A Critical Look at the Standard Cosmological Picture

Abstract

The discovery that the Universe is accelerating in its expansion has brought the basic concept of cosmic expansion into question. An analysis of the evolution of this concept suggests that the paradigm that was finally settled into prior to that discovery was not the best option, as the observed acceleration lends empirical support to an alternative which could incidentally explain expansion in general. I suggest, then, that incomplete reasoning regarding the nature of cosmic time in the derivation of the standard model is the reason why the theory cannot coincide with this alternative concept. Therefore, through an investigation of the theoretical and empirical facts surrounding the nature of cosmic time, I argue that an enduring three-dimensional cosmic present must necessarily be assumed in relativistic cosmology—and in a stricter sense than it has been. Finally, I point to a related result which could offer a better explanation of the empirically constrained expansion rate.

Appendix: Concerning Schwarzschild-de Sitter as a Cosmological Solution

During the Essay Contest discussions, the critical remarks on this essay that were most important for me, and were by far the most probing, were those offered by George Ellis. Professor Ellis’ criticism of the final section indicated, first of all, that the brief mention I made there of a result from my Ph.D thesis was too underdeveloped to pique much interest in it—and in fact that, stated as it was there, briefly and out of context of the explicit analysis leading from Eq. (7.4) to (7.5), the point was too easily missed. He wrote that the model is ‘of course spatially inhomogeneous,’ when the spatial slices are actually homogeneous, but rather are anisotropic; and when I pointed out to him that this is so because, in the cosmological form of the SdS solution $r>0$ is timelike and $t$ is forever spacelike, he replied that ‘the coordinate notation is very misleading’.

So, one purpose of this appendix is to provide the intermediate calculation between Eqs. (7.4) and (7.5), that had to be left out of the original essay due to space limitations—and, in developing a familiarity with the common notation, through the little calculation, to ensure that no confusion remains in regard to the use of $r$ as a timelike variable and $t$ as a spacelike one. For the notation is necessary both in order to be consistent with every other treatment of the SdS metric to date, and because, regardless of whether $r$ is timelike or spacelike in Eq. (7.4), it really does make sense to denote the coordinate with an ‘$r$’ because the space-time is isotropic (i.e. ‘radially’ symmetric) in that direction.

With these ‘bookkeeping’ items out of the way (after roughly the first four pages), the appendix moves on to address Professor Ellis’ two more substantial criticisms, i.e. regarding the spatial anisotropy and the fact that the model has no dynamic matter in it; for, as he noted, the model ‘is interesting geometrically, but it needs supplementation by a dynamic matter and radiation description in order to relate to our cosmic history’. These important points were discussed in the contest forum, but were difficult to adequately address in that setting, so the problem is given more proper treatment in the remaining pages of this appendix once the necessary mathematical results are in-hand. Specifically, in the course of developing a physical picture in which the SdS metric provides the description of a universe that would appear isotropic to fundamental observers who measure the same rate of expansion that we do (viz. as given by Eq. (7.5)), we will come to a possible, consistent resolution to the problem of accounting for dynamic matter, which leads to a critical examination of the consistency and justification of some of the most cherished assumptions of modern physics, thus further questioning its foundations.

We begin by writing down the equations of motion of ‘radial’ geodesics in the SdS geometry, using them to derive a description of the SdS cosmology that would be appropriate to use from the perspective of fundamental observers who evolve as they do, beginning from a common origin at $r=0$, always essentially because of the induced field potential. It will be proved incidentally that the observed cosmological redshifts, in this homogeneous universe which is not orthogonal to the bundle of fundamental geodesics—and is therefore precluded by the a priori assumptions of standard FLRW cosmology—must evolve through the course of cosmic time, as a function of the proper time of fundamental observers, with the precise form of the flat $\Lambda$CDM scale-factor—i.e., with exactly the form that has been significantly constrained through observations of type Ia supernovae, baryon acoustic oscillations, and CMBR anisotropies [15–26].

Since the Lagrangian,

$$\begin{aligned} L=-\frac{r-2M-\frac{\Lambda }{3}r^3}{r}\left( \frac{\mathrm {d}t}{\mathrm {d}\tau }\right) ^2+\frac{r}{r-2M-\frac{\Lambda }{3}r^3}\left( \frac{\mathrm {d}r}{\mathrm {d}\tau }\right) ^2=-1, \end{aligned}$$

(7.6)

for timelike $(r,t)$-geodesics with proper time $\tau$ in the SdS geometry is independent of $t$, the Euler-Lagrange equations indicate that

$$\begin{aligned} E\equiv -{1 \over 2}{\partial {L} \over \partial (\mathrm {d}t/\mathrm {d}\tau )}=\frac{r-2M-\frac{\Lambda }{3}r^3}{r}\left( \frac{\mathrm {d}t}{\mathrm {d}\tau }\right) \end{aligned}$$

(7.7)

is conserved ($-2\mathrm {d}E/\mathrm {d}\tau \equiv \mathrm {d}L/\mathrm {d}t=0$). Substituting Eq. (7.7) into (7.6), then, we find the corresponding equation of motion in $r$:

$$\begin{aligned} \left( \frac{\mathrm {d}r}{\mathrm {d}\tau }\right) ^2=E^2-\frac{r-2M-\frac{\Lambda }{3}r^3}{r}. \end{aligned}$$

(7.8)

While the value of $E$ may be arbitrary, we want a value that distinguishes a particular set of geodesics as those describing particles that are ‘fundamentally at rest’—i.e., we’ll distinguish a preferred fundamental rest frame by choosing a particular value of $E$ that meets certain physical requirements. In order to determine which value to use, we first note that where $r$ is spacelike, Eq. (7.8) describes the specific (i.e., per unit rest-mass) kinetic energy of a test-particle, as the difference between its (conserved) specific energy and the gravitational field’s effective potential,

$$\begin{aligned} V_{\mathrm {eff}}(r)\equiv \frac{r-2M-\frac{\Lambda }{3}r^3}{r}. \end{aligned}$$

(7.9)

Then, a reasonable definition sets the ‘fundamental frame’ as the one in which the movement of particles in $r$ and $t$ is essentially caused by the non-trivial field potential—i.e., so that $\mathrm {d}r/\mathrm {d}\tau =0$ just where the gravitational potential is identically trivial ($V_{\mathrm {eff}}\equiv 1$), and the line-element, Eq. (7.4), reduces to that of Minkowski space. From Eq. (7.8), this amounts to setting $E^2=1$; therefore, a value $E^2>1$ corresponds to a particle that would not come to rest at $r=-\root 3 \of {6M/\Lambda }$, where $V_{\mathrm {eff}}\equiv 1$, but has momentum in $r$ beyond that which would be imparted purely by the field.

As a check that the value $E^2=1$ is consistent with our aims, we can consider its physical meaning another way. First of all, note that where $V_{\mathrm {eff}}\equiv 1$, at $r=-\root 3 \of {6M/\Lambda }$, $r$ is spacelike and $t$ is timelike regardless of the values of $M$ and $\Lambda$; therefore, it is consistent in any case to say that a particle with $E^2>1$ has non-vanishing spatial momentum there. Indeed, from Eq. (7.7), we find that $t=\tau$ at $r=-\root 3 \of {6M/\Lambda }$ if, and only if, $E=1$—so the sign of $E$ should in fact be positive for a particle whose proper time increases with increasing $t$ in the absence of gravity.

Furthermore, note that when $\Lambda =0$, Eq. (7.8) reduces to

$$\begin{aligned} \left( \frac{\mathrm {d}r}{\mathrm {d}\tau }\right) ^2=E^2-1+\frac{2M}{r}. \end{aligned}$$

(7.10)

As such, Misner, Thorne and Wheeler describe $E$ as a particle’s ‘specific energy at infinity’, where the effective potential is trivial [42]. It is relevant to note their statement (on p. 658), that the conservation of 4-momentum ‘allows and forces one to take over the [term] $E=$“energy at infinity”..., valid for orbits that do reach to infinity, for an orbit that does not reach to infinity.’ More generally, we should describe $E$ for arbitrary $M$ and $\Lambda$ as the ‘energy at vanishing field potential’ even when $r=-\root 3 \of {6M/\Lambda }$ is a negative mathematical abstraction that lies beyond a singularity at $r=0$. In particular, we take $E=1$ to be the specific energy of a test-particle that is at rest with respect to the vanishing of the potential. It’s simply a matter of algebraic consistency.

Thus, we have $E=1$ as the specific energy of particles that would come to rest in the absence of a gravitational field, which are therefore guided purely through the effective field potential. We therefore use the geodesics with $E=1$ to define a preferred rest frame in the SdS geometries, and say that any particle whose world line is a geodesic with $E\ne 1$ is one that has uniform momentum relative to the fundamental rest frame.

We can now write the SdS line-element, Eq. (7.4), in the proper frame of a bundle of these fundamental geodesics, which evolve through $t$ and $r$ all with the same proper time, $\tau$, and occupy constant positions in ‘space’. Since $\Lambda$ must be positive in order to satisfy the requirement, ${\Lambda }M^2>1/9$, for $r>0$ to be timelike—i.e. the requirement for the SdS line-element to be cosmological rather than a local solution—it is more convenient to work with scale-invariant parameters $r\rightarrow {r'}=\sqrt{\frac{\Lambda }{3}}r$, $t\rightarrow {t'}=\sqrt{\frac{\Lambda }{3}}t$, $\tau \rightarrow {\tau '}=\sqrt{\frac{\Lambda }{3}}\tau$, $M\rightarrow {M'}=\sqrt{\frac{\Lambda }{3}}M$, etc., normalising all dimensional quantities by the cosmic length-scale $\sqrt{3/\Lambda }$ (see, e.g., § 66 in [4] or § 66 in [43] for interesting discussions of this length parameter). This normalisation ultimately amounts to striking out the factor $\Lambda /3$ from the $r^3$-term in the line-element $\left( \mathrm {since}\ \frac{\sqrt{\frac{\Lambda }{3}}r-2\sqrt{\frac{\Lambda }{3}}M-\left( \sqrt{\frac{\Lambda }{3}}r\right) ^3}{\sqrt{\frac{\Lambda }{3}}r}\rightarrow \frac{r-2M-r^3}{r}\right)$, or e.g. writing the flat $\Lambda$CDM scale-factor, Eq. (7.5), as

$$\begin{aligned} r(\tau )\propto \sinh ^{2/3}\left( 3\tau /2\right) , \end{aligned}$$

(7.11)

and the corresponding Hubble parameter as

$$\begin{aligned} H\equiv \dot{r}/r=\coth (3\tau /2), \end{aligned}$$

(7.12)

which exponentially approaches $H=1$ on timescales $\tau \sim 2/3$.

The evolution of each geodesic through scale-invariant $t$ and $r$ is then given, through Eqs. (7.7) and (7.8) with $E=1$, as

$$\begin{aligned} \partial _{\tau }{t}\equiv \frac{\partial {t}}{\partial \tau }=\frac{r}{r-2M-r^3}, \end{aligned}$$

(7.13)

$$\begin{aligned} (\partial _{\tau }r)^2\equiv \left( \frac{\partial {r}}{\partial \tau }\right) ^2=\frac{2M+r^3}{r}. \end{aligned}$$

(7.14)

Eq. (7.14) can be solved using $\int \left( u^2+a\right) ^{-1/2}\mathrm {d}u=\ln \left( u+\sqrt{u^2+a}\right)$ after substituting $u^2=2M+r^3$. Taking the positive root (so $\tau$ increases with $r$), we have,

$$\begin{aligned} \tau =\int \limits ^{r(\tau )}_{r(0)}\sqrt{\frac{r}{2M+r^3}}\mathrm {d}r=\left. \frac{2}{3}\ln \left( \sqrt{2M+r^3}+r^{3/2}\right) \right| ^{r(\tau )}_{r(0)}, \end{aligned}$$

(7.15)

where the lower limit on $\tau$ has been arbitrarily set to 0. Thus, in this frame we can express $r$ as a function of each observer’s proper time $\tau$ and an orthogonal (i.e. synchronous, with constant $\tau =0$) spatial coordinate, $r(0)$, which may be arbitrarily rescaled without altering the description in any significant way.

Then, as long as $M$ is nonzero, a convenient set of coordinates from which to proceed results from rescaling the spatial coordinate as³

$$\begin{aligned} r(0)\equiv (2M)^{1/3}\sinh ^{2/3}\left( \frac{3}{2}\chi \right) ;~M\ne 0, \end{aligned}$$

(7.16)

from which we find, after some rearranging of Eq. (7.15),⁴

$$\begin{aligned} r(\tau ,\chi )=(2M)^{1/3}\sinh ^{2/3}\left( \frac{3}{2}[\tau +\chi ]\right) , \end{aligned}$$

(7.17)

which immediately shows the usefulness of rescaling the $r(0)$ as in Eq. (7.16), since it allows Eq. (7.15) to be solved explicitly for $r(\tau ,\chi )$. As such, we immediately have the useful result (cf. Eq. (7.8)),

$$\begin{aligned} \partial _{\chi }r\equiv \frac{\partial {r}}{\partial \chi }=\partial _{\tau }r=\sqrt{\frac{2M+r^3}{r}}. \end{aligned}$$

(7.18)

The transformation, $t(\tau ,\chi )$, may then be calculated from

$$\begin{aligned} \frac{\mathrm {d}t}{\mathrm {d}r}=\frac{\partial _{\tau }t}{\partial _{\tau }r}=\frac{r}{r-2M-r^3}\sqrt{\frac{r}{2M+r^3}}. \end{aligned}$$

(7.19)

Then, to solve for $t(\tau ,\chi )$, we can gauge the lower limits of the integrals over $t$ and $r$, at $\tau =0$, by requiring that their difference, defined by

$$\begin{aligned} t(\tau ,\chi )=\int \limits ^{r(\tau ,\chi )}\frac{r}{r-2M-r^3}\sqrt{\frac{r}{2M+r^3}}\mathrm {d}r-F(\chi ), \end{aligned}$$

(7.20)

sets

$$\begin{aligned} 0=g_{\chi \tau }=g_{tt}\partial _{\tau }t\partial _{\chi }t+g_{rr}\partial _{\tau }r\partial _{\chi }r. \end{aligned}$$

(7.21)

(Thus, $\chi$ will be orthogonal to $\tau$.) This calculation is straightforward⁵:

$$\begin{aligned} 0&= -\frac{r}{r-2M-r^3}+\partial _{\chi }F(\chi )+\frac{r}{r-2M-r^3}\frac{2M+r^3}{r} \end{aligned}$$

(7.22)

$$\begin{aligned}&= \partial _{\chi }F(\chi )-1, \end{aligned}$$

(7.23)

so that $F(\chi )=\chi$.

Now, it is a simple matter to work out the remaining metric components as follows: our choice of proper reference frame immediately requires

$$\begin{aligned} g_{\tau \tau }=g_{tt}(\partial _{\tau }t)^2+g_{rr}(\partial _{\tau }r)^2=-1, \end{aligned}$$

(7.24)

according to the Lagrangian, Eq. (7.6); and by direct calculation, we find

$$\begin{aligned} g_{\chi \chi }&= g_{tt}(\partial _{\chi }t)^2+g_{rr}(\partial _{\chi }r)^2 \end{aligned}$$

(7.25)

$$\begin{aligned}&= -\frac{r-2M-r^3}{r}\left( \frac{2M+r^3}{r-2M-r^3}\right) ^{2}+\frac{r}{r-2M-r^3}\frac{2M+r^3}{r} \end{aligned}$$

(7.26)

$$\begin{aligned}&= (\partial _{\chi }r)^2. \end{aligned}$$

(7.27)

But this result is independent of any arbitrary rescaling of $\chi$; for if we replaced $\chi =f(\xi )$ in Eq. (7.16), we would then find the metric to transform as $g_{\xi \xi }=g_{\chi \chi }(d\chi /d\xi )^2=(\partial _{\xi }r)^2$, the other components remaining the same.

Therefore, the SdS metric in the proper frame of an observer who is cosmically ‘at rest’, in which the spatial coordinates are required, according to an appropriate definition of $F(\chi )$, to be orthogonal to $\tau$,⁶ can generally be written,

$$\begin{aligned} \mathrm {d}s^2=-\mathrm {d}\tau ^2+\left( \partial _{\chi }r\right) ^2\mathrm {d}\chi ^2+r^2\left( \mathrm {d}\theta ^2+\sin ^2\theta {\mathrm {d}\phi }^2\right) . \end{aligned}$$

(7.28)

This proves Lemaître’s result from 1949 [41],—that slices $\mathrm {d}\tau =0\left( \Rightarrow \left( \partial _{\chi }r\right) ^2\mathrm {d}\chi ^2=\right. \left. \left( \mathrm {d}r/\mathrm {d}\chi \right) ^2\mathrm {d}\chi ^2=\mathrm {d}r^2\right)$ are Euclidean, with line-element,

$$\begin{aligned} \mathrm {d}\sigma ^2=\mathrm {d}r^2+r^2\left( \mathrm {d}\theta ^2+\sin ^2\theta {\mathrm {d}\phi }^2\right) . \end{aligned}$$

(7.29)

However, in the course of our derivation we have also found that Lemaître’s physical interpretation—that the ‘geometry is Euclidean on the expanding set of particles which are ejected from the point singularity at the origin’—is wrong.

It is wrong to interpret this solution as describing the evolution of synchronous ‘space’ which always extends from $r=0$ to  $r=+\infty$ along lines of constant $\tau$, being truncated at the $r=0$ singularity at $\chi =-\tau$ from which particles are continuously ejected as $\tau$ increases. But this is exactly the interpretation one is apt to make, who is accustomed to thinking of synchronous spacelike hypersurfaces as ‘space’ that exists ‘out there’, regardless of the space-time geometry or the particular coordinate system used describe it.

As we noted from the outset, the ‘radial’ geodesics that we have now described by the lines $\chi =const.$, along which particles all measure their own proper time to increase with $\tau$, describe the world lines of particles that are all fundamentally at rest—i.e., at rest with respect to the vanishing of the effective field potential. Therefore, these particles should not all emerge from the origin at different times, and then somehow evolve together as a coherent set; but by Weyl’s principle they should all emerge from a common origin, and evolve through the field that varies isotropically in $r$, together for all time. In that case, space will be homogeneous, since the constant cosmic time ($\mathrm {d}r=0$) slices of the metric can be written independent of spatial coordinates; so every fundamental observer can arbitrarily set its spatial position as $\chi =0$ and therefore its origin in time as $\tau =0$.

The spaces of constant cosmic time should therefore be those slices for which $r(\tau +\chi )=const.$—i.e., we set $\bar{\tau }=\tau +\chi$ as the proper measure of cosmic time in the fundamental rest frame of the universe defined by this coherent bundle of geodesics, so that Eq. (7.17) becomes

$$\begin{aligned} r(\bar{\tau })=(2M)^{1/3}\sinh ^{2/3}\left( 3\bar{\tau }/2\right) . \end{aligned}$$

(7.30)

The spacelike slices of constant $\bar{\tau }$ are at $45^{\circ }$ angles in the $(\tau ,\chi )$-plane, and are therefore definitely not synchronous with respect to the fundamental geodesics. However, given this definition of cosmic time, the redshift of light that was emitted at $\bar{\tau }_e$ and is observed now, at $\bar{\tau }_0$, should be

$$\begin{aligned} 1+z=\frac{r(\bar{\tau }_0)}{r(\bar{\tau }_e)}, \end{aligned}$$

(7.31)

where $r(\bar{\tau })$ has exactly the form of the flat $\Lambda$CDM scale-factor (cf. Eq. (7.11)), which is exactly the form of expansion in our Universe that has been increasingly constrained over the last fifteen years [15–26].

Now, in order to properly theoretically interpret this result for the observed redshift in our SdS cosmology, it should be considered in relation to FLRW cosmology—and particularly the theory’s basic assumptions. As noted in Sect. “Implications for Cosmology”, the kinematical assumptions used to constrain the form of the line-element are: i. a congruence of geodesics, ii. hypersurface orthogonality, iii. homogeneity, and iv. isotropy. Assumptions i. and ii. have a lot to do with how one defines ‘simultaneity’, which I have discussed both in the context of special relativity in Sect. “The Cosmic Present”, and now in the context of the SdS cosmology, in which simultaneous events that occur in the course of cosmic time are not synchronous even in the fundamental rest frame. As the discussion should indicate, the definition of ‘simultaneity’ is somewhat arbitrary—and it is an assumption in any case—and should be made with the physics in mind. Einstein obviously had the physics in mind when he proposed using an operational definition of simultaneity [33]; but it has since been realised that even special relativity, given this definition, comes to mean that time can’t pass, etc., as noted in Sect. “The Cosmic Present”.

Special relativity should therefore be taken as an advance on Newton’s bucket argument, indicating that not only should acceleration be absolute, as Newton showed (see, e.g., [44] for a recent discussion of Newton’s argument), but velocity should be as well, since time obviously passes—which it can’t do, according to special relativity, if motion isn’t always absolute. Usually, however, the opposite is done, and people who have been unwilling or unable to update the subjective and arbitrary definitions of simultaneity, etc., from those laid down by Einstein in 1905, have simply concluded that Physical Reality has to be a four-dimensional Block in which time doesn’t pass, and the apparent passage of time is a stubborn illusion; see, e.g., Sect. 5 in [45] for a popular account of this, in addition to Refs. [34–38]. The discussion in Sect. “The Cosmic Present” shows how to move forward with a realistic, physical, and most importantly a relativistic description of objective temporal passage, which can be done only when ‘simultaneity’ is not equated with ‘synchronicity’ a priori; and another useful thought-experiment along those lines, which shows how perfectly acceptable it is to assume objective temporal passage in spite of relativistic effects, is presented in my more recent FQXi essay [46].

In contrast to the hardcore relativists who would give up temporal passage in favour of an operational definition of simultaneity, Einstein was the first relativist to renege on truly relative simultaneity when he assumed an absolute time in constructing his cosmological model [28]; and despite immediately being chastised by de Sitter over this [29], he never did balk in making the same assumption whenever he considered the cosmological problem [12, 31, 32]—as did just about every other cosmologist who followed, with very few notable exceptions (e.g., de Sitter [3, 29] was one, as there was no absolute time implicit in his model).

But whenever the assumption of absolute time has been made in cosmology, it has been made together with special relativity’s baggage, as the slices of true simultaneity have been assumed to be synchronous in the fundamental rest frame. Now we see that, not only is the operational definition wrong in the case of special relativity (since it comes to require that time does not pass, which is realistically unacceptable), but here we’ve considered a general relativistic example in which equating ‘simultaneity’ and ‘synchronicity’ makes even less sense in terms of a reasonable physical interpretation of the mathematical description, since the interpretation is causally incoherent—i.e. Lemaître’s interpretation, that the line-element Eq. (7.28) should describe an ‘expanding set of particles which are ejected from the point singularity at the origin’ represents abominable physical insight. The main argument of this essay was therefore, that while assumption i. of FLRW cosmology is justified from the point of view that relative temporal passage should be coherent, assumption ii. is not, and this unjustified special relativistic baggage should be shed by cosmologists—and really by all relativists, as it leads to further wrong interpretations of the physics.

Assumption iii. hardly requires discussion. It is a mathematical statement of the cosmological principle—that no observer holds a special place, but the Universe should look the same from every location—and is therefore as fundamental an assumption as the principle of relativity. Furthermore, our SdS universe is homogeneous, so there is no problem.

The final assumption, however, is a concern. The isotopy of our Universe is an empirical fact—it looks the same to us in every direction, and the evidence is that it must have done since its beginning. In contrast, the spatial slices of the cosmological SdS solution are not isotropic: they are a 2-sphere with extrinsic radius of curvature $r$, multiplied by another dimension that scales differently with $r$. Furthermore, by Eq. (7.19) we know that all these fundamental world lines move uniformly through this third spatial dimension, $t$, as $r$ increases.

The SdS cosmology therefore describes a universe that should be conceived as follows: First of all suppressing one spatial dimension, the universe can be thought of as a 2-sphere that expands from a point, with all fundamental observers forever motionless along the surface; then, the third spatial dimension should be thought of as a line at each point on the sphere, through which fundamental observers travel uniformly in the course of cosmic time.

Since it is a general relativistic solution, the distinction between curvature along that third dimension of space and motion through it is not well-defined. However, a possibility presents itself through an analogy with the local form of the SdS solution. As with all physically meaningful solutions of the Einstein field equations, this one begins with a physical concept from which a general line-element is written; then the line-element gets sent through the field equations and certain restrictions on functions of the field-variables emerge, allowing us to constrain the general form to something more specific that satisfies the requisite second-order differential equations. This is, e.g., also how FLRW cosmology is done—i.e., first the RW line-element satisfying four basic physical/geometrical requirements is written down, and then it is sent through the field equations to determine equations that restrict the form of the scale-factor’s evolution, under a further assumption that finite matter densities in the universe should influence the expansion rate. The local SdS solution, too, is derived as the vacuum field that forever has spacelike radial symmetry about some central gravitating body, and the field equations are solved to restrict the form of the metric coefficients in the assumed coordinate system. But then, as Eddington noted [4],

We reach the same result if we attempt to define symmetry by the propagation of light, so that the cone $ds=0$ is taken as the standard of symmetry. It is clear that if the locus $ds=0$ has complete symmetry about an axis (taken as the axis of $t$) $ds^2$ must be expressable by [the radially isotropic line-element with general functions for the metric coefficients].

Therefore, the local SdS metric corresponds to the situation in which light propagates isotropically, and its path in space-time is described by the null lines of a Lorentzian metric. Prior to algebraic abstraction (i.e. the assumption of a Lorentzian metric and a particular coordinate system), the geometrical picture is already set; and it is upon that basic geometrical set-up that the algebraic properties of the general relativistic field are imposed.

This construction of the local SdS solution through physical considerations of light-propagation can be used analogously in constructing a geometrical picture upon which the cosmological SdS solution can be based; however, a some more remarks are necessary before coming to that. First of all, as our discussion of the local SdS and FLRW solutions indicates, in general much of the physics enters into the mathematical description already in defining the basic geometrical picture and the corresponding line-element, which broadly sets-up the physical situation of interest. Only then is the basic physical picture further constrained by requiring that it satisfy the specific properties imposed by Einstein’s field law. In fact, when it comes to the cosmological problem, and we begin as always by assuming what will be our actual space, and how it will roughly evolve as actual cosmic time passes—i.e. by first assuming prior kinematical definitions of absolute space and time, and then constructing an appropriate cosmological line-element, which is finally constrained through the dynamical field law—there is really a lot of room to make it as we like it.

But now we have a particular line-element in mind (vizṫhe SdS cosmological solution), and we can use it in guiding our basic kinematical definitions. In particular, we have the description of a universe that is a two-dimensional sphere that expands as a well-defined function of the proper time of fundamental observers who all remain absolutely at rest, multiplied by another dimension through which those same observers are moving at a rate that varies through the course of cosmic time. According to the equivalence principle, it may be that the gravitational field is non-trivial along that particular direction of space, and therefore guides these fundamental observers along—or it could be that this direction is uniform as well, and that the fundamental observers are moving along it, and therefore describe it relatively differently.

What is interesting about this latter possibility, is that there would be a fundamental background metric describing the evolution of this uniform space, and that the metric used by fundamental observers to describe the evolution of space-time in their proper frames shouldn’t necessarily have to be the same fundamental metric transformed to an appropriate coordinate system. The metric itself might be defined differently from the background metric, for other physical reasons. In this case, an affine connection defining those world lines as fundamental geodesics may not be compatible with the more basic metric, and could be taken as the covariant derivative of a different one. The picture starts to resemble teleparallelism much more closely than it does general relativity; but since the two theories are equivalent, and we have recognised that in any case the kinematical definitions bust be made first—i.e. since we must set-up the kinematical definitions in the first place, according to the physical situation we want to describe, before ensuring that the resulting line-element satisfies the field equations—we’ll press on in this vein.

Let us suppose a situation where there is actually no gravitational mass at all, but fundamental inertial observers—the constituent dynamical matter of our system—are really moving uniformly through a universe that fundamentally is isotropic and homogeneous, and expands through the course of cosmic time. The fundamental metric for this universe should satisfy even the RW line-element’s orthogonality assumption, although the slices of constant cosmic time would not be synchronous in the rest frame of the fundamental observers. Since space, in the two-dimensional slice of the SdS cosmology through which fundamental observers are not moving, really is spherical, the obvious choice is an expanding 3-sphere, with line-element

$$\begin{aligned} \mathrm {d}s^2=-\mathrm {d}T^2+R(T)^2{\mathrm {d}\Omega _3}^2, \end{aligned}$$

(7.32)

where the radius $R(T)$ varies, according to the vacuum field equations, as

$$\begin{aligned} R(T) = C_1 e^{\sqrt{\Lambda \over 3}T}+C_2 e^{-\sqrt{\Lambda \over 3}T};\ C_1C_2=\frac{3}{4\Lambda }. \end{aligned}$$

(7.33)

In particular, because a teleparallel theory would require a parallelisable manifold, we note that this is true if

$$\begin{aligned} C_1=C_2=\frac{1}{2}\sqrt{\frac{3}{\Lambda }}, \end{aligned}$$

(7.34)

and the de Sitter metric is recovered (cf. Eq. (7.3)). While there may be concern because in this case $R(0)=\sqrt{3/\Lambda }>0$ is a minimum of contracting and expanding space, I will argue below that this may actually be an advantage.

This foliation of de Sitter space is particularly promising for a couple of reasons: i. the bundle of fundamental geodesics in Eq. (7.3) are world lines of massless particles, i.e. the ones at $r=0$ for all $t$ in Eq. (7.4) with $M=0$; and ii. unlike e.g. the 2-sphere (or spheres of just about every dimension), the 3-sphere is parallelisable, so it is possible to define an objective direction of motion for the dynamic matter.

Now we are finally prepared to make use, by analogy, of Eddington’s remark on the derivation of the local SdS metric in terms of light propagation along null lines. In contrast, we are now beginning from the description of a universe in which particles that are not moving through space are massless, but we want to write down a different Lorentzian metric to describe the situation from the perspective of particles that are all moving through it at a certain rate, who define null lines as the paths of relatively moving masselss particles—so, we will write down a new metric to use from the perspective of particles that all move along null lines pointing in one direction of de Sitter space, describing the relatively moving paths of massless particles that actually remain motionless in the 3-sphere, as null lines instead. This new line-element can be written,

$$\begin{aligned} \mathrm {d}s^2=-A(r,t)\mathrm {d}r^2+B(r,t)\mathrm {d}t^2+r^2\mathrm {d}\Omega ^2, \end{aligned}$$

(7.35)

where $r$ points in the timelike direction of the universe’s increasing radius, and $t$ describes the dimension of space through which the fundamental particles are moving. Solving Einstein’s field equations proves the Jebsen-Birkhoff theorem—that $A$ and $B$ are independent of the spacelike variable $t$—and leads to the cosmological SdS solution, Eq. (7.4), as the abstract description of this physical picture.

Thus, we have come full circle to a statement of the line-element that we started with. Our analysis began with a proof that in this homogeneous universe, redshifts should evolve with exactly the form that they do in a flat $\Lambda$CDM universe; and in the last few pages we have aimed at describing a physical situation in which this line-element would apply in the proper reference frame of dynamical matter, and the observed large scale structure would be isotropic. And indeed, in this universe, in which the spatial anisotropy in the line-element is an artifact of the motion of fundamental observers through homogeneous and isotropic expanding space and their definition of space-time’s null lines, this would be so—for as long as these fundamental observers are uniformly distributed in space that really is isotropic and homogeneous, all snapshots of constant cosmic time (and therefore the development, looking back in time with increasing radius all the way to the cosmological horizon) should appear isotropic, since these uniformly distributed observers would always be at rest relative to one another.

Having succeeded in showing how the SdS metric can be used to describe a homogeneous universe in which the distribution of dynamical matter appears isotropic from each point, and which would be measured to expand at exactly the rate described by the flat $\Lambda$CDM scale-factor—i.e. with exactly the form that has been observationally constrained [15–26], but which has created a number of problems because, from a basic theoretical standpoint, many aspects of the model are not what we expect—we should conclude with a brief discussion of potential implications emerging from the hypothesis that the SdS cosmology accounts for the fundamental background structure in our Universe.

The most obvious point to note is that the hypothesis would challenge what Misner, Thorne, and Wheeler called ‘Einstein’s explanation for “gravitation”’—that ‘Space acts on matter, telling it how to move. In turn, matter reacts back on space, telling it how to curve’ [42]. For according to the SdS cosmology, the structure of the absolute background should remain unaffected by its matter-content, and only the geometry of space-time—the four-dimensional set of events that develops as things happen in the course of cosmic time—will depend on the locally-inertial frame of reference in which it is described.

When I mentioned to Professor Ellis in the essay contest discussions, that I think the results I have now presented in this appendix may provide some cause to seriously reconsider the assumption that the expansion rate of the Universe should be influenced by its matter-content—a fundamental assumption of standard cosmology, based on ‘Einstein’s explanation for “gravitation”’, which Professor Loeb also challenged in his submission to the contest—his response was, ‘Well its not only of cosmology its gravitational theory. It describes solar system dynamics, structure formation, black holes and their interactions, and gravitational waves. The assumption is that the gravitational dynamics that holds on small scales also holds on large scales. It’s worked so far.’ And indeed, it has worked so far—but that is not a good reason to deny consideration of alternate hypotheses. In fact, as noted by Einstein in the quotation that began this essay, ‘Concepts that have proven useful in the order of things, easily attain such an authority over us that we forget their Earthly origins and accept them as unalterable facts...The path of scientific advance is often made impassable for a long time through such errors.’ From Einstein’s point of view, such a position as ‘It’s worked so far’ is precisely what becomes the greatest barrier to scientific advance.

After making this point, Einstein went on to argue that it is really when we challenge ourselves to rework the basic concepts we have of Nature that fundamental advances are made, adding [1],

This type of analysis appears to the scholars, whose gaze is directed more at the particulars, most superfluous, splayed, and at times even ridiculous. The situation changes, however, when one of the habitually used concepts should be replaced by a sharper one, because the development of the science in question demanded it. Then, those who are faced with the fact that their own concepts do not proceed cleanly raise energetic protest and complain of revolutionary threats to their most sacred possessions.

In this same spirit, we should note that not only our lack of a technical reason for why the Universe should ever have come to expand—i.e. the great explanatory deficit of modern cosmology described in Sect. “The Cosmic Present”—but also both the cosmological constant problem [47] and the horizon problem [48] are significant problems under the assumption that cosmic expansion is determined by the Universe’s matter-content. The former of these two is the problem that the vacuum does not appear to gravitate: an optimistic estimate, that would account for only the electron’s contribution to the vacuum energy, still puts the theoretical value $10^{30}$ larger than the dark energy component that cosmologists have measured experimentally [49]. The latter problem is that we should not expect the observable part of a general relativistic big bang universe to be isotropic, since almost all of what can now be seen would not have been in causal contact prior to structure formation—and indeed, the light reaching us now from antipodal points of our cosmic event horizon has come from regions that still remain out of causal contact with each other, since they are only just becoming visible to us, at the half-way point between them.

While there is no accepted solution to the cosmological constant problem, the horizon problem is supposed to be resolved by the inflation scenario [48]—an epoch of exponential cosmic expansion, proposed to have taken place almost instantly after the Big Bang, which would have carried off our cosmic horizon much faster than the speed of light, leaving it at such a great distance that it is only now coming back into view. Recently, provisional detection of B-mode polarisation in the CMBR [50] that is consistent with the theory of gravitational waves from inflation [51–53] has been widely lauded as a ‘proof’ of the theory. However, in order to reconcile apparent discrepancies with measurements of the CMBR’s anisotropy signature from the Planck satellite, BICEP2 researchers have suggested that ad hoc tweaks of the $\Lambda$CDM model may be necessary [54].

Details involving the emergence of dynamical matter in the SdS cosmology have not been worked out; however, there is no reason to suspect ab initio that the gravitational waves whose signature has potentially been preserved in the CMBR, would not also be produced in the scenario described here. More importantly, though, the SdS cosmology provides a description that precisely agrees with the observed large-scale expansion of our Universe, and does so without the need to invent any ad hoc hypotheses in order to ‘save the appearances’ that we have found to be so very different from our prior theoretical expectations. The theory simultaneously solves the expansion problem outlined in Sect. “On Cosmic Expansion” (since expansion must proceed at an absolute rate, regardless of the universe’s dynamical matter-content) and subverts the major issues associated with the assumption that cosmic expansion is determined by the Universe’s matter-content.

In the SdS cosmology, the vacuum can very well gravitate locally without affecting the cosmic expansion rate, and the universe should appear, from every point, to be isotropic out to the event horizon. Even the flatness problem—viz. the problem that the curvature parameter has been constrained very precisely around zero, when the expectation, again under the assumption that the Universe’s structure should be determined by its matter-content, is that it should have evolved to be very far from that—which inflation is also meant to resolve, is subverted in this picture. For indeed, the universe described here, despite being closed, would be described by fundamental observers to expand exactly according to the flat $\Lambda$CDM scale-factor, Eq. (7.30).

Additionally, the SdS cosmology provides a fundamentally different perspective on the so-called ‘coincidence problem’—viz. that the matter and dark energy density parameters of the standard FLRW model are nearly equal, when for all our knowledge they could easily be hundreds of orders of magnitude different. Indeed, if we write down the Friedman equation for a flat $\Lambda$CDM universe using Eq. (7.12) with dimensionality restored,

$$\begin{aligned} H^2=\frac{\Lambda }{3}\coth ^2\left( \frac{\sqrt{3\Lambda }}{2}\tau \right) =\frac{1}{3}(8\pi \rho +\Lambda ), \end{aligned}$$

(7.36)

we find that the matter-to-dark energy density ratio, $\varepsilon \equiv 8\pi \rho /\Lambda$, while infinite at the big bang, should approach zero exponentially quickly. For example, given the measured value $\Lambda \sim 10^{-35}~\mathrm {s}^{-2}$, we can write

$$\begin{aligned} \tau =\frac{2}{\sqrt{3\Lambda }}\mathrm {arcoth}\,\sqrt{\varepsilon +1}\approx 11~\mathrm {Gyr}\cdot \mathrm {arcoth}\,\sqrt{\varepsilon +1}. \end{aligned}$$

(7.37)

From here, we find that the dark energy density was $1\,\%$ of the matter density at $\tau (\varepsilon =100)=1$ billion years after the big bang, and that the matter density will be $1\,\%$ of the dark energy density at $\tau (\varepsilon =0.01)=33$ billion years after the big bang.

At first glance, these results may seem to indicate that it is not so remarkable that $\varepsilon$ should have a value close to 1 at present. However, from an FLRW perspective, the value of $\Lambda$ could really have been anything—and if it were only $10^4$ larger than it is (which is indeed still far less than our theoretical predictions), $\varepsilon$ would have dropped below $1\,\%$ already at $\tau =0.3$ billion years, and the Universe should now, at 13.8 billion years, be nearly devoid of observable galaxies (so we would have trouble detecting $\Lambda$’s presence). On the other hand, if $\Lambda$ were smaller by $10^4$, it would only become detectable after 100 billion years. Therefore, it is indeed a remarkable coincidence that $\Lambda$ has the particular order of magnitude that it has, which has allowed us to conclusively detect its presence in our Universe.

In contrast to current views, within the FLRW paradigm, on the problem that we know of no good reason why $\Lambda$ should have the particularly special, very small value that it has—which has often led to controversial discussions involving the anthropic principle and a multiverse setting—the SdS cosmology again does not so much ‘solve’ the problem by explaining why the particular value of $\Lambda$ should be observed, but really offers a fundamentally different perspective in which the same problem simply does not exist. For indeed, while the mathematical form of the observed scale-factor in the SdS cosmology should equal that of a flat $\Lambda$CDM universe, the energy densities described in Friedmann’s equations are only effective parameters within the former framework, and really of no essential importance. And in fact, as the analysis in the first part of this Appendix indicates (and again, cf. the sections by Eddington [4] and Dyson [43]), $\Lambda$ fundamentally sets the scale in the SdS universe: from this perspective, $\sqrt{3/\Lambda }$, which has an empirical value on the order of 10 billion years, is the fundamental timescale. It therefore makes little sense to question what effect different values of $\Lambda$ would have on the evolution of an SdS universe, since $\Lambda$ sets the scale of everything a priori; thus, the observable universe should rescale with any change in the value of $\Lambda$. But for the same reason, it is interesting to note that the present order of things has arisen on roughly this characteristic timescale. From this different point-of-view, then, the more relevant question to ponder is: Why should the structure of the subatomic world be such that when particles did coalesce to form atoms and stars, those stars evolved to produce the atoms required to form life-sustaining systems, etc., all on roughly this characteristic timescale?

Despite the SdS cosmology’s many attractive features, it may still be objected that the specific geometrical structure of the SdS model entails a significant assumption on the fundamental geometry of physical reality, for which there should be a reason; i.e., the question arises: if the geometry is not determined by the world-matter, then by what? While a detailed answer to this question has not been worked out (although, see Sect. 3.3 in [40]), it is relevant to note that the local form of the SdS solution—which is only parametrically different from the cosmological form upon which our analysis has been based; i.e. ${\Lambda }M^2<1/9$ rather than ${\Lambda }M^2>1/9$—is the space-time description outside a spherically symmetric, uncharged black hole, which is exactly the type that is expected to result from the eventual collapse of every massive cluster of galaxies in our Universe—even if it takes all of cosmic time for that collapse to finally occur. In fact, there seems to be particular promise in this direction, given that the singularity at $r=0$ in the SdS cosmology is not a real physical singularity, but the artifact of a derivative metric that must be ill-defined there, since space must actually always have a finite radius according to the fundamental metric, Eq. (7.32). This is the potential advantage that was noted above, of the finite minimum radius of the foliation of de Sitter space defined in Eq. (7.3). And as far as that goes, it should be noted that the Penrose-Hawking singularity theorem ‘cannot be directly applied when a positive cosmological constant $\lambda$ is present’ [55], which is indeed our case. For all these reasons, we might realistically expect a description in which gravitational collapse leads to universal birth, and thus an explanation of the Big Bang and the basic cosmic structure we’ve had to assume.

Along with such new possibilities as an updated description of collapse, and of gravitation in general, that may be explored in a relativistic context when the absolute background structure of cosmology is objectively assumed, the SdS cosmology, through its specific requirement that the observed rate of expansion should be described exactly by the flat $\Lambda$CDM scale-factor, has the distinct possibility to explain why our Universe should have expanded as it evidently has—and therein lies its greatest advantage.