Celebratio Mathematica

Colin P. Rourke

Colin Rourke’s new paradigm for the universe

by R. S. MacKay

1. Introduction

Figure 1. The Zeeman building at the centre of the universe.

Colin’s in­terest in cos­mo­logy came to my at­ten­tion when I saw that he would give the first War­wick Math­em­at­ics In­sti­tute col­loqui­um in our new build­ing (Decem­ber 2003), with title “A new paradigm for the uni­verse”. In the end, the safety checks had not yet been com­pleted, so he gave it in the old build­ing. Non­ethe­less, to mark the oc­ca­sion, the Phys­ics De­part­ment kindly presen­ted a pho­to­graph show­ing the new build­ing at the centre of the uni­verse (ac­tu­ally, the centre of the M31 galaxy) (Fig­ure 1).

To sum­mar­ise Colin’s talk, here is the ab­stract for the pa­per he put on arX­iv the month be­fore [1]:

A new paradigm for the struc­ture of galax­ies is pro­posed. The main hy­po­thes­is is that a nor­mal galaxy con­tains a hy­per­massive black hole at its centre which gen­er­ates the spir­al arms. The paradigm gives sat­is­fact­ory ex­plan­a­tions for:

  • the ro­ta­tion curve of a galaxy;
  • the spher­ic­al bulge at the centre of a nor­mal galaxy;
  • the spir­al struc­ture and long-term sta­bil­ity of a nor­mal galaxy;
  • the age and or­bits of glob­u­lar clusters;
  • the ori­gin and pre­val­ence of sol­ar sys­tems;
  • (highly spec­u­lat­ive) the ori­gin of life.

The paradigm is com­pat­ible with dir­ect ob­ser­va­tions but not with many of the cur­rent in­ter­pret­a­tions of these ob­ser­va­tions. It is also in­com­pat­ible with large swathes of cur­rent cos­mo­lo­gic­al the­ory and in par­tic­u­lar with the ex­pand­ing uni­verse and hot big-bang the­or­ies. A tent­at­ive new ex­plan­a­tion for the ob­served red­shift of dis­tant ob­jects is giv­en which is com­pat­ible with a stat­ic mod­el for the uni­verse. This pa­per is be­ing cir­cu­lated in a very pre­lim­in­ary form in the hope that oth­ers will work both on in­ter­pret­ing ob­ser­va­tion­al data in the light of the new paradigm and on the ob­vi­ous gaps in un­der­ly­ing the­ory.

Far reach­ing! Sub­sequently, Colin de­veloped his ideas fur­ther, in­clud­ing firm­ing up the prom­ised ex­plan­a­tion of red­shift. It cul­min­ated in a book The Geo­metry of the Uni­verse [6], which forms the basis for this re­view.

I found Colin’s talk some­what whacky, but sev­er­al as­pects res­on­ated with me, in par­tic­u­lar that the big-bang story is over-simplist­ic, that red­shift could have al­tern­at­ive ex­plan­a­tions, and that dark mat­ter is wild spec­u­la­tion. So I took up his in­vit­a­tion “to help com­plete the work” [1] and we began talk­ing about it.

2. Galaxies

Figure 2. Gas cloud around Sagittarius A\( ^* \); the dark region is said to correspond to a black hole (from [e20]).

Colin’s paradigm starts with galax­ies. In par­tic­u­lar, he pro­posed that they have a hy­per­massive black hole at their centre. This might not sound re­volu­tion­ary, be­cause it was pro­posed more than 50 years ago [e6]. Fur­ther­more, im­ages have been pro­duced re­cently of the gas cloud around pro­posed black holes at the centres of the Messi­er 87 galaxy and our own one (Fig­ure 2). But al­though the mass of the ob­served black hole Sgr A\( ^* \) in the Milky Way is es­tim­ated at 4 mil­lion sol­ar masses, Colin’s pro­pos­al is that there is one of around \( 10^{11}-10^{12} \) sol­ar masses in a dif­fer­ent loc­a­tion (re­vised down from his earli­er es­tim­ate of \( 10^{15} \) sol­ar masses but still much lar­ger than the stand­ard one).

Figure 3. A CEERS2 image from the Webb space telescope with two successive blowups to show Maisie’s galaxy (from Science News).

He pro­posed that galax­ies with spir­al arms are around \( 10^{12}-10^{16} \) years old, longer than the time since the big bang (stand­ardly es­tim­ated at \( 13.8 \times 10^9 \) years ago), thereby chal­len­ging the big-bang story. Amus­ingly, ob­ser­va­tions from the re­cently launched James Webb space tele­scope are show­ing dis­tant galax­ies as they were 400–500 mil­lion years after the big bang, for ex­ample, [e21] and Fig­ure 3, but con­tain­ing stel­lar pop­u­la­tions that ac­cord­ing to some look over a bil­lion years old [e22] (though much scorn has been poured on that art­icle). So Colin is not alone today in ques­tion­ing the big-bang story.

A key in­gredi­ent of Colin’s new paradigm for galax­ies is an in­er­tial-drag ef­fect due to ro­ta­tion of the cent­ral black hole. Ein­stein’s gen­er­al re­lativ­ity is known to pro­duce in­er­tial-drag ef­fects, that is, Cori­ol­is and cent­ri­fu­gal forces in frames that do not ap­pear to be ro­tat­ing. In par­tic­u­lar, the Lense–Thirr­ing ef­fect makes an ef­fect­ive frame-ro­ta­tion rate \( \omega= \mathrm{G}J/(\mathrm{c}^2r^3) \) at dis­tance \( r \) from a ro­tat­ing black hole of an­gu­lar mo­mentum \( J \) (where \( \mathrm{G} \) is the grav­it­a­tion­al con­stant and \( c \) the speed of light), and this has been con­firmed by ob­ser­va­tions [e19]. But Colin’s pro­posed ef­fect has \( \omega = \mathrm{G}M\Omega/(\mathrm{c}r) \) with mass \( M \) and an­gu­lar ve­lo­city \( \Omega \) of the black hole. He ar­gues for it from Sciama’s ver­sion of Mach’s prin­ciple, which I’ll re­view shortly.

Colin uses this in­er­tial drag law to ex­plain the ro­ta­tion curves of galax­ies and to re­in­ter­pret quas­ars. I’ll ad­dress the ro­ta­tion curves first.

The conun­drum of the ro­ta­tion curves of galax­ies is that ac­cord­ing to New­to­ni­an grav­ity the tan­gen­tial speed of mat­ter out­side the main mass of a galaxy should de­crease like \( r^{-1/2} \) where \( r \) is the dis­tance from the centre, where­as in prac­tice it is al­most con­stant (at about \( 10^{-3} \) times the speed of light) (see Fig­ure 4). This sug­ges­ted that there is a lot of in­vis­ible mass fur­ther out than the vis­ible core. Named “dark mat­ter”, it is an in­noc­u­ous pro­pos­al (why should all mat­ter be vis­ible?) ex­cept that no-one has man­aged to de­tect any of it yet, apart from via its grav­it­a­tion­al ef­fect (though again, how else might one hope to de­tect it?). Dark mat­ter had been pro­posed in 1933 by Zwicky to ex­plain fail­ure of ob­ser­va­tions of vis­ible mat­ter to fit the viri­al the­or­em (though with later ob­ser­va­tions it seems the dis­crep­ancy is much smal­ler). It has also been found to be an es­sen­tial in­gredi­ent for the stand­ard mod­el of the uni­verse: to fit ob­ser­va­tions of red­shift as a func­tion of lu­min­os­ity dis­tance (Hubble’s law) and the cos­mic mi­crowave back­ground, \( 27\% \) of the mass in the uni­verse is pro­posed to be dark mat­ter (see [e15], for ex­ample).

Figure 4. Rotation curves of several galaxies (from [e13]).

Sciama’s ver­sion of Mach’s prin­ciple is that “the state­ment that the Earth is ro­tat­ing and the rest of the uni­verse is at rest should lead to the same dy­nam­ic­al con­sequences as the state­ment that the uni­verse is ro­tat­ing and the Earth is at rest” [e2]. Thus, ro­ta­tion of bod­ies in the uni­verse should pro­duce Cori­ol­is ef­fects. Mach’s prin­ciple is plaus­ible but ap­pears in­com­pat­ible with Ein­stein’s gen­er­al re­lativ­ity, des­pite hav­ing been part of the in­spir­a­tion for it. So Colin is mak­ing a modi­fic­a­tion to stand­ard phys­ics here. But he is in good com­pany: Sciama was an out­stand­ing phys­i­cist, PhD stu­dent of Dir­ac and su­per­visor of many in­flu­en­tial cos­mo­lo­gists, in­clud­ing Steph­en Hawk­ing. On the oth­er hand, Mach’s prin­ciple has also been ad­dressed by Piet­ronero [e8], who de­rived a frame-drag­ging ef­fect \( \omega = {\mathrm{G}M^{\prime}\Omega/\mathrm{c}} \) at the ax­is of a ro­tat­ing cyl­in­der of mass \( M^{\prime} \) per unit length from gen­er­al re­lativ­ity and ar­gued that this is a real­isa­tion of Mach’s prin­ciple. A nat­ur­al ques­tion is what is the ef­fect out­side the cyl­in­der? Does it de­cay like \( 1/r \) (just as in New­to­ni­an grav­ity, the po­ten­tial in­side a spher­ic­al shell is con­stant and de­cays like \( 1/r \) out­side)? Still, let us fol­low the im­plic­a­tions of Colin’s pro­posed frame-drag­ging law.

Colin’s ex­plan­a­tion of ro­ta­tion curves gives asymp­tot­ic tan­gen­tial speed \( \mathrm{G}M\Omega/\mathrm{c}^2 \) (in units of \( \mathrm{c} \)) for large \( r \). Ob­ser­va­tions show that it has much the same value (around \( 10^{-3} \)) for all galax­ies, so a re­main­ing chal­lenge is to ex­plain this value of \( M\Omega \). Note that \( \Omega \) is not an ob­serv­able quant­ity in stand­ard the­ory of black holes: only the mass \( M \) and an­gu­lar mo­mentum \( J \) are defined, not its an­gu­lar ve­lo­city \( \Omega \). Yet, there are nat­ur­al no­tions of the ra­di­us of a black hole, for ex­ample, the Schwar­z­schild ra­di­us \( R_s = 2\mathrm{G}M/\mathrm{c}^2 \). So Colin quite reas­on­ably writes \( J = M\Omega k^2 R_s^2 \) for some factor \( k \) of or­der 1, and hence de­duces \[ \Omega = \frac{\mathrm{c}^4 J}{4\mathrm{G}^2M^2k^2}. \] Then Colin’s tan­gen­tial speed is \( \mathrm{c}^2 J/(4k^2\mathrm{G}M) \). Not­ing the lim­it \( J \le \mathrm{G}M^2/\mathrm{c}^2 \) for Kerr solu­tions with an event ho­ri­zon, Colin’s the­ory could cor­res­pond to black holes choos­ing their an­gu­lar mo­mentum to be a factor about \( 10^{-3} \) from the Kerr lim­it. But is there a reas­on? Fur­ther­more, it is claimed from ob­ser­va­tions [e9] that the asymp­tot­ic speed for the ro­ta­tion curve is pro­por­tion­al to the fourth root of the lu­min­os­ity; Colin does not give much weight to this fit, but could it be ex­plained by his the­ory?

3. Quasars

Next, Colin used his in­er­tial drag law to give a new in­ter­pret­a­tion of quas­ars. These “quas­istel­lar ra­dio sources” are, ac­cord­ing to stand­ard the­ory, large black holes (up to \( 10^{11} \) sol­ar masses) sur­roun­ded by a gaseous ac­cre­tion disk that ra­di­ates elec­tro­mag­net­ic en­ergy con­ver­ted from part of the kin­et­ic en­ergy of in­fall­ing gas. They are con­sidered to be at large dis­tances, thereby ex­plain­ing their large red­shifts. Colin pro­poses (re­viv­ing a view dis­cussed 60 years ago but re­jec­ted [e4]) that they might be closer and con­sid­er­ably less massive, with their red­shift res­ult­ing from the ra­di­ation tak­ing place from a small ra­di­us about the black hole and be­ing red­shif­ted as it climbs out (grav­it­a­tion­al red­shift). Colin’s key idea is that his frame-drag­ging law al­lows the black hole to ab­sorb an­gu­lar mo­mentum from the gas, which oth­er­wise was a known obstacle to the­or­ies along this line. He thereby pro­posed that to ob­tain a rough idea of the pro­cess it is enough to con­sider ra­di­al in­fall onto a spher­ic­ally sym­met­ric black hole and to sup­pose that the kin­et­ic en­ergy of in­fall­ing gas is con­ver­ted to ra­di­ation com­ing prin­cip­ally from the Ed­ding­ton ra­di­us, which is the loc­a­tion where ra­di­ation pres­sure provides the ac­cel­er­a­tion ne­ces­sary to stop the in­fall. Colin re­futes the ob­jec­tions to this in­ter­pret­a­tion in con­sid­er­able de­tail.

Colin draf­ted a pa­per on his mod­el for quas­ars [5], ask­ing our joint PhD stu­dent Rosem­berg Toalá-Enríquez and me to fill in some parts. I still haven’t man­aged the parts Colin asked me to do (mak­ing a real­ist­ic mod­el for the pro­files of dens­ity, tem­per­at­ure and ion­isa­tion frac­tion, to find the prin­cip­al loc­a­tion from which the ra­di­ation comes), nor did I un­der­stand suf­fi­ciently Colin’s pro­pos­al about ig­nor­ing an­gu­lar mo­mentum, and I felt some­what un­com­fort­able hav­ing my name on the draft, so res­isted for a while but he pre­vailed. Re­read­ing it now, I un­der­stand his ar­gu­ment bet­ter. It would still be good to ex­am­ine his pro­pos­al in more de­tail (see the Ap­pendix to this re­view).

4. The HLSW law

As can be in­ferred from his ab­stract, an­oth­er im­port­ant top­ic of Colin’s work is re­in­ter­pret­ing Hubble’s law. Hubble’s law is that the ra­di­ation re­ceived from ob­jects is red­shif­ted by an amount pro­por­tion­al to their lu­min­os­ity dis­tance.

First, Colin makes some com­ments on no­men­clature. It was ac­know­ledged by the In­ter­na­tion­al As­tro­nom­ic­al Uni­on in 2018 that cred­it for Hubble’s law should be shared with Lemaître. Colin and I cer­tainly share this view and I have fol­lowed some of Lemaître’s his­tory. Fig­ure 5 shows the col­lege in Leuven where he lived around this time (the Kath­olieke Uni­versiteit had a col­lege sys­tem with a his­tory to rival Oxbridge’s). Colin ar­gues that the dis­cov­ery of the law goes back to Sli­pher and Weyl, so he refers to it as the HLSW law. Cer­tainly, Lemaître used Sli­pher’s ob­ser­va­tion­al data.

Figure 5. Holy Ghost College, Katholieke Universiteit Leuven.

The HLSW law is where our col­lab­or­a­tion began. First we played around with the thought that it could be a cu­mu­lat­ive res­ult of null geodesics passing through fluc­tu­ations in the met­ric; we didn’t reach a con­clu­sion, but see Sec­tion 7. Then after a study in spher­ic­ally sym­met­ric space-times [3], we found that the HLSW law is a fea­ture of all space-times hav­ing what we called a “nat­ur­al ob­serv­er field” [2] (note that al­though writ­ten after [3], it was pub­lished earli­er). Per­haps there is a bet­ter name, but we defined a nat­ur­al ob­serv­er field to be a time­like vec­tor field whose in­teg­ral curves are geodes­ic and whose per­pen­dic­u­lar plane field is in­teg­rable. This gen­er­al­ises the concept of the “Hubble flow”, the vec­tor field on a Fried­mann uni­verse point­ing purely in the time dir­ec­tion (in “co­mov­ing co­ordin­ates”), along which all mat­ter is as­sumed (ap­prox­im­ately) to flow (and would be bet­ter called the Fried­mann flow since it was part of Fried­mann’s mod­el). In par­tic­u­lar, a nat­ur­al ob­serv­er field in­duces a co­her­ent no­tion of time: con­stant on the per­pen­dic­u­lar space-slices and whose dif­fer­ence between two space-slices is the prop­er time along any field­line.

For any ob­ject, ra­di­ation emit­ted at fre­quency \( \omega_e \) in its frame is re­ceived by us at a fre­quency \( \omega_r \) pro­por­tion­al to \( \omega_e \), by con­ven­tion writ­ten as \( \omega_r = \omega_e/(1+z) \), with \( z \) called the red­shift. For space-times with a nat­ur­al ob­serv­er field we proved that if we are mov­ing along the ob­serv­er field, the red­shift \( z \) of an ob­ject also mov­ing along the ob­serv­er field is giv­en by \[ \log(1+z) = \int_{t_e}^{t_r} \rho(v(t))\, dt, \] where \( \rho(v) \) is the rate of ex­pan­sion of space in spa­tial dir­ec­tion \( v \) along the ob­serv­er field, \( v(t) \) is the spa­tial ve­lo­city of the null geodes­ic from the ob­ject to us at co­her­ent time \( t \) and \( t_e, t_r \) are the times of emis­sion and re­cep­tion, re­spect­ively. The HLSW law relates the red­shift to the lu­min­os­ity dis­tance \( d_L \). This is the dis­tance such that the re­ceived power per unit per­pen­dic­u­lar area (in the re­ceiv­er frame) is the emit­ted power per unit sol­id angle (in the emit­ter frame) di­vided by \( ((1+z)d_L)^2 \), the factor that one would ob­tain from the in­verse square law and the re­duc­tion of power by red­shift. For space-times with a nat­ur­al ob­serv­er field we ob­tained \[ d_L = \int_{t_e}^{t_r} (1+z(t))\, dt, \] where \( z(t) \) is the red­shift that an ob­serv­er mov­ing along the ob­serv­er field would see on cross­ing the null geodes­ic at its time \( t \). If \( \rho(v(t)) \) is ap­prox­im­ately a con­stant \( \rho_0 \) along the null geodesics then we de­rived \( z \approx \rho_0 d_L/\mathrm{c} \), the HLSW law with HLSW con­stant \( H_0 = \rho_0/\mathrm{c} \).

We found de Sit­ter space to provide an in­struct­ive ex­ample [2], [3]. De Sit­ter space is a solu­tion of Ein­stein’s equa­tions with pos­it­ive cos­mo­lo­gic­al con­stant \( \Lambda \) and neg­li­gible mat­ter. It has ex­pand­ing nat­ur­al ob­serv­er fields, namely the ve­lo­city field of any bundle of time­like geodesics all back­wards asymp­tot­ic to each oth­er. In­deed, Weyl [e1] de­clared that all mat­ter must flow along such a field, else for each ob­ject there would be a time up to which it would be in­vis­ible to us and after which it would be­come vis­ible, which he felt to be in­con­sist­ent with ideas of de­term­in­ism (more on this later). These ob­serv­er fields have con­stant ex­pan­sion rate \( \sqrt{\Lambda/3} \) in any dir­ec­tion. Our uni­verse is be­lieved to be in a de Sit­ter phase [e17], with \( \sqrt{\Lambda/3} \) equal to the HLSW time \( H_0^{-1} \approx 14.4 \times 10^9 \) years, so it is a reas­on­able mod­el. To in­clude some mat­ter we ex­ten­ded our il­lus­tra­tion to Kot­tler space (also known as Schwar­z­schild–de Sit­ter space), which can be con­sidered to be de Sit­ter space with a single non­ro­tat­ing black hole (or more pre­cisely an an­ti­pod­al pair of black holes) [2], [3]. In Kot­tler space the ex­pan­sion rate de­pends on dir­ec­tion.

Then we dropped the con­straint of sup­pos­ing that all mat­ter flows along a nat­ur­al ob­serv­er field. Cal­cu­lat­ing ar­bit­rary time-like geodesics in de Sit­ter space, and the null geodesics from one to the oth­er, we con­firmed that for an emit­ter-re­ceiv­er pair in gen­er­al po­s­i­tion there is a re­ceiv­er time \( t^* \) be­fore which no light is re­ceived from the emit­ter (as an­ti­cip­ated by Weyl), and found there is an in­ter­val \( (t^*,t^*+t_B) \) dur­ing which light is re­ceived blue­shif­ted, and on \( (t^*+t_B,+\infty) \) light is re­ceived red­shif­ted and asymp­tot­ic­ally sat­is­fy­ing the HLSW law with HLSW con­stant \( \sqrt{\Lambda/3} \). So the HLSW law could be the res­ult of pos­it­ive cos­mo­lo­gic­al con­stant and an ob­serv­er se­lec­tion-bi­as: an emit­ter is more likely to be seen with a red­shift near the HLSW law. A sim­il­ar con­clu­sion ap­plies to per­turb­a­tions of de Sit­ter space.

5. Gamma-ray bursts

Our study of de Sit­ter space had a re­mark­able spin-off, namely, it sug­ges­ted a pos­sible kin­emat­ic ex­plan­a­tion for gamma-ray bursts (GRB) [4], [7]. These are bursts of in­tense elec­tro­mag­net­ic ra­di­ation of cos­mic ori­gin that start ab­ruptly, dy­ing out over a few seconds or sev­er­al minutes (Fig­ure 6), whose fre­quency starts in the gamma range and de­creases. Stand­ard ex­plan­a­tions are in terms of cata­clys­mic events like a black hole eat­ing a neut­ron star, but we pro­pose a simple kin­emat­ic ex­plan­a­tion.

Figure 6. 12 GRB light curves; GRB_BATSE_12lightcurves.png in the public domain, created by Daniel Perley (see\_burst).

We already had the hint from the ana­lys­is that re­ceived light in de Sit­ter space starts blue-shif­ted. In­deed, it starts in­fin­itely blue-shif­ted (so, even high­er fre­quen­cies than gamma-rays). What about its in­tens­ity? Call the “in­tens­i­fic­a­tion” the ra­tio of re­ceived power per unit area to emit­ted power per unit sol­id angle. We com­puted the in­tens­i­fic­a­tion and found that it starts in­fin­ite. Both blue-shift and in­tens­i­fic­a­tion sub­sequently de­cay (with the blue-shift turn­ing to red after the blue­shift peri­od \( t_B \)). The typ­ic­al blue­shift peri­od is an HLSW time, which is far too long com­pared to the ob­served dur­a­tions, but we showed that the in­tens­i­fic­a­tion is much stronger for short­er \( t_B \), and the dis­tri­bu­tion of \( t_B \) for emit­ter geodesics chosen uni­formly with re­spect to the nat­ur­al Haar meas­ure has dens­ity like \( t_B^{-5} \). Thus we can ex­pect to see very short \( t_B \). The ob­ser­va­tions peak around \( t_B \) of the or­der of 20 seconds and fall off for short­er dur­a­tions, but this cutoff could have vari­ous ex­plan­a­tions, on which more later.

More gen­er­al space-times have sim­il­ar ef­fects (though not the Fried­mann uni­verses with big bang). So we pro­pose that many GRBs are just the nat­ur­al way that any emit­ter com­ing “over the ho­ri­zon” should look in a non-big-bang uni­verse. This fits with many ob­ser­va­tions, though is not ac­cep­ted by the main­stream as­tro­phys­ics com­munity.

The non­ac­cept­ance of our ex­plan­a­tion of GRBs was re­flec­ted in our at­tempts to pub­lish. We had a pa­per on GRB ready in 2011 but it was re­jec­ted by the journ­als we tried over sev­er­al years. To at­tempt to get some vis­ib­il­ity, we also pro­duced a punchy short ver­sion, but it was again re­jec­ted by all the places we tried (it is in­cluded as Ap­pendix F in [6]). Even­tu­ally the long ver­sion was ac­cep­ted as [4]. When I real­ised I could also ex­plain ob­ser­va­tions that GRBs are not po­lar­ised, I re­wrote the short ver­sion to in­clude that, but the journ­als I tried re­jec­ted it, and to cap it all, so did arX­iv! The best I found so far was to put it on Re­searchG­ate [7].

6. Big bang

As men­tioned near the be­gin­ning, a cent­ral fea­ture of Colin’s work is to re­ject the big bang. Colin bases this on a num­ber of ar­gu­ments. One is that there are glob­u­lar clusters that are dated to be 15 bil­lion years old, thus be­fore the big bang. An­oth­er is that the Hubble ul­tradeep field im­ages show fully formed galax­ies so far in the past that he says they did not have time to form.

My take on the big bang is that even if the con­di­tions for the Pen­rose–Hawk­ing sin­gu­lar­ity the­or­ems are sat­is­fied, the the­or­ems do not say that everything came out of a sin­gu­lar­ity, only that there are some in­com­plete back­ward geodesics. The strongest ver­sion I have seen is that there is a 3-para­met­er fam­ily of in­com­plete time­like back­ward geodesics [e7]. That leaves plenty of room for the rest of the uni­verse to keep go­ing back­wards as long as it likes.

I be­lieve that the idea that all came from a sin­gu­lar­ity is due to over-re­li­ance on the Fried­mann solu­tions of gen­er­al re­lativ­ity (start­ing with the “primev­al atom” the­ory of Lemaître, who re­dis­covered them). They are a spe­cial class of solu­tions in which there is a pre­ferred time-co­ordin­ate and the spa­tial slices are ho­mo­gen­eous and iso­trop­ic. Un­der plaus­ible phys­ic­al con­di­tions, they shrink to a sin­gu­lar­ity in fi­nite back­wards time. Fried­mann’s solu­tions were a very im­port­ant step in the un­der­stand­ing of the cos­mo­lo­gic­al con­sequences of gen­er­al re­lativ­ity (in keep­ing with their im­port­ance, they were re­dis­covered by not only Lemaître but also Robertson and then Walk­er), but gen­er­al re­lativ­ity has many more solu­tions. In par­tic­u­lar, a key con­sequence of Ein­stein’s geni­us was to lib­er­ate us from the concept of an ab­so­lute time and it is a pity that cos­mo­lo­gists put it back in. In par­tic­u­lar, even if there is a pre­ferred ve­lo­city field in space-time, in gen­er­al the per­pen­dic­u­lar plane field is not in­teg­rable (the Frobeni­us con­di­tion fails) so it does not define spa­tial slices nor an ab­so­lute time. We need to go bey­ond Fried­mann’s solu­tions (and not just to the vari­ous Bi­an­chi uni­verses [e11]).

7. Cosmic microwave background

The largest thing that re­mains for Colin or suc­cessors to firm up is an ex­plan­a­tion of the cos­mic mi­crowave back­ground (CMB) without a big bang. The stand­ard big-bang story is that the uni­verse was filled by a hy­dro­gen plasma and elec­tro­mag­net­ic ra­di­ation in ap­prox­im­ate equi­lib­ri­um, but as the uni­verse ex­pan­ded, the dens­ity and tem­per­at­ure fell, and at a tem­per­at­ure of about \( 3000K \) the plasma com­bined in­to hy­dro­gen atoms, the coup­ling between the mat­ter and ra­di­ation was lost, and the res­ult­ing ra­di­ation was red­shif­ted by fur­ther ex­pan­sion by a factor around 1100 to be­come what we see now as the al­most iso­trop­ic \( 2.7K \) cos­mic mi­crowave back­ground field. In­stead, we pro­pose that it is the res­ult of therm­al­isa­tion of ra­di­ation by in­homo­gen­eit­ies in the met­ric, per­haps in the form of grav­it­a­tion­al waves, as out­lined in Colin’s book.

Con­sider the bundle of null geodesics that reach us in some sol­id angle of dir­ec­tions over the product of some trans­verse sur­face (which I will call “aper­ture”) and time in­ter­val. Fol­low them back­wards. It is con­ven­tion­al to talk about go­ing back­wards along light rays for a giv­en time or a giv­en dis­tance, but that re­quires choos­ing a ref­er­ence frame. In a Fried­mann uni­verse (or with a nat­ur­al ob­serv­er field) there is a pre­ferred no­tion of time, but for a gen­er­al Lorent­zi­an met­ric there is not. For time-like geodesics one has an in­trins­ic no­tion of time between two points along them, namely the prop­er time, but the in­trins­ic time between any two points along a null geodes­ic is zero. Non­ethe­less, there is a no­tion of af­fine para­met­er for a null geodes­ic, that is, a para­met­risa­tion that is in­trins­ic up to shift of ori­gin and lin­ear scal­ing. We choose to fix the ori­gin of an af­fine para­met­risa­tion to be where the null geodes­ic crosses the product of the aper­ture and time in­ter­val, and choose the scal­ing to make the para­met­risa­tion agree with our prop­er time in a loc­al frame.

In gen­er­al, the bundle of null geodesics di­verges as one fol­lows it back­wards in af­fine para­met­er. At first it di­verges lin­early be­cause of the sol­id angle of dir­ec­tions, but fur­ther away one can ex­pect typ­ic­al nearby pairs of null geodesics to sep­ar­ate ex­po­nen­tially be­cause of de­vi­ations of the met­ric from a simple one. This is be­cause the equa­tions for null geodes­ic flow on a Lorent­zi­an space-time \( M \) with met­ric \( g \) are a Hamilto­ni­an sys­tem in 7D (the flow on the zero-level of the met­ric \( g^{-1}:T^*M\to \mathbb{R} \) on the co­tan­gent bundle), with the only gen­er­al sym­metry be­ing scal­ing of the co­tan­gent vec­tors, so we can ex­pect to see a pos­it­ive Lya­pun­ov ex­po­nent. In­deed, one can ex­pect the bundle of null geodesics to form 5D ver­sions of whorls and tendrils and mix trans­versely, just like time-de­pend­ent stir­ring of a lay­er of flu­id. So the bundle of null geodesics comes from a spread-out sample of points at early af­fine para­met­er, and what we see is an av­er­age over such a set.

Fur­ther­more, the fre­quency spec­trum we see is the res­ult of some ran­dom red­shift­ing. The red­shift­ing can be de­scribed by par­al­lel-trans­port­ing the emit­ter ve­lo­city (or the ve­lo­city of a frame in which the fre­quency dis­tri­bu­tion is known) along the null geodes­ic to the re­ceiv­er, where it can be writ­ten as \( (\gamma, \gamma v) \) for some \( v \in \mathbb{R}^3 \) and \( \gamma = \sqrt{1-|v|^2/c^2} \). The red­shift \( z \) is giv­en by \[ 1+z = \gamma \biggl(1+\frac{|v|}{c}\cos\theta\biggr) \] where \( \theta \) is the angle between \( v \) and the dir­ec­tion of re­ceipt of the null geodes­ic. Could the res­ult per­haps be a black-body spec­trum, with tem­per­at­ure \( T \) cor­res­pond­ing to the ra­di­ation dens­ity \( U \) by the stand­ard for­mula \( U=(4/\mathrm{c})\sigma T^4 \) (where \( \sigma \) is Stefan’s con­stant)? Even if one con­siders a very small sol­id angle of ini­tial dir­ec­tions, cor­res­pond­ing to look­ing in a pre­cise dir­ec­tion, and if one takes a small aper­ture, what we re­ceive can come from a very spread-out set. Colin makes the ana­logy with fog, where light paths are scattered a long way from a straight line.

One thing that needs es­tim­at­ing is the Lya­pun­ov time, the af­fine para­met­er change for one e-fold­ing in sep­ar­a­tion of null geodesics, that de­term­ines how fast the mix­ing oc­curs in back­wards time. There is plenty of evid­ence for op­tic­al dis­tor­tions, some of which is sur­veyed by Colin in ([6], Ap­pendix E), gen­er­ally at­trib­uted to pas­sage of light rays near something par­tic­u­larly massive or through dark mat­ter dis­tri­bu­tions, but some of which could be from grav­it­a­tion­al waves. Fig­ure 7 shows a more re­cent ex­ample. These dis­tor­tions are already seen at red­shifts of or­der 1, but more rel­ev­ant is prob­ably so-called “weak-lens­ing”, which is dis­tor­tion due to pas­sage through gen­er­al in­homo­gen­eit­ies. Ob­jects can be iden­ti­fied to at least \( z=10 \). So one could ex­pect a Lya­pun­ov time some­what more than this.

Figure 7. A famous early image from the James Webb telescope (from

After that, the big ques­tion is why does the res­ult­ing av­er­aged ra­di­ation have the ob­served tem­per­at­ure of \( 2.7 K \)? The tem­per­at­ure would cor­res­pond to that of the ra­di­ation at the op­tic­al depth, mod­i­fied by the red­shift from there to us (note that it seems now stand­ard to use the term “op­tic­al depth” for the log­ar­ithm of the in­cid­ent to trans­mit­ted power, but I learnt it as the depth to which one can dis­tin­guish ob­jects in the visu­al field, like “vis­ib­il­ity” in met­eor­o­logy, thus cor­res­pond­ing to trans­mit­ted power be­ing some frac­tion like \( 1/e \) of in­cid­ent).

And can we ex­plain the ob­served slight an­iso­trop­ies of the CMB? A big suc­cess of stand­ard cos­mo­lo­gic­al the­ory is to ex­plain them in terms of (prin­cip­ally) acous­tic os­cil­la­tions in the plasma be­fore the trans­ition to atoms [e15], [e18]. Al­though the the­ory of the amp­litude of acous­tic os­cil­la­tions de­pends on the hy­po­thet­ic­al no­tion of in­fla­tion, we have a steep chal­lenge to do bet­ter! Nev­er­the­less, per­haps the spec­trum of an­iso­trop­ies from our mech­an­ism is gov­erned by the same law; the Sachs–Wolfe an­iso­tropy ef­fects of grav­it­a­tion­al in­homo­gen­eit­ies (and a vari­ant pro­posed by Rees and Sciama) are called sec­ond­ary in the stand­ard the­ory.

8. Other topics

There is a lot more in Colin’s pa­per [1] and in his book [6], for ex­ample on the struc­ture of galax­ies, the abund­ance of ele­ments, and on quas­ars and galax­ies as life forms. I don’t feel com­pet­ent to re­view those, but re­com­mend in­ter­ested read­ers to study the book.

It is worth men­tion­ing that there are oth­er books chal­len­ging stand­ard cos­mo­logy, even by lead­ers in the field like Ro­ger Pen­rose [e16] and Stein­hardt & Tur­ok [e14]. It seems per­mit­ted to such au­thors to make far wilder spec­u­la­tions than Rourke’s.

9. Colin

I close with some per­son­al re­marks about Colin. He is very ar­tic­u­late (I re­mem­ber him des­troy­ing oth­er people’s ar­gu­ments in staff meet­ings). He is ready to ques­tion re­ceived dogma: we need more like him. And it is fun to dis­cuss ideas with him.

Appendix: Comments on Appendix C of “The geo­metry of the uni­verse” on quasars

As stated in Ap­pendix C of [6], the work on quas­ars re­por­ted there is still in pro­gress and is based on a draft pa­per [5] that we in­tend to get in­to a pub­lish­able form one day.

Ex­pand­ing on what is in­dic­ated in his Ap­pendix C, the main thing we felt is needed is a suit­able mod­el of the in­fall­ing gas to de­term­ine pro­files of dens­ity, tem­per­at­ure and ion­isa­tion frac­tion as func­tions of ra­di­al co­ordin­ate. Mestel [e3] gives an idea of what is in­volved but sim­pli­fies by pro­pos­ing two shells of con­stant tem­per­at­ure. Our pur­pose is to find the loc­a­tion from where the out­go­ing ra­di­ation can be con­sidered as com­ing. It is giv­en by \( r \) for which \[ \sigma_T \int_r^\infty \frac{n_e}{\sqrt{Q}}\, dr \] is of or­der 1 (in treat­ments of op­tic­al depth it seems con­ven­tion­al to choose \( \tfrac23 \)), where \( n_e \) is the dens­ity of elec­trons and \( \sigma_T \) is the Thom­son cross-sec­tion for scat­ter­ing of light by an elec­tron (or pos­sibly mod­i­fied ver­sions to take in­to ac­count oth­er scat­ter­ing pro­cesses). The factor \( \sqrt{Q} = \sqrt{1-2M/r} \) is to con­vert changes in \( r \) to changes in ra­di­al dis­tance (\( r \) is the Schwar­z­schild co­ordin­ate such that the sphere at con­stant \( r \) has area \( 4\pi r^2 \)).

I think there is a sim­pler prob­lem that needs fix­ing too, namely, (C.13) needs modi­fy­ing to \( E(r) = mc^2(1-\sqrt{Q}) \), be­cause the en­ergy re­ceived out­side is re­duced by a factor only \( 1/(1+z) \). This agrees with [e5], for ex­ample. It is power that is re­duced by \( 1/(1+z)^2 \). But I don’t think this cor­rec­tion has any ef­fect out­side sec­tion C.4.

I be­lieve also that in (C.6) both the ac­cre­tion rate \( A \) and the Ed­ding­ton lu­min­os­ity need di­vid­ing by \( \sqrt{Q} \), be­cause the ac­cre­tion rate in prop­er time at ra­di­us \( r \) is in­creased by a factor of \( 1/\sqrt{Q} \) and so is the re­lativ­ist­ic Ed­ding­ton lu­min­os­ity (see, for ex­ample, [e10], [e12]). But this makes no dif­fer­ence to the out­come of (C.6).


[1] C. Rourke: A new paradigm for the uni­verse. Pre­print, 2003. ArXiv abs/​astro-​ph/​0311033 techreport

[2] R. S. MacK­ay and C. P. Rourke: “Nat­ur­al ob­serv­er fields and red­shift,” J. Cos­mol. 15 (2011), pp. 6079–​6099. article

[3] R. S. MacK­ay and C. Rourke: “Nat­ur­al flat ob­serv­er fields in spher­ic­ally sym­met­ric space-times,” J. Phys. A, Math. The­or. 48 : 22 (2015), pp. 225204, 11. Id/No 225204. MR 3355220 Zbl 1318.​83002 article

[4] R. S. Mack­ay and C. Rourke: “Are gamma-ray bursts op­tic­al il­lu­sions?,” Palest. J. Math. 5 (2016), pp. 175–​197. MR 3477629 Zbl 1346.​83071 article

[5] C. Rourke, R. Toalá-En­ríquez, and R. S. MacK­ay: Black holes, red­shift and quas­ars. Pre­print, 2017. techreport

[6]C. Rourke: The geo­metry of the uni­verse. Series on Knots and Everything 71. World Sci­entif­ic Pub­lish­ing (Hack­en­sack, NJ), 2021. MR 4375354 book

[7] R. S. MacK­ay and C. P. Rourke: Po­lar­isa­tion of gamma-ray bursts, 2021. pre­print. misc