Vacuum Energy and the Economical Universe

Abstract

We revisit a suggestion, first made by Haas and Jordan but often attributed to Tryon and others, that the universe could have zero net energy since a test body''s positive rest energy is approximately cancelled out by its negative gravitational binding energy with respect to the rest of the matter in the universe. If the universe is not matter- but vacuum-dominated, as observations now indicate, then the balance between rest and binding energies is substantially altered. We consider whether this may make the Newtonian case for a universe from nothing more plausible than before.

Full text

Vacuum energy and the economical universe
James Overduin∗and Hans-J¨org Fahr
Institut f¨ur Astrophysik und Extraterrestrische Forschung, Universit¨at Bonn
Auf dem H¨ugel 71, D-53121 Bonn, Germany
We revisit a suggestion, first made by Haas and Jordan but often attributed to Tryon
and others, that the universe could have zero net energy since a test body’s positive
rest energy is approximately cancelled out by its negative gravitational binding energy
with respect to the rest of the matter in the universe. If the universe is not matter-
but vacuum-dominated, as observations now indicate, then the balance between rest
and binding energies is substantially altered. We consider whether this may make the
Newtonian case for a “universe from nothing” more plausible than before.
The idea of a universe with zero net energy played an important role in stimulating early work
in quantum cosmology(1,2) and is usually traced back to an influential 1973 paper by Tryon.(3) †
It should be credited historically to Arthur Haas, who introduced it in 1936,(6) and to Pascual
Jordan, who subsequently made it a central feature of his “empirical cosmology,” writing in 1947:
“The sum of the individual energies of all the elementary particles in the world is within an order
of magnitude of their mutual gravitational potential energy; and the possibility suggests itself of
tightening this empirical order-of-magnitude equality by hypothesizing that the total energy of
∗Present address: Astrophysics and Cosmology Group, Department of Physics, Waseda University,
Okubo 3-4-1, Shinjuku-ku, Tokyo 169-8555, Japan
†Related ideas were discussed in the same year by Albrow(4) and Fomin,(5) with publication delayed in the latter’s
case until 1975. Because these two papers (especially that of Albrow) are so persistently mis-cited in the literature,
we give the references in full in the bibliography.
1

the universe is zero.”(7)‡This hypothesis has received support from calculations within general
relativity which indicate that total energy (including gravitational energy) vanishes to within a
constant of integration for homogeneous and isotropic cosmological models.(10,11)
The plausibility of the zero-energy idea can be demonstrated on straightforward dimensional
grounds within Newtonian gravity, based only on the expectation that a test particle’s gravitational
potential energy Ushould be expressible in the usual form
U=−GMm
d,(1)
where Mand drepresent a suitably defined mass and “effective distance” of the rest of the matter
in the universe. These quantities are presumably cosmological, so that Mcan be related to the
mean large-scale matter density ρM. Assuming that this is close to the present critical density,
ρcrit ≡3H2/(8πG), and defining ΩM≡ρM/ρcrit as usual, one has M=4
3π d 3ρM= ΩMH2d3/2G
where Gand Hare Newton’s and Hubble’s constants respectively. Substitution into (1) gives
U=−ΩM
2Hd
c2
mc2.(2)
Reasonable choices for dwould be of the same order as the present Hubble distance dh≡c/H,
which sets the length scale of most physical distance measures in relativistic as well as Newtonian
cosmology. The particle horizon distance dph, for instance, is perhaps the most straightforward
relativistic analog of the “size of the universe,” and is given by dph = 2c/H in the simplest case of
a matter-filled universe of critical density (ΩM= 1).§If dcan be identified with length scales like
‡Translation by the authors. Jordan’s original reads: “Die Summe der Einzelenergien aller Elementarteilchen der
Welt ist gr¨oßenordnungsm¨aßig gleich dem Betrage ihrer wechselseitigen Gravitationsenergie; und es liegt nahe, diese
empirische Gr¨oßenordnungsgleichheit hypothetisch zu versch¨arfen zu der Annahme einer Gesamtenergie Null f¨ur das
Weltall.” See Ref. 8 for a historical review. Gamow(9) has related that Einstein was so struck on hearing of this idea
that he stopped in his tracks in the middle of a Princeton street and was nearly run over.
§The particle horizon distance is the distance travelled by photons since the time of the big bang and hence
represents an upper limit on what might be called the “causal size” of the universe, whose total physical size is of
course infinite in flat models.
2

this, then Eq. (2) shows that Uis indeed within a factor of two of −mc2, plausibly cancelling out
the rest energy E=mc2of the test particle and implying that a Newtonian universe made up of
such particles might well have a total energy of zero.(3)
The gravitational binding energy of a test body with respect to the rest of the universe, however,
depends critically on the latter’s equation of state. Experimental data from the magnitude-redshift
relation for Type Ia supernovae(12,13) and and the angular power spectrum of fluctuations in the
cosmic microwave background (CMB)(14) now indicate that the universe is dominated, not by
matter, but by vacuum energy with an effective cosmological constant
Λeff = 3H2ΩΛ/c2,(3)
where ΩΛ≡ρvac/ρcrit and ρvac = (Λbarec2/8πG) + ρvev is the physical vacuum energy density (with
contributions from a bare cosmological constant Λbare and the vacuum expectation values of all
other fields). Let us see what this Λeff-term implies for the hypothesis that the rest mass energy E
of cosmic fluid within some spherical region of radius dis precisely cancelled by its gravitational
binding energy U. We will take this fluid to consist only of a pressureless matter component
(pM= 0) and a vacuum energy component whose pressure, from the first law of thermodynamics,
must be equal and opposite in sign to its energy density so that pvac =−ρvacc2. Net density and
pressure are then ρ=ρM+ρvac and p=pM+pvac =−ρvacc2respectively.
The nonzero pressure term leads to significant departures from Newtonian gravity on large
scales. Poisson’s equation ∇2φG=−4πGρ, for instance, must be modified to read(15)
∇2φG=−4πG(ρ+ 3p/c2),(4)
where φG(r) is the scalar gravitational potential. (Here it should be borne in mind that this is
a non-relativistic equation and may not be an accurate approximation when applied to situations
where the magnitude of the pressure term is comparable to that of the density, as is now thought
to be the case on large scales in cosmology.) Inserting ρand pfor the matter-vacuum energy fluid,
3

and integrating twice with respect to the radial coordinate r, one obtains
φG(r) = −1
2H2(ΩM−2ΩΛ) ( 1
2r2−A
r+B),(5)
where A, B are integration constants. We set A= 0 in order to avoid the possibility of nonphysical
short-range forces (of the form FG=−m∇φG) and choose B= 0 so that φG(0) = 0 (keeping B6= 0
would give us an additional free parameter, but one without obvious physical significance). This
leaves us with
φG(r) = −1
4H2(ΩM−2ΩΛ)r2,(6)
The total gravitational binding energy of matter in a spherical region (of radius dand mass M)
may then be obtained in the usual way by integrating over spherical mass shells
U=ZφG(r)dM , (7)
where dM = 4πρMr2dr = (3M /d 3)r2dr is the mass of the shell between rand r+dr. We find
U=−3ΩM
20 Hd
c2
1−2ΩΛ
ΩMMc2,(8)
where we have again chosen the reference potential so that U= 0 for d= 0. Eq. (8) differs from
the dimensional estimate (2) by a multiplicative factor of 3
10 (1 −2ΩΛ/ΩM).
Pressure also modifies the total mass, or rest energy of the spherical region.(16) For bounded,
stationary systems this is given by the Tolman formula(17)
E=Z(ρ c2+ 3p)√−g dV . (9)
Inserting the density ρand pressure pof the combined matter-vacuum fluid as before, and inte-
grating, we find that E= (1 −2ρvac/ρM)c2RρM√−g dV , or
E=1−2ΩΛ
ΩMMc2,(10)
which differs from what one might expect in a matter-only universe by the factor (1 −2ΩΛ/ΩM).
4

Let us now implement the Haas-Jordan idea:
E+U= 0 .(11)
Inserting Eqs. (8) and (10) puts this into the form (assuming Mc26= 0):
1−2ΩΛ
ΩM"1−3ΩM
20 Hd
c2#= 0 ,(12)
which has two roots. There are thus two scenarios in which the Newtonian universe could have
sprung into existence “from nothing.” The first is obtained if
ΩM= 2ΩΛ,(13)
This condition, if satisfied, would imply that both Uand Evanish, Eqs. (8) and (10), which is
counterintuitive but follows from the fact that contributions from vacuum pressure and energy
density are opposite in sign. It might also be suggestive of a scaling-type relationship between the
effective cosmological constant Λeff and the mean matter density ρM, although that would be an
extrapolation from the static picture considered here.
For a spatially flat universe (as indicated by the CMB data, and as implicitly assumed in the
Newtonian approach), Eq. (13) would imply that ΩM∼2/3 and ΩΛ∼1/3. The corresponding
value of the effective cosmological constant is given by (3) as Λeff ∼(H/c)2. While these numbers
are in somewhat better agreement with observation than those which have often been held on
theoretical grounds (e.g., the Einstein-de Sitter model with ΩM= 1 and ΩΛ= 0), they are only
marginally compatible with the latest experimental evidence, which favors the reverse situation
with ΩM≈0.3±0.2 and ΩΛ≈0.8±0.2.(14)
The second zero-energy scenario is that in which
ΩM=20
3Hd
c−2
.(14)
Since 2GM = ΩMH2d3, this can be rearranged in the form 3GM/(10 c2d) = 1. This is reminiscent
of the “perfect dragging condition” obtained many years ago by Thirring,(18) who showed that the
5

inertial frame of a test body would co-rotate with that of a spherical shell of mass Mat radius d
if the relation 4GM/(3 c2d) = 1 were satisfied. The Machian properties of Newtonian models with
vanishing total energy (and total quantities in general) have been noted by others over the years
[see Ref. (19) for review].
Numerically, if we identify dfor argument’s sake with length scales of order the particle horizon
distance, then a critical-density universe with no vacuum energy would have d∼2c/H, as above.
This in (14) gives ΩM∼5/3, which is excessively large and throws some doubt on the viability of
the zero-energy hypothesis, in this version at least.
Characteristic length scales in vacuum-dominated cosmological models, however, are larger than
those in models containing only pressureless matter, and this could bring the idea of an “economical
universe” of zero net energy back within reach. In a flat model with the observationally-preferred
value of ΩΛ∼0.8, for instance, the particle horizon distance increases to dph ≈4c/H. This in
Eq. (14) would imply ΩM∼0.4, which is more consistent with the values actually seen (as well as
with spatial flatness, as implicitly assumed in the Newtonian approach adopted here). While the
numerical examples given here should not be pushed too far, the implication is that regions whose
size is of the order of the cosmological horizon come closer to having zero net energy if the universe
contains significant amounts of vacuum energy. This suggests that a vacuum-dominated universe
might be more likely to fulfil the Haas-Jordan hypothesis.
The two conditions (13) and (14) are moreover independent, which raises the third possibility,
at least in principle, that both might be satisfied simultaneously. If this were the case then the
derivative of Eq. (12) with respect to dwould also vanish, corresponding to a universe which was
not only economical now, but remained so for all time.
6

Acknowledgements
Thanks go to W. Priester for comments, and to the Alexander von Humboldt Stiftung and the Japan Society for the
Promotion of Science for support.
References
1. R. Brout, F. Englert and E. Gunzig, “The creation of the universe as a quantum phenomenon,” Ann. Phys.
(NY) 115, 78 (1978).
2. A. Vilenkin, “Creation of universes from nothing,” Phys. Lett. 117B, 25 (1982).
3. E. P. Tryon, “Is the universe a vacuum fluctuation?” Nature 246, 396 (1973).
4. M. G. Albrow, “CPT conservation in the oscillating model of the universe,” Nature: Physical Science 241, 56
(1973).
5. P. I. Fomin, “Gravitational-instability of vacuum and cosmological problem,” Doklady Akademii Nauk Ukrain-
skoi S.S.R. 9A, 831 (1975).
6. A. Haas, “An attempt to a purely theoretical derivation of the mass of the universe,” Phys. Rev. 49, 411
(1936).
7. P. Jordan, Die Herkunft der Sterne (Wissenschaftliche Verlagsgessellschaft M.B.H., Stuttgart, 1947), p. 16.
8. H. Kragh, Cosmology and Controversy (Princeton Univ. Press, New Jersey, 1996), pp. 71-72.
9. G. Gamow, My World Line (Viking Press, New York, 1970), p. 150.
10. N. Rosen, “The energy of the universe,” Gen. Rel. Grav. 26, 319 (1994).
11. F. I. Cooperstock and M. Israelit, “The energy of the universe,” Found. Phys. 25, 631 (1995).
12. A. G. Riess et al., “Observational evidence from supernovae for an accelerating universe and a cosmological
constant,” Astron. J. 116, 1009 (1998).
13. S. Perlmutter et al., “Measurements of Ω and Λ from 42 high-redshift supernovae,” Astrophys. J. 517, 565
(1999).
14. A. H. Jaffe et al., “Cosmology from MAXIMA-1, BOOMERANG & COBE/DMR CMB observations,” Phys.
Rev. Lett. 86, 3475 (2000).
15. J. A. Peacock, Cosmological Physics (Cambridge Univ. Press, Cambridge, 1999), p. 25.
7

16. N. Rosen and F. I. Cooperstock, “The mass of a body in general relativity,” Class. Quant. Grav. 9, 2657
(1992).
17. R. Tolman, Relativity, Thermodynamics and Cosmology (Clarendon, Oxford, 1934), p. 235.
18. H. Thirring, 1918, “ ¨
Uber die Wirkung rotierender, ferner Massen in der Einsteinschen Gravitationstheorie,”
Phys. Zeits. 19, 33 (1918); err. Phys. Zeits. 22, 29 (1921).
19. J. M. Overduin and H.-J. Fahr, “Matter, spacetime and the vacuum,” Naturwissenschaften 88, 491 (2001).
8