Celebratio Mathematica

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.

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${}^{\ast }$ 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).

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 $\omega =\mathrm{G}J/({\mathrm{c}}^2{r}^3)$ 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 $\omega =\mathrm{G}M\mathrm{\Omega }/(\mathrm{c}r)$ with mass $M$ and an­gu­lar ve­lo­city $\mathrm{\Omega }$ 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, $27\mathrm{\%}$ of the mass in the uni­verse is pro­posed to be dark mat­ter (see [e15], for ex­ample).

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 $\omega =\mathrm{G}{M}^{\prime }\mathrm{\Omega }/\mathrm{c}$ 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 $\mathrm{G}M\mathrm{\Omega }/{\mathrm{c}}^2$ (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 $M\mathrm{\Omega }$. Note that $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$. 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 $J=M\mathrm{\Omega }{k}^2{R}_s^2$ for some factor $k$ of or­der 1, and hence de­duces $$\mathrm{\Omega }=\frac{{\mathrm{c}}^{4}J}{4{\mathrm{G}}^2{M}^2{k}^2}.$$ 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?

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).

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.

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 $\mathrm{\Lambda }$ 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 $\sqrt{\mathrm{\Lambda }/3}$ in any dir­ec­tion. Our uni­verse is be­lieved to be in a de Sit­ter phase [e17], with $\sqrt{\mathrm{\Lambda }/3}$ 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 $t^{\ast }$ be­fore which no light is re­ceived from the emit­ter (as an­ti­cip­ated by Weyl), and found there is an in­ter­val $(t^{\ast },t^{\ast }+t_B)$ dur­ing which light is re­ceived blue­shif­ted, and on $(t^{\ast }+t_B,+\mathrm{\infty })$ light is re­ceived red­shif­ted and asymp­tot­ic­ally sat­is­fy­ing the HLSW law with HLSW con­stant $\sqrt{\mathrm{\Lambda }/3}$. 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.

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.

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].

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]).

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 $g^{-1}:T^{\ast }M\to \mathbb{R}$ 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 $$1+z=\gamma (1+\frac{|v|}{c}\cos \theta )$$ 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.

After that, the big ques­tion is why does the res­ult­ing av­er­aged ra­di­ation have the ob­served tem­per­at­ure of $2.7K$? 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.

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.

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.

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 $${\sigma }_T{\int }_r^{\mathrm{\infty }}\frac{n_e}{\sqrt{Q}}dr$$ is of or­der 1 (in treat­ments of op­tic­al depth it seems con­ven­tion­al to choose $\frac{2}{3}$), 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 $E(r)=m{c}^2(1-\sqrt{Q})$, 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).