Something Out of Almost Nothing. Drunken Risibility.

Kant’s first antinomy makes the error of the excluded third option, i.e. it is not impossible that the universe could have both a beginning and an eternal past. If some kind of metaphysical realism is true, including an observer-independent and relational time, then a solution of the antinomy is conceivable. It is based on the distinction between a microscopic and a macroscopic time scale. Only the latter is characterized by an asymmetry of nature under a reversal of time, i.e. the property of having a global (coarse-grained) evolution – an arrow of time – or many arrows, if they are independent from each other. Thus, the macroscopic scale is by definition temporally directed – otherwise it would not exist.

On the microscopic scale, however, only local, statistically distributed events without dynamical trends, i.e. a global time-evolution or an increase of entropy density, exist. This is the case if one or both of the following conditions are satisfied: First, if the system is in thermodynamic equilibrium (e.g. there is degeneracy). And/or second, if the system is in an extremely simple ground state or meta-stable state. (Meta-stable states have a local, but not a global minimum in their potential landscape and, hence, they can decay; ground states might also change due to quantum uncertainty, i.e. due to local tunneling events.) Some still speculative theories of quantum gravity permit the assumption of such a global, macroscopically time-less ground state (e.g. quantum or string vacuum, spin networks, twistors). Due to accidental fluctuations, which exceed a certain threshold value, universes can emerge out of that state. Due to some also speculative physical mechanism (like cosmic inflation) they acquire – and, thus, are characterized by – directed non-equilibrium dynamics, specific initial conditions, and, hence, an arrow of time.

It is a matter of debate whether such an arrow of time is

1) irreducible, i.e. an essential property of time,

2) governed by some unknown fundamental and not only phenomenological law,

3) the effect of specific initial conditions or

4) of consciousness (if time is in some sense subjective), or

5) even an illusion.

Many physicists favour special initial conditions, though there is no consensus about their nature and form. But in the context at issue it is sufficient to note that such a macroscopic global time-direction is the main ingredient of Kant’s first antinomy, for the question is whether this arrow has a beginning or not.

Time’s arrow is inevitably subjective, ontologically irreducible, fundamental and not only a kind of illusion, thus if some form of metaphysical idealism for instance is true, then physical cosmology about a time before time is mistaken or quite irrelevant. However, if we do not want to neglect an observer-independent physical reality and adopt solipsism or other forms of idealism – and there are strong arguments in favor of some form of metaphysical realism -, Kant’s rejection seems hasty. Furthermore, if a Kantian is not willing to give up some kind of metaphysical realism, namely the belief in a “Ding an sich“, a thing in itself – and some philosophers actually insisted that this is superfluous: the German idealists, for instance -, he has to admit that time is a subjective illusion or that there is a dualism between an objective timeless world and a subjective arrow of time. Contrary to Kant’s thoughts: There are reasons to believe that it is possible, at least conceptually, that time has both a beginning – in the macroscopic sense with an arrow – and is eternal – in the microscopic notion of a steady state with statistical fluctuations.

Is there also some physical support for this proposal?

Surprisingly, quantum cosmology offers a possibility that the arrow has a beginning and that it nevertheless emerged out of an eternal state without any macroscopic time-direction. (Note that there are some parallels to a theistic conception of the creation of the world here, e.g. in the Augustinian tradition which claims that time together with the universe emerged out of a time-less God; but such a cosmological argument is quite controversial, especially in a modern form.) So this possible overcoming of the first antinomy is not only a philosophical conceivability but is already motivated by modern physics. At least some scenarios of quantum cosmology, quantum geometry/loop quantum gravity, and string cosmology can be interpreted as examples for such a local beginning of our macroscopic time out of a state with microscopic time, but with an eternal, global macroscopic timelessness.

To put it in a more general, but abstract framework and get a sketchy illustration, consider the figure.

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Physical dynamics can be described using “potential landscapes” of fields. For simplicity, here only the variable potential (or energy density) of a single field is shown. To illustrate the dynamics, one can imagine a ball moving along the potential landscape. Depressions stand for states which are stable, at least temporarily. Due to quantum effects, the ball can “jump over” or “tunnel through” the hills. The deepest depression represents the ground state.

In the common theories the state of the universe – the product of all its matter and energy fields, roughly speaking – evolves out of a metastable “false vacuum” into a “true vacuum” which has a state of lower energy (potential). There might exist many (perhaps even infinitely many) true vacua which would correspond to universes with different constants or laws of nature. It is more plausible to start with a ground state which is the minimum of what physically can exist. According to this view an absolute nothingness is impossible. There is something rather than nothing because something cannot come out of absolutely nothing, and something does obviously exist. Thus, something can only change, and this change might be described with physical laws. Hence, the ground state is almost “nothing”, but can become thoroughly “something”. Possibly, our universe – and, independent from this, many others, probably most of them having different physical properties – arose from such a phase transition out of a quasi atemporal quantum vacuum (and, perhaps, got disconnected completely). Tunneling back might be prevented by the exponential expansion of this brand new space. Because of this cosmic inflation the universe not only became gigantic but simultaneously the potential hill broadened enormously and got (almost) impassable. This preserves the universe from relapsing into its non-existence. On the other hand, if there is no physical mechanism to prevent the tunneling-back or makes it at least very improbable, respectively, there is still another option: If infinitely many universes originated, some of them could be long-lived only for statistical reasons. But this possibility is less predictive and therefore an inferior kind of explanation for not tunneling back.

Another crucial question remains even if universes could come into being out of fluctuations of (or in) a primitive substrate, i.e. some patterns of superposition of fields with local overdensities of energy: Is spacetime part of this primordial stuff or is it also a product of it? Or, more specifically: Does such a primordial quantum vacuum have a semi-classical spacetime structure or is it made up of more fundamental entities? Unique-universe accounts, especially the modified Eddington models – the soft bang/emergent universe – presuppose some kind of semi-classical spacetime. The same is true for some multiverse accounts describing our universe, where Minkowski space, a tiny closed, finite space or the infinite de Sitter space is assumed. The same goes for string theory inspired models like the pre-big bang account, because string and M- theory is still formulated in a background-dependent way, i.e. requires the existence of a semi-classical spacetime. A different approach is the assumption of “building-blocks” of spacetime, a kind of pregeometry also the twistor approach of Roger Penrose, and the cellular automata approach of Stephen Wolfram. The most elaborated accounts in this line of reasoning are quantum geometry (loop quantum gravity). Here, “atoms of space and time” are underlying everything.

Though the question whether semiclassical spacetime is fundamental or not is crucial, an answer might be nevertheless neutral with respect of the micro-/macrotime distinction. In both kinds of quantum vacuum accounts the macroscopic time scale is not present. And the microscopic time scale in some respect has to be there, because fluctuations represent change (or are manifestations of change). This change, reversible and relationally conceived, does not occur “within” microtime but constitutes it. Out of a total stasis nothing new and different can emerge, because an uncertainty principle – fundamental for all quantum fluctuations – would not be realized. In an almost, but not completely static quantum vacuum however, macroscopically nothing changes either, but there are microscopic fluctuations.

The pseudo-beginning of our universe (and probably infinitely many others) is a viable alternative both to initial and past-eternal cosmologies and philosophically very significant. Note that this kind of solution bears some resemblance to a possibility of avoiding the spatial part of Kant’s first antinomy, i.e. his claimed proof of both an infinite space without limits and a finite, limited space: The theory of general relativity describes what was considered logically inconceivable before, namely that there could be universes with finite, but unlimited space, i.e. this part of the antinomy also makes the error of the excluded third option. This offers a middle course between the Scylla of a mysterious, secularized creatio ex nihilo, and the Charybdis of an equally inexplicable eternity of the world.

In this context it is also possible to defuse some explanatory problems of the origin of “something” (or “everything”) out of “nothing” as well as a – merely assumable, but never provable – eternal cosmos or even an infinitely often recurring universe. But that does not offer a final explanation or a sufficient reason, and it cannot eliminate the ultimate contingency of the world.

Conformal Factor. Metric Part 3.

Part 1 and Part 2.

Suppose gab is a metric on a manifold M, ∇ is the derivative operator on M compatible with gab, and Rabcd is associated with ∇. Then Rabcd (= gam Rmbcd) satisfies the following conditions.

(1) Rab(cd) = 0.

(2) Ra[bcd] = 0.

(3) R(ab)cd = 0.

(4) Rabcd = Rcdab.

Conditions (1) and (2) follow directly from clauses (2) and (3) of proposition, which goes like

Suppose ∇ is a derivative operator on the manifold M. Then the curvature tensor field Rabcd associated with ∇ satisfies the following conditions:

(1) For all smooth tensor fields αa1…arb1 …bs on M,

2∇[cd] αa1…arb1 …bs = αa1…arnb2…bs Rnb1cd +…+ αa1…arb1…bs-1n Rnbscd – αna2…arb1…bs Ra1ncd -…- αa1…ar-1nb1…bs Rarncd.

(2) Rab(cd) = 0.

(3) Ra[bcd] = 0.

(4) ∇[mRa|b|cd (Bianchi’s identity).

And by clause (1) of that proposition, we have, since ∇agbc = 0,

0 = 2∇[cd]gab = gnbRnacd + ganRnbcd = Rbacd + Rabcd.

That gives us (3). So it will suffice for us to show that clauses (1) – (3) jointly imply (4). Note first that

0 = Rabcd + Radbc + Racdb

= Rabcd − Rdabc − Racbd.

(The first equality follows from (2), and the second from (1) and (3).) So anti-symmetrization over (a, b, c) yields

0 = R[abc]d −Rd[abc] −R[acb]d.

The second term is 0 by clause (2) again, and R[abc]d = −R[acb]d. So we have an intermediate result:

R[abc]d = 0 —– (1)

Now consider the octahedron in the figure below.

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Using conditions (1) – (3) and equation (1), one can see that the sum of the terms corresponding to each triangular face vanishes. For example, the shaded face determines the sum

Rabcd + Rbdca + Radbc = −Rabdc − Rbdac − Rdabc = −3R[abd]c = 0

So if we add the sums corresponding to the four upper faces, and subtract the sums corresponding to the four lower faces, we get (since “equatorial” terms cancel),

4Rabcd −4Rcdab = 0

This gives us (4).

We say that two metrics gab and g′ab on a manifold M are projectively equivalent if their respective associated derivative operators are projectively equivalent – i.e., if their associated derivative operators admit the same geodesics up to reparametrization. We say that they are conformally equivalent if there is a map : M → R such that

g′ab = Ω2gab

is called a conformal factor. (If such a map exists, it must be smooth and non-vanishing since both gab and g′ab are.) Notice that if gab and g′ab are conformally equivalent, then, given any point p and any vectors ξa and ηa at p, they agree on the ratio of their assignments to the two; i.e.,

(g′ab ξa ξa)/(gab ηaηb) =  (gab ξa ξb)/(g′ab ηaηb)

(if the denominators are non-zero).

If two metrics are conformally equivalent with conformal factor, then the connecting tensor field Cabc that links their associated derivative operators can be expressed as a function of Ω.