Does the cosmological constant imply the existence of a minimum mass?
ABSTRACT We show that in the framework of the classical general relativity the presence of a positive cosmological constant implies the existence of a minimal mass and of a minimal mean density in nature. These results rigorously follow from the generalized Buchdahl inequality in the presence of a cosmological constant.
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ABSTRACT: The physical properties of the astronomical compact objects are explored on a theoretical and observational basis in a textbook designed for a one-semester beginning-graduate-level astrophysics course. Overlapping topics from solid-state, nuclear, and particle physics, relativity theory, and hydrodynamics are considered. Subjects discussed include star deaths and the formation of the compact objects, the cold equation of state below and above neutron dip, white dwarfs and their cooling, perturbations of fluid configurations, rotation and magnetic fields, pulsars, compact X-ray sources, accretion onto compact objects, gravitational radiation, supermassive stars, stellar collapse, and supernova explosions.Research supported by the National Science Foundation. New York, Wiley-Interscience, 1983, 663 p. 01/1983;
- 01/1990; Pergamon Press.
arXiv:gr-qc/0509110v2 5 Oct 2005
Preprint ESI 1712, TUW–05–14
Does the cosmological constant imply the existence of a minimum
C. G. B¨ ohmer∗
The Erwin Schr¨ odinger International Institute for Mathematical Physics,
Boltzmanngasse 9, A-1090 Wien, Austria and
Institut f¨ ur Theoretische Physik, Technische Universit¨ at Wien,
Wiedner Hauptstr. 8-10, A-1040 Wien, Austria
Department of Physics, The University of Hong Kong,
Pokfulam Road, Hong Kong SAR, P. R. China
(Dated: February 7, 2008)
We show that in the framework of the classical general relativity the presence of a positive cosmo-
logical constant implies the existence of a minimal mass and of a minimal density in nature. These
results rigorously follow from the generalized Buchdahl inequality in the presence of a cosmological
PACS numbers: 03.70.+k, 11.90.+t, 11.10.Kk
∗Electronic address: email@example.com
†Electronic address: firstname.lastname@example.org
One of the most important characteristics of compact relativistic astrophysical objects
is their maximum allowed mass. The maximum mass is crucial for distinguishing between
neutron stars and black holes in compact binaries and in determining the outcome of many
astrophysical processes, including supernova collapse and the merger of binary neutron stars.
The theoretical values of the maximum mass and radius for white dwarfs and neutron stars
were found by Chandrasekhar and Landau and are given by Mmax ≈
and Rmax≤ (?/mc)(?c/Gm2
either electron or neutron . Thus, with the exception of composition-dependent numerical
B)1/2, where mBis the mass of the baryons and m the mass of
factors, the maximum mass of a degenerate star depends only on fundamental physical
constants. For non-rotating neutron stars with the central pressure at their center tending
to the limiting value ρcc2an upper bound of around 3M⊙has been found . The maximum
mass of different types of astrophysical objects (neutron stars, quark stars etc.) under
different physical conditions, including rotation and magnetic fields, was considered by using
both numerical and analytical methods (, and references therein).
With the use of the gravitational field equations for a static equilibrium configuration,
Buchdahl  obtained an absolute limit of the mass-radius ratio of a stable compact object,
given by 2GM/c2R ≤ 8/9. This limit has been generalized in the case of scalar-tensor
theories , for charged fluid spheres , and for the Schwarzschild-de Sitter geometries in
the presence of a cosmological constant .
If the problem of the maximum mass of compact objects had been considered in great
detail, the more fundamental question of the possible existence of a minimum mass in
the framework of general relativity had been investigated at a much lesser extent. The
minimum mass of neutron stars or of white dwarfs can be derived qualitatively from en-
ergy considerations . A lower limit for the radius of the neutron stars of the form
R ≥ (3.1125 − 0.44192x + 2.3089x2− 0.38698x3), with x = M/M⊙and 1 ≤ x ≤ 2.5 has
been found in .
At a microscopic level two basic quantities, the Planck mass mPand the Planck length lP
are supposed to play a fundamental physical role. The Planck mass is derived by equating
the gravitational radius 2Gm/c2of a Schwarzschild mass with its Compton wavelength
?/mc. The corresponding mass mPl= (c?/2G)1/2is of the order mPl≈ 1.5 × 10−5g. The
Planck length is given by lPl= (?G/c3)1/2≈ 1.6×10−33cm and at about this scale quantum
gravity will become important for understanding physics. The Planck mass and length are
the only parameters with dimension mass and length, respectively, which can be obtained
from the fundamental constants c, G and ?.
The observations of high redshift supernovae  and the Boomerang/Maxima data ,
showing that the location of the first acoustic peak in the power spectrum of the microwave
background radiation is consistent with the inflationary prediction Ω = 1, have provided
compelling evidence for a net equation of state of the cosmic fluid lying in the range −1 ≤
w = p/ρ < −1/3. To explain these observations, two dark components are invoked: the
pressure-less cold dark matter (CDM) and the dark energy (DE) with negative pressure.
CDM contributes Ωm∼ 0.25, and is mainly motivated by the theoretical interpretation of
the galactic rotation curves and large scale structure formation. DE provides ΩDE ∼ 0.7
and is responsible for the acceleration of the distant type Ia supernovae. The best candidate
for the dark energy is the cosmological constant Λ, which is usually interpreted physically
as a vacuum energy. Its size is of the order Λ ≈ 3 × 10−56cm−2. In some theoretical
models is is assumed that the cosmological constant can be derived from the reduction to
4D of higher-dimensional unified theories . Since at least 70% of the Universe consists
of vacuum energy, it is natural to consider Λ as a fundamental constant. Therefore we can
chose as the set of fundamental constants (c,G,?,Λ).
By using dimensional analysis Wesson  has found two different masses which can
be constructed from this set of constants. The mass mP relevant at the quantum scale is
mP= (?/c)?Λ/3 ≈ 3.5 × 10−66g while the mass mPErelevant to the cosmological scale is
mPE= (c2/G)?3/Λ ≈ 1 × 1056g.
The interpretation of the mass mPEis straightforward: it is the mass of the observable
part of the universe, equivalent to 1080baryons of mass 10−24g each. The interpretation of
the mass mPis more difficult. By using the dimensional reduction from higher dimensional
relativity and by assuming that the Compton wavelength of a particle cannot take any value,
Wesson  proposed that the mass is quantized according to the rule m = (n?/c)?Λ/3.
Hence mP is the minimum mass corresponding to the ground state n = 1.
These results about the fundamental mass have been obtained by using a phenomeno-
logical approach. It is the purpose of the present Letter to give a rigorous proof on the
existence of a minimum mass in general relativity. The existence of such a mass is a di-
rect consequence of the presence of a non-zero cosmological constant in the gravitational
field equations. Therefore these two quantities are strongly inter-related. In order to prove
the existence of a minimum mass we follow the approach introduced by Buchdahl  and
generalized to the case of a non-zero Λ in .
The present Letter is organized as follows. The limiting density and mass for a general
relativistic object is derived in the next Section. We conclude our results in the last section.
II. LOWER MASS AND DENSITY BOUNDS FOR STATIC GENERAL RELA-
We assume that the spherically symmetric general relativistic mass distribution is de-
scribed by the metric (in the present Section we set c = 1):
ds2= −eν(r)dt2+ eλ(r)dr2+ r2(dθ2+ sin2θdφ2).(1)
Static and spherically symmetric perfect fluids in general relativity are described by three
independent field equations (for four unknown functions) that imply conservation of energy-
momentum. Eliminating the function ν(r) from the field equations yields the well known
Tolman-Oppenheimer-Volkoff equation in the presence of a cosmological constant Λ .
Let us introduce Buchdahl variables, defined by 
y2= e−λ= 1 − 2w(r)r2−Λ
3r2,ζ = eν/2,x = r2, (2)
where w(r) is the mean density up to r, w(r) = m(r)/r3and m(r) is the mass inside radius
r, m(r) = 4π?r
Eliminating the pressure function from the field equations, one obtains the following
0ρ(r′)r′2dr′, with ρ the mass density of the compact object with radius R.
differential equation [4, 7]
Eq. (3) can be used to compare solutions with decreasing energy density with ones having
constant density, for which the second term in Eq. (3) vanishes. In the latter case one can
integrate Eq. (3) and compare it with a decreasing solution, which then yields the generalized
Buchdahl inequality in the presence of the cosmological constant :
Eq. (4) provides a lower bound for the mass and density of general relativistic objects.
To prove this result, we start by squaring Eq. (4), multiplying it by G2M2, eliminating the
density on the right-hand side by ρ = 3M/(4πR3) and taking all terms to the left-hand side.
Then we obtain the following expression
9R3?2R ≥ 0, (5)
which can be written as a product of three terms
Dividing by the factor (−2) reverses the inequality sign. Hence either one or all three
terms of the product must have a negative sign in order to fulfill the latter equation.
With Λ = 0 we can easily find the correct signs. The first term is strictly positive for
Λ = 0 (it reads GM), hence only one of the remaining terms is negative. The second term
for Λ = 0 also yields GM. Therefore, the last term must be negative, which for vanishing Λ
gives 2GM ≤ 8R/9, which is nothing but the standard Buchdahl inequality .
Since the signs of the three terms are now known, let us analyze Eq. (6) with a non-zero
cosmological constant. We shall consider separately the cases of a positive (Λ > 0) and of
a negative (Λ < 0) cosmological constant. For Λ > 0 the analysis of the signs of Eq. (6)
gives the following algebraic conditions to be satisfied by the mass and radius of the matter
distribution and by the cosmological constant.
(i) Positivity of the first term of (6) implies
GM ≥ −Λ
For positive Λ this is trivially fulfilled.
(ii) Positivity of the second term yields
which as before gives a lower bound on the mass.
(iii) Finally, negativity of the last term of the product (6) reads