxies.32 These trajectoriesare extremely simple: ηcom obs(τc) = constant, θcom obs(τc) =constant, ϕcom obs (τc) = constant. Starlight and Timeaffirms that this is so.35 Such observers are often calledcomoving because they move with or co-move withthe expansion of the matter of the Universe; the coordinates(η, θ, ϕ) are called comoving coordinates for the samereason. For such observers, the differential coordinatetrajectories are given by dηcom obs = 0, dθcom obs = 0, dϕcom obs =0. Substitution of these coordinate trajectories into themetric gives us the proper time interval measured by suchobservers:

ds c d c dcom obs com obs c 2

2= =2 2 2τ τ

ord dcom obs cτ τ =

This equation may be trivially integrated from the beginningof the expansion of the Universe to yield

τ τ τ τcom obs com obs beginning c c beginning − = −, ,

In other words, the physical time measured by a comoving

observers clock since the beginning of the Universe isidentical to the cosmic time elapsed since the beginning ofthe Universe. It should be noted that the location (η, θ, ϕ)of the observer in the matter sphere does not enter into thisequation all observers measure the same dependenceof their local physical clock time on the local radius ofcurvature a of the Universe. If one defines a surface ofcosmic simultaneity as the space-time surface on whichthe radius of curvature a is everywhere the same (this isthe standard definition of cosmic simultaneity in Big Bangtheory and is the definition Humphreys adopts26), then allphysical clocks associated with Earth and distant galaxies,regardless of their location on this surface, will read thesame time elapsed since the beginning of the expansion.This equation does not tell us what that time passage is forany specific stage of the expansion of the Universe. Todetermine that, it is necessary to determine or adopt anequation of state for the matter of the Universe and solvethe dynamical equations of cosmology.36 In all cases, theelapsed dynamical time is a function only of the cosmicscale parameter a. That is, there is no dependence of theelapsed cosmic or comoving clock time on a comovingclocks location within the matter-filled region. For thematter-dominated, pressureless, positive curvature (k = +1 in equation (3)) case which is used for illustrativepurposes in Starlight and Time, the cosmic time is givenin differential form by

d dda

c

a

a acom obs cτ τ = = ±−max

and in integral form by

τ τcom obs ca a

a

c

a

a

a

a

a

a

=

( ) = ( )

− −

−max

max max max

sin 12

2

where a is the radius of curvature of the Universe, amaxbeing its value at the moment of maximum expansion (thisexpression is equivalent to Humphreys equation (19)37),except that his equation is missing a factor of π/2.6 It shouldbe noted that equation (10) applies to both the boundedand unbounded k = + 1 universes. This is so because thedynamical equations which govern the time behaviour ofa(τc) depend only on the local properties of the Universe,and these are the same for the bounded and unboundedcases.

Equation (10) is the elapsed time on a comoving clockand hence is the age of the Universe as measured by aphysical observer residing on Earth or on a distant galaxy.There is no hint in this expression of differential timepassage on Earth and on distant galaxies the elapsedcomoving clock proper time is independent of the location(η, θ, ϕ) of the clock. There is no differential time passagefor real, physical observers located on Earth and in distant

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galaxies in a locally homogeneous and isotropic universe.Humphreys claims that consideration of time passage

using his preferred Schwarzschild coordinate systemindicates otherwise. We will now consider this. As weswitch over to Schwarzschild coordinates, it is importantto keep in mind the question we wish to answer using them:How much time is observed to pass on a physical clocklocated on Earth or in a distant galaxy as the Universeexpands? We are still thinking about the space-timetrajectories of comoving clocks, since these clocks are theones which tell how much time elapses on Earth and indistant galaxies. When we switch to Schwarzschildcoordinates, the metric coefficients will be different fromthe Robertson-Walker metric, but the space-time trajectorydescription will also be different. Both effects must betaken into account when computing the elapsed proper timein the new coordinate system.

The transformation from comoving coordinates to theSchwarzschild-like coordinate system employed byHumphreys was derived by Klein:38

r a= η

tt

b

b

b

b

bo= ±

+ ++−

1 12

3

2 lnζζ

b

b+

++ +

+−

1

1 3

1 22

2

2 arctanζζ

π ζ

ta

c

a

a a

b

edge

edge

edge

edge

οη

ζηη

η

η

≡−

≡−

−−

−

≡−

max

max

max

1

1

11

1

2

2

2

2

In these expressions, ηedge is the comoving radial coordinateof the edge of the matter sphere. The + sign is appropriatefor a collapsing matter sphere and the sign for anexpanding one. It should be noted that there is an error inthe corresponding expression in Starlight and Time forthe coordinate time t (equation (20)39). That expression ismissing the leading (1 + b2)-1 and has a spurious sign inthe denominator of the middle term. In this new coordinatesystem, the metric is

ds r t c dt r t dr r d2 2 2 2 2 2= ( ) − ( ) −β α, , Ω

α r ta r

a

,max

( ) =−

1

12

3

β

η

η

ηη

η

r t

a

a

a

a

a

a

edge

edge

,

max

max max

( ) =

− −−( )

−

−

− −

−( )−

1 11

1

1 1 11

1

2 3 2

2

2

22

2

3

In order to use the transformed metric of equation (14)to compute the proper time elapsed on a comoving clock,it is necessary to transform the comoving clock trajectoryfrom η,τc coordinates to r, t coordinates. Recall that thecomoving coordinate trajectory of a comoving observer isgiven by the spatial coordinates fixed: dηcom obs = dθcom obs =dϕcom obs = 0. The θ and ϕ coordinates are the same for thecomoving and Schwarzschild coordinate systems, so wewill confine our attention to r and t. Differentiation ofequation (11) subject to the constraint dηcom obs = 0 gives

dr dacom obs com obs = η

As the Universe expands, comoving observers moveoutward from the coordinate origin. The Schwarzschildtime coordinate is given in equation (13) as a function of aand η. For a comoving observer, a is variable (since theUniverse expands) but η is fixed. Therefore, the differentialelement of Schwarzschild time elapsed along a comovingclock trajectory is

dt dat a

a

tb

a

a ada

Schw com obs

oedge

com obs

,

max

max

,

= ( )

= −+( ) −( )

−−

−( )

=

∂ η∂

ζ

ζ ζ

ηη

η η

3

2 2 2 2

2

2

1 2

21

1

1

Now we can substitute the comoving observerdifferential coordinate trajectories, equations (16) and (17)into the Klein metric, equation (14), to obtain thedifferential element of proper time elapsed along acomoving observers trajectory:

ds c d

r t dt r t dr

com obs

com obs com obs com obs com obs com obs com obs

2 2

2 2

=

( ) − ( )τ

β α

2

= , ,

(as before, dθcom obs = dϕcom obs = 0). The evaluation of thisexpression is straightforward but tedious. The reader mayperform it for himself or refer to the Supplement, wherewe work it out step by step. The result is

dda

c

a

a adcom obs cτ τ = ±

−=

max

which is identical to the result (9) derived from theRobertson-Walker metric and the dynamical equations ofcosmology.

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(13)

(14)

(15)

(16)

(17)

(18)

(19)

(15)

where

where

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Appeal to Schwarzschild coordinates cannot changethe amount of time which passes on a physical clock; thatis determined by the trajectory of the clock and not by thecoordinate system. This exercise is an illustration of thefact that the proper time elapsed along a space-timetrajectory is a scalar invariant. When one uses either theKlein form of the metric or the Robertson-Walker form ofthe metric (or any other form, for that matter, of the metricof this space-time geometry) to answer the question, Howmuch time elapses on Earth clocks (or distant galaxy clocks)as the Universe expands? one finds that the elapsed timeis equal to the cosmic time and is the same, regardless ofthe location of the clock inside the matter sphere.

Before moving on to consideration of whatSchwarzschild time is, it is worth noting that Oskar Klein,who devised38 the Schwarzschild-like coordinate systememployed by Humphreys in Starlight and Time, does notuse the coordinate time t as the time coordinate in his ownbounded cosmological model.2 There, to the extent thathe discusses time at all, he employs the conventional cosmictime/comoving clock proper time of standard cosmologytheory. He adopts an age of 13 billion years as a reasonableestimate of the age of his bounded model, based on thedata available in 1971.14

3.3 What is Schwarzschild Time and Why isAppeal to It Made in Starlight and Time ?

Humphreys acknowledges that Schwarzschild timedoes not correspond to the time kept by comoving clocks,that is, physical clocks on Earth and on distant galaxies.24

He also acknowledges, as the reader may verify byinspection of equation (12), that for certain values of ηand a the Schwarzschild time has an imaginary component.What kind of clocks would keep Schwarzschild time? Whatare the space-time trajectories of hypothetical clocks whichkeep Schwarzschild time? It is easy to show that the space-time trajectories of such hypothetical clocks are, for muchof the history of the Universe, physically impossibletrajectories, moving faster than the speed of light.

3.3.1 The Space-time Structure ofConstant tSchwarzschild

Hypersurfaces andSchwarzschild Clock Trajectories

In order to visualize constant tSchwarzschild surfaces andSchwarzschild clock trajectories in a way which makestheir physical absurdity manifest, it is desirable to recastthe coordinate description of a bounded cosmos into a formwhich makes the trajectories of light and of physicalobservers obvious. We do this by appealing to the ψ, χcoordinate system. The angle χ is an alternate comovingradial coordinate defined by η = sinχ.40 The radius ofcurvature a and cosmic time τc of the bounded sphere maybe expressed in terms of the parametric equations

τ ψ ψ ψ

ψ ψ ψ

ca

c

aa

a

( ) = −( )

( ) −( ) =

max

maxmax

sin

cos sin

2

21

22

where ψ is an angle which ranges from 0 at the beginningof the expansion to 2π at the end of the recollapse.6

Substitution of these into the k = + 1 Robertson-Walkermetric yields the simple form

ds a d d d2 2 2 2 2 2= − −( )ψ χ χsin Ω

The χ, ψ coordinate system yields a simple space-timediagram (see Figure 3) which closely resembles theMinkowski diagrams of Special Relativity. In thiscoordinate system, horizontal lines are hypersurfaces ofconstant spatial curvature a and constant comoving clockproper time τcom obs and cosmic time τc. These lines terminateat χedge = sin-1 ηedge, which is π/6 for the example used inStarlight and Time;6,17 vertical lines are the trajectoriesof comoving particles. Successive horizontal lines thusrepresent the Universe at successive moments in itsexpansion. Radial light trajectories, ds2 = 0, dϕ2 = dθ 2 = 0,manifestly obey the relation dχ = ±dψ, so radiallypropagating light rays travel on trajectories tilted at 45° tothe vertical, as shown in Figure 3.

Surfaces of constant tSchwarzschild are defined by ζconstant, or equivalently, ζ 2 constant. The relation η =sinχ along with equation (20) may be substituted into theexpression for ζ (equation (13)) to yield the equation ofconstant tSchwarzschild surfaces in the ψ, χ plane. With a littlesimplification, the result is

cos cosχ ψ ηζ

22

22

1

1=

−

+edge

A sample of representative constant tSchwarzschild surfaces isplotted as the dashed curves in Figure 4. Constant tSchwarzschildsurfaces are not horizontal and so do not coincide withconstant τc or τcom obs or constant a surfaces. It is also clearfrom this diagram that surfaces of constant tSchwarzschild arenot space-like for all a and η, since parts of them lie insidethe future light cone (less than 45° to the vertical). Theslope of the constant t surfaces is given by

d

dconst tSchw

χψ

ψ

χ

= −

tan

tan2

When this slope is steeper than 1, the surfaces are time-like. The boundary between the time-like and space-likeparts of the constant tSchwarzschild surfaces is plotted as thedotted line in Figure 4. In the regions of space-time belowthis curve, the constant tSchwarzschild surfaces are time-like there is an unambiguous future-past relationship betweenadjacent points on these parts of the constant tSchwarzschild

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surfaces. This means that these surfaces cannot besimultaneous surfaces for any physical clocks (this isdiscussed more fully in the Supplement).

The succession of constant tSchwarzschild hypersurfacesdefines the constant time surfaces in the Schwarzschildcoordinate system. The space-time trajectories orthogonalto these surfaces are the trajectories of the Schwarzschildclocks. These clock trajectories can be determined byfinding the radial trajectories dTµ = (dψSchw cl , dχSchw cl) whichare orthogonal to the tangents dSµ = (dψconst t Schw, dχconst t Schw)of the constant tSchwarzschild surfaces. The orthogonalitycriterion is given by the metric: gµνdTµdSν = 0. Using themetric appropriate to the (ψ, χ) coordinate system, we have

0 = −d d d dSchw cl const t Schw cl const tSchw Schwψ ψ χ χ

or

d

d

d

dSchw cl const tSchw

χψ

χψ

χψ

=

= −

tan

tan2

This equation may be integrated to yield the equation whichdefines the Schwarzschild clock trajectories:

sin sin

tan

max

max

χ ψ ηSchw clSchwcl

Schw cl

Schw cl

a

a

acons t

=r

2

2=

=

This leads to the unsurprising result that Schwarzschildclocks are stationary with respect to the Schwarzschildradius coordinate r. Since η = r/a, Schwarzschild clocks

Constant tSchwarzschild surfaces and Schwarzschild clock trajectoriesbounded k = +1 cosmos (ηedge = 0.5, χedge = π/6)

(24)

(25)

(26)

EH

A

Comoving radial angle χ

Exp

ansi

on a

ngle

ψ1.4

1.2

1

0.8

0.6

0.4

0.2

0

Figure 4. ψ, χ spacetime diagram of the Starlight and Time boundedcosmos showing surfaces of constant Schwardschildcoordinate time (dashed lines) and the trajectories ofSchwarzschild clocks (solid lines). The dotted line is theboundary below which the constant tSchwarzschild surfaces andSchwarzschild clocks are physically impossible. The heavydash-dot line is the β = 0 surface, which clearly does notcoincide with the event horizon (wavy line labelled ‘EH’).The wavy line ‘A’ is a radial inward light trajectory whichleaves the surface at the beginning of the expansion,arriving at Earth long before the event horizon.

0 0.2 0.4

Comoving radial angle χ = sin-1η0 0.1 0.2 0.3 0.4 0.5

A

Exp

ansi

on A

ngle

ψ =

cos

-1(1

- 2

a/a m

ax)

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

B

Constant τcomoving surfaces, comoving clock trajectories and lighttrajectories, k = +1 cosmos in ψ, χ coordinates

Figure 3. Space-time diagram in ψ, χ coordinates for the boundedcosmos discussed in Starlight and Time (k = +1, ηedge =0.5). The ψ coordinate parametrises the expansion andcomoving time/cosmic time passage via equations (20).ψ ranges from 0 to 2π; the range 0 to π/4 is shown here.Surfaces of constant spatial curvature a and constantcomoving clock time and cosmic time are horizontal lines(shown dashed). Comoving clock trajectories are verticallines (shown solid). Radial light trajectories travel at + 45°to the vertical. An inward (A) and outward (B) propagatinglight ray is shown by the labelled wavy lines.

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are not stationary with respect to the matter of the Universe.Representative Schwarzschild clock trajectories are plottedas the thin solid lines in Figure 4. The unphysical characterof Schwarzschild clocks is seen by the fact that part of thetrajectory (the part below the dotted line) of such clocksinvolves the clocks moving faster than the speed of lightwith respect to the matter of the Universe, somethingimpossible for real, physical clocks. From this diagram, itis obvious that Schwarzschild clocks travel inward withrespect to the matter of the Universe (which travels onvertical trajectories in this coordinate system) and that, inthe lower right part of the diagram, Schwarzschild clockstravel faster than the speed of light. In the same portion ofspace-time in which the constant tSchwarzschild surfaces aretime-like, the Schwarzschild clock trajectories are space-like.

The properties of time-like constant t surfaces andspace-like clock trajectories are impossible for any physicaltimekeeping system. Physical clocks are constrained totravel along time-like trajectories (within 45° of the verticalin the ψ, χ plane) and physically possible surfaces ofsimultaneity must be space-like (within 45° of thehorizontal in the ψ, χ plane).6 The time-like constanttSchwarzschild surfaces and space-like Schwarzschild clocktrajectories imply clocks which travel, with respect to thematter of the Universe, faster than the speed of light. Thisis a physical impossibility, and Schwarzschild clocks areconsequently physically impossible. If one confines onesattention to the space-time region above the dotted line inFigure 4, the Schwarzschild clock trajectories and constanttSchwarzschild surfaces are physically possible, but they areirrelevant, because they do not correspond to the real,physical clocks we are interested in, which are clocks onEarth and clocks associated with distant galaxies.

With this understanding ofSchwarzschild time, we are in a positionto evaluate Humphreys reliance onSchwarzschild time as the correct timemeasure in a bounded Universe.Humphreys rightly notes that comovingclocks and Schwarzschild clocks do notkeep the same time. It is obvious thatthis must be the case since Schwarzschildclocks move with respect to comovingclocks. Which of these clocks is moresuitable as a measure of how much timehas passed in the Universe since thebeginning of the expansion? During theearly part of the expansion of theUniverse, Humphreys favouredSchwarzschild clocks travel faster thanthe speed of light and thus do notcorrespond to physically possible clocks.Comoving clocks, on the other hand,travel along with the matter of theUniverse. Schwarzschild clocks

measure the time elapsed along space-time trajectorieswhich, early in the expansion, are physically impossibleand which at all times do not coincide with the trajectoriesof matter particles in the Universe. Comoving clocksmeasure the proper time elapsed along the trajectorieswhich matter particles actually follow in the Universe. Itis clear that comoving clocks are the correct answer to thequestion, Which clock tells how much time an observeron Earth or on a distant galaxy measures to elapse fromthe beginning of the expansion of the Universe to somelater moment in the expansion?Finally, we consider the claim (quote # 3 above) that, in anunbounded Universe, the Schwarzschild time would matchthe cosmic time or comoving clock proper time. Theunbounded k = + 1 Universe occurs when we take the edgeof the bounded Universe to be χedge = π.41 The trajectoriesof Schwarzschild clocks are still given by rSchw cl = aηSchw cl= constant, so relation (26) still gives the trajectories ofSchwarzschild clocks. The constant tSchwarzschild surfaces arestill orthogonal to the Schwarzschild clock trajectories, soit follows that these are given by

± ( ) ( ) =cos cos / tanχ ψconst t const tSchw Schwcons t

2 2

(+ for χ < π/2, - for χ > π/2). Schwarzschild clocktrajectories and constant tSchwarzschild surfaces for theunbounded case are shown in Figure 5. The shapes ofthese surfaces and trajectories are identical to the boundedUniverse surfaces and trajectories except that they extendall the way to χ = π, whereas the bounded case stopped atχ = π/6 (or χedge = sin-1 ηedge if a different choice of ηedge ismade). It is obvious from this figure that constant tSchwarzschildsurfaces do not correspond to constant τc surfaces (whichare horizontal) and that the trajectories of Schwarzschild

Constant tSchwarzschild surfaces and Schwarzschild clock trajectoriesunbounded k = +1 cosmos (χedge = π)

Exp

ansi

on a

ngle

ψ

1.5

1

0.5

00

0.5 1 1.5 2 2.5 3Comoving radial angle χ

Figure 5. ψ, χ space-time diagram of an unbounded (χedge = π) k = +1 cosmos, showingsurfaces of constant Schwarzschild time and the trajectories of Schwarzschildclocks. Schwarzschild clock trajectories are physically impossible in the regionbetween the dotted lines.

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clocks do not correspond to the trajectories of physicalcomoving clocks (which are vertical). Thus, contrary toHumphreys claim, it is not the case that the Schwarzschildtime coincides with the cosmic time in the unbounded BigBang cosmos.

3.3.2 Earth Standard Timeand Schwarzschild Time

The appeal to Schwarzschild time is puzzling in viewof the fact that in several places in Starlight and Time, itis explicitly affirmed that the relevant time measure is Earthproper time (which is also called Earth Standard Time).42

Further, Starlight and Time shows a plot of the expansionof its example bounded sphere model37 which clearlyindicates that this bounded model requires the same billionsof years (of order amax/c, where amax = 4 x 1010 light years)of proper time to expand as an unbounded model ofidentical interior matter density. Thus, in spite of theextensive appeal to Schwarzschild coordinate time, it isexplicitly demonstrated within the pages of Starlight andTime that the introduction of a boundary does not modifythe Earth proper time required for the Universe to expandto its present size.

3.4 ConclusionSchwarzschild time, appealed to so extensively in

Starlight and Time, is an irrevelant time coordinate whichcorresponds to the time kept by a set of hypothetical clockswhich move with respect to the matter of the Universe atall times, and at faster than light-speed at early times. Thistime coordinate tells us nothing about the time which ismeasured by real, physical clocks in the Universe. Despitethe fact that Humphreys asserts in a number of places thatthe relevant time quantity in the Universe is the time keptby real, physical, comoving clocks, when the need arisesto compute time passage on such real, physical clocks, heveers away from physical reality and focusses instead onthe physically irrelevant Schwarzschild coordinate time.It is true that, if one fails to notice that Schwarzschild clocksare not real, physical clocks on Earth and on distantgalaxies, this focus on Schwarzschild time can make itappear that the centre of the Universe is younger than theedge, but when one returns to reality and asks the question,how much time has elapsed on Earth clocks and on clockslocated in distant galaxies, one must drop Schwarzschildtime and compute the proper time elapsed on these clocks.This is a trivially easy calculation the metric tells usthat τcomoving = τcosmic. Real, physical clocks on Earth and ingalaxies located in different parts of a bounded Universeall keep the same time, cosmic time, and all remainsynchronised with each other throughout the expansion ofthe Universe (this permanent synchronisation is readilyverified by inspection of Figure 3). The dynamicalequations which govern the expansion of a boundedUniverse are identical to those which govern an unboundedUniverse with identical interior properties, so that the

expansion time-scales are identical for the two cases.It follows from this that, if a young-Universe bounded

cosmos model could be devised, an unbounded Universewith the same interior matter properties would also yield ayoung-Universe time-scale. However, as we show below,realistic young-Universe relativistic cosmological modelsare impossible.

4. MISUNDERSTANDINGS ABOUTTHE NATURE AND EFFECTS OF EVENT

HORIZONS IN THE UNIVERSE

The alleged effects of an event horizon in an expandingbounded Universe occupy a prominent place in the physicssections of Starlight and Time. These argue that thearrival of a shrinking event horizon at Earth would allowdistant regions of the Universe to age billions of yearsduring the passage of a single day of Earth time.43 However,the discussion of event horizons is as flawed as the rest ofthe book. The mistakes include: event horizons occurwhere the time-time metric component, gtt, vanishes,17,37

the reason for the absence of an event horizon in anunbounded cosmos is the absence of a large-scale patternof gravitational force,44 an event horizon stops the passageof time on physical clocks traversing it,37 and matter andlight cannot travel inward inside a past event horizon.45

4.1 What Event Horizons AreIn cosmology theory, the term horizon refers to a

space-time boundary, typically between regions of space-time which are accessible to an observer and those whichare not. An event horizon, in particular, a future eventhorizon, is a space-time boundary between regions fromwhich light can propagate to great distances and regionsfrom which it cannot. Space-time events within theboundary cannot be observed by observers far awaybecause light signals conveying information about theseevents cannot propagate to the distant regions. Becausean event horizon is the boundary between light trajectorieswhich can and cannot propagate to great distance, it followsthat the event horizon, considered through time, is itself alight-like three-surface, which is one in which at each pointthere is one light-like direction (gµνdxµdxν = 0) within thesurface, and all other directions within the surface are space-like (gµνdxµdxν < 0). Thus, a necessary though not sufficientcondition that a space-time trajectory xµ (λ) (where λ is aquantity which parametrises the trajectory) coincide withan event horizon is that gµνdxµdxν = 0 for all λ.

How can one locate the event horizon of a sphericallysymmetric geometry? In a static geometry, such as theSchwarzschild geometry, the event horizon will bestationary. In such a case, and adopting coordinates suchthat the metric is diagonal, it follows that gtt = 0 and dxeh

i = 0(i = 1, 2, 3) satisfies the null requirement g dx dxeh ehµυ

µ ν = 0 .In a dynamic geometry, the event horizon may not bestationary this is the case in the bounded cosmos

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proposed in Starlight and Time. In that case, the fact thatdxeh

i ≠ 0 implies that gtt = 0 cannot define the location ofthe event horizon, since those two conditions areincompatible with the null requirement g x xeh ehµυ

µ ν = 0(unless grr fortuitously vanishes wherever gtt vanishes,which it does not in the model in question).

4.2 Why There is an Event Horizon in theBounded Cosmos But Not in the

Unbounded CosmosStarlight and Time is correct in affirming that there

is an event horizon in the bounded cosmos which is absentin the unbounded cosmos, but the reason offered inexplanation of this is mistaken. It is claimed, falsely, thatthe reason is that there is a radial pattern of gravitationalforce in the bounded matter sphere which is absent fromthe corresponding region of an unbounded Universe. Weshowed in section 2 that the gravitational field is identicalfor the interior regions of the bounded and unboundedUniverse. The correct reason is that event horizons areglobal properties of the geometry which depend on theexistence of distant observers recall that the horizon isthe boundary between space-time regions which can andcannot be seen by distant observers. In the boundedcosmos, there is an external asymptotically flat space-timein which distant observers can reside, and from which adistinction can be made between space-time regions in theinterior of the matter sphere from which light can or cannotescape to the exterior. In the unbounded cosmos, on theother hand, there is no outside for light to escape to andconsequently no event horizon.6

4.3 The Location and Effects of the Past EventHorizon of an Expanding Bounded Universe

It will be easier to think about the past event horizonout of which an expanding bounded cosmos emerges afterfirst considering the future event horizon into which acollapsing bounded cosmos would fall. This event horizonappears, at some value of the expansion parameter a, atthe centre of the matter sphere and thereafter expandsoutward, eventually arriving at the surface as the surfacefalls inside its own Schwarzschild radius.6 To determinethe trajectory of the expanding event horizon inside thematter sphere, we need only keep in mind the meaning ofthe event horizon: the boundary between light rays whichcan and cannot escape to great distance. Clearly, anyradially outward-travelling light ray which arrives at thesurface before the surface falls within the event horizonwill escape, and any light ray which arrives at the surfaceafter the surface falls inside the horizon will not escape.The boundary between these two cases is the outward-travelling radial light trajectory which arrives at the surfaceexactly at the moment the surface falls inside itsSchwarzschild radius. This defines the event horizon inthe interior of the matter sphere.

If, in our Minkowski-like coordinate system (ψ, χ) we

label the moment the future event horizon arrives at thesurface by ψsurf, χsurf, the trajectory of the event horizoninside the matter is simply given by the outward radial lighttrajectory (dχ = dψ) which intersects this point: χeh (ψ) =ψ + χsurf ψsurf, where the range of ψ during which thehorizon propagates to the surface is (ψsurf χsurf < ψ < ψsurf).

The inward-travelling past event horizon which occursduring the expansion of a bounded cosmos is simply thetime-reversal of the future event horizon: it is the set of allradially-inward-travelling light trajectories which departfrom the surface at the moment the surface arrives at theSchwarzschild radius. The equation of the past eventhorizon in the interior of the matter sphere is given by χeh(ψ)= χsurf ψ + ψ surf. We show in the Supplement that, for thesample bounded cosmology employed in Starlight andTime (amax = 4 x 1010 light years, ηedge = 0.5, χedge = π/6) ,the event horizon reaches the surface at ψsur f = π/3, a =0.25amax. Figure 4 shows the past event horizon (the wavyline which heads inward from the surface at ψ = π/3). Thelocus on which β, the gtt component of the Klein metric,vanishes is shown as the heavy dash-dot line. The β = 0curve is clearly not coincident with the event horizon,illustrating the point made previously that gtt = 0 does notdefine the event horizon in a dynamic geometry.

4.4 What Effects Does the Event HorizonHave on Observers Traversing It?

Is it true, as claimed in Starlight and Time, that theevent horizon has radical effects on the passage of time foran observer traversing it or that, as the past event horizonreaches Earth, distant regions of the Universe ageenormously during the passage of a brief period of time onEarth? It is easy to see, by examination of Figure 4, thatneither of these claims is true.

From the previous discussion, it is clear that the pastevent horizon of an expanding bounded Universe is simplya particular light-like surface: that generated by the familyof inward-travelling radial light trajectories which leaveeach point of the surface of the matter at the moment thesurface expands to its Schwarzschild radius. Each pointin the interior of the matter lies on one of these trajectories,and the arrival of the event horizon at the position of aparticular comoving observer in the interior of the mattersimply corresponds to the arrival of one of these radiallight rays at his location. The light ray arrives at thecomoving observer and passes on toward the centre. Thatis the event horizon, as seen by such an observer. Thislight trajectory manifestly has no effect on the passage oftime for this observer. His comoving clock continues totick in concert with the expansion of the Universe and insynchrony with every other comoving clock on his constanta surface.

Starlight and Time asserts the contrary, but this isdue to a misunderstanding. The r, t part of the metric inSchwarzschild coordinates is

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ds c d c dt dr2 2 2 2 2 2= = −τ β α

Humphreys wrongly alleges that the vanishing of β(mistakenly identified with the event horizon) will causedτ to vanish, resulting in the momentary stopping ofnatural clocks37 during which, it is claimed, exterior partsof the Universe rapidly age. This claim is in error: thisequation gives dτ = 0 for finite dt only if dr = 0simultaneously. However, comoving clocks do not, ingeneral, have dr = 0. As we showed in section 3, comovingclocks have fixed η, so that dr = ηda = sinχda. In addition,at the moment β = 0, the coordinate time diverges to -∞,6,17,39 so that dt at this moment is arbitrarily large. Thus,the arrival of the β = 0 surface at any finite comovingradius χ will not stop comoving clocks at that radius:dτcomoving ≠ 0. As we showed previously, throughout theexpansion of the Universe, the elapsed proper time oncomoving clocks is given by

da

a a

da

ccomovingτ =−max

The vanishing of β is due to a singularity in Kleinscoordinate system. The coordinate time tSchwarzschild divergesto -∞ along a certain surface in the interior (wronglyidentified with the event horizon in Starlight and Time)and the gtt metric component must vanish on this surfaceto keep the proper time interval finite. One could redefinethe time coordinate in the interior of the matter sphere toremove this singularity or even move the singularity sothat it coincides with the true event horizon.6 However,due to the scalar invariant nature of the proper time,whatever modification of the coordinate system employed,it would still be the case that the time elapsed on physical,comoving clocks would be

da

a a

da

ccomovingτ =−max

Are any unusual aging effects in the distant Universeobserved by an Earthbound observer as the event horizonarrives at Earth? To determine this, we need to considerhow Earth-based observers receive information about thepassage of time at distant comoving clocks in a boundedUniverse. This information obviously arrives by means oflight signals. As shown in Figure 6, two signals emittedfrom some comoving radius χem at different values of theexpansion parameter ψ, ψem and ψem + dψem, will arrive atEarth at ψrec and ψrec + dψrec, respectively. Since an inward-travelling ray obeys the relation dχ = dψ and the two rayshave to travel the same ∆χ to arrive at Earth, it is clear thatthe dψrec = dψem. Recalling the coordinate transformationbetween τc and ψ,

da

cd a dcτ ψ ψ ψ ψ= −( ) = ( )max cos

21

gives dτc,em = a(ψem)dψem and dτc,rec = a(ψrec)dψrec. Sincedψrec = dψem, it follows that

d da

ac em c emrec

em

τ τψψ, ,= ( )

( )This means that the proper time interval separating thereception of the two signals dτc,rec will differ from the propertime interval separating the emission of the two signals,dτc,em by the factor a(ψrec)/a(ψem), the ratio of the size ofthe Universe when the signals arrive at Earth to the size ofthe Universe when the signals were emitted. In other words,distant clocks will be seen by an Earth-based observer torun slow (since the Universe is expanding, a(ψem) < a(ψrec)).Distant comoving observers would likewise observe Earthclocks to run slow. This is just the well-known redshifteffect. The same thing is true for signals arriving at Earthas the event horizon arrives: they will simply be redshifted(the two signals shown in Figure 6 have been chosen tobracket the event horizon, shown as the dash-dots line).Contrary to Starlight and Time, Earth observers will notwitness any rapid aging of the distant Universe as the eventhorizon arrives at Earth.

Ingoing light trajectories in a bounded k = +1 cosmos

∆ψrec

1.6

1.5

1.4

1.3

1.2

1.1

1

Comoving radial angle χ0 0.1 0.2 0.3 0.4 0.5

Exp

ansi

on a

ngle

ψ

∆ψem

EH

Figure 6. Earth observations of distant clock behaviour areunaffected by the event horizon of the bounded cosmos.The time interval at a distant galaxy between the emissionof two signals leads to a time interval at Earth between thereception of the two signals. Starlight and Time claimsthat the reception time interval shrinks to zero for signalsarriving near the arrival of the event horizon, resulting inan apparent speed-up of distant clocks when the eventhorizon arrives at Earth. As shown in the text, this is notso. Earth observers observe distant clocks to run slowerthan Earth clocks throughout the expansion of the boundedcosmos.

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4.5 Is It Impossible for Light to Travel InwardInside the Matter Sphere Until the Event Horizon

Has Shrunk to the Centre?Humphreys claims that light cannot travel inward in

the interior of the past event horizon of a white hole,which is what he calls the expanding bounded matterdistribution discussed in Starlight and Time.46 Thisassertion, coupled with the claim that the event horizondid not arrive at Earth until Day Four of Genesis 1, plays aminor role in the correlation of his cosmological proposalto the Genesis 1 narrative: it accounts for the absence ofvisible starlight from distant galaxies on Earth prior to DayFour.

Examination of the space-time diagrams we have beenusing demonstrates that this assertion is untrue. Figure 4shows the past event horizon (labelled EH) and an inwardradial light ray emitted from the edge of the matter sphereat the beginning of the expansion (labelled A). Clearly,light rays from the edge of the matter as well as from regionsin the interior can and do travel inward and reach Earthlong before the event horizon arrives at Earth. The lightray A arrives at Earth at ψ = π/6, or τcomoving = 0.47 x 109

yr, while the event horizon does not arrive at Earth until ψ= π/2, or τcomoving = 11.4 x 109 yr (where we have adoptedHumphreys choice of amax = 4 x 1010 light years).

4.6 Conclusion: Event HorizonsThe physics of event horizons plays a prominent role

in the new cosmological model proposed in Starlight andTime. There it is claimed that the shrinking event horizonof an expanding bounded Universe stops clocks as it passesthem, producing differential aging from centre to edge ofthe matter distribution. This effect is so powerful, it isclaimed, that the distant Universe ages billions of yearsduring the course of a single Earth day as the event horizonarrives at Earth. In addition, it is claimed that light cannottravel inward to Earth until after the event horizon arrivesat Earth.

These assertions are false. Humphreys understandingof the meaning and effects of event horizons is mistaken.Starlight and Time gets the location and consequencesof the past event horizon of its model cosmology all wrong.The past event horizon is simply a particular inward-travelling light-like surface. It has no consequences at allfor the passage of time on physical clocks in the interior ofa bounded Universe and no effect on the ability of light totravel inward in the interior of a white hole. Time passeson physical clocks in a bounded Universe in exactly thesame way as it does in an unbounded Universe, and lighttrajectories are likewise identical for the two cases.

5. OBSERVATIONAL REFUTATION OFSTARLIGHT AND TIME

In response to earlier analysis22 which pointed outincompatibilities between the model proposed in Starlight

and Time and the observed properties of the Universe,Humphreys has pleaded lack of time to develop this modelfully.20 In this section we show that no amount ofdevelopment can fix this models problems. Observationswhich thoroughly rule out young-Universe relativisticcosmologies are already available.

5.1 The Observed Invariance of ExtragalacticRedshifts is Incompatible With Expansion asRapid as That Required by Young-Universe

TheoryIt is easy to show that, in a Robertson-Walker

cosmology, a galaxy or QSO with cosmological redshift Zwill exhibit a present rate of change in its redshift of 6,47

dZ

dZ H H

oo em oτ

τ τ= +( ) − ( )( )1

where τ0 is the present cosmic time, H0 is the present valueof the Hubble parameter, τem(τ0) is the emission time forlight now arriving at Earth with redshift Z, and H(τem) isthe value of the Hubble parameter at that emission time.The value of H0 has been measured to be close to10-10 yr-1. Any young-Universe cosmology which acceptscosmic expansion as the cause of redshifts will require thatthe past value of H(τ) be large in order to expand theUniverse to its present size in a few thousand years or less.Humphreys model, for example, expands the Universemuch of the way to its present size in a period of six daysor less.48,49 Other proposals have been made,50 and thecommon feature of all these proposals is that H(τ) is verylarge in the past.

The simplest physical model which can produce rapidpast expansion is the supposition of a large value of thecosmological constant (this is proposed in Starlight andTime, for example) or an equation of state with slowlyvarying energy density (cosmic inflation51). In this case,the Hubble parameter is constant, Hlarge, and the cosmicscale factor varies as a(τc) ∝ exp(τcHlarge). We may obtaina lower estimate for the value of Hlarge from the fact thatthe largest redshift observed for discrete objects is at presentabout Zmax = 5, implying that the Universe has expandedby a factor of 1 + Zmax = 6 since the light from these objectswas emitted. If we assume that the rapid expansion of theUniverse from one sixth its present size to its present sizetook place during some Earth time period ∆τrapid, we canobtain an expression for the value of the Hubble parameterduring this period

HZ

l erapid rapid

argmaxln .=

+( ) ≈1 18

∆τ ∆τ

The length of the rapid expansion period, ∆τrapid, is afree parameter of the theory. A lower limit on the value ofHlarge may be obtained from the fact that ∆τrapid cannotexceed the age of the Universe. For a young Universemodel with ∆τrapid < 6000yr, Hlarge > 3 x 10-4yr-1

so that

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distant galaxies and QSOs should now exhibit redshiftchange rates with magnitudes in excess of 3 x 10-4 per year:

dZ

dyr

oτ< − × − −3 10 4 1

This is just a lower limit on the magnitude of the predictedpresent redshift change rate. Starlight and Time calls fora much larger expansion factor (about 1010 from theprimordial nucleosynthesis state52 in a much shorter time(six days or less), which would imply a present dZ/dτ of ≈1400yr-1, a manifestly absurd prediction.

Astronomical observations (see the discussion in theSupplement) rule out any changes as large as those ofequation (31), which in turn implies that the Universecannot have expanded to its present size as rapidly asrequired by young-Universe theology. Figure 7 shows aseries of measurements of dZ/dτ for a number of objectswith different redshifts. The measurements are allconsistent with zero observed change (the changes impliedby the standard model, ≈ 10-10yr-1, are undetectibly small)and are all incompatible with the large rates of changerequired by any young-Universe model based on relativity.

It might appear possible, in principle, to wriggle outof this argument by claiming, the Universe did not expandin a smooth fashion. Rather, it expanded in a series ofjerks, between which it was static. The observations whichshow ∆Z/∆τ ≈ 0 only show that the Universe was notexpanding during the time period between the emissiontimes of those observations. In such a case, a(τ) wouldhave the form of a series of steps or ramps and H(τ)would be a series of Dirac delta functions or top-hatfunctions at the times corresponding to the steps/ramps.This is an unsatisfactory response, however. The redshiftchange rate of equation (29) can be integrated to yield theobserved redshift of a source at arbitrary Earth time:

Z Z d Z H H emτ τ τ

ττ τ τ τ τ

2 1 1

2 1( ) ( )= + + ( )( ) ( ) − ( )( )[ ]∫ ' ' ' '

where the integration variable τ′ is the Earth observer timebetween two measurements of the redshift of an object(made at Earth times τ1 and τ2) and τem(τ′) is the emissiontime for a signal which arrives at Earth at Earth time τ′.Noting that the present Hubble parameter, H(τ′), isessentially zero and assuming that H(τem(τ′)) is zeroeverywhere except during the steps/ramps, the integralsimplifies to

Z Z Hem i l e ii

N

τ τ2 11

( ) ( )=

= − ∑∆τ , arg ,

where ∆τem,i is the length of the ith period of expansion (ofN total) which occurs between τem(τ1) and τem(τ2), and Hlarge,iis the value of the Hubble parameter during that period. Ifthe expansion is confined to a set of discrete steps or ramps,the values of Hlarge,i must be correspondingly larger thanthe smooth value Hlarge in order to expand the Universe toits present size. If any of these brief, but very rapid,

expansions occurs between the emission times of the firstand second light ray, there will be a difference in theobserved redshifts of the two rays. In order to expand theUniverse six-fold, since the emission of light from Z = 5quasars, it is necessary that the sum Σ∆τiHlarge,i over all thediscrete expansions occuring between the emission timeof Z = 5 light and the present must be at least 1.79. Thus,it is not possible to hide the expansion in a limited numberof very large jerks if there are many pairs of observationsZ1, Z2 for different objects at different redshifts whichindicate no redshift change.

In the astronomical literature there are thousands ofpairs of redshift observations of nearby and distant galaxiesand quasars separated by Earth time intervals of years todecades. The emission time intervals corresponding to theobservation dates of these measurements would denselyblanket a 6,000-year expansion history. No statisticallysignificant redshift variation has been detected in the 80+years of such measurements, ruling out the jerky 6,000-year expansion scenario. Most of these observations arenot sufficiently precise to rule out smooth expansion spreadover 6,000 years, but there are a few which can be used totest for this. Figure 7 shows the highest quality (that is,smallest uncertainty in dZ/dτ) data the authors have foundin the literature.6 These rule out the rapid expansionscenario proposed by Humphreys and any smoothexpansion scenario for which the expansion time is lessthan about 106 years.

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Present redshift lapse rate vs redshift

A

B

0

-5

-10

-15

log 10

– d

Z/d

τ Ear

th(y

r-1)

S1

S2

log10

Z

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5

Figure 7. Observed and predicted redshift change rates for young-Universe cosmologies and the standard model. Curve Ais the prediction of Starlight and Time , assuming 1010-fold expansion in six days. Curve B is the lower limitpossible in a 6000-year expansion, given observedredshifts as large as Z = 5. Curves S1 and S2 are theprecitions of standard Big Bang models with k = 0, Ω0 = 1and H0 = 100 and 40 km/s/Mpc, respectively. The datapoints show 2σ upper liits on the observed dZ/dτ for veryhigh precision measurements of several distant galaxiesand quasars.6 The observed limits on dZ/dτ are 100 to1000 times too small to be compatible with the young-Universe scenario.

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The measured upper limit on dZ/dτ sets an upper limiton the value of Hlarge (assuming expansion spread out over6,000 years, which makes Hlarge as small as it can be in theyoung-Universe scenario):

H yrl earg < − −10 6 1

This limit on Hlarge in turn sets a lower limit on how long ithas taken the Universe to expand from one sixth its presentsize, (that is, from the size it was when the light from Z = 5QSOs was emitted) to its present size:

Hl e fold ansionarg exp ln .∆τ 6 6 1 79− = =

implies that

∆τ 6618 10− > ×fold ansion yr exp .

The quantity ∆τ6-fold expansion is also the light travel time froma Z = 5 object. Thus, the tiny upper limits on observedredshift changes imply long expansion times and long lighttravel times.

5.2 The Light Travel Time from DistantGalaxies Places a Firm Upper Limit on the

Distance to Those GalaxiesThe whole premise of Starlight and Time is that it is

possible to construct a locally homogeneous and isotropiccosmological model in which light travels great distancesduring the passage of short periods of time on Earth clocks.It is straightforward to show that this is impossible theEarth time elapsed during the light travel gives a hard upperbound on the distance to the object from which the light istravelling. Conversely, the distance to a distant objectplaces a hard lower bound on the amount of Earth timewhich elapses as light travels from that object to Earth.This fact can be shown by a short mathematical proof. Wepresent a terse form of the proof below. Readers interestedin more elaboration are referred to our supplementary paper.

5.2.1 The Dyer-Roeder EquationIn a Universe which is homogeneous and isotropic in

the mean, and in which a fraction α of the total matter(assumed to be non-relativistic, an excellent approximation)is distributed smoothly (a fraction (1 - α) is clumped intogalaxies, clusters, etc.), the angular diameter distance53 DAobeys the differential equation

d D

dZ DA

o A

2

253

21

λαΩ= − +( )

where λ is the affine parameter along the light trajectoryfrom the source to Earth54 (λ = 0 at the beginning of thelight trajectory at the source at redshift Z and is λEarth(Z) atthe end of the trajectory when it reaches Earth) and Ω0 isthe present value of the cosmic density parameter.55 Theterminal conditions on equation 37 are DA(λEarth(Z)) = 0

and dDA/dλλEarth(Z) = -c/H0. 6

The angular diameter distance of any object may bedetermined by integrating equation (37) backwards fromλEarth(Z) to λ = 0 twice with respect to λ. Short of doingthis for all Z, α, and Ω0, we can derive a general relationbetween DA and λEarth(Z) as follows. From equation (37),it is obvious that DA(λ) curves toward the λ axis and fromthe terminal conditions, DA(λ) intersects the λ axis withslope -c/H0 (see Figure 8). By definition, DA is a positivequantity, so it follows that the angular diameter distanceof an object at redshift Z must obey the inequality

00

≤ ( ) ≤ ( )D Zc

HZA Earthλ

The affine parameter λ is related to cosmic time τc by54

d

d

a

H a Hc

c

τλ τ

= ( ) ≥0

0 0

1

where the the inequality follows since the past scale factorof the Universe a(τc) is smaller than the present scale factora0 in an expanding Universe. Given that a(τc) < a0 forcosmic times earlier than the present, it is obvious that

λτEarth

ccZ

c

H ac Z( ) ( ) ≤ ( )

0

∆τ

where ∆τc(Z) is the cosmic time elapsed during light travelfrom an object with redshift Z. Inequalities (38) and (40)together imply that

D Z

cZA

c( ) ≤ ( )∆τ

We have already shown that the elapsed cosmic time ∆τcis identical, to within a few parts in 106, to the elapsed

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(37)

DA

DA(Z)

0

cH Earth Z

λ b g

slope = -c/H0

0 λEarth(Z) λ

Figure 8. Diagram illustrating how the Dyer-Roeder equation(equation (37)) provides an upper limit on the angulardiameter distance in terms of the affine parameter λ. Fromthe diagram, it is obvious that DA(Z) < λEarth(Z)c/H0.

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Earth time, so that without loss of accuracy we may rewritethis as

D Z

cZA

Earth( ) ≤ ( )∆τ

where ∆τEarth(Z) is the Earth proper time elapsed duringlight travel from an object with redshift Z. Thus, in anarbitrary on-average locally homogeneous and isotropicexpanding Universe, the angular diameter distance providesan absolute lower bound to the light travel time as measuredby physical clocks on Earth.

Angular diameter distances are difficult to measure,but are related by DA(Z) = DL(Z)/(1+Z)2 to a more easilymeasured distance quantity called the luminosity distance,DL.56 Based on large numbers of measured luminositydistances and confirmed with a much smaller number ofmeasured angular diameter distances, it is found that theangular diameter distance to objects with redshifts muchless than unity is

D Z Zc

HZ light yearsA( ) = = ×

0

915 10

where we have taken H0 = 64 km/s/Mpc.57,58 The lighttravel time from low redshifts objects is consequently

∆τ Earth Z Z years( ) ≥ ×15 109

So, for example, the light travel time of an object withredshift Z = 0.1 must exceed 1.5 billion years.

The converse of this theorem is that, if the light-traveltime from an object must be small, the angular diameterdistance of that object cannot be large: DA(Z) < c∆τEarth(Z).This is an observable consequence of the Starlight andTime model: there should be no visible objects in theUniverse with an angular diameter distance of greater than6,000 light years. This is a preposterous proposition, andone which Humphreys himself rejects.59

This argument is strictly valid only if the visibleUniverse is homogeneous and isotropic in the mean, oneof the key assumptions of standard cosmology theory andone shared by Starlight and Time. However, it is possibleto generalise this argument to an arbitrary inhomogeneousUniverse (provided only that the Universe undergoesexpansion in the past). A proof of this is presented in theSupplement.

5.3 Conclusion: Observable Implications ofStarlight and Time

The recent-creation, rapid expansion approach tocosmology of Starlight and Time has two profoundobservable implications not recognised in that book. Thefirst of these is that extragalactic objects should exhibitlarge annual changes in redshift. The second is that novisible object in the Universe should have an angulardiameter distance greater than 6,000 light years. Both ofthese predictions are found to be false. No significant

changes in redshift of extragalactic objects have beenobserved in the almost 100 years of such measurements.The most precise measured upper limits on redshift changesimply that the Universe has been expanding for at leastone million years of Earth time, not the 6,000 or lessrequired by Starlight and Time. No extragalactic angulardiameter distances as small as 6,000 light years have everbeen observed. Indeed, the centre of the Milky Way galaxyis about four times this distance from Earth.60 These resultsapply, regardless of the details of the expansion. Thus, noadjustment of the expansion history can avoid the force ofthese consequences. These two phenomena rule out allpossible variations of young-Earth relativistic cosmologywhich possess the properties of past expansion and largepresent size, properties which Humphreys concedes doapply to the real Universe.

6. CONCLUSION

The arguments of Starlight and Time are profoundlyflawed and by no means offer a solution to the light-travel-time problem. The central premise of the book, that thelong time-scale implications of standard cosmology theoryare a consequence of the historically unbounded characterof the standard model, is mistaken; in fact, the presence orabsence of boundaries is irrelevant to the standard model,provided that the observable Universe is approximatelyhomogeneous and isotropic. The appeal to Schwarzschildcoordinate time is ill-conceived and avoids the centralquestion of which clocks are relevant and how much timeelapses on them as the Universe expands; the relevantclocks in a bounded Universe, as in an unbounded one, arecomoving clocks, which remain synchronised to cosmictime throughout the expansion of the Universe. The claimof profound time distortions accompanying the passage ofa past event horizon in an expanding bounded cosmos ismistaken; in fact, the past event horizon of an expandingbounded Universe produces no effects detectible byobservers inside the Universe. The rapid-expansionhypothesis required to bring the Universe to its presentsize in only a few thousand years leads to observableconsequences, such as rapid changes in redshift and a verylow upper limit on the observed distance to extragalacticobjects, which clearly do not occur in the real universe;conversely, the observed staticity of redshifts and theobserved large distances of extragalactic objects imply thatthe Universe has been expanding and light from distantgalaxies has been travelling toward Earth for spans of timefar longer than the brief time-scale permitted by young-Universe interpretation of the Bible.

The bounded cosmology approach of Starlight andTime leads to cosmological models which are identical tothe standard Big Bang model in their time-scaleimplications. It should not be promoted in the Church as ascientifically sound young-Universe alternative to Big Bangcosmology.

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REFERENCES

1. Humphreys, D. R., 1994. Starlight and Time: Solving the Puzzle ofDistant Starlight in a Young Universe, Master Books.

2. Klein, O., 1971. Arguments concerning relativity and cosmology.Science, 171:339345.

3. Misner, C. W., Thorne, K. S. and Wheeler, J. A., 1973. Gravitation,W. H. Freeman, pp. 10961132.

4. Humphreys, Ref. 1, pp. 11, 4349, 91, 100.5. Conner, S. R. Dynamical Problems of the Norman-Setterfield

Hypothesis, in preparation.6. Conner, S. R. and Page, D. N., 1998. The Big Bang Cosmology of

Starlight and Time, unpublished manuscript, 200+ pages, referred toin the text as the Supplement. Readers interested in obtaining theSupplement may contact Mr Conner by mail at 10 Elmwood Avenue,Vineland, NJ 08360, USA, or by email at <CERS_corresp@hotmail.com>.

7. Kolb, E. W. and Turner, M. S., 1990. The Early Universe, Frontiersin Physics, Vol. 69, Addison-Wesley, p. 8.

8. Peebles, P. J. E., 1993. Principles of Physical Cosmology, PrincetonSeries in Physics, Princeton University Press, p. 15.

9. Kolb and Turner, Ref. 7, pp. 314315.10. Davies, P., 1995. Are We Alone? Philosophical Implications of the

Discovery of Extraterrestrial Life, Basic Books, pp. 148149.11. Linde, A., Linde, D. and Mezhlumian, A., 1994. From the big bang

theory to the theory of a stationary universe. Physical Review D,49(4):1784.

12. Linde, Linde and Mezhlumian, Ref. 11, p. 1791.13. Humphreys, D. R., 1995. Seeing distant stars in a young Universe: a

new creationist cosmology. Bible Science News, 33(4):1417.14. Klein, Ref. 2, p. 342.15. Kolb and Turner, Ref. 7, pp. 812.16. Weinberg, S., 1972. Gravitation and Cosmology, John Wiley and

Sons, pp. 252255.17. Humphreys, Ref. 1, p. 119.18. Humphreys, Ref. 1, pp. 19, 100.19. Humphreys, Ref. 13, pp. 1617.20. Humphreys, D. R., 1995. How we can see a young Universe: a reply

to Conner and Page. Bible Science News, 33(7):12, 1619.21. Weinberg, Ref. 16, p. 338.22. Conner, S. R. and Page, D. N., 1995. Light travel time in Starlight

and Time. Bible Science News, 33(7):1216.23. Humphreys, Ref. 1, p. 91.24. Humphreys, Ref. 1, p. 92.25. Humphreys, Ref. 20, p. 17.26. Humphreys, Ref. 20, pp. 1719.27. Byl, J., 1997. On time dilation in cosmology. Creation Research

Society Quarterly, 34(1):2632.28. Humphreys, D. R., 1997. Its just a matter of time. Creation Research

Society Quarterly, 34(1):3234.29. Humphreys, Ref. 1, pp. 8990.30. Humphreys, Ref. 1, p. 89.31. Humphreys, Ref.1, pp. 91, 114.32. If one were calculating with extremely high precision, one could take

into account the motion of Earth with respect to the centre of mass ofthe Milky Way galaxy, the effect of the Galactic potential on Earthclocks and the motion of the Milky Way galaxy with respect to thelocal rest frame of the matter of the Universe. The correction to theresults cited below is tiny, only about one part in 105.33,34 For thepurposes of this discussion, the microscopic distinction between Earth

clocks and truly comoving clocks is ignored.33. Rees, M., 1995. Perspectives in Astrophysical Cosmology,

Cambridge University Press, p. 4.34. Smoot, G. F., 1996. Of cosmic background anistropies. In: Examining

the Big Bang and Diffuse Background Radiations: Proceedings ofthe 168th Symposium of the International Astronomical Union, M.Kafatos and Y. Kondo (eds), International Astronomical Union, p. 38.

35. Humphreys, Ref. 1, pp. 9192, 116.36. Peebles, Ref. 8, pp. 7576, 312.37. Humphreys, Ref. 1, p. 117.38. Klein, O., 1961. Werner Heisenberg und die Physik Unserer Zeit,

F. Vieweg and Sohn, Braunschweig, pp. 5872.39. Humphreys, Ref. 1, p. 118.40. Humphreys, Ref. 1, p. 94.41. Peebles, Ref. 8, pp. 298300.42. Humphreys, Ref. 1, pp. 29, 74, 120.43. Humphreys, Ref. 1, pp. 7879, 83, 126.44. Humphreys, Ref. 1, pp. 2627.45. Humphreys, Ref. 1, pp. 24, 108.46. Humphreys, Ref. 1, p. 108.47. Weinberg, Ref. 16, p. 451.48. Humphreys, Ref. 1, pp. 7679.49. Humphreys, Ref. 1, pp. 124126.50. DeYoung, D. B., 1995. The Hubble Law. Creation Ex Nihilo

Technical Journal, 9(1):711.51. Kolb and Turner, Ref. 7, pp. 270272.52. Humphreys, Ref. 1, pp. 123124.53. This distance is the ratio of the physical size L of an object to its apparent

angular extent ∆θ as seen from Earth: DA = L/∆θ.54. Dyer, C. C. and Roeder, R. C., 1973. Distance-redshift relations for

universes with some intergalactic medium. Astrophysical Journal,180:L31L34.

55. Peebles, Ref. 8, pp. 100, 312.56. Weinberg, Ref. 16, p. 423.57. Riess, A. G., Press, W. H. and Kirshner, R. P., 1996. A precise distance

indicator: type Ia supernova multicolor light-curve shapes.Astrophysical Journal, 473:88109.

58. Kundic;, T., Turner, E. L., Colley, W., Gott III, R., Rhoads, J., Wang,Y., Bergeron, L., Gloria, K., Long, D., Malhotra, S. and Wambsganns,J., 1997. A robust determination of the time delay in 0957 +561A, Band a measurement of the global value of Hubbles constant.Astrophysical Journal, 482:7582.

59. Humphreys, Ref. 1, pp. 10, 46, 84.60. Burton, W. B., 1988. Structure of our galaxy from observations of HI.

In: Galactic and Extragalactic Radio Astronomy, Springer-Verlag,pp. 295358 (p. 312).

Samuel R. Conner is a non-resident Ph.D. candidate inthe Physics Department at Massachusetts Institute ofTechnology, Boston, USA. His Ph.D. research project ison the subject of gravitational lensing and cosmology.

Don N. Page is a Professor of Physics in the TheoreticalPhysics Institute, University of Alberta, Edmonton, Alberta,Canada, and a cosmology researcher in the CosmologyProgram of the Canadian Institute for Advanced Research.