Excerpts from the final chapter of
astrophysicists Charles W. Misner, Kip S. Thorne and John Archibald Wheeler’s

Gravitation

(S.F.: W.H. Freeman & Co., 1970-71)

Conclusions in Chapter 44

« Beyond the End of Time »

Box 44.2  Three Levels of Gravitational Collapse (p. 1201)

  1. Universe
  2. Black hole
  3. Fluctuations at the Planck scale of distances
Recontraction and collapse of the universe is a kind of mirror image of the “big bang,” on which one already has so much evidence.
Collapse of matter to form a black hole is most natural at two distinct levels:  (a) collapse of the dense white-dwarf core of an individual star (when that core exceeds the critical mass ~1M or ~2M, at which a neutron star is no lonber a possible stable end-point for collapse) and (b) coalescence one by one of the stars in a galactic nucleus to make a black hole of mass up to 10⁶M or even 10⁹M.
In either case, no feature of principle about matter falling into the black hole is more interesting than the option that the observer has (symbolized by the branching arrow in the inset).  He can go along with the infalling matter, in which case he sees the final stages of collapse, but only at the cost of his own demise.  Or he can stay safely outside, in which case even after indefinitely long time he sees only the first part of the collapse, with the infalling matter creeping up more and more slowly to the horizon. Y
In the final stages of the collapse of a closed model universe, all black holes present are caught up and driven together, amalgamating one by one.  No one has any way to look at the event from safely outside;  one is inevitably caught up in it oneself.
Collapse at the Planck scale of distances is taking place everywhere and all the time in quantum fluctuations in the geometry and, one believes, the topology of space.  In this sense, collapse is continually being done and undone, modeling the undoing of the collapse of the universe itself, summarized in the term, “the reprocessing of the universe” (see text).

§44.3.  Vacuum Fluctuations:  Their Prevalence and Final Dominance (p. 1202)

If Einstein’s theory thus throws light on the rest of physics, the rest of physics also throws light on geometrodynamics.  No point is more central than this, that empty space is not empty.  It is the seat of the most violent physics.  The electromagnetic field fluctuates (Chapter 43).  Virtual pairs of positive and negative electrons, in effect, are continually being created and annihilated, and likewise pairs of mu mesons, pairs of baryons, and pairs of other particles.  All these fluctuations coexist with the quantum fluctuations in the geometry and topology of space.  Are they additional to those geometrodynamic zero-point disturbances, or are they, in some sense not well-understood, mere manifestations of them?

Put the question in other words.  Recall Clifford, inspired by Riemann, speaking to the Cambridge Philosophical Society on February 21, 1870, “On the Space Theory of Matter” [Clifford (1879), pp. 244 and 322;  and (1882), p. 21], and saying, “I hold in fact (1) That small portions of space are in fact of a nature analogous to little hills on a surface which is on the average flat;  namely, that the ordinary laws of geometry are not valid in them.  (2) That this property of being curved or distorted is continually being passed on from one portion of space to another after the manner of a wave.  (3) That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or etherial.  (4) That in the physical world nothing else takes place by this variation, subject (possibly) to the law of continuity.”  Ask if there is a sense in which one can speak of a particle as constructed out of geometry.  Or rephrase the question in updated language:  “Is a particle a geometrodynamic exciton?”  What else is there out of which to build a particle except geometry itself?  And what else is there to give discreteness to such an object except the quantum principle?

§44.5.  Pregeometry as the Calculus of Propositions (pp. 1208f.)

Paper in white the floor of the room, and rule it off in one-foot squares.  Down on one’s hands and knees, write in the first square a set of equations conceived as able to govern the physics of the universe.  Think more overnight.  Next day put a better set of equations into square two.  Invite one’s most respected colleagues to contribute to other squares.  At the end of these labors, one has worked oneself out into the door way.  Stand up, look back on all those equations, some perhaps more hopeful than others, raise one’s finger commandingly, and give the order “Fly!”  Not one of those equations will put on wings, take off, or fly.  Yet the universe “flies.” 

Some principle uniquely right and uniquely simple must, when one knows it, be also so compelling that it is clear the universe is built, and must be built, in such and such a way, and that it could not possibly be otherwise.  But how can one discover that principle?  If it was hopeless to learn atomic physics by studying work-hardening and dislocations, it may be equally hopeless to learn the basic operating principle of the universe, call it pregeometry or call it what one will, by any amount of work in general relativity and particle physics.

Thomas Mann (1937), in his essay on Freud, utters what Niels Bohr would surely have called a great truth (“A great truth is a truth whose opposite is also a great truth”) when he says, “Science never makes an advance until philosophy authorizes and encourages it to do so.”  If the equivalence principle (Chapter 16) and Mach’s principle §21.9) were the philosophical godfathers of general relativity, it is also true that what those principles do mean, and ought to mean, only becomes clear by study and restudy of Einstein’s theory itself.  Therefore it would seem reasonable to expect the primary guidance in the search for pregeometry to come from a principle both philosophical and powerful, but one also perhaps not destined to be wholly clear in its contents or its implications until some later day.

Among all the principles that one can name out of the world of science, it is difficult to think of one more compelling than simplicity;  and among all the simplicities of dynamics and life and movement, none is starker [Werner (1969)] than the binary choice yes—no or true—false.  It in no way proves that this choice for a starting principle is correct, but it at least gives one some comfort in the choice, that Pauli’s “nonclassical two-valuedness” or “spin” so dominates the world of particle physics.

It is one thing to have a start, a tentative construction of pregeometry;  but how does one go on?  How not to go on is illustrated by Figure 44.3 [of a sewing machine fashioning a chain mail sweater out of rings according to instructions on a tape being fed into the machine].  The “sewing machine” builds objects of one or another definite dimensionality, or of mixed dimensionalities, according to the instructions that it receives on the input tape in yes—no binary code.  Some of the difficulties of building up structure on the binary element according to this model, or any one of a dozen other models, stand out at once.  (1) Why N = 10,000 building units?  Why not a different N ?  And if one feeds in one such arbitrary number at the start, why not fix more features “by hand?”  No natural stopping point is evident, nor any principle that would fix such a stopping point.  Such arbitrariness contradicts the principle of simplicity and rules out the model.  (2) Quantum mechanics is added from outside, not generated from inside (from the model itself).  On this point too the principle of simplicity speaks against the model.  (3) The passage from pregeometry to geometry is made in a too-literal-minded way, with no appreciation of the need for particles and fields to appear along the way.  The model, in the words used by Bohr on another occasion, is “crazy, but not crazy enough to be right.”

Noting these difficulties, and fruitlessly trying model after model of pregeometry to see if it might be free of them, one suddenly realizes that a machinery for the combination of yes—no or true—false elements does not have to be invented.  It already exists.  What else can pregeometry be, one askes oneself, than the calculus of propositions?  (Box 44.5)

Box 44.5  “Pregeometry as the Calculus of Propositions” (pp. 1211f.)

A sample proposition taken out of a standard text on logic selected almost at random reads [Kneebone (1963), p. 40]

[X ⟶ ((XX ) ⟶ Y )] & (XZ ) eq (X  ᐯ Y  ᐯ Z ) & (X  ᐯ Y  ᐯ Z ) & (X  ᐯ Y  ᐯ Z ) & (X  ᐯ Y  ᐯ Z ).

The symbols have the following meaning:

A, Not A ;
A  ᐯ B A  or B  or both (“A vel B ”)
A  & B A  and B ;
AB  A  implies B  (“if A, then B ”)
AB  B  is equivalent to A  (“B  if and only if A ”)

Propositional formula 𝕬 is said to be equivalent (“eq”) to propositional formula 𝕭 if and only if 𝕬 ⟷ 𝕭 is a tautology.  The letters A, B, etc. serve as connectors to “wire together” one proposition with another.  Proceeding in this way, one can construct propositions of indefinitely great length.

A switching circuit [see, for example, Shannon (1938) or Hohn (1966) is isomorphic to a proposition.

Compare a short proposition or an elementary switching circuit to a molecular collision.  No idea seemed more preposterous than that of Daniel Bernoulli (1733), that heat is a manifestation of molecular collisions.  Moreover, a three-body encounter is difficult to treat, a four-body collision is more difficult, and a five- or more molecule system is essentially intractable.  Nevertheless, mechanics acquires new elements of simplicity in the limit in which the number of molecules is very great and in which one can use the concept of density in phase space.  Out of statistical mechanics in this limit come such concepts as temperature and entropy.  When the temperature is well-defined, the energy of the system is not a well-defined idea;  and when the energy is well-defined, the temperature is not.  This complementarity is built inescapably into the principles of the subject.  Thrust the finger into the flame of a match and experience a sensation like nothing else on heaven or earth;  yet what happens is all a consequence of molecular collisions, early critics notwithstanding.

Any individual proposition is difficult for the mind to apprehend when it is long;  and still more difficult to grasp is the content of a cluster of propositions.  Nevertheless, make a statistical analysis of the calculus of propositions in the limit where the number of propositions is great and most of them are long.  Ask if parameters force themselves on one’s attention in this analysis (1) analogous in some small measure to the temperature and entropy of statistical mechanics but (2) so much more numerous, and everyday dynamic in character, that they reproduce the continuum of everyday physics.

Nothing could seem so preposterous at first glance as the thought that nature is built on a foundation as ethereal as the calculus of propositions.  Yet, beyond the push to look in this direction provided by the principle of simplicity, there are two pulls.  First, bare-bones quantum mechanics lends itself in a marvelously natural way to formulation in the language of the calculus of propositions, as witnesses not least the book of Jauch (1968).  If the quantum principle were not in this way already automatically contained in one’s proposed model for pregeometry, and if in contrast it had to be introduced from outside, by that very token one would conclude that the model violated the principle of simplicity, and would have to reject it.  Second, the pursuit of reality seems always to take one away from reality.  Who would have imagined describing something so much a part of the here and now as gravitation in terms of curvature of the geometry of spacetime?  And when later this geometry came to be recognized as dynamic, who would have dreamed that geometrodynamics unfolds in an arena so ethereal as superspace?  Little astonishment there should be, therefore, if the description of nature carries one in the end to logic, the ethereal eyrie at the center of mathematics.  If, as one believes, all mathematics reduces to the mathematics of logic, and all physics reduces to mathematics, what alternative is there but for all physics to reduce to the mathematics of logic?  Logic is the only branch of mathematics that can “think about itself.”

“An issue of logic having nothing to do with physics” was the assessment by many of a controversy of old about the axiom, “parallel lines never meet.”  Does it follow from the other axioms of Euclidean geometry or is it independent?  “Independent,” Bolyai and Lobachevsky proved.  With this and the work of Gauss as a start, Riemann went on to create Riemannian geometry.  Study nature, not Euclid, to find out about geometry, he advised;  and Einstein went on to take that advice and to make geometry a part of physics.

“An issue of logic having nothing to do with physics” is one’s natural first assessment of the startling limitations on logic discovered by Gödel (1931), Cohen (1966), and others [for a review, see, for example, Kae and Ulam (1968)].  The exact opposite must be one’s assessment if the real pregeometry of the real physical world indeed turns out to be identical with the calculus of propositions.

“Physics as manifestation of logic” or “pregeometry as the calculus of propositions” is as yet [Wheeler (1971a)] not an idea, but an idea for an idea.  It is put forward here only to make it a little clearer what it means to suggest that the order of progress may not be

physics ⟶ pregeometry

but

pregeometry ⟶ physics.

§44.6.  The Black Box:  The Reprocessing of the Universe (pp. 1209-1217)

No amount of searching has ever disclosed a “cheap way” out of gravitational collapse, any more than earlier it revealed a cheap way out of the collapse of the atom.  Physicists in that earlier crisis found themselves in the end confronted with a revolutionary pistol, “Understand nothing — or accept the quantum principle.”  Today’s crisis can hardly force a lesser revolution.  One sees no alternative except to say that geometry fails and pregeometry has to take its place to ferry physics through the final stages of gravitational collapse and on into what happens next.  No guide is evident on this uncharted way except the principle of simplicity, applied to drastic lengths.

Whether the whole universe is squeezed down to the Planck dimension, or more or less, before reexpansion can begin and dynamics can return to normal, may be irrelevant for some of the questions one wants to consider.  Physics has long used the “black box” to symbolize situations where one wishes to concentrate on what goes in and what goes out, disregarding what takes place in between.

At the beginnng of the crisis of electric collapse one conceived of the electron as headed on a deterministic path toward a point-center of attraction, and unhappily destined to arrive at a condition of infinite kinetic energy in a finite time.  After the advent of quantum mechanics, one learned to summarize the interactions between center of attraction and electron in a “black box:”  fire in a wave-train of electrons traveling in one direction, and get electrons coming out in this, that, and the other direction with this, that, and the other well-determined probability amplitude (Figure 44.4 [illustrating the scattering of an electron propagating through superspace to alternative new histories]).  Moreover, to predict these probability amplitudes quantitatively and correctly, it was enough to translate the Hamiltonian of classical theory into the language of wave mechanics and solve the resulting wave equation, the key to the “black box.”

A similar “black box” view of gravitational collapse leads one to expect a “probability distribution of outcomes.”  Here, however, one outcome is distinguished from another, one must anticipate, not by a single parameter, such as the angle of scattering of the electron, but by many.  They govern, one foresees, such quantities as the size of the system at its maximum of expansion, the time from the start of this new cycle to the moment it ends in collapse, the number of particles present, and a thousand other features.  The “probabilities” of these outcomes will be governed by a dynamic law, analogous to (1) the Schrödinger wave equation for the electron, or, to cite another black box problem, (2) the Maxwell equations that couple together, at a wave-guide junction, electromagnetic waves running in otherwise separate wave guides.  However, it is hardly reasonable to expect the necessary dynamic law to spring forth as soon as one translates the Hamilton-Jacobi equation of general relativity (Chapter 43) into a Schrödinger equation, simply because geometrodynamics, in both its classical and its quantum version, is built on standard differential geometry.  That standard geometry leaves no room for any of those quantum fluctuations in connectvity that seem inescapable at small distances and therefore also inescapable in the final stages of gravitation collapse.  Not geometry, but pregeometry, must fill the black box of gravitational collapse.

Little as one knows the internal machinery of the black box, one sees no escape from this picture of what goes on:  the universe transforms, or transmutes, or transits, or is reprocessed probabilistically from one cycle of history to another in the era of collapse.

However straightforwardly and inescapably this picture of the reprocessing of the universe would seem to follow from the leading features of general relativity and the quantum principle, the two overarching principles of twentieth-century physics, it is nevertheless fantastic to contemplate.  How can the dynamics of a system so incredibly gigantic be switched, and switched at the whim of probability, from one cycle that has lasted 1011 years to one that will last only 106 years?  At first, only the circumstance that the system gets squeezed down in the course of this dynamics to incredibly small distances reconciles one to a transformation otherwise so unbelievable.  Then one looks at the upended strata of a mountain slope, or a bird not seen before, and marvels that the whole universe is incredible:

mutation of a species,
metamorphosis of a rock,
chemical transformation,
spontaneous transformation of a nucleus,
radioactive decay of a particle,
reprocessing of the universe itself.

If it cast a new light on geology to know that rocks can be raised and lowered thousands of meters and hundres of degrees, what does it mean for physics to think of the universe as being from time to time “squeezed through a knothole,” drastically “reprocessed,” and started out on a fresh dynamic cycle?  Three considerations above all press themselves on one’s attention, prefigured in these compressed phrases:

destruction of all constants of motion in collapse;
particles, and the physical “constants” themselves, as the
      “frozen-in part of the meteorology of collapse;”
“the biological selection of physical constants.”

The gravitational collapse of a star, or a collection of stars, to a black hole extinguishes all details of the system (see Chapters 32 and 33) except mass and charge and angular momentum.  Whether made of matter or antimatter or radiation, whether endowed with much entropy or little entropy, whether in smooth motions or chaotic turbulence, the collapsing system ends up as seen from outside, according to all indications, in the same standard state.  The laws of conservation of baryon number and lepton number are transcended [Chapter 33;  also Wheeler (1971b)].  No known means whatsoever will distinguish between black holes of the most different provenance if only they have the same mass, charge, and angular momentum.  But for a closed universe, even these constants vanish from the scene.  Total charge is automatically zero because lines of force have nowhere to end except upon charge.  Total mass and total angular momentum have absolutely no definable meaning whatsoever for a closed universe.  This conclusion follows not least because there is no asymptotically flat space outside where one can put a test particle into Keplerian orbit to determine period and precession.

Of all principles of physics, the laws of conservation of charge, lepton number, baryon number, mass, and angular momentum are among the most firmly established.  Yet with gravitational collapse the content of these conservation laws also collapses.  The established is disestablished.  No determinant of motion does one see left that could continue unchanged in value from cycle to cycle of the universe.  Moreover, if particles are dynamic in origin, no option would seem left except to conclude that the mass spectrum is itself reprocessed at the time when “the universe is squeezed through a knot hole.”  A molecule in this piece of paper is as “fossil” from photochemical synthesis in a tree a few years ago.  A nucleus of the oxygen in this air is a fossil from thermonuclear combustion at a much higher temperaure in a star a few 10⁹ years ago.  What else can a particle be but a fossil from the most violent event of all, gravitational collapse?

That one geological stratum has one many-miles long slope, with marvelous linearity of structure, and another stratum has another slope, is either an everyday triteness, taken as for granted by every passerby, or a miracle, until one understands the mechanism.  That an electron here has the same mass as an electron there is also a triviality or a miracle.  It is a triviality in quantum electrodynamics because it is assumed rather than derived.  However, it is a miracle on any view that regards the universe as being from time to time “reprocessed.”  How can electrons at different times and places in the present cycle of the universe have the same mass if the spectrum of particle masses differs between one cycle of the universe and another?

Inspect the interior of a particle of one type, and magnify it up enormously, and in that interior see one view of the whole universe [compare the concept of monad of Leibniz (1714), “The monads have no window through which anything can enter or depart”];  and do likewise for another particle of the same type.  Are particles of the same pattern identical in any one cycle of the universe because they give identically patterned views of the same universe?  No acceptable explanation for the miraculous identity of particles of the same type has even been put forward.  That identity must be regarded, not as a triviality, but as a central mystery of physics.

Not the spectrum of particle masses alone, but the physical “constants” themselves, would seem most reasonably regarded as reprocessed from one cycle to another.  Reprocessed relative to what?  Relative, for example, to the Planck system of units,

L*  = (ℏG/c³)½ = 1.6 ⨉ 10-33 cm,
T*  = (ℏG/c⁵)½ = 5.4 ⨉ 10-44 sec,
M*  = (ℏ/G )½ = 2.2 ⨉ 10-5 g,

the only system of units, Planck (1899) pointed out, free, like black-body radiation itself, of all complications of solid-state physics, molecular binding, atomic constitution, and elementary particle structure, and drawing for its background only on the simplest and most universal principles of physics, thelaws of gravitation and black-bod radiation.  Relative to the Planck units, every constant in every other part of physics is expressed as a pure number.

No pure numbers in physics are more impressive than ℏc ² = 137.0360 and the so-called “big numbers” [Eddington (1931, 1936, 1946);  Dirac (1937, 1938);  Jordan (1955, 1959);  Dicke (1959b, 1961, 1964b);  Hayakawa (1965a,b);  Carter (1968b)]:

~1080 particles in the universe,*
~1040 ~ (1028cm/10-12cm)  ~
    [(radius of universe at maximum expansion)*/(“size” of an elementary particle)],
~1040 ~ e²/GmM  ~ [(electric forces)/(gravitational forces)],
~1020 ~ [(e²/mc²)/(ℏG/c³)½]  ~
    [(“size” of an elementary particle)/(Planck length)],
~1020 ~ [(number of photons in universe)/(number of baryons in universe)].
* Values based on the “typical cosmological model” of Box 27.4;  subject to much uncertainty, in the present state of astrophysical distance determinations, not least because the latitude in these numbers is even enough to be compatible wth an open universe.

Some understanding of the relationships between these numbers has been won [Carter (1968b)].  Never has any explanation appeared for their enormous magnitude, nor will there ever, if the view is correct that reprocessing the universe reprocesses also the physical constants.  These constants on that view are not part of the laws of physics.  They are part of the initial-value data.  Such numbers are freshly given for each fresh cycle of expansion of the universe.  To look for a physical explanation for the “big numbers” would thus seem to be looking for the right answer to the wrong question.

In the week between one storm and the next, most features of the weather are ever-changing, but some special patterns of the wind last the week.  If the term “frozen features of the meteorology” is appropriate for them, much more so would it seem appropriate for the big numbers, the physical constants and the spectrum of particle masses in the cycle between one reprocessing of the universe and another.

A per cent or so change one way in one of he “constants,” ℏc/e ², will cause all stars to be red stars;  and a comparable change the other way will make all stars to be blue stars, according to Carter (1968b).  In neither case will any star like the sun be possible.  He raises the question whether life could have developed if the determinants of the physical constants had differed substantially from those that characterize this cycle of the universe.

Dicke (1961) has pointed out that the right order of ideas may not be, here is the universe, so what must man be;  but here is man, so what must the universe be?  In other words:  (1) What good is a universe without awareness of that universe?  But:  (2) Awareness demands life.  (3) Life demands the presence of elements heavier than hydrogen.  (4) The production of heavy elements demands thermonuclear combustion.  (5) Thermonuclear combustion normally requires several 10⁹ years of cooking time in a star.  (6) Several 10⁹ years of time will not and cannot be available in a closed universe, according to general relativity, unless the radius-at-maximum expansion of the universe is several 10⁹ light years or more.  So why on this view is the universe as big as it is?  Because only so can man be here!

In brief, the considerations of Carter and Dicke would seem to raise the idea of the “biological selection of physical constants.”  However, to “select” is impossible unless there are options to select between.  Exactly such options would seem for the first time to be held out by the only over-all picture of the gravitational collapse of the universe that one sees how to put forward today, the pregeometry black-box model of the reprocessing of the universe.

Proceeding with all caution into uncharted territory, one must nevertheless be aware that the conclusions one is reaching and the questions one is asking at a given stage of the analysis many be only stepping stones on the way to still more penetrating questions and an even more remarkable picture.  To speak of “reprocessing and selection” may only be a halfway point on the road toward thinking of the universe as Leibniz did, as a world of relationships, not a world of machinery.  Far from being brought into its present condition by “reprocessing” from earlier cycles, may the universe in some strange sense be “brought into being” by the participation of those who participate?  On this view the concept of “cycles” would even seem to be altogether wrong.  Instead the vital act is the act of participation.  “Participator” is the incontrovertible new concept given by quantum mechanics;  it strikes down the term “observer” of classical theory, the man who stands safely behind the thick glass wall and watches what goes on without taking part.  It can’t be done, quantum mechanics says.  Even with the lowly electron one must participate before one can give any meaning whatsoever to its position or its momentum.  Is this firmly established result the tiny tip of a giant iceberg?  Does the universe also derive its meaning from “participation”?  Are we destined to return to the great concept of Leibniz, of “preestablished harmony” (“Leibniz logic loop”), before we can make the next great advance?

Rich prospects stand open for investigation in gravitation physics, from neutron stars to cosmology and from post-Newtonian celestial mechanics to gravitational waves.  Einstein’s geometrodynamics exposes itself to destruction on a dozen fronts and by a thousand predictions.  No predictions subject to early test are more entrancing than those on the formation and properties of a black hole, “laboratory model” for some of what is predicted for the universe itself.  No field is more pregnant with the future than gravitational collapse.  No more revolutionary views of man and the universe has one ever been driven to consider seriously than those that come out of pondering the paradox of collapse, the greatest crisis of physics of all time.

All of these endeavors are based on the belief that existence shoulld have a
completely harmonious structure.  Today we have less ground than ever before for
allowing ourselves to be forced away from this wonderful belief.


EINSTEIN (1934)

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