IL NUOVO CIMENTO

VOL. 109 B, N. 1

Gennaio 1994

Is T h e r e a F u n d a m e n t a l A c c e l e r a t i o n ? C. MASSA

Via FrateUi Manfredi 55, 42100 Reggio Emilic~ Italia (ricevuto il 10 Maggio 1993; approvato il 27 Luglio 1993)

- - The acceleration ao - 10 -s cm s-2 appears insistently in physics and seems to play a fundamental role at the cosmological, astrophysical and microphysical level. It is the surface gravity of all mesons, of a typical protostar, of a typical galaxy and of the entire Universe; it is also the minimal acceleration an electric charge must have to radiate, the minimal centripetal acceleration of the quarks trapped in a spinning meson, the minimal acceleration a frame must have to feel the Unruh temperature, the minimal value a gravitational field must have to induce transitions between states of opposite parity in the hydrogen atom, the acceleration of any particle of rest mass h/Rc (R = the radius of the observable Universe) such as the graviton; from the existence of the fundamental acceleration a0 one can derive the Weinberg relation (connecting microphysics to cosmology) and the well-known rough equality (so far unexplained) between the energy density of the cosmological microwave radiation, of the mean galactic magnetic field, and of the starlight in the Milky Way. Summary.

PACS 98.80 - Cosmology. PACS 14.40 - Mesons and meson resonances. PACS 95.30 - Fundamental aspects of astrophysics.

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

There are several reasons to think the acceleration a o - 1 0 -s c m s -2 plays a fundamental role in the Universe; it is of the order of: A) The surface gravity Gm/r 2 of a pion (mass - 10 -25 g, radius - 10 -13 cm). B) The surface gravity of a typical protostar (mass - 10~ g, radius - 1017 cm) and protoplanet (mass - 102s g, radius - 1014 cm). C) The -

surface

gravity of a typical spiral

galaxy ( m a s s - 10~ g, radius

10 ~ c m ) .

D) The (,surface gravity,, of the observable Universe (mass- 10~ g, radius - i 0 ~ cm); with such incorrect but vivid expression I mean the acceleration (relative to a comoving Robertson-Walker observer) of a test particle trapped in the Hubble expansion near the event horizon at proper distance R of a Friedmann model 25

26

c. M A S S A

with flat space, mass density D and observable mass M, I/~] = ( 4 r : / 3 ) G D R = = G M / R 2 ~ ao where D - 10-s~ g cm-3 and R = 2 c / H , where H is the Hubble constant, 1 / H = 20 billion years. The same result holds in curved space, but for a factor of order unity, and also in the Newtonian cosmology where the expression ,,surface gravity of the Universemay have a more direct meaning. E) The minimal acceleration an electric charge must have to radiate (the transition radius from the near field to the radiation zone is c 2 / a and, if a < a0, the radiation zone lies beyond the cosmological horizon R). Another approach to the subject is the following: the minimal observable energy in a universe with age t is h / t [1] (h = the quantum of action) and the minimal observable power output is h/t2; then, according to classical electrodynamics, the minimal acceleration a charge q must have to radiate is (c/qt)(hc) 1/2 that equals ao if t - 1 / H and q - - t h e electron charge. F) The minimal acceleration a frame must have to measure the Unruh temperature T = (h/2r~ck)a. If a < a0, then T < 10-8~ and the mean energy of a photon in the thermal field is smaller than 10-45erg=h/t, and then unobservable. G) The minimal gravitational field able to induce transitions between states of opposite parity in the hydrogen atom; the energy scale for the gravity-induced transition, that is a relativistic quantum gravitational effect explored in [2], is ( a h / c ) and is unobservable (less than h / t ) if a < ao. H) The minimal centripetal acceleration undergone by the quarks inside a spinning meson in the bag model (see sect. 3). I) The acceleration of a levion (I call levions, from the Latin adjective levis = light, hypothetical particles with rest energy of the order of the minimal energy h / t ; their existence is quite possible, e.g. gravitons, if massive, should be levions [3]). A levion should move with acceleration c/t ~ ao relative to any observer

frame (whether inertial or not) just as photons move with speed c relative to any observer; this is suggested by the constraints obeyed by any particle of rest mass m[1], (1)

( h / m t 3 ) 1/2 < a < mcS / h .

Inequality (1) springs from the generalized Heisenberg time-energy uncertainty relation. Set mc 2 = h / t and get a = c / t - ao. According to points from D) to I), the fundamental acceleration is closely connected to cosmology (more precisely, to the f'mite value R of the radius of the observable Universe) via (2)

ao = c 2 / R

(but for a factor of the order of unity).

From points A) and D) we get (3a)

M / m = ( R / L )2 ,

(3b)

m "~= h 2 / G R ,

27

IS THERE A FUNDAMENTAL ACCELERATION?

where m = t h e pion mass, L = h / m c = t h e pion radius, M = t h e mass of the observable Universe. Equations (3) are well-known empirical relations connecting microphysics to cosmology [4]. Note, by points B) and C), that such relation (3a) can be enlarged to protoplanets, protostars and galaxies, so that it can be written in the compact form (4)

M ~ / M b = R e2/ R b 2 ,

where M means mass, R means radius, and subscripts a~ b run (independently of each other) from 0 to 4; numbers 0, 1, 2, 3, 4 refer, respectively, to a typical hadron, protoplanet, protostar, galaxy and the Universe. But for hadrons, such structures are all gravity-dominated systems (shortly, G-bodies). Hadrons, however, could be considered G-bodies of the strong-gravity field mediated by the heavy graviton (that is the f-meson), see [5]. Stars and planets are not G-bodies because pressure, viscosity, thermal and solid-state effects play an important role in their structure; protostars and protoplanets, on the contrary, are certainly G-bodies and obey eq. (4). Equation (4) fits well some speculations [6] which consider hadrons, galaxies and the Universe as physical systems internally governed by similar laws (equation (4) suggests that all G-bodies are similar systems); such similarity prompts to extend to all G-bodies the well-known relations valid for hadrons: J - K M 2 / c (Regge slope) and ,~ ~ ( J / c ) K 1/2 connecting spin J to mass M and to the intrinsic magnetic moment ~ via the strong gravitational coupling K; replace K by G and get the well-known empirical relations of Brosche and Schuster, which are obeyed by a wide variety of celestial bodies [7]. A dynamical ground for such similarity has been suggested by Tassie who assumes that all astronomical objects have evolved through a hierarchical breaking of rotating pieces of cosmic strings (which eventually transformed by a phase transition into ordinary matter)[8]. From points A) to D), the close connection between the fundamental acceleration and the G-bodies self-similarity is clear. In sect. 2, I show that the well-known rough equality between the energy density of the 3 K relic radiation, of the mean galactic magnetic field, and of the starlight in the Milky Way springs from points C) and D) and from the Brosche and Schuster relations; in sect. 3, I show that ao is the surface gravity of any meson and the minimal acceleration of its quarks. Planck units are employed throughout, unless otherwise indicated. 2. - The ,~coincidence- o f the e n e r g y densities.

The energy per unit volume of starlight in a typical galaxy is (5)

~L -- P / A ,

where A is the surface area of the galaxy, and P is the electromagnetic (e.m.) power radiated from the entire galaxy, given by (6)

P - N(s)

P(s),

the number of (typical) stars in the galaxy, P ( s ) = the e.m. power radiated from each star, given by

N(s) =

(7)

P(s) - E ( s ) / T ( s ) ,

28

c. MASSA

E ( s ) = energy radiated by a star during its entire lifetime; T(s) = the lifetime of a typical star, of the order of 1 0 1 7 S - - t. The coincidence T(s) ~ t is easily explained by

eq. (3b) and by the standard model of the inner constitution of stars, that give, respectively, (8a)

t m 3 ~ 1,

(8b)

T(s) m e m ~ - 1,

mo = the electron mass, mp = the proton mass, m = the pion m a s s - ( m e m O ) 1/3, t-R.

Hence T ( s ) -

R, and eq. (7) reads

(9)

P(s) - E ( s ) / R = t i M ( s ) / R ,

M ( s ) = the mass of a typical star, and ti = 7.10 -a is the mass-energy conversion

factor, (if i g of hydrogen is turned into helium, 7.10 -3 g of material are converted into energy). As the galaxy surface gravity equals the fundamental acceleration a0 = l / R , we have R = r 2 / M (in this section M and r indicate the galactic mass and radius, respectively) and eqs. (9) and (6) give, respectively, (10a)

P ( s ) - tiM(s) M / r 2 ,

(10b)

P - ti(M/r) 2 ,

(because N ( s ) M ( s ) =

M ) . Then, eq. (5) gives

(11)

PL -- ( M / r 2 ) 2 (ti/4~)

(because A = 47:r 2). The galactic mean magnetic field is B = t~/r a and the intrinsic magnetic moment t~ is related to the galacting spin J by the Schuster relation [8], (12)

fz - J - M V r - M ( M r ) 1/2 ,

where V - ( M / r ) 1/2 is the rotation speed. The magnetic energy density is .oB = B2/8r: and then (13)

PB /~ L -- M / ( t i r ) .

Insert the Brosche relation (14) into eq. (12) and find M / r (15)

J = M2/a ;

~ = constant - 10 2

a2 which for eq. (13) gives ~B/PL -- a2/ti.

Thus ~:~/,zL has a constant value not far from unity. A group-theoretical analysis by Wesson[7] suggests a is the free-structure constant equal to 1/137 (and the fundamental dimensionless parameter of microphysics could be also the fundamental parameter of gravitational physics).

29

IS T H E R E A F U N D A M E N T A L A C C E L E R A T I O N ?

Therefore, the puzzling coincidence ,~L ~ ?B could be explained as a simple consequence of three facts: a) the surface gravity of the galaxy is of the order of the fundamental acceleration 1 / R; b) the galaxy obeys the Schuster and the Brosche relation; c) the E-factor is of the order of the free-structure constant a. Facts a) and b) are related to the self-similarity of G-bodies; fact c) requires no explanation because there is no mystery in the coincidence of two numbers which are both not very far from unity. From M / r ~ ~ 1 / R we get M R 3//r 4 ~ ( R / r ) 2 ~ M ( u ) / M , where M ( u ) is the mass of the observable Universe, and eq. (4) has been employed. We get at once M ( u ) / R 3 - ( M / r 2 ) 2 - 4r~Lfl (for eq. (11)). Therefore, (16)

~D - ,oi,

where D is the mean mass density of the Universe. Thus, there is a constant ratio of the order of fl - 10 -a between the starlight energy density of the galaxy and the mass density of the Universe. The latter is about one thousand times the energy density ,OT of the cosmological microwave radiation and then (roughly) ?L ~ T

9

(It is rather pointless to question why fl ~ .~T/D as the equality of two numbers not very far from unity needs no explanation, it can be a play of whimsical chance.) In conclusion, the equality between the energy density of starlight, of the interstellar magnetic field, and of the cosmological 3 K radiation field is not surprising if one considers G-bodies as similar systems, with surface gravity equal to the fundamental acceleration a0. 3. - F u n d a m e n t a l

acceleration

and

mesons.

In the bag model for spinning mesons a quark and an antiquark are confined to a bubble elongated by the centrifugal force. The transverse dimension s (measured by a co-rotating observer) and the length L of the bubble (with L > s) are (17a)

s N ? -1/4,

(17b)

L - pro,

where ~ (a universal constant) is the pressure the vacuum exerts upon the bubble, m is the meson mass, and (18a)

p - b-1,

(18b)

b --

( a ~ ) 1/2 ,

where a is the strong-coupling constant and b (another universal constant) is the energy of the tube per unit length along the axis; namely, b is the constant force which exerts among the quarks trapped in the bag.

30

c.

MASSA

The model leads to the spin-mass relationship [9] (19a)

J = pm 2 ,

(19b)

J - Lm.

For a slowly rotating bubble, /~e. for a low-spin meson ( J - 1 ) gives (2O)

L-

s - 1/m

eq. (19b) (J-

1).

The gravitational potential V and the gravitational field g at distance r from the centre of the bubble and measured by a co-rotating observer lying on the transversal axis of the tube at r I> s are easily computed in Newton's theory, (21)

V(r) = - (m/L)

(21a)

In ((X + 1 ) ( X - 1)-1),

X = (1 + ( 2 r / L ) 2 ) 1/2 ,

(22)

g ( r ) = - ( m / r 2)(1 + ( L / 2 r ) 2 ) - 1/2.

As s is a universal constant (it depends essentially on p via eq. (17a)) all mesons with spin of order unity have more or less the same mass m0 and, for eq. (20), L - s N 1/rao, their surface gravity is then (23)

g(s) -

- mos-2 ~ _ 1/s 3 .

Interestingly, the surface gravity of a heavy high-spin meson (m>>m0, J > > l ) is given still by eq. (23); in fact, L/s ~ pm/s

~ pmmo ~ pm~ (m/mo) ~ J(mo)m/mo

where eqs. (6b), (8), (9) have been employed. Therefore L > > s and for eq. (22), g ( s ) - - m ( L s ) -

~ m / m o >> 1,

-1 and, for eq. (17b), g ( s ) -

1/(ps).

For eq. (19a) p ~ 1 / m ~ ~ s 2 and then g ( s ) ~ 1 / s 3. To conclude: all mesons (independently of their mass, spin and shape) have the same surface gravity g ( s ) = = 1 / s 8 and by setting g ( s ) = ao ~ 1 / R one finds (24)

s - R 1/3 ,"

set R - 10~cm and find the quite reasonable value s - 10-13cm, that for eqs. (17a), (18a), (18b) gives p - 10~ erg/cm 3, b - 109dyne, p - 1021 cm 2 s -1 g-1. The length of the bubble cannot exceed the radius R of the observable Universe, then L < R which for eqs. (17b), (18a) gives (25)

m < Rip - bR,

(26)

I V ( s ) l < ( l / p ) ln(R 2 + s2)s -2 .

The force among the quarks trapped in the bag is b = ma, where a = the centripetal acceleration; then eq. (25) leads to (27)

a ~> 1/R ~ a o .

IS THERE A FUNDAMENTAL ACCELERATION?

31

So, ao is also the minimal acceleration undergone by the quarks trapped in a spinning meson; any acceleration less than ao would cause the meson to break because the couple quark-antiquark would be separated by the cosmological event horizon. If R - 102Scm, then m ~< 1017 g and IV(s)l ~ 10 -15 cm 2 s -2. Such bounds are of course more of speculative than of practical interest because it is very difficult to stretch the bubble up to L = R, the transition from a one-meson to a two-meson state becoming more and more probable as L increases. (Incidentally, the constraint L < R looks interesting in a radiation-dominated universe where the temperature of the cosmic fluid T N R-1/2 creates (for photon-photon scattering process) particles of mass < T; for mesons, p m = L < R and then (28)

m < p -~/a - 10 -18 g N 106 GeV.

Note, such bound is independent of T.) The bound m < 1017g applies to mesons, because they are elongated bubbles according to the bag model; it does not apply to particles with a spherically symmetric gravitational field; such particles obey the much more severe constraint: rest mass < the Planck mass - 2.10-5g, otherwise [10] their wave packet of size (mass) -1 would be less than their gravitational radius: a situation general relativity forbids. Mesons cannot be black holes, whatever their mass may be, because their surface gravitational potential V(s), constrained by inequality (26), is < c 2. As final remark, I raise the question whether equality surface gravity = ao applies to leptons; leptons are usually considered point particles, thus we do not know how to define their surface gravity. Strong gravity, however, gives a natural scale of length for any particle, leptons included. According to [11], the photon ~, is coupled to the massive ,o-meson which has the same quantum numbers as ~, and is hence called the heavy photon. The p-meson couples (strongly) to the f-meson and then in an electromagnetic (e.m.) field a probability of finding a non-zero density of f-mesons does exist. The f-meson is a massive particle with the same quantum numbers as the massless graviton and is hence called the heavy graviton. Just as the massless graviton transmits the usual gravitational field described by Einstein's equations with coupling constant G = = 6.7.10-Scm~g -1 s - 2 = 1 in Planck units, so the heavy graviton f transmits the strong-gravity (s.g.) field described again by Einstein's equations, with the constant G replaced by the new constant K - 103e cm3 g - i s-2 _ 104o in Planck units [10]. In conclusion, (29)

e.m. field -* ~, --~ p --) f --~ s.g. field,

where each arrow reads ,,produces,. Chain (29) shows that any e.m. field is an indirect source of strong gravity, and that one can speak of a strong-gravity influence upon the e.m. field of a particle, whether the electric charge is carried by a lepton or by a hadron. Detailed calculations show [12] the existence of a critical value r~ = q(2K) ~/2 of the radial coordinate r for which the electrostatic potential of any charged particle has an equilibrium point and the electric field vanishes. As q2 _ 10 for a hadron and - 1/137 for a lepton, the hadronic characteristic radius is about 10 -14 cm, and the leptonic one about 10-15 cm. I recall that the Salam-Weinberg electroweak model reveals a close connection between the photon V and the neutral weak Z-boson so it is not unreasonable to

32

c. MASSA

conjecture the existence of a Z-p mixing which in turn (owing to the ~-f coupling) causes a Z-f mixing and the consequent chain (30)

weak field ~ Z ~ p ---)f---) s.g. field.

By paralleling ref.[12] (and by neglecting the f'mite range R ( w ) N 10-16cm of the weak interaction) one fmds that any neutral lepton carrying the weak charge q has the characteristic radius (31)

r e - qK 1/2 .

Observations [4] give q 2 10-13, then the characteristic neutrino radius turns out to be about 10 .20 cm; this is much smaller than R(w), thus our approximation looks justified. If we define now as surface gravity of any particle (hadron or lepton) its gravitational field at r = re, (32)

g - m / r ~ - m ( q 2 K ) -1

and if we set g = ao - 1/R, we get (33)

m - Kq 2 / R .

In the strong-gravity theory [5] K is related to the mass mo of a typical light hadron, such as the pion, by K = 1 / m ~ ; the meson mass, in turn, is given by (see eq. (3b)) m ~ R -1/3, and then (34)

K - R 2/3 = c(RG )2/3h - 1/3,

that, for eq. (33), leads to (35)

m-

q2 R-1/3 = (h2 /GR)l/a(q2 /hc) ,

where c, h~ G have been displayed. Equation (35) generalizes eq. (3b) to any particle with ,,main- charge q; for a proton q 2 = 14.5, for a pion q 2 = 2, for an electron q 2 = 1/137, for a neutrino q2 = 10-13. Set R = 101~ light years = 10~cm and find, respectively, the masses 10 - u g ,

10 - ~ g ,

10 -27g,

10 - 2 7 g - 1 0 -4eV.

This rather small value for the neutrino mass is characteristic of many GUT models.

REFERENCES [1] C. MASSA:Lett. Nuovo Cimento, 44, 609 (1985). [2] E. FISCHBACH: Tests of general relativity at the quantum level, in Cosmology and Gravitation, edited by P. G. BERGMANNand V. DE SABBATA(Plenum Press, New York, N.Y., 1980), p. 359. [3] C. MASSA:Lett. Nuovo Cimento, 44, 695 (1985) and references therein.

IS THERE A FUNDAMENTAL ACCELERATION?

33

[4] P. CALDIROLA,M. PAVSIC and E. RECAMI: Nuovo Cimento B, 48, 205 (1978), see pp. 212, 216, 233. [5] C. SWARAM and K. P. SINHA: Phys. Rep., 51, 111 (1979). [6] See ref. [4], [1] and references therein. [7] For the Schuster relation, see: S. P. SmAG: Nature (London), 278, 535 (1979); C. MASSA: Ann. Phys. (Leipzig), 46, 156 (1989) and references therein. For the Brosche relation, see: P. S. WESSOS: Phys. Rev. D, 23, 1730 (1981) and references therein; V. TRIMBLE: Comments Astrophys., 10, 27 (1984). [8] L. J. TASSIE: Nature, 40, 323 (1986). [9] K. GOTTFRIED and V. F. WEISSKOPF: Concepts of Particle Physics (Oxford, 1984), 2nd volume, p. 404. [10] See ref. [5] and references therein; see also H.-J. TREDER: Found. Phys., 15, 161 (1985). [11] C. IS~hM, )a SALAM and J. STRATHDEE: Phys. Rev. D, 3, 867 (1971). [12] V. DE SABBATAand M. GASPERISI: Lett. Nuovo Cimento, 24, 215 (1979).