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Qualitative analysis of the Tolman metrics within the unimodular framework

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We investigate the behaviour of the Tolman metrics within the formalism of the trace-free (or unimodular) gravity. While this approach is similar to the standard Einstein field equations, some subtlety arises. The effective number of independent field equations is reduced by one on account of the density and pressure appearing as an inseparable entity the inertial mass density. Further energy is not conserved within the trace-free theory but the conservation law may be supplemented to the field equations. This presentation of the field equations offers a different avenue to determine the density and pressure explicitly. It turns out that an extra integration constant is always in evidence. While this constant has little impact on the dynamics and energy conditions, it makes a significant impact on the gravitational mass and equation of state. Graphical plots are generated to analyse the behaviour of physical quantities qualitatively.
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Qualitative analysis of the Tolman metrics within the unimodular
framework
Sudan Hansraj
Astrophysics and Cosmology Research Unit, University of KwaZulu Natal
Njabulo Mkhize
UKZN
(Dated: June 12, 2018)
Abstract
We investigate the behaviour of the Tolman metrics within the formalism of the trace-free (or
unimodular) gravity. While this approach is similar to the standard Einstein field equations, some
subtlety arises. The effective number of independent field equations is reduced by one on account
of the density and pressure appearing as an inseparable entity the inertial mass density. Further
energy is not conserved within the trace-free theory but the conservation law may be supplemented
to the field equations. This presentation of the field equations offers a different avenue to determine
the density and pressure explicitly. It turns out that an extra integration constant is always in
evidence. While this constant has little impact on the dynamics and energy conditions, it makes a
significant impact on the gravitational mass and equation of state. Graphical plots are generated
to analyse the behaviour of physical quantities qualitatively.
1
arXiv:1806.04076v1 [gr-qc] 11 Jun 2018
INTRODUCTION
Unimodular gravity, also known as trace-free Einstein gravity, originated from Einstein
himself as a method to simplify the analysis of the field equations of general relativity by
fixing a coordinate system with a constant volume element. The idea was revived by Wein-
berg [1] as a potential paradigm to explain the phenomenon of vacuum energy. Then the
proposal lay dormant until Ellis [2, 3] realised that the large discrepancy in the value of
the cosmological constant predicted by quantum field theory and that of observation and
measurement may be explained by invoking the trace-free field equations. In this formula-
tion the cosmological constant is reduced to merely a constant of integration instead of an
innocuous object inserted by hand to address the accelerated expansion of the universe prob-
lem. Moreover, Ellis [2] demonstrated that for compact objects the usual Isreal–Darmois
boundary conditions are preserved. Note that unimodular gravity and the Einstein standard
theory are completely equivalent. Further important treatments of unimodular gravity may
be found in [4–6].
Because the equations of motion are a coupled system of up to ten partial differential
equations, the method of finding exact solutions is important. It has been shown by Hansraj
et al [7] that unravelling the Einstein equations through the trace-free paradigm, offered a
different solution generating algorithm. Indeed the old solutions are still valid, however, the
solution methods sometimes yield more general behaviour in the solution. For example, the
exterior metric in the unimodular scenario should be the Schwarzschild metric by the Jebsen–
Birkhoff theorem. The theorem asserts that the Schwarzschild solution is a consequence
strictly of spherical symmetry and independent of whether the distribution is static or not.
Through the trace-free algorithm a de-Sitter term in r2,rbeing the radial parameter, appears
in the solution of the differential equation. The implication is that although the cosmological
constant has been hidden in the field equations, it still lives on in the solution space.
In view of the foregoing, if indeed the unimodular framework constitutes a viable theory
of gravity, it is important to ask what the behaviour of astrophysical compact objects would
be like in this scenario. Usually the effect of the cosmological constant is negligible if not zero
when constructing models of stars. However, in this formulation, the hidden cosmological
constant has the potential to alter the physics as was demonstrated by Hansraj et al [7].
While the trace–free equations and the standard Einstein field equations are equivalent,
2
their presentation as a system of nonlinear partial differential equations leave room for
variation in behaviour. Whereas in the standard theory, there exists a system of at most
10 differential equations in the space and dynamical variables, there are 9 in the trace–free
version since the density and pressure are inextricably linked as the inertial mass density
ρ+p. In the Einstein equations, the law of energy conservation arises from the vanishing
divergence of the energy momentum tensor and generates the equation of hydrodynamical
equilibrium. This equation conveys no new information compared to the standard Einstein
field equations and may substitute one of the field equations. In contrast, in the trace–free
theory the energy conservation does not arise in the same way and must be inserted by
hand. That is the 9 field equations must be supplemented by the conservation law for a
complete system of equations governing the gravitational field. This is where the potential
for some variation in the dynamics exists. In the work of Hansraj et al [7] it was shown that
the well known Finch–Skea [8] stellar model was supplemented by additional terms which
affected the profiles of the energy density and pressure and consequently the sound speed,
stellar mass and surface redshift. The Finch–Skea model was shown to be consistent with
the astrophysical theory of Walecka [9].
Alternate or extended theories of gravity have aroused considerable interest recently. The
principal motivation to modify Einstein’s equations is the observed accelerated expansion
of the universe that is not a consequence of the standard model. Therefore, proposals
to include higher derivative or higher curvature terms have been invoked. In particular,
Einstein–Gauss–Bonnet (EGB) theory has proved promising in this regard. Strong support
for involving the Gauss–Bonnet term lies in the fact that this term appears in the effective
low energy action of heterotic string theory [10]. The Gauss–Bonnet term is the second
order term in the more general Lovelock polynomial [11, 12] which is constructed from
terms polynomial in the Ricci tensor, Ricci scalar and Riemann tensor. The Lovelock action
is the most general action generating at most second order equations of motion. A drawback
of this theory is that the dynamical behaviour is only impacted for spacetime dimensions
higher than 4 but the standard theory is regained for orders less than or equal to 4. In
Starobinksy’s [13] f(R) theory the action that is proposed is a polynomial in the Ricci
scalar. While this idea has the potential to account for the late time accelerated expansion
of the universe, it suffers from the severe drawback of yielding derivatives of orders higher
than two (ghosts) in the equations of motion. It is usually expected that gravitational
3
behavior is characterized by up to second order equations of motion and that the Newtonian
theory would be regained in the appropriate limit. Recently f(R) theory has been shown to
be conformal to scalar tensor theory.
In this work we analyze the effect of trace–free gravity on exact solutions found by Tolman
[14] from his study of static spherically symmetric perfect fluid field equations. The Tolman
metrics were derived after writing the equation of pressure isotropy in the special form
d
dr eλ1
r2!+d
dr eλν0
2r!+eλνd
dr eνν0
2r!= 0.(1)
As the system of partial differential equations is underdetermined, choices for one of the
metric potentials were made on the basis of the vanishing of some of the terms in the
isotropy equation in the form given by Tolman. In each of Tolman solutions, we review
the original assumptions and the consequent metric potentials. In the dynamical quantities
found by Tolman, we then proceed to insert the Tolman metric components into our trace–
free algorithm in order to probe the dynamical quantities.
TRACE–FREE FIELD EQUATIONS
In order to facilitate a direct comparison with the work of Tolman, we follow his conven-
tions. The static spherically symmetric spacetime in coordinates (t, r, θ, φ) is taken as
ds2=eν(r)dt2eλ(r)dr2r22+ sin2θ2(2)
where the gravitational potentials νand λare functions of the radial coordinate ronly. We
utilise a comoving fluid 4-velocity ua=eν/2δa
0a perfect fluid source with energy momentum
tensor Tab = (ρ+p)uaubpgab in geometrized units setting the gravitational constant G
and the speed of light cto unity. The quantities ρand pare the energy density and pressure
respectively.
The trace–free Einstein field equations are given by
Rab 1
4Rgab =Tab 1
4T gab (3)
where Tis the trace of the energy momentum tensor. The trace–free components of the
energy-momentum tensor are given by
ˆ
Tab = 3
4(ρ+p)e2ν,1
4(ρ+p)e2λ,r2
4(ρ+p),r2sin2θ
4(ρ+p)!(4)
4
from which the coupling of density (ρ) and pressure (p) is readily apparent. We follow the
notation of [2] and the hat symbol refers to trace-less quantities. Ordinarily in the regular
Einstein field equations the Gtt and Ttt components are free of the pressure variable.
The trace–free field equations may now be expressed as
(2ν00 +ν02ν0λ0) + 4
r(ν0+λ0) + 4
r2(eλ1) = 6(ρ+p)eλ(5)
4
r(ν0+λ0)(2ν00 +ν02ν0λ0)4
r2(eλ1) = 2(ρ+p)eλ(6)
(2ν00 +ν02ν0λ0) + 4
r2(eλ1) = 2(ρ+p)eλ.(7)
These three equations are not independent. Subtracting three times equation (6) from
(5) and equating (6) and (7) give the master set of field equations
2ν00 +ν02ν0λ02
r(ν0+λ0) + 4
r2(eλ1) = 0 (8)
ν0+λ0
r= (ρ+p)eλ(9)
p0+1
2(ρ+p)ν0= 0 (10)
where the last equation (10) is the conservation equation Tab
;b= 0. Note that the conservation
law is necessary in the trace–free field equations since the divergence of ˆ
Tab does not vanish
in general.
While the trace–free version of the field equations are equivalent to the standard Einstein
system, the presentation as a system of partial differential equations is manifestly different.
This presentation raises the question whether more general behaviour in the known metrics
may be found on solving the system of equations. The equation of pressure isotropy (5)
is identical to the standard version so any metric known to solve the standard Einstein
equations may be utilised. The trace–free system offers a useful algorithm to find exact
solutions. Once a metric is selected, the components may be substituted into (6) to find
the inertial mass density ρ+p. This quantity may then be substituted into (7) to reveal
the pressure profile explicitly. Finally removing the pressure from the inertial mass density
generates the energy density. We shall implement this algorithm in what follows.
5
When constructing models of stellar distributions composed of perfect fluid matter, the
following conditions are usually imposed in order that the model is physically reasonable.
The energy density (ρ) and pressure (p) profiles are expected to be positive definite with the
pressure vanishing for some radial value that demarcates the boundary of the fluid according
to the Israel–Darmois junction conditions. Generally it is preferred that both functions are
monotonically decreasing from the centre outwardly although this requirement may be too
strict in compact matter. The sound speed should be subluminal and obey the causality
criterion 0 <dp
<1. The interior metric must smoothly match the exterior Schwarzschild
solution across the boundary hypersurface. The energy conditions must be satisfied. That
is the (i) weak energy condition: ρp > 0, (ii) strong energy condition: ρ+p > 0 and
(iii) dominant energy condition: ρ+ 3p > 0. For static fluid spheres with a monotonically
decreasing and positive pressure profile, the surface redshift zr=R=1
g00 1 = eν/21
should be less than 2. The Buchdahl [16] limit mass
radius <4
9governing the mass-radius ratio
ensures the stability of the sphere must be satisfied.
TOLMAN I METRIC (EINSTEIN UNIVERSE)
Following Einstein, Tolman began with the assumption eν= const. as the simplest pre-
scription of a variable to solve the field equations. The metric potentials are then found to
be eλ=1
1r2
R2
and eν=c2for some constants cand R. Accordingly the dynamical quantities
work out to ρ=3
R2and p=1
R2. In the trace–free situation the density and pressure are
calculated as ρ=2
R2Kand p=K. The energy conditions ρp=2
R22K,ρ+p=2
R2
and ρ+ 3p=2
R2+ 2K. This is the well–known static Einstein universe solution providing
a non-zero energy density and pressure. The severe defect in this model is the constant
value of the density and pressure which does not conform to observation. For this case, the
density and pressure coincide with Tolman I so no new insight is gained.
TOLMAN II METRIC (SCHWARZSCHILD–DE SITTER)
Based on his arrangement of the master field equation, Tolman made the choice eλν=
constant and obtained the potentials eλ=12m
rr2
R21and eν=c212m
rr2
R2.The
density and pressure emerge as ρ=3
R2and p=3
R2.Within the unimodular framework,
6
the dynamics are ρ=Kand p=Kfor some constant K. The above metrics lead to
uniform energy density and pressure as in the standard theory. The density and pressure
are constant and are related by the equation of state ρ+p= 0. This is a characteristic
of dark matter – a speculative idea proposed to explain the observed accelerated expansion
of the universe. The expansion requires a negative pressure and this is the case if p=ρ.
The term 12m
r, belongs to the Schwarzchild exterior metric while the term quadratic
in rarises on account of the cosmological constant which is believed to be the generator of
vacuum energy in empty space through curvature.
TOLMAN III METRIC: SCHWARZSCHILD INTERIOR
Introducing the relationship eλ= 1 r2
R2, the Tolman metric potentials evaluate to
eλ=1
1r2
R2
and eν=
AB 1r2
R2!1
2
2
(11)
while the dynamical quantities have the form ρ=3
R2and p=1
R2
3Bq1r2
R2A
ABq1r2
R2
where A,
Band Rare constants. The TFE algorithm generates the same results. When A= 0 the
Tolman I metric is obtained. Note that the Schwarzschild potentials produce a constant
density fluid. However, the converse is not necessarily true. In [7] it was demonstrated that
beginning with the requirement of a constant density yielded metric potentials that included
a Nariai term [15]. When this term is suppressed the usual Schwarzschild metric is regained.
TOLMAN IV METRIC
This is the first of the new solutions of Tolman that exhibited previously unknown physical
behaviour. The assumption eνν0
2r= const. generated the metric potentials
eλ=1 + 2r2
A2
1 + r2
A21r2
R2and eν=B2 1 + r2
A2!
where A,Band Rare constants. In the trace–free algorithm the dynamical quantities are
expressed as
ρ=(3A2+ 2r2) (A2+ 2R2)
2R2(A2+ 2r2)2K(12)
7
p=A2+ 2R2
2R2(A2+ 2r2)+K. (13)
These are equivalent to the Tolman quantities except for the appearance of an extra constant
of integration which indeed plays a role in the dynamical evolution of the fluid. This is an
artefact of the trace–free equations. The sound speed index has the form
dp
=A2+ 2r2
5A2+ 2r2=1+2r
A2
5+2r
A2.(14)
It is clear that dp
0 while demanding dp
1, leads to 4
5+2(r
A)2>0, which is always time.
Hence the sound speed is subluminal for all values of radii as well as constants within the
Tolman IV metric.
The energy conditions work out to
ρp=A4(1 2KR2)+2A2R2(1 4Kr2)8Kr4R2
R2(A2+ 2r2)2(15)
ρ+p=2 (A2+r2) (A2+ 2R2)
R2(A2+ 2r2)2(16)
ρ+ 3p=A4(2KR2+ 3) + A2(r2(8KR2+ 4) + 6R2)+8r2R2(Kr2+ 1)
R2(A2+ 2r2)2.
(17)
and these may be studied with the aid of graphical plots. A barotropic equation of state
exists. Solving for r2in equation (12) and substituting in (13) generates a functional de-
pendence of pon ρ. This is a usual expectation of perfect fluids and is given by
p(ρ) = 2 (A2+ 2R2) (K+ρ)
q(A2+ 2R2) (A2(16KR2+ 16ρR2+ 1) + 2R2) + A2+ 2R2+K. (18)
The gravitational mass of the star is computed as
m(r) = r3(A2(3 2K R2)+2R2(3 2K r2))
6R2(A2+ 2r2)(19)
in geometric units.
The compactification parameter expresses the ratio of mass to radius throughout the distri-
bution and has the form
Λ = m(r)
r=r2(A2(3 2KR2)+2R2(3 2Kr2))
6R2(A2+ 2r2).(20)
8
012345
0.5
1.0
1.5
2.0
2.5
FIG. 1. Plot of energy density (ρ) versus radius (r).
while the gravitational surface redshift z=1
Brr2
A2+11 is expected to be less than 2 at
the boundary r=R.
9
12345
0.1
0.2
0.3
0.4
0.5
0.6
FIG. 2. Plot of pressure (p) versus radius (r).
12345
0.3
0.4
0.5
0.6
0.7
0.8
FIG. 3. Plot of sound speed dp
versus radius (r).
Analysis of the plots: A very slight deviation on pressure profile for this case is noted.
Now for the purpose of analyzing our model graphically we make the following parameter
choices: A= 2; B= 0.5; K=0.1 and R= 1. Figure 1 portrays the behavior of the
energy density versus the radial coordinate (r). The plot clearly shows a positive definite
and a monotonically decreasing energy density with increasing radius reverywhere within
the spherical distribution. Figure 2 is the plot of the isotropic pressure versus radial r
coordinate. The plot evidently shows that the pressure is positive at the origin, inside the
boundary. At around radius r= 3.6, it vanishes indicating that the distribution has a finite
radius. Evidently, Figure 2 also demonstrates monotonic decrease with increasing radius r
1 2 3 4 5
1
2
3
4
FIG. 4. Plot of energy conditions (ρp,ρ+pand ρ+ 3p) versus radius (r).
10
01234
0.5
1.0
1.5
2.0
2.5
FIG. 5. Plot of equation of state (p=p(ρ)) versus radius (r).
1 2 3 4 5
2
4
6
8
10
FIG. 6. Plot of gravitational mass (m(r)) versus radius (r).
Figure 3 depicts the sound of speed value versus the radial rcoordinate. The profile satisfies
the condition 0 dp
1, as demanded for causality. Figure 4 exhibits the energy conditions
which are all positive inside the sphere. In figure 5 we provide a plot for the equation of
state, expressing the pressure as a function of the density. It is a smooth singularity free
function within the star’s radius. A plot of the gravitational mass is displayed in figure
6. This reveals a smooth increasing function with the increasing radius as is expected.
The compactification parameter (Figure 7) plot is an increasing function most importantly
satisfying the inequality mass
radius <4
9. Finally, we consider the redshift profile (Figure 8)
0.2 0.4 0.6 0.8
0.1
0.2
0.3
0.4
FIG. 7. Plot of compactification parameter (Λ) versus radius (r).
11
which is less than 2 units close to the radius r= 3.6. Therefore this model does satisfy the
elementary requirements for realistic behavior.
TOLMAN V METRIC
Tolman’s prescription ev= const.r2nresulted in the potentials
eλ=1+2nn2
1(1 + 2nn2)r
RNand eν=B2r2n(21)
where n,N=2(1+2nn2)
n+1 and Bare constants. Setting M=2(n2+1)
n+1 the trace–free
algorithm yields
ρ=(2 n)n
(1 + 2nn2)r2(2n+ 1)(n3)r2r
RM
(n+ 1)R4K(22)
p=K(2n+ 1)r2r
RM
R4+n2
(1 + 2nn2)r2.(23)
for the energy density and pressure respectively. Again we note that the expressions are
equivalent to Tolman however a significant constant Kappears on account of the process of
unravelling the trace–free field equations. This must be a manifestation of the cosmological
constant that the trace–free equations sought to conceal. The sound speed is given by
dp
=(n1)(n+ 1)(2n+ 1)(1 + 2nn2)r4n(n+ 1)2R4r
RM
(n3)(n1)(2n+ 1)(1 + 2nn2)r4(n2)(n+ 1)2R4r
RM.(24)
and the energy conditions are expressed as
ρp=4(2n+ 1)r2r
RM
(n+ 1)R42n(n1)
(1 + 2nn2)r22K(25)
ρ+p=2 (1 + n2n2)r2r
RM
(n+ 1)R4+2n
(1 + 2nn2)r2(26)
ρ+ 3p= 2K4n(2n+ 1)r2r
RM
(n+ 1)R4+2n(n+ 1)
(1 + 2nn2)r2.(27)
The gravitational mass function assumes the form
m(r) = r5r
RM
R4+(n2)nr
n22n1Kr3
3.(28)
12
while the compactification parameter is given by
Λ = r4r
RM
R4+(n2)n
n22n1Kr2
3(29)
The redshift expression z=1
Brn1 must be smaller than 2 at the boundary. In the main, the
additive integration constant Kacts to shift the values of the density, pressure, sound speed
and energy profiles. However, in the case of the active mass a new term in r3is introduced
and this influences the overall mass of the star. Note that when n= 0, the density ρbecomes
constant and the result is reduced to the Tolman I solution (Einstein universe) which was
covered in the previous chapter. In this solution Tolman decided to study the case n=1
2
and to examine the physical properties of the solution. Note that an equation of state may
be explicitly obtained depending on what value of nis chosen. To provide a wider treatment
of the behaviour of the model for various nvalues we select n=1
2(small dashes), n= 0
(dotted line), n=1
2(dotted and dashed), n= 1 (thick), n= 2 (big dashes)and n= 3 (solid
thin curve). These values are suggested by the expressions for the density and pressure since
in each case some rdependence is suppressed.
13
1 2 3 4 5 6
2
3
4
5
6
7
FIG. 8. Energy density (ρ) versus radius (r).
1 2 3 4 5 6
-6
-4
-2
2
FIG. 9. Pressure (p) versus radius (r).
Physical Analysis: The plots represent the physical quantities for various values of n.
From Fig. 8 it is clear that the energy density is positive for all values of nselected. The
pressure (Fig. 9) vanishes in the cases n=1
2at about r= 0.4, n= 1 at r= 0.65 and n= 2
at r= 1.8 units. The remaining plots may now be ignored as they do not meet the basic
requirement of a finite radius. Interestingly the n=1
2case (Fig. 10) meets the causality
criterion within the bounded distribution while the case n= 1 appears demonstrate the
extreme square of sound speed dp
= 1 which is characteristic of stiff fluid matter. The
case n= 2 does not support a subluminal sound speed and may now be eliminated from
1 2 3 4 5 6
-8
-6
-4
-2
2
4
6
FIG. 10. Sound speed index dp
versus radius (r).
14
1 2 3 4 5 6
5
10
FIG. 11. Weak energy condition (ρp) versus radius (r).
1 2 3 4 5 6
-4
-2
2
4
6
FIG. 12. Strong energy condition (ρ+p) versus radius (r).
the analysis. The case n=1
2satisfied all the energy requirements (Figs. 11, 12 and 13)
within its radius while the case n= 2 violates the dominant energy condition in the interval
1< r < 1.8. Finally the profile of the mass (Fig. 14) conforms to what is expected
for all cases of n: that is a smooth increasing function to the boundary layer. The plot
of the compactification parameter Λ (Fig. 15) suggests that the case n=1
2does satisfy
the Buchdahl limit m
r<4
9everywhere within the sphere and specifically at the boundary.
Therefore we may conclude that the case n=1
2meets the elementary requirements for
physical plausibility. This happens to be the only case studied by Tolman in detail.
123456
-15
-10
-5
5
FIG. 13. Dominant energy condition (ρ+ 3p) versus radius (r).
15
123456
50
100
150
FIG. 14. Gravitational mass (m(r)) versus radius (r).
123456
5
10
15
20
25
30
FIG. 15. Compactification parameter (Λ) versus radius (r).
GENERALIZED TOLMAN VI METRIC
The choice eλ= const. = 1
2n2suggests metric potentials eλ= 2 n2and eν=
(Ar1nBrn+1)2for some constants n,Aand B. Within the trace-free framework we
obtain
ρ=n21
(n22) r2K(30)
p=K(n+ 1)24An
ABr2n
(n22) r2.(31)
for the dynamical variables. These correspond to Tolman’s calculation and as usual an extra
additive constant is in attendance. The sound speed index has the form
dp
=(A(n1) + B(n+ 1)r2n)2
(n21) (ABr2n)2.(32)
while the energy conditions are governed by the expressions
ρp=2n(A(n1) B(n+ 1)r2n)
(n22) r2(ABr2n)2K(33)
16
ρ+p=2 (A(n1) + B(n+ 1)r2n)
(n22) r2(ABr2n)(34)
ρ+ 3p=2B(n+ 1)(n+ 2)r2n2A(n2)(n1)
(n22) r2(ABr2n)+ 2K. (35)
The equation of state may be explicitly determined as
p(ρ) = K(K+ρ)
n21
(n+ 1)24An
ABn21
(n22)(K+ρ)2n
.(36)
for all values of n. Note that the new constant Karising from the unimodular approach
exerts some influence on the equation of state. We have found all the necessary quantities to
analyse the model. However, the Tolman VI solution has been criticized for being irregular.
For example, there are singularities in the metric and dynamical quantities for n=±2
and for r=rB
A2.
While Tolman utilised the form eλ= 2n2, the simple prescription eλ=βfor a constant
βallows us to obtain the remaining potential as eν=(r2α+c1)2c2
r2(α1) where α=2eβ,β,c1
and c2are constants. In this notation the trace–free algorithm results in
p=eβ
r23+2α4αc1
r2α+c11
r2+K(37)
ρ=sinh βcosh β+ 1
r2K(38)
for the density and pressure and where Kis a constant of integration. The sound speed has
the form
dp
=2eβ1c1r2α2α+eβ3c2
1+2αeβ+ 3r4α
(eβ1) (r2α+c1)2.(39)
while the energy conditions expressions are
ρp=eβ
r24αc1
r2α+c12α3+2
r2+sinh βcosh β
r22K(40)
ρ+p=eβ
r23+2α4αc1
r2α+c1+sinh βcosh β
r2(41)
ρ+ 3p=2eβ
r24+3α6αc1
r2α+c12
r2+ 2K. (42)
17
Kuchowicz [17] exploited a similar approach to examine the Tolman VI solution and found
the general solution as above but with K= 0. On account of the many singularities present
in this model we do not carry out any further study of this case. It is worth noting that
K= 0 results in an inverse square law fall-off of the density and the other constants may
also be suitably picked so that the pressure has this same behaviour. In this case isothermal
fluid spheres result [18].
EXTENSION OF THE TOLMAN VII METRIC
Commencing with the ansatz eλ= 1 r2
R2+4r4
A4the remaining metric potential evaluates
to eν=B2
sin
log eλ
2+2r2/A2A2/4R2
C1
2
2
as per Tolman. Despite the polynomial as-
sumption imposed by Tolman, the dynamical quantities became unwieldy. For this reason,
they were omitted in his work. More importantly the form for νis not the most general. To
find the expanded solution, we substitute λfrom the ansatz into the isotropy equation (8)
and obtain
eν=c2(c1cos f+ sin f)2(43)
where c1and c2are integration constants and
f= log r4RqA4(R2r2)+4r4R2+ 2r2RA4. The density and pressure are found to
be
ρ=3
R220r2
A4(44)
p=eλ
4R(cot fc1)
qA4(R2r2)+4r4R2(c1cot f+ 1) +1
r2
1
r2(45)
In the case c1= 0 we regain the original incomplete solution of Tolman. It is now straight-
forward but tedious to generate the other physical quantities however, we omit these lengthy
expressions.Note that a barotropic equation of state may be found since r2may be expressed
in terms of ρand this may be plugged into (45) to determine p(ρ). There is also the prospect
of a bounded distribution as it is possible to solve p(r=R) = 0.
18
TOLMAN VIII METRIC
This the last solution Tolman investigated. Tolman’s assumption eλ= const.r2beν
generates the potentials eλ=HA
rnrq
Fand eν=B2r2beλwhere a sequence of
constants are defined by H=2
qn and A= (2m)n; with q=ab,n=a+ 2b1 and Fis a
constant. Within the trace–free framework the dynamical quantities are obtained as
ρ= (b((n+ 2)rn((b2)F H (q2) + (2b+ 3q4)rq)
AF (q2)(2b3n4))) /F(n+ 2)(q2)rn+2K(46)
p=b
r2 A(n2b)
rn(n+ 2) +(2b+q)rq
F(q2) +bH!+K. (47)
and the familiar integration constant Kre-appears. The sound speed indicator is given by
dp
=AF (n2b) + rn(2bF H (2b+q)rq)
AF (2b3n4) + rn((2b+ 3q4)rq2(b2)F H ).(48)
while the energy conditions are expressed by
ρp= 2 (n+ 2)rnF(q2) (b1)bH +Kr2
+2b(b+q1)rq)2AbF d)) /F(n+ 2)(q2)rn+2(49)
ρ+p=2b(rn(F H rq)AF )
F rn+2 (50)
ρ+ 3p= 2 (n+ 2)rnF(q2) b(b+ 1)H+Kr2
+2b(b+ 1)rq)2AbF 2(b+ 1)(q2)
/(n+ 2)(q2)rn+2(51)
where d= (q2)(bn1).
We observe that in this case the solution given by Tolman is recovered except for the new
constant K. A number of special cases covered earlier may be regained for certain values of
nand q. The equation of state may also be determined in some special cases, however, it
cannot be found in general. Accordingly we neglect a complete study of this case although
the additive constant Kintroduced through the unimodular approach, will have some effect
on the dynamics.
19
DISCUSSION
The trace–free Einstein field equations offer an alternative route to establish the energy
density and pressure relevant to a perfect fluid distribution. Removing the trace of the energy
momentum destroys the energy conservation property however, the coupling of density and
pressure reduces the number of independent field equations by one. To accommodate this the
conservation law may be added on. We examine the impact on following this presentation
of the field equations by investigating the well known Tolman metrics. It is found that an
additional constant of integration always appears and this has some bearing on the dynamical
behaviour of the fluid. In the case of the dynamical quantities and energy conditions, a
mere shift is introduced, however, in the active gravitational mass and equation of state a
significant contribution emerges. We have analysed some of these models with the aid of
graphical plots. In some cases we have extended and generalised the incomplete solutions
provided by Tolman.
hansrajs@ukzn.ac.za
mkhizen18@gmail.com
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  • G F R Ellis
  • J Van Elst
  • J-P Murugan
  • Uzan
G F R Ellis, H van Elst, J. Murugan and J-P Uzan Class. Quantum Grav. 28 225007 (2011)
  • G F R Ellis
G F R Ellis, Gen. Relativ. Gravit. 46 1619 (2014)
  • A D R Finkelstein
  • J E Galiautdinov
  • Baugh
D R Finkelstein, A A Galiautdinov and J E Baugh J. Math. Phys. 42 340 (2001) [arXiv:grqc/0009099v1]
  • L Smolin
L Smolin Phys. Rev. D 80 084003 (2009) [arXiv:0904.4841v1 [hep-th]]
  • S Hansraj
  • R Goswami
  • G F R Mkhize
  • Ellis
S. Hansraj, R. Goswami, N Mkhize and G. F. R. Ellis Phys.Rev. D 96, 044016 (2017) arXiv:1703.06326
  • M R Finch
  • J E F Skea
M R Finch and J E F Skea Class. Quantum Grav. 6 (1989) 467