IL NUOVO CIMENTO VOL. 107 B, N. 2 Febbraio 1992

Holes in Space-Time.

D. K. Ross

Physics Department, Iowa State University - Ames, IA 50011, USA

(ricevuto il 14 Febbraio 1991; approvato il 21 Maggio 1991)

Summary. - - We explore the possibility that the topology of space-time may not be that of Minkowski's space-time but may include many topological holes. We show that these holes should behave like massless scalar particles with Lorentz invariant cross-sections which depend upon their size. We argue that they should have a thermal energy distribution with a temperature the same as the cosmic neutrino background. A photon interacting with this sea of holes is red-shifted in an energy-dependent way. This allows us to put severe constraints on their cross-sections using radio and optical quasar data.

PACS 04.90 - Other topics in relativity and gravitation. PACS 98.70.Vc - Background radiations.

1 . - I n t r o d u c t i o n .

Einstein's gravitational-field equations determine the geometry of space-time but not the topology[l]. It is thus an open question what this topology might be. Isham [2] and Avis and Isham [3] have looked at the effects of a nontrivial global space-time topology on classical and quantum fields evolving on that topology. In the present paper we want to explore the simplest case of a nontrivial topology, namely a space-time with many holes present. We want to see how small localized topological holes behave in their interaction with ordinary matter . We assume that our holes are regions excised from the 3-geometry of the usual space-time manifold (a paracompact, connected c ~ Hausdorff manifold without boundary with an affine connection and a metric with Lorentzian signature defined and with a topology equivalent to Minkowski's space). Our holes in 4-dimensional space-time are thus extended regions composed of the world line swept out by the excised region of the 3-geometry. By considering the presence of holes in Minkowski's space, we are really changing our concept of what constitutes a priori space-time.

We can write down the Euler-Poincar~ characteristic for our manifold with holes as [4-6].

(1) x = f J 1 s~y~ s,v~a I ~ Ry~:a d4x. 128 Vg

1 4 - l l Nuovo Cimento B 203

204 D . K . ROSS

This is a topological invariant and can be written in terms of Betti numbers. Betti numbers can be defined and are finite in the present case since homology theory can be extended to noncompact Hausdorff spaces[7]. The Betti numbers are also topological invariants. If the number of topological holes changes, so do the Betti numbers. Thus the number of holes is fixed, and we will assume our holes are primordial in origin and indestructible.

Holes or other structures in space-time could also be important because they may be related to our most fundamental constants such as h and c. c, for example, has its ultimate existence in the topologically invariant signature of space-time which in turn is related to the light cone structure. Do other fundamental constants such as h ultimately also find their origin in the geometry of space-time? Might new fundamental lengths appear in the a priori topology of space-time? The fact that quantum field theories are divergent at small distance scales suggests that our conception of space-time may be wrong there. String theory [8-10] is one possibility but there may be others. Part of the impetus for the present paper was to see if topological holes could play the role of a fundamental constant and/or make quantum field theories finite. We will see below that the answer to both questions is no. We will also see, however, that topological holes have interesting properties in their own right and affect propagating photons and other particles in a way that allows us to put severe limits on their parameters.

We will discuss the properties of holes in sect. 2 below and their behavior in interacting with other particles in sect. 3. In sect. 4 we look at constraints on their parameters arising primarily from quasar spectra, and we discuss our results briefly in sect. 5.

2. - Propert ies o f holes in space- t ime.

We now want to discuss the properties of our topological holes. First, since a hole is simply a place where Minkowski's space-time is missing, we expect space-time not to be strongly warped in the vicinity. This implies from general relativity that our holes should have no rest mass. Also, if a hole had a rest mass, there would exist a frame where it was at rest. To avoid the existence of special frames in Minkowski's space, it is therefore desirable that our holes be massless and always travel at the velocity of light. To preserve Lorentz invariance, our holes must move randomly in direction. The topological holes, we are discussing here, are clearly quite different from the massive Planck mass scale black-holes which arise in the quantum gravitational vacuum [11]. Our holes do not arise dynamically but rather arise from the a priori topology of space-time we start with in doing physics. This does not have to be a Minkowski topology as pointed out in the introduction.

Topological holes will interact with ordinary particles and therefore will acquire energy momentum, k,. As usual in quantum mechanics, this k, is related to the wavelength and frequency of the quantum mechanical wave describing the probability of where our hole is to be found. Our holes are also characterized by an invariant ,,size, or cross section (see the following section). Space-time will have a slight curvature near a hole arising purely from the k, they may acquire. Both k, and this curvature are small as we see below. Now let us assume that at some point in the very early universe our holes were in thermal equilibrium with the rest of the universe. They would have a black-body distribution of energies at that point with

HOLES IN SPACE-TIME 205

some high temperature T. As the universe expands, the temperature T ~ R -1, where R is the scale of the universe [12]. When our holes later thermally decouple from the universe, this decoupling will have no effect on their distribution function. As the universe expands, the holes will still have a black-body distribution function of energies characterized by a steadily dropping temperature. They will in fact behave exactly like neutrinos and will have the present neutrino temperature[12] (2.8 K)(4 / l l ) 1/3. The photon temperature [13] of 2.8 K is higher because of reheating of the photons by later e+-e - annihilations. Our holes thus are characterized by an average energy of k0 = 1.72"10-4eV at the present epoch. This argument actually works bet ter for our holes than for neutrinos, since the number of our holes is a topological invariant. The number of neutrinos can change slightly due to subsequent interactions.

The inside of a topological hole is simply a place where space-time itself does not exist from our fundamental assumption about their nature. Thus wavefunctions of all ordinary particles are rigidly excluded from the inside of the hole, and our holes behave like a rigid sphere in some sense in their interactions. The interaction is universal like gravitation, since the structure of the ordinary particle is irrelevant. Schiff[14] shows that the scattering from a rigid sphere of radius a gives a total cross-section of 4~a 2 for low-energy incident particles (~particle>>27za) and a cross-section of 2=a 2 for high-energy incident particles. For our holes we will find below that the cross-sections are very small. Thus we might expect spherically symmetric scattering to lowest order and a differential cross-section which is not a function of scattering angles. We are also in the low-energy regime and all ordinary particles should see the same total cross-section. We are interested in the effect that a sea of Nv0 holes per cubic centimeter of average energy k0 has on the propagation of an ordinary particle or photon. To lowest order, we can consider Nvo/6 holes to be headed in each of 3 perpendicular directions. Only collisions from holes hitting from head on or from the back of the propagating particle will matter, since collisions from the side will cancel. We can characterize the hole by a Lorentz-invariant cross-section A0, which is related to its actual cross-sectional area as in Schiff above. In any Lorentz frame moving in the direction of the propagating particle, we will see the same cross-sectional area and the same A0, since transverse directions do not change in a Lorentz boost. Also, since holes move at the velocity of light, a particle of any velocity has the same relative velocity to the hole. A0 thus is a well-defined Lorentz invariant quantity for any head-on or rear collision, and these are the only collisions we need consider. Holes may have a distribution of cross-sections and therefore of A0, of course, but we will characterize them all by an average or effective A0.

Our holes are just excised regions of space-time. Thus, as space-time expands so should our holes. If they have total cross-section A0 now, they would have

(2) A -- A0

at other times, where R is the scale factor of the universe. Note that the topological invariant associated with the holes do not change if the holes change shape or size provided that the continuity and connectivity of space-time do not change. Since A increases as R 2 (t) our holes are different from other particles whose cross-sections do not change with time. Also, since A keeps changing, it cannot be used as a new ,~fundamental constant, associated with the structures of space-time as alluded to in

206 D.K. ROSS

the introduction. Holes are created at the Planck-mass era when the universe is 10 -3z cm and composed of a topological foam [15]. They have been expanding with the universe ever since.

A hole should have zero spin since again it is just a place where Minkowski's space is missing. The spin certainly must be 0, 1, or 2. Spin 2 is already identified with the gravitation and spin 1 with the photon. Also the universality of the interaction rules out spin 1. Spin zero is also consistent with a universal cross-section A0 with no spin effects present as above.

The total number of holes in the universe is fixed, but the number density Nv changes as the universe expands according to

(3) Nv = N ~ R ~ / R ~ ,

where N~o has an upper limit imposed by requiring it to be less than the critical energy density, 3 H ~ C 2 / 8 ~ k necessary to close the universe. If we use [16-18] H0 = = 58 k m / s Mpc = (16.8. 109years) - ' , this gives

(4) koNvo -- 3580Z eV/cm 3 ,

with Z a number ~< 1. Z is the fraction of the total energy in the universe in our holes. We can estimate the maximum effect holes can have by letting • = 1 if we choose. It is conceivable that holes could make a substantial contribution to the energy of the universe.

3. - K inemat i c s o f co l l i s ions wi th space- t ime holes .

We are interested in the effect that massless holes of energy k0 will have on propagating massive particles and photons. As mentioned above we are interested in the cases where the hole hits the particle either head-on or from the back. In a head-on collision, the final energy of a particle of mass m, energy E, and momentum P after hitting a hole of energy k head-on is

(5) <Ef~on t > -- E + k - ( P k + E k ) ( m 2 + 2 E k + 2 P k ) -1/~ ,

where we have averaged over the final scattering angle of the hole. If the hole hits the particle directly from the back, we get

(6) (Egack > = E + k - ( - P k + E k ) ( m 2 + 2 E k - 2 P k ) -1/2 .

For each collision from the front we will have a collision from the back so that the energy lost by the particle is <Ef~ont> - <E~ack>.

If a particle travels through a cloud of holes, the probability it will interact in a distance dx is

No (7) A - ~ d x ,

where A is the Lorentz-invariant cross-section related to the frontal area of the hole as above. We divide by 6 since N ~ / 6 holes are approaching from head-on at any instant at least in terms of momentum components. The energy loss rate for a particle

HOLES IN SPACE-TIME 207

interacting with holes using (5) and (6) is then

dE N v A (8) - - -

dx 6 - - [ - k ( E + P ) (m 2 + 2kE + 2kP) -1/2 + k (E - P ) ( m ~ + 2kE - 2kP)-1/2].

For photons interacting with holes, E = P and m s = 0 so this becomes

(9) dE _ N v A (_Vrff-E). dx 6

4. - Observational l imits on the properties o f holes.

We want to see what limits we can place on our holes now using observations of distant sources. If we integrate (8) or (9), we see that the most stringent upper limit can be placed on N v A V ~ if we use data on low-energy photons traveling long distances. This suggests using data from quasar spectra. Since quasars are so far away we need to take into account that N~, A and k for our holes all change as the universe expands. We have (2), (3), and k = k o R o R - 1 , where Nvo, Ao , and ko refer to the present epoch. Using these in (9) then gives

(10) dE I d x holes 6

Now the expansion of the universe also Doppler shifts the energy of the photon and we have approximately

(11) Ro E R Eo '

where E is the changing energy of the photon along its path and Eo is the observed energy of the photon on the Earth. This Eo is red-shifted from the energy emitted Ee at the quasar due to both the expansion of the universe and to the effects of our topological holes. We shall see below that the holes are constrained to have a minor effect and most of the red-shift is due to the expansion of the universe. Thus (11) is still approximately true even in the presence of holes. Using (11), (10) can be rewritten as

Nvo Ao ~ o E 2 (12) ~xx holes-- 6 E~/2

Now consider a photon at distance x from a quasar. The usual Hubble red-shift can be written

(13) Ho__XX = E e - E c E '

where E is the energy of the photon at that point on its path. We again assume that most of the red-shift is due to expansion of the universe and find this amply verified

208 D .K. ROSS

below. Differentiating (13) with respect to x gives

I E2Ho (14) d E ] _ d x I expansion Ee c

Now we can obtain the effects of our holes on the spectra by letting the total d E / d x be the sum of (12) and (14). We then integrate the energy from Ee to Eo and integrate x over the distance to the quasar, D. Using

(15) --E~ _= 1 + z E0

to eliminate E~ from the result gives finally

(16) z + (1 + z) = D. 6V o

In interpreting the spectrum, we would assume we had the usual Hubble relation, and so we have an effective red-shift

(17) z~ff = 1 + ~(1 + z)Eo 1/2 '

where

c N~ oAo V~0 (18) ~ - ,

H0 6

z is the actual red-shift due to expansion and Zeff is the observed effective red-shift due to both expansion and energy loss due to our topological holes. They are equal if ~=0 .

Now we notice the crucial fact that zeff depends on the energy E0 of the observed photon. This means that the two ends of the same spectrum will have different Zeff if holes are present. This places severe constraints on our holes and means that most of the red-shift must arise from the usual expansion. We also notice from (17) that low-energy photons such as radio waves have a larger contribution from the hole term high-energy photons. We will exploit this below.

To put limits on ~ in (18) let us first look at the optical spectrum of TON 1530. This is typical of many quasars. Lower-resolution spectra were taken by Bahcall, Osmer and Schmidt [19] and higher-resolution spectra over a narrower range were taken by Morton and Morton[20]. The earlier work saw strong Ly-~ 1215.67 A and strong CIV 1548.188 A at z= 1.9365. Let us assume that the measured z is accurate to_+0.0004 and assume that Ly-a give Zeff, = 1.9369 and appears at Eol = 3.472 681 eV while CIV at the other end of the spectrum give Zeff2 = 1. 9365 and appears at E02 = 2. 727 194 eV. This is the maximum differential z our holes could give and still be within the error bars of the measurements. We are assuming that 5z from one end of the spectrum to the other and part of the total z arise from our holes. Putt ing zeffl and Eo, into (17) gives one equation for the actual z and for L Putting z,ff~ and E0~ gives another equation. We

H O L E S I N S P A C E - T I M E 209

can thus solve for z and for ~. We find z= 1.9400 and ~ = 1 .015.10-3eV 1/2. Note that most of the red-shift is due to expansion and not to the holes.

A much more severe constraint arises from the da ta for the quasar PKS 1157 + 014. Wolfe et al. [21] saw a radio line at 482.537 MHz corresponding to the 21 cm line red-shifted by z = 1.9436. This agrees v e ry well wi th a low ionization red-shift sys tem at z = 1.9438 seen in the optical by Wrigh t et al. [22]. Both the radio and optical lines apparent ly arise in an ex t r eme ly opaque HI region lying be tween us and the quasar. Now we have spectra covering a much wider r ange of energy. This will put much t igh ter constraints on an ene rgy-dependen t Zeff and on our holes. Allowing again for possible e r ro r bars let us assume the 21 cm line appears at zeff-- = 1.9436 with E01 = 1.995 628.10 .6 eV and the 1548.188/~ CIV line appears optically at Zeff = 1. 9440 with E02 = 4.119550 eV. Put t ing these two sets of values into (17) as above and solving the result ing two equations in two unknowns gives z ~ 1.9440 and an upper limit ~ < 0.987.10-TeV 1/2 . This is a much more s t r ingent limit on ~ than using optical data alone.

Le t us see how this limit on ~ t rans la tes into a limit on the cross-sect ion A0 of our holes. We use the definition (18) with H0 = 58 k m / s Mpc, k0 = 1 .72-10-4eV from above, and koNvo = 3.580z eV/cm 3, where x measures the fract ion of the total en e rg y of the universe in our holes. This gives

(19) Aoz < 1.4.10-4~ 2 .

Our holes must be physically quite small and have a small cross-section for interactions if they are not to violate the constraints imposed by quasar spectra .

5 . - D i s c u s s i o n .

We have found that topological holes are an in teres t ing possible modification of Minkowski's space providing a new a priori stage on which to do physics. Quasar spectra grea t ly res t r ic t their physical size and thus thei r cross-sections. Holes behave in many ways like massless scalar particles with universal cross-sections. We expec t their size and hence their cross-section A to increase as R 2 as the universe expands unlike the cross-sections associate with ordinary particles. This also makes A unsuitable as a new fundamental constant.

We originally hoped that topological holes might make QED and o ther quantum field theories finite by modifying the short-distance proper t ies of space-time. In order to stop high-energy particles in a short distance, we need AE ~ -Eparticle in a distance Ax ~ ~'particle or dE/dx ~ - E 2 / h c . d E / d x for our holes has too weak an

E-dependence and quasar data res t r ic t NvoAo V~oo to be much too small for holes to have an effect in cutt ing off quantum field theories.

R E F E R E N C E S

[1] J. A. WHEELER: Geometrodynamics (Academic Press, New York, N.Y., 1962). [ 2 ] C. J. ISHAM: Proc. R. Soc. London, Ser. A, 362, 383 (1978); 364, 591 (1978). [3] S. J. AvIs and C. J. ISHAM: Proc. R. Soc. London, Ser. A, 363, 581 (1978). [4] C. B. ALLENDOERFER: Am. J. Math., 62, 243 (1940).

210 D.K. ROSS

[5] S.-S. CHERN: J. Soc. Ind. Appl. Math., 10, 751 (1962). [6] T. EGUCHI and P. G. O. FREUND: Phys. Rev. Lett., 37, 1251 (1976). [7] W. S. MASSEY: Homology and Cohomology Theory (M. Dekker, New York, N.Y., 1978), p.

267. [8] D. J. GROSS, J. A. HARVEY, E. MARTINEC and R. ROHM: Phys. Rev. Lett., 54, 502 (1985);

Nucl.Phys. B, 256, 253 (1985). [9] P. CANDELAS, G. T. HOROWITZ, A. STROMINGER and E. WITTEN: Nucl. Phys. B, 258, 46

(1985). [10J E. WITTEN: Phys. Lett. B, 155, 151 (1985); Nucl. Phys. B, 258, 75 (1985). [11] S. W. HAWKING: Phys. Rev. D, 18, 1747 (1978). [12] S. WEINBERG: Gravitation and Cosmology (Wiley, New York, N.Y., 1972). [13J A. A. PENZIAS and R. W. WILSON: Astrophys. J., 142, 419 (1965). [14] L. I. SCHIFF: Quantum Mechanics, 2nd edition (McGraw-Hill, New York, N.Y. 1955),

p. 110. [15] C. W. MISNER, K. S. THORNE and J. A. WHEELER: Gravitation (Freeman, San Francisco,

Cal., 1973), chapt. 43. [16J G. DE VAUCOULEURS: Nature, 299, 303 (1982). [17] G. A. TAMMANN and A. SANDAGE: Preprint of the Astronomical Institute of the Univ. of

Basel no. 6 (1985). [18] W. D. ARNETT, D. BRANCH and J. C. WHEELER: Nature, 314, 337 (1985). [19] J. N. BAHCALL, P. S. OSMER and M. SCHMIDT: Astrophys. J. Lett., 156, L1 (1969). [20] W. A. MORTON and D. C. MORTON: Astrophys. J., 178, 607 (1972). [21] A. M. WOLFE, F. H. BRIGGS and D. L. JAUNCEY: Astrophys. J., 248, 460 (1981). [22] A. E. WRIGHT, D. C. MORTON, B. A. PETERSON and D. L. JAUNCEY: Mon. Not. R. Astron.

Soc., 189, 611 (1979).