Radiant gravitational collapse with anisotropy in pressures and bulk viscosity A. C. Mesquitaand M. F. A. da Silvay

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Radiant gravitational collapse with anisotropy in pressures and

bulk viscosity

A. C. Mesquita∗and M. F. A. da Silva†

Departamento de F´ısica Te´orica, Universidade do Estado do Rio de Janeiro,

Rua S˜ao Francisco Xavier 524, Maracan˜a,

CEP 20550–013, Rio de Janeiro – RJ, Brazil

(Dated: October 27, 2022)

Abstract

We model a compact radiant star that undergoes gravitational collapse from a certain initial

static conﬁguration until it becomes a black hole. The star consists of a ﬂuid with anisotropy

in pressures, bulk viscosity, in addition to the radial heat ﬂow. A solution of Einstein’s ﬁeld

equations with temporal dependence was presented to study the dynamic evolution of physical

quantities, such as the mass-energy function, the luminosity seen by an observer at inﬁnity and

the heat ﬂow. We checked the acceptability conditions of the initial static conﬁguration to obtain

a range of mass-to-radius ratio in which the presented star model is physically reasonable. The

energy conditions were analyzed for the dynamic case, in order to guarantee that the model is

composed of a physically acceptable ﬂuid within the range of the mass-to-radius ratio obtained

for the static conﬁguration or if they will be modiﬁed during the collapse.

PACS numbers: 04.20.Dw, 04.20.Jb, 04.70.Bw, 97.60.Jd, 26.60.-c

∗Electronic address: arthurcamara2007@hotmail.com

†Electronic address: mfasnic@gmail.com

arXiv:2210.14288v1 [gr-qc] 25 Oct 2022

I. INTRODUCTION

Neutron stars can be detected through optical and X-ray observations, which reveal

properties crucial for understanding their structure and evolution, such as surface radius

and temperature [3]. The internal structure of a neutron star depends on its equation of

state, that is, a relationship between density and pressure within it. However, as we have

not yet been able to produce such high densities in laboratories, we are not aware of the

equation of state that best describes this matter, making its theoretical modeling diﬃcult.

Analogous to a white dwarf, this type of star has an upper limit of mass at which it would

fall out of equilibrium and continue to collapse. Oppenheimer and Volkoﬀ [4], using the

theory of general relativity, established an upper limit of 0.7M, which became known as

the Tolman-Oppenheimer-Volkoﬀ limit, or simply TOV limit. Modern estimates shift this

upper bound to about 2M[5]. Soon after the TOV limit was established, Oppenheimer

and Snyder [6] studied the cataclysmic behavior for neutron stars with masses greater

than this limit. The star will contract until its surface radius approaches r=2m(mis the

mass of the central object), the Schwarzschild radius. When it exceeds this Schwarzschild

radius, no information is transmitted to the region outside r. Thus, the fate of a neutron

star whose mass exceeds the TOV limit is a black hole.

The observational scenario has shown to be very promising in bringing us new possi-

bilities for the study of these compact objects. From the discovery of the spiralization

and coalescence of a binary neutron star system (GW170817), by the Laser Interferometer

Gravitational Waves Observatory (LIGO, VIRGO) (on August 17, 2017), a new alterna-

tive for accessing the equation of state emerged of such stars at high densities [7, 8], at least

to exclude some of them. This new astrophysical observation window, accessed through

gravitational waves, has brought us many surprises and new challenges. In another work

[9], it is observed what appears to be the coalescence of a binary system involving a black

hole of about 22.2 - 24.3 Mand a compact object of approximately 2.50 - 2.67 M, the

latter having a mass too small to be a black hole, but larger than expected so far for a

neutron star.

Expanding our knowledge about the behavior of ﬂuids under strong self-gravitation in

light of the TGR is crucial for the interpretation of results like this, which emerge from

the new observations.

In this article we are interested in investigating the temporal evolution, along the

collapse process, of certain physical quantities, starting from an initial static conﬁguration

to the formation of a black hole, through the introduction of a temporal dependence in

the metric. We consider a spherically symmetrical distribution of a ﬂuid with anisotropy

in pressures, with heat ﬂux and viscosities, governed by a non-local equation of state,

originally proposed by H`ernandez and N´u˜nez [12].

Our article is organized as follows. In section 2, we present a description of the

geometry of spacetime and the energy-momentum tensor. In section 3, we present a spe-

cial metric, time dependent, to study the evolution of an initial static conﬁguration, for

which the ﬁeld equations lead to a non-local equation of state for ﬂuids with anisotropic

pressures. Then, in section 4, we investigate the evolution from collapse to black hole for-

mation from a solution given by a density proﬁle proposed by Wyman [20] that is written

in a similar way to the one presented in Hern´andez and N´u˜nez [12]. We also corrected

some results obtained from the latter, exploring the graphic behavior of quantities such

as mass-energy enclosed in the surface of the distribution, luminosity for an observer at

inﬁnity, eﬀective surface temperature, adiabatic index eﬀective, heat ﬂux and scalar ex-

pansion. In section 5, the energy conditions for the dynamic case are analyzed. Finally,

in section 7 we present our ﬁnal remarks. We include an Appendix presenting the energy

conditions considered here.

II. EINSTEIN’S FIELD EQUATION

In order to study the gravitational collapse problem, we need to separate spacetime into

three regions: the ﬁrst consists of the interior region, that is, the spherically symmetric

distribution of matter. The second, an outer region which is fullﬁled by null radiation,

emitted by the matter distribution. Finally, the third of them refers to a Σ junction

hypersurface that separates these last two.

Let gij be the metric intrinsic to the hypersurface Σ, which takes into account the

description in comoving coordinates of the inner spacetime, that is

ds2

Σ=gijdξidξj=−dτ2+R2(τ)dΩ2,(1)

where dΩ2=dθ2+sen2θdφ2is the angular element and τrepresents the proper time, with

ξi= (τ, θ, φ) representing the coordinates intrinsic to Σ.

On the other hand, the interior space-time of the matter distribution, is described

by a spherically symmetric metric in the most general way possible, using comoving

coordinates, given by

ds2

−=g−

αβdχα

−dχβ

−=−A2(r, t)dt2+B2(r, t)dr2+C2(r, t)dΩ2,(2)

where χα

−= (χ0

−, χ1

−, χ2

−, χ3

−)=(t, r, θ, φ) are the coordinates of the interior space-time.

The energy-momentum tensor, describing the matter that ﬁlls such space-time is rep-

resented by

T−

αβ = (ρ+P⊥)uαuβ+P⊥gαβ + (Pr−P⊥)XαXβ+qαuβ(3)

+qβuα−2ησαβ −ζΘ(gαβ +uαuβ),

where ρis the energy density of the ﬂuid, Pris the radial pressure, P⊥is the tangential

pressure, Xαis a unit 4-vector along the radial direction, uαis the 4-velocity and qαis the

radial heat ﬂux vector, which satisfy qαuα= 0, XαXα= 1, Xαuα= 0 and uαuα=−1.

The 4-vectors are given by uα=δα

0/A,qα=qδα

1and Xα=δα

1/B. The amounts η > 0

and ζ > 0 are the shear viscosity and volume viscosity coeﬃcients, respectively. Whereas

σαβ and Θ are, respectively, the shear tensor and the expansion scalar.

On the other hand, let us now consider the outer spacetime described by the Vaidya

metric [13], written as

ds+2=gαβ+dχα

+dχβ

+=−1−2m(v)

rdv2−2dvdr+r2dΩ2(4)

where χα

+= (χ0

+, χ1

+, χ2

+, χ3

+) are the coordinates of outer spacetime and m(v) represents

the total ﬂuid energy stored within the hypersurface Σ as a function of the time delay v.

The energy-momentum tensor for the outer region, representing a pure radiation ﬁeld,

is given by

T+

αβ =ekαkβ,(5)

where kαis a null vector and eis the radiation energy density measured locally by an

observer over Σ. Hereafter we use the indices ”+” or ”-” to represent quantities referring

to outer and inner spacetime, respectively.

We can write the expansion scalar, the shear tensor, and the shear scalar, respectively,

Θ = uα

;α=1

A ˙

B+2˙

C!,(6)

σαβ =u(α;β)+ ˙u(αuβ)−1

3Θ(gαβ +uαuβ),(7)

σ=−1

3A ˙

B−˙

C!,(8)

where the parentheses in the subscripts in (7) mean symmetrization, the dot in (6) and

(8) represents ∂/∂t e ˙uα=uα;βuβ.

Combining the metric (2) with the energy-momentum tensor (3), the Einstein’s ﬁeld

equations for the inner region are given by

−A

B2"2C00

C+C0

C2

−2C0

B#+A

C2

+˙

C ˙

C+ 2 ˙

B!=kA2ρ , (9)

CC0

C+ 2A0

A−B

C2

−B

A2

2¨

C+ ˙

C!2

−2˙

C



=kB2(Pr+ 4ησ −ζΘ) ,(10)

C

B2A00

A+C00

C−A0

B+A0

C−B0

C

+C

A2"−¨

B−¨

C+˙

B+˙

C−˙

=kC2(P⊥−2ησ −ζΘ) ,(11)

2A0

C+ 2 ˙

C−2˙

C=−kAB2q , (12)

where k= 8πin the geometric coordinate system (c=G= 1), the dot represents ∂/∂t,

while the line represents ∂/∂r.

III. DYNAMIC SOLUTION OF FIELD EQUATIONS

Just like in [19] and [22], we introduce a time dependent function in the static metric

proposed by Hern´andez and N´u˜nez [12] in order to study the evolution of gravitational

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RadiantgravitationalcollapsewithanisotropyinpressuresandbulkviscosityA.C.MesquitaandM.F.A.daSilvayDepartamentodeFsicaTeorica,UniversidadedoEstadodoRiodeJaneiro,RuaS~aoFranciscoXavier524,Maracan~a,CEP20550{013,RiodeJaneiro{RJ,Brazil(Dated:October27,2022)AbstractWemodelacompactradiantstarthatunder...

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Radiant gravitational collapse with anisotropy in pressures and bulk viscosity A. C. Mesquitaand M. F. A. da Silvay

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