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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 e−λ−1

r2!+d

dr e−λν0

2r!+e−λ−νd

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)dt2−eλ(r)dr2−r2dθ2+ sin2θdφ2(2)

where the gravitational potentials νand λare functions of the radial coordinate ronly. We

utilise a comoving ﬂuid 4-velocity ua=e−ν/2δa

0a perfect ﬂuid source with energy momentum

tensor Tab = (ρ+p)uaub−pgab 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 +ν02−ν0λ0) + 4

r(ν0+λ0) + 4

r2(eλ−1) = 6(ρ+p)eλ(5)

4

r(ν0+λ0)−(2ν00 +ν02−ν0λ0)−4

r2(eλ−1) = 2(ρ+p)eλ(6)

(2ν00 +ν02−ν0λ0) + 4

r2(eλ−1) = 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 +ν02−ν0λ0−2

r(ν0+λ0) + 4

r2(eλ−1) = 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

dρ <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: ρ−p > 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 = e−ν/2−1

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

1−r2

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

R2−Kand p=K. The energy conditions ρ−p=2

R2−2K,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 e−λ−ν=

constant and obtained the potentials eλ=1−2m

r−r2

R2−1and eν=c21−2m

r−r2

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 1−2m

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 e−λ= 1 −r2

R2, the Tolman metric potentials evaluate to

eλ=1

1−r2

R2

and eν=

A−B 1−r2

R2!1

2

2

(11)

while the dynamical quantities have the form ρ=3

R2and p=1

R2

3Bq1−r2

R2−A

A−Bq1−r2

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

A21−r2

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)2−K(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

dρ =A2+ 2r2

5A2+ 2r2=1+2r

A2

5+2r

A2.(14)

It is clear that dp

dρ ≥0 while demanding dp

dρ ≤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

ρ−p=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+1−1 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

dρ 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 (ρ−p,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

dρ ≤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+2n−n2

1−(1 + 2n−n2)r

RNand eν=B2r2n(21)

where n,N=2(1+2n−n2)

n+1 and Bare constants. Setting M=−2(n2+1)

n+1 the trace–free

algorithm yields

ρ=(2 −n)n

(1 + 2n−n2)r2−(2n+ 1)(n−3)r2r

RM

(n+ 1)R4−K(22)

p=K−(2n+ 1)r2r

RM

R4+n2

(1 + 2n−n2)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

dρ =(n−1)(n+ 1)(2n+ 1)(1 + 2n−n2)r4−n(n+ 1)2R4r

RM

(n−3)(n−1)(2n+ 1)(1 + 2n−n2)r4−(n−2)(n+ 1)2R4r

RM.(24)

and the energy conditions are expressed as

ρ−p=4(2n+ 1)r2r

RM

(n+ 1)R4−2n(n−1)

(1 + 2n−n2)r2−2K(25)

ρ+p=2 (1 + n−2n2)r2r

RM

(n+ 1)R4+2n

(1 + 2n−n2)r2(26)

ρ+ 3p= 2K−4n(2n+ 1)r2r

RM

(n+ 1)R4+2n(n+ 1)

(1 + 2n−n2)r2.(27)

The gravitational mass function assumes the form

m(r) = r5r

RM

R4+(n−2)nr

n2−2n−1−Kr3

3.(28)

12

while the compactiﬁcation parameter is given by

Λ = r4r

RM

R4+(n−2)n

n2−2n−1−Kr2

3(29)

The redshift expression z=1

Brn−1 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

dρ = 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

dρ versus radius (r).

14

1 2 3 4 5 6

5

10

FIG. 11. Weak energy condition (ρ−p) 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 e−λ= const. = 1

2−n2suggests metric potentials eλ= 2 −n2and eν=

(Ar1−n−Brn+1)2for some constants n,Aand B. Within the trace-free framework we

obtain

ρ=n2−1

(n2−2) r2−K(30)

p=K−(n+ 1)2−4An

A−Br2n

(n2−2) 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

dρ =−(A(n−1) + B(n+ 1)r2n)2

(n2−1) (A−Br2n)2.(32)

while the energy conditions are governed by the expressions

ρ−p=2n(A(n−1) −B(n+ 1)r2n)

(n2−2) r2(A−Br2n)−2K(33)

16

ρ+p=2 (A(n−1) + B(n+ 1)r2n)

(n2−2) r2(A−Br2n)(34)

ρ+ 3p=2B(n+ 1)(n+ 2)r2n−2A(n−2)(n−1)

(n2−2) r2(A−Br2n)+ 2K. (35)

The equation of state may be explicitly determined as

p(ρ) = K−(K+ρ)

n2−1

(n+ 1)2−4An

A−B√n2−1

√(n2−2)(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 e−λ= 2−n2, the simple prescription eλ=βfor a constant

βallows us to obtain the remaining potential as eν=(r2α+c1)2c2

r2(α−1) where α=√2−eβ,β,c1

and c2are constants. In this notation the trace–free algorithm results in

p=e−β

r23+2α−4αc1

r2α+c1−1

r2+K(37)

ρ=sinh β−cosh β+ 1

r2−K(38)

for the density and pressure and where Kis a constant of integration. The sound speed has

the form

dp

dρ =2eβ−1c1r2α−2α+eβ−3c2

1+2α−eβ+ 3r4α

(eβ−1) (r2α+c1)2.(39)

while the energy conditions expressions are

ρ−p=e−β

r24αc1

r2α+c1−2α−3+2

r2+sinh β−cosh β

r2−2K(40)

ρ+p=e−β

r23+2α−4αc1

r2α+c1+sinh β−cosh β

r2(41)

ρ+ 3p=2e−β

r24+3α−6αc1

r2α+c1−2

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 e−λ= 1 −r2

R2+4r4

A4the remaining metric potential evaluates

to eν=B2

sin

log e−λ

2+2r2/A2−A2/4R2

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(R2−r2)+4r4R2+ 2r2R−A4. The density and pressure are found to

be

ρ=3

R2−20r2

A4(44)

p=e−λ

4R(cot f−c1)

qA4(R2−r2)+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 e−λ= const.r−2beν

generates the potentials e−λ=H−A

rn−rq

Fand eν=B2r2be−λwhere a sequence of

constants are deﬁned by H=2

qn and A= (2m)n; with q=a−b,n=a+ 2b−1 and Fis a

constant. Within the trace–free framework the dynamical quantities are obtained as

ρ= (−b((n+ 2)rn((b−2)F H (q−2) + (2b+ 3q−4)rq)

−AF (q−2)(2b−3n−4))) /F(n+ 2)(q−2)rn+2−K(46)

p=b

r2 A(n−2b)

rn(n+ 2) +(2b+q)rq

F(q−2) +bH!+K. (47)

and the familiar integration constant Kre-appears. The sound speed indicator is given by

dp

dρ =AF (n−2b) + rn(2bF H −(2b+q)rq)

AF (2b−3n−4) + rn((2b+ 3q−4)rq−2(b−2)F H ).(48)

while the energy conditions are expressed by

ρ−p= 2 −(n+ 2)rnF(q−2) (b−1)bH +Kr2

+2b(b+q−1)rq)−2AbF d)) /F(n+ 2)(q−2)rn+2(49)

ρ+p=2b(rn(F H −rq)−AF )

F rn+2 (50)

ρ+ 3p= 2 (n+ 2)rnF(q−2) b(b+ 1)H+Kr2

+2b(b+ 1)rq)−2AbF 2(b+ 1)(q−2)

/(n+ 2)(q−2)rn+2(51)

where d= (q−2)(b−n−1).

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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