Nuclear Physics B (Proc. Suppl.) 6 (1989) 345-348 345 North-Holland, Amsterdam

LATE STAGES OF CRUNCH

ASGHAR QADIR

Physics Department, The University of Texas, Austin, Texas 78712, USA, and Mathematics Department, Quaid-i-Azam University, Islamabad, Pakistan

JOHN ARCHIBALD WHEELER

Physics Departments, Princeton University, Princeton, New Jersey 08544, and The University of Texas, Austin, Texas 78712, USA

Originally filling a thin gap between two expanding clouds of dust in a closed model universe, Schwarzschild geometry in the final stage of collapse grows into a ever-lengthening corridor that represents a gravity wave (zero Ricci curvature, nonzero Weyl curvature) and totally dominates the crunch.

Of all questions about a model universe expanding

out of a big bang, none attracts more immediate atten-

tion than this: Will it expand forever? Or will it

end in a big crunch? Crunch is inescapable---it is well

known--when

(1) the connectivity of the manifold is that of a

3-sphere,

(2) the dynamics of the geometry follows Einstein's

equation with zero cosmological constant, and

(3) the stress-energy tensor obeys a certain simple

condition of normegativity.1

Crunch, then, yes; but crunch everywhere at the same

time?

Collapse will occur sooner in some regions than in

others? Sooner in a region occupied by a thick pow-

ering of zero-pressure matter, or "dust," than in a do-

main where the dust has a lower density (top diagram

in Fig. 1)? Yes and no. Yes, a dust particle in the thick

cloud has a shorter life from bang to crunch than one

in the thin cloud; yes, when by time we mean proper

time, time as measured by a clock carried along by the

particle in question. But no, when time is York crunch

time, 2

crunch = K = York time = TrK

= (trace of the extrinsic curvature tensor)

= (fractional rate of contraction of the 3-volume per unit advance in proper time (cm) normal to the hypersurface).

In terms of crunch time the collapse occurs every-

where simultaneously.

Everywhere simultaneously, yes. Figure 1 illustrates

successive stages of the crunch of the thick-and-thin

suture-model universe as calculated by us 3,4'5 for four

values of crunch time with the help of Benjamin

Skrainka and John M. Wheeler, Princeton classes of

1989 and 1990.

Collapse of the thick-and-thin suture-model universe,

as depicted in successive K = constant time frames--

the later ones of which appear in Fig. 1--has these

features:

(1) The smaller, thicker region runs through its ex-

pansion and starts to contract before the larger,

thinner domain has even reached maximum dis-

tention.

(2) The geodesic proper distance in the 3-geometry

from the N-pole or center of the thin region to

the S-pole or center of the thick region peaks at

a time not greatly different from the times in (1).

0920-5632/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

346 A. Qadir, J.A. Wheeler / Late stages o /crunch

""*') .r'.'. " ,:.:'.~..: ;I.~e:" d

0 O0

- 0 0 I

o +GO

TrK

FIGURE 1 The 3-geometry of four constant-crunch-time slice~ through the spaeetime of the Qadir-J.A.W. thick-and- thin suture model universe, from near the phase of maximum expansion(first, or top, frame)to near crunch (fourth frame); detailed calculations as summarized in free-hand sketches. An axis of 2-sphere symmetry, not shown, is to be imagined as cutting horizontally through each frame. Bottom: Proper distance, as mea- sured in the 3-geometry, from pole to pole of the model cosmology, in its dependence on crunch time.

Subsequently, however, this N - S distance falls

to a rninimmn and then rises to infinity at K = ~ .

(3) The empty-space suture between the two dust

clouds, as part and parcel of its pure Schwarzsctfild

3-geometry, has at each end a flange, by virtue

of which it joins smoothly to the pure Friedmann

3-geometry of the clouds.

(4) This Schwarzschild region, possessing zero Ricci

curvature and nonzero Weyl curvature, is not

only dynamic6; it represents a gravity wave.

(5) This suture, possessing the topology of the

2-sphere times a finite line segment, develops into

an ever longer, ever thinner corridor. In the final

stages of crunch this corridor's 3-volume goes to

zero even as its length goes to infinity.

(6) Tile corridor dominates the crunch.

Belinsky, Khalatnikov, and Lifshitz r long ago em-

phasized tile dominance of gravity waves over effec-

tive energy of every other form in the final stages of

crunch. This point the present example exhibits in

striking form. The dust-containing fractions of the

3-geometry shrink to unimportance. The corridor wins

dominance. Moreover, it curves itself up into 2-sphere

closure by virtue of its content, not of dust (for it con-

tains none) but of effective gravity-wave energy.

A multiplicity of such corridors we have to expect to

develop in the collapse of the generic model universe.

These corridors, these bristles on the hedgehog, pos-

sessing though they will the connectivity of S 2 # (line

segment), typically will lack the geometric symmetry

of the 2-sphere. Instead. they will manifest something

kin to mixmaster oscillation.

Will the corridors reconnect, as do magnetic lines

of force? What possibilities do they hold out for the

communication of experimental findings and the es-

tablishment of meaning?

Late stages of crunch? W'e stand today at the be-

ginning of the questions, not at the end of the answers[

To Eugene Wigner, we and many other colleagues

A. Qadir, J.A. Wheeler / Late stages o f crunch 347

owe thanks for penetrating insights. Moreover, as Wolf-

gang Pauli used to say, "When Wigner asks a question,

listen very carefully." Among the many Wigner ques-

tions that we all prize are three of special relevance to

the points touched on here:

(1) What technique offers itself to measure space-

time geometry? s'9

(2) The most immediately evident scheme measures

the geometrodynamics (GMD) field quantities to

a precision far short of expectation. 8 In contrast,

the electrodynamic (EM) field quantities, with

sufficient cleverness, as Bohr and Rosenfeld

showed, 1° can be measured with all the precision

called for by quantum theory. The discrepancy

with quantum theory in the case of GMD: Is it

nonexistent? 11 Or does it betoken want of imagi-

nation in the technique of measurement? ]2,13 Or

is it insuperable and therefore indicative of some

as-yet-unappreciated flaw of principle in the very

concept of quantum geometrodynamics?

(3) Interaction with the environment is one term, re-

peated measurement another, for the much stud-

ied process (Mott, Zeh, Joos, Wootters, Zurek)

that hands over the track of an alpha particle,

the orbit of the moon, and the path of a dust

particle from one mode of description, quantum

theory, to another, classical theory. Wigner has

recently 14 asked whether repeated measurements

of geometry do not likewise hand spacetime over

to classical analysis. This question, guide to to-

morrow, is one of the many reminders of all we

owe Eugene Wigner for yesterday and today.

REFERENCES

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J. A. Wheeler, Geometrodynamic steering prin- ciple reveals the determiners of inertia, Intnl. J. of Mod. Phys. A3 (1988) 2207-2247 p. 2214.

A. Qadir and J. A. Wheeler, York's cosmic time versus proper time as relevant to changes in the dimensionless "constant", K-meson decay, and the unity of black hole and big crunch, in: From SU(3) to Gravity, eds. E. Gotsman and G. Tauber (Cambridge University Press, Cambridge, 1985) pp. 383-394.

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A. Qadir and J. A. Wheeler, detailed report in preparation.

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V. A. Behnsky, I. M. Khalatnikov, and E. M. Lifshitz, Oscillatory approach to a singular point in the relativistic cosmology, Usp. Fiz. Nauk 102 (1970) 463-500; English tranlation in Soy. Phys.- JETP 32, pp. 169-172.

H. Saiecker and E. P. Wigner, Phys. Rev. 109 (1958) pp. 571-577.

MTW, Box 16.4, pp. 397-399.

N. Bohr and L. Rosenfeld, Zur Frage der Mess- barkeit der elektromagnetischen FeldgrSssen, Kgl. Danske Videnskab. Sels. Mat.-fys. Medd. 12 (1933) No. 8; English translation in: Quantum The- ory and Measurement, eds. J. A. Wheeler and W. H. Zurek (Princeton University Press, Prince- ton, 1983).

B. S. DeWitt , a section on measurement of ge- ometry in: Relativity, Groups and Topology, eds. C. M. DeWitt and B. S. DeWitt (Gordon and Breach, New York, 1969).

L. Rosenfeld, a chapter in: Niels Bohr and the Development of Physics, eds. L. Rosenfeld and W. Pauli (McGraw-Hill, New York, 1955) pp. 70-95.

348 A. Qadir. ,LA. Wheeler / Late stages o f crunch

13. J. A. Wheeler, Quantum Gravity: The question of measurement, in: Quantum Theory of Grav- ity: Essays in honor of the 60th birthday of Bryce S. DeWitt, ed. S. M. Christensen (Hilger, Bristol, 1984) pp. 224-233.

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