Disk-Haloes in Einstein-Maxwell gravity

Disk-haloes inEinstein-Maxwell gravity

Antonio C. Gutiérrez-Piñeres

Universidad Tecnológica de Bolívar, Colombia

UNILA, Brazil

November 14, 2014

A. C. Gutiérrez-Piñeres Disk-haloes in Einstein-Maxwell gravity

Physical Relevance

1 Disks may be used to model

• Accretion disks• Galaxies in thermodynamic equilibrium• Superposition of galaxies and black holes• + Magnetic field: neutron stars, white dwarfs, galaxies

formation2 Disks with Halos

• A large number of galaxies and other astrophysical systemhave extended distribution of matter surrounded by a“material” halo

• The flat rotation curves problem and the DARK MATTERassumption


Physical Relevance

1 Disks may be used to model• Accretion disks

• Galaxies in thermodynamic equilibrium• Superposition of galaxies and black holes• + Magnetic field: neutron stars, white dwarfs, galaxies





Physical Relevance

1 Disks may be used to model• Accretion disks• Galaxies in thermodynamic equilibrium

• Superposition of galaxies and black holes• + Magnetic field: neutron stars, white dwarfs, galaxies





Physical Relevance

1 Disks may be used to model• Accretion disks• Galaxies in thermodynamic equilibrium• Superposition of galaxies and black holes

• + Magnetic field: neutron stars, white dwarfs, galaxiesformation

2 Disks with Halos• A large number of galaxies and other astrophysical system

have extended distribution of matter surrounded by a“material” halo



Physical Relevance

1 Disks may be used to model• Accretion disks• Galaxies in thermodynamic equilibrium• Superposition of galaxies and black holes• + Magnetic field: neutron stars, white dwarfs, galaxies

formation

2 Disks with Halos• A large number of galaxies and other astrophysical system

have extended distribution of matter surrounded by a“material” halo



Physical Relevance

1 Disks may be used to model• Accretion disks• Galaxies in thermodynamic equilibrium• Superposition of galaxies and black holes• + Magnetic field: neutron stars, white dwarfs, galaxies





Mathematical Relevance

1 Exact Solution of Einstein-Maxwell equations

• Pure Exact solutions ambit• Conceptual basis in the study of other topics and research

areas

2 Distributional theory• Shell problem and General Relativity• The Electromagnetic potential

(Aharonov-Bohm effect in GR)• Covariant jump• The electro-geodesic and the electromagnetic problem



1 Exact Solution of Einstein-Maxwell equations• Pure Exact solutions ambit

• Conceptual basis in the study of other topics and researchareas





1 Exact Solution of Einstein-Maxwell equations• Pure Exact solutions ambit• Conceptual basis in the study of other topics and research

areas






areas

2 Distributional theory• Shell problem and General Relativity

• The Electromagnetic potential(Aharonov-Bohm effect in GR)

• Covariant jump• The electro-geodesic and the electromagnetic problem




areas


(Aharonov-Bohm effect in GR)

• Covariant jump• The electro-geodesic and the electromagnetic problem




areas


(Aharonov-Bohm effect in GR)• Covariant jump

• The electro-geodesic and the electromagnetic problem




areas




Finite axisymmetric charged dust disks in conformastaticspacetimes

[1]

Conformastatic disk-haloes in Einstein-Maxwell gravity.[2]

Conformastationary disk-haloes in Einstein-Maxwell gravity .[3]

Conformastationary disk-haloes in Einstein-Maxwell gravity II:the physical interpretation of the halo.

[4]Variational thermodynamics of relativistic thin disks

[5]



[1]Conformastatic disk-haloes in Einstein-Maxwell gravity.

[2]

Conformastationary disk-haloes in Einstein-Maxwell gravity .[3]



[5]




[2]Conformastationary disk-haloes in Einstein-Maxwell gravity .

[3]



[5]





[3]Conformastationary disk-haloes in Einstein-Maxwell gravity II:the physical interpretation of the halo.

[4]

Variational thermodynamics of relativistic thin disks[5]





[3]Conformastationary disk-haloes in Einstein-Maxwell gravity II:the physical interpretation of the halo.


[5]


The Einstein-Maxwell equations andthe thin disk-haloes

WE ASSUME THAT:• The coordinates are quasicylindrical: xa = (t, ϕ, r, z).

• There exists an infinitesimally thin disk, located at thehypersurface z = 0, so that the components of the metrictensor gab and the components of the electromagnticpotential Aa are symmetrical functions of z and their firstderivatives have a finite discontinuity at z = 0.


The Einstein-Maxwell equations andthe thin disk-haloes

WE ASSUME THAT:• The coordinates are quasicylindrical: xa = (t, ϕ, r, z).• There exists an infinitesimally thin disk, located at the

hypersurface z = 0, so that the components of the metrictensor gab and the components of the electromagnticpotential Aa are symmetrical functions of z and their firstderivatives have a finite discontinuity at z = 0.


The metric and the electromagneticpotentials

REFLEXION SYMMETRY z = 0

gab(r, z) = gab(r,−z), Aa(r, z) = Aa(r,−z),gab,z(r, z) = −gab,z(r,−z) Aa,z(r, z) = −Aa,z(r,−z).

CONTINUITY z = 0

[gab] = gab|z=0+− gab|z=0−

= 0, (2a)[Aa] = Aa|z=0+

−Aa|z=0−= 0, (2b)

γab = [gab,z], (2c)ζa = [Aa,z]. (2d)


The metric and the electromagneticpotentials

gab = g+abθ(z) + g−ab{1− θ(z)}, (3a)

Aa = A+a θ(z) +A−a {1− θ(z)}, (3b)


The Ricci, energy-momentum tensor and the electric currentdensity can be expressed as

Rab = R+abθ(z) +R−ab{1− θ(z)}+Habδ(z), (4a)

Tab = T+abθ(z) + T−ab{1− θ(z)}+Qabδ(z), (4b)

Ja = J+a θ(z) + J−a {1− θ(z)}+ Iaδ(z), (4c)

Hab =1

2{γzaδzb + γzb δ

za − γccδzaδzb − gzzγab}, (5)

T±ab and J±ab are the energy-momentum tensors and electriccurrent density of the z ≥ 0 and z ≤ 0 regions. Qab and Ia givesthe part of the energy-momentum tensor and the electriccurrent density corresponding to the disklike source.


The Einstein-Maxwell equations andthe thin disk-halos

G±ab = R±ab −1

2gabR

± = T±ab, (6)

Hab −1

2gabH = Qab, (7)

F ab± ;b = Ja±, (8)

[F ab]nb

= Ia, (9)

T±ab = E±ab +M±ab,

Eab =1

4π

{FacF

cb −

1

4gabFcdF

cd

},

Fab = Ab,a −Aa,b.


The Einstein-Maxwell equations andthe thin disk-halos

G±ab = R±ab −1

2gabR

± = T±ab, (6)

Hab −1

2gabH = Qab, (7)

F ab± ;b = Ja±, (8)

[F ab]nb

= Ia, (9)

T±ab = E±ab +M±ab,

Eab =1

4π

{FacF

cb −

1

4gabFcdF

cd

},

Fab = Ab,a −Aa,b.A. C. Gutiérrez-Piñeres Disk-haloes in Einstein-Maxwell gravity

General conformastationary electromagnetized disk-haloes

Inspired by inverse method techniques, let us assume that thesolution has the general form [6, 7]

ds2 = −f2(dt+ ωdϕ)2 + Λ4[dr2 + dz2 + r2dϕ2], (10)

where we have introduced the cylindrical coordinatesxα = (t, r, z, ϕ) in which the metric function f,Λ and ω and theelectromagnetic potential, Aα = (At, 0, 0, Aϕ) depend only onrand z .


The surface energy-momentum tensor of the disk Sαβ is

Sαβ ≡∫Qαβδ(z)dsn =

√gzzQαβ, (11)

Stt =1

r2Λ6{−f2ωω,z + 8r2Λ3Λ,z},

Stϕ = − 1

r2Λ6f{−4r2ωfΛ3Λ,z + ω2f3ω,z + 2r2Λ4ωf,z + r2fΛ4ω,z},

Srr =2

Λ3f{2fΛ,z + Λf,z},

Sϕt =f2

r2Λ6ω,z,

Sϕϕ =1

r2Λ6f{2r2Λ4f,z + 4r2Λ3fΛ,z + ωf3ω,z},

where dsn =√gzz dz is the “physical measure” of length in the

direction normal to the z = 0 surface.


The surface electric current density of the disk J α is

J α ≡∫Iαδ(z)dsn =

√gzzIα (13)

J t =1

r2Λ6f2{ωf2 [Aϕ,z] + (r2Λ4 − f2ω2) [At,z]},

J ϕ =1

r2Λ6{ω [At,z]− [Aϕ,z]}.


The electric current density of the halo is

J t± =1

Λ6f∇ · {Λ2f−1∇At

+ r−2ωfΛ−2(∇Aϕ − ω∇At)}, (15a)

Jϕ± =1

Λ6f∇ · {r−2fΛ−2(∇Aϕ − ω∇At)}. (15b)


The non-zero components of the energy-momentum tensor ofthe halo,

M±αβ = G±αβ − E±αβ, (16)


M±tt =1

4r2Λ8{3f4∇ω · ∇ω − 16r2f2Λ3∇2Λ

− 2f2∇Aϕ · ∇Aϕ + 4f2ω∇At · ∇Aϕ− 2(r2Λ4 + f2ω2)∇At · ∇At}, (17)


M±tϕ =1

4r2Λ8{−4r2f2Λ3∇ω · ∇Λ + 3f4ω∇ω · ∇ω

− 16r2f2Λ3ω∇2Λ + 6r2Λ4f∇ω · ∇f+ 2Λ4r2f2∇2ω − 4rf2Λ4∇ω · ∇r − 2ωf2∇Aϕ · ∇Aϕ+ 2(ωΛ4r2 − f2ω3)∇At · ∇At+ 4(f2ω2 − r2Λ4∇At · ∇Aϕ)} = M±ϕt, (18)


M±rr = − 1

4r2Λ4f2{−f4(ω2

r − ω2z)− 4Λ4r2f∇2f

+ 4r2fΛ4frr − 8r2f2Λ3∇2Λ + 8r2f2Λ3Λrr

− 16r2Λ3ff,rΛ,r + 8f2Λ2r2Λ2,z − 16f2r2Λ2Λ2

,r

+ 2f2(A2ϕ,r −A2

ϕ,z)− 2(Λ4r2 − f2ω2)(A2t,r −A2

t,z)

− 4ωf2(At,rAϕ,r −At,zAϕ,z)}, (19)


M±rz =1

2r2Λ4f2{4r2fΛ3(Λ,rf,z + Λ,zf,r)

+ 12r2f2Λ2Λ,rΛ,z + f4ω,rω,z − 4r2f2Λ3Λ,rz

− 2r2Λ4ff,rz − 2(f2ω2 − Λ4r2)At,rAt,z

+ 2f2ω(Aϕ,rAt,z +At,rAϕ,z)− 2f2Aϕ,rAϕ,z}, (20)


M±zz =1

4r2Λ4f2{−f4(ω2

,r − ω2,z) + 4r2Λ4f∇2f

− 4r2Λ4ff,zz + 8r2f2Λ3∇2Λ− 8r2f2Λ3Λ,zz

− 8r2f2Λ2Λ2,r + 16r2Λ3ff,zΛ,z + 16r2f2Λ2Λ2

,z

+ 2f2(A2ϕ,r −A2

ϕ,z)− 4ωf2(At,rAϕ,r −At,zAϕ,z)− 2(Λ4r2 − f2ω2)(A2

t,r −A2t,z)}, (21)


M±ϕϕ =1

4r2Λ8f2{4r4fΛ8∇2f − 4r3Λ8f∇f · ∇r

+ f4(Λ4r2 + 3ω2f2)∇ω · ∇ω − 16r2f4ω2Λ3∇2Λ

+ 4r2Λ4f4ω∇2ω − 8r4f4ω∇ω · ∇r + 12Λ4r2ωf3∇ω · ∇f− 8r2ωf4Λ3∇ω · ∇Λ + 8f2Λ7r4∇2Λ− 8f2Λ7r3∇Λ · ∇r− 8r4f2Λ6∇Λ · ∇Λ− 2(f2Λ4r2 + f4ω2)∇Aϕ · ∇Aϕ− 4(f2ωΛ4r2 − ω3f4)∇Aϕ · ∇At− 2(Λ8r4 − 2Λ4r2f2ω2 + f4ω4)∇At · ∇At}. (22)


Conformastationary magnetizeddisk-haloes

ds2 = −e2φ(dt+ ωdϕ)2 + e−2βφ(dr2 + dz2 + r2dϕ2), (23)Jα± = 0, Aα = (0, 0, 0, Aϕ). (24)

e(1+β)φ =1

1− U , ω = kωU, (25)

Aϕ,r = − rkU,z, Aϕ,z =

r

kU,r. (26)

∇2U = 0. (27)


THE DISK:

Sαβ = (µ+ P )VαVβ + Pgαβ +QαVβ +QβVα + Παβ, (28)

{V α, Iα,Kα, Y α} ≡ {h α(t) , h

α(r) , h

α(z) , h

α(ϕ)}, (29)

{Vα, Iα,Kα, Yα} ≡ {−h(t)α, h(r)α, h(z)α, h(ϕ)α}. (30)


µ =4βU,z

(β + 1)(1− U)2β+1β+1

, (31a)

P =(1− β)

3βµ, (31b)

Qα = − kωU,z1− U δ

ϕα , (31c)

Πrr =2(1− β)U,z

3(1 + β)(1− U)1

1+β

, (31d)

Πϕϕ = r2Πrr, (31e)

j = − [U,r]

k(1− U)3β1+β

, (31f)


U = −N∑n=0

bnPn(z/R)

Rn+1, Pn(z/R) = (−1)n

Rn+1

n!

∂n

∂zn

(1

R

),(32)

Aϕ = −1

k

N∑n=0

bn(−1)n

n!

∂n

∂zn

( zR

). (33)


µ(r) =4β∑N

n=0 bn(n+ 1)Pn+1(a/Ra)R−(n+2)a

(1 + β)(

1 +∑N

n=0 bnPn(a/Ra)R−(n+1)a

)(2β+1)/(β+1),

P = (1− β)µ/(3β),


Qα =kωδ

ϕα∑N


1 +∑N


,

Πrr =2(1− β)

∑Nn=0 bn(n+ 1)Pn+1(a/Ra)R

−(n+2)a

3(1 + β)(

1 +∑N


)1/(1+β) ,Πϕϕ = r2Πrr,

j = − 2r∑N

n=0 bnP′n+1(a/Ra)R

−(n+3)

k(1 +∑N

n=0 bnPnR−(n+1))2β/(1+β)

.


w

m0

rw

Figure: Dimensionless surface energy densities µ̃ as a function of r̃.We plot µ̃0(r̃) for different values of the parameter β with b̃0 = 1

s


w

m0

rw


s A. C. Gutiérrez-Piñeres Disk-haloes in Einstein-Maxwell gravity

rw

w

m1



rw

wj0

rw

wj1

(a) (b)

Figure: Dimensionless current of magnetization µ̃ as a function of r̃.In each case, we plot µ̃0(r̃) and µ̃1(r̃) for different values of theparameter β with b̃0 = 1 and b̃0 = 0.5


THE HALO:

M±tt =U2,r + U2

,z

(1 + β)2e4(1+β)φ

{(β2 + 2β)

− 1

2k2(1 + β)2e−2φ +

3k2ω4

(1 + β)2r−2}


M±tϕ =kω(U2

,r + U2,z)

(1 + β)2e4(1+β)φ

×{

1

2(3 + β)(1 + β)e−(1+β)φ + U

(3k2ω4

(1 + β)2r−2 + β2 + 2β

)}− kωe2(1+β)φ

(1

2k2e2βφU(U2

,r + U2,z) + r−1U,r,

)


M±rr = −U2,r − U2

,z

(1 + β)2e2(1+β)φ

((2− β2)− (1 + β)2

2k2e−2φ

)+

(1− β)

(1 + β)2e2(1+β)φ

(2U2

,r − (1 + β)e−(1+β)φU,rr

)+k2ω4r−2e2(1+β)φ(U2

,r − U2,z)


M±rz = − 2U,rU,z(1 + β)2

e2(1+β)φ(

(2− β2)− (1 + β)2

2k2e−2φ

)+

(1− β)

(1 + β)2e2(1+β)φ

(2U,rU,z − (1 + β)e−(1+β)φU,rz

)+

1

2k2ωr

−2e2(1+β)φU,rU,z,


M±zz =U2,r − U2

,z

(1 + β)2e2(1+β)φ

((2− β2)− (1 + β)2

2k2e−2φ

)+

(1− β)

(1 + β)2e2(1+β)φ

(2U2

,z − (1 + β)e−(1+β)φU,zz

)− 1

4k2ωr

−2e2(1+β)φ(U2,r − U2

,z),


M±ϕϕ =r2(U2

,r + U2,z)

(1 + β)2e2(1+β)φ

((2− β2)− (1 + β)2

2k2e−2φ

)− (1− β)rU,r

(1 + β)e(1+β)φ + k2ωe

2(1+β)φK,

K =U2,r + U2

,z

(1 + β)2

{(β2 + 2β)U2e2(1+β)φ + (3 + β)(1 + β)Ue(1+β)φ

+(1 + β)2U2

2e2βφ

(3

2k2ωr

−2 − 1

k2

)+

(1 + β)2

4

}− 2r−1UU,r.


M±αβ = (µ± + P±)VαVβ + P±gαβ +Q±αVβ +Q±β Vα + Π±αβ, (34)

{V α, Iα,Kα, Y α} ≡ {h α(t) , h

α(r) , h

α(z) , h

α(ϕ)}, (35)

{Vα, Iα,Kα, Yα} ≡ {−h(t)α, h(r)α, h(z)α, h(ϕ)α}. (36)


µ± =(U2

,r + U2,z)e

2(1+2β)φ

(1 + β)2r2

×{

(2β + β2)r2 − (1 + β)2

2k2r2e−2φ +

3k2ω(1 + β)2

4

},


P± =(U2

,r + U2,z)e

2(1+2β)φ

3(1 + β)2r2

{(4− 2β − β2)r2 − (1 + β)2

2k2r2e−2φ

+k2ω(1 + β)2

4

(1 + 3k2ωr

−2U2e2βφ(1− e2φ))}

,


Q±α =kωe

(1+2β)φ

2(1 + β)r

{2(1 + β)U,r − (3 + β)r(U2

,r + U2,z)e

(1+β)φ

}δϕα .


Π±αβ = P±r IαIβ + P±z KαKβ + P±ϕ YαYβ + 2P±T I(αKβ),

P±r = e2βφΠ±rr,

P±z = e2βφΠ±zz,

P±ϕ =e2βφ

r2Π±ϕϕ,

P±T = e2βφΠ±rz.

P± ≡ P±r + P±z + P±ϕ = 0, Π±αα = 0.


Π±rr =e2(1+β)φ

3(1 + β)2r2

{(k2ω(1 + β)2

2+

2(1 + β)2

k2r2e−2φ

− 4(1 + β − β2)r2 − 3k4ω(1 + β)2

4r−2U2e2βφ(1− e2φ)

)U2r

+

(− k2ω(1 + β)2 − (1 + β)2

k2r2e−2φ + 2(1 + β − β2)r2

− 3k4ω(1 + β)2

4r−2U2e2βφ(1− e2φ)

)U2z

− 3(1− β2)r2e−(1+β)φU,rr},


Π±zz =e2(1+β)φ

3(1 + β)2r2

{(− k2ω(1 + β)2 − (1 + β)2

k2r2e−2φ

+ 2(1 + β − β2)r2 − 3k4ω(1 + β)2

4r−2U2e2βφ(1− e2φ)

)U2r

+

(k2ω(1 + β)2

2+

2(1 + β)2

k2r2e−2φ − 4(1 + β − β2)r2

− 3k4ω(1 + β)2

4r−2U2e2βφ(1− e2φ)

)U2z

− 3(1− β2)r2e−(1+β)φU,zz},


Π±ϕϕ =(U2

,r + U2,z)e

2(1+β)φ

3(1 + β)2

{2(1 + β − β2)r2 − (1 + β)2

k2r2e−2φ

+k2ω(1 + β)2

2

(1 + 3k2ωr

−2U2e2βφ(1− e2φ))}

− (1− β)

1 + βre(1+β)φU,r


Π±rz =e2(1+βφ)

(1 + β)2

{(− 2(1 + β − β2) +

(1 + β)2

k2e−2φ

+k2ω(1 + β)2

2r2

)U,rU,z − (1− β2)e−(1+β)φU,rz

}.


U = −N∑n=0

bnPn(z/R)

Rn+1, Pn(z/R) = (−1)n

Rn+1

n!

∂n

∂zn

(1

R

),(37)

Aϕ = −1

k

N∑n=0

bn(−1)n

n!

∂n

∂zn

( zR

). (38)


30.1 30.1

220.30.3

22

110.50.5

zr zr00

44

0.70.7 − 1− 1

66

− 2− 2

88

1010

(a)

Figure: Dimensionless surface plot of the energy density µ̃±0

on theexterior halo as a function of r̃ and z̃ with parameters β = 0.5, b̃0 = 1,k = 1 and kω = 1


30.1 30.1

220.30.3

22

110.50.5

zr zr00

44

0.70.7 − 1− 1

66

− 2− 2

88

1010

(a) (b)

Figure: Dimensionless surface plot and level curves of the energydensity µ̃±

0on the exterior halo as a function of r̃ and z̃ with

parameters β = 0.5, b̃0 = 1, k = 1 and kω = 1


30.1 30.1

22

55

0.30.311

0.50.5

1010

zzrr00

− 1− 10.70.7

1515

− 2− 2

2020

2525

(a) (b)

Figure: Dimensionless surface plot and level curves of the energydensity µ̃±

1on the exterior halo as a function of r̃ and z̃ with

parameters β = 0.5, b̃0 = 1, b̃1 = 0.5, k = 1 and kω = 1


0.1 30.1 3

220.30.3

2020

110.50.5r zzr

00

4040

− 10.7 − 10.7

6060

− 2− 2

8080

100100

(a) (b)

Figure: Dimensionless surface plot and level curves of the pressureP̃±

0on the exterior halo as a function of r̃ and z̃ with parameters

β = 0.5, b̃0 = 1, k = 1 and kω = 1


30.1 30.1

22

55

0.30.311

0.50.5

1010

zzrr00

− 1− 10.70.7

1515

− 2− 2

2020

2525

(a) (b)

Figure: Dimensionless surface plot and level curves of the pressureP̃±

1on the exterior halo as a function of r̃ and z̃ with parameters

β = 0.5, b̃0 = 1, b̃1 = 0.5, k = 1 and kω = 1


Perspectives

• Orbit of test particles on the halo

• Orbit of test particles in the disk• Confrontation of the model with observational data• Conformastationary thick disk with halos• Interpretation of the EMT of the disk (halo) from the ZAMO• Stability and Interpretation of the EMT of the disk (halo)• Covariant thermodynamics of the conformastationary

disk-haloes.


Perspectives

• Orbit of test particles on the halo• Orbit of test particles in the disk

• Confrontation of the model with observational data• Conformastationary thick disk with halos• Interpretation of the EMT of the disk (halo) from the ZAMO• Stability and Interpretation of the EMT of the disk (halo)• Covariant thermodynamics of the conformastationary

disk-haloes.


Perspectives

• Orbit of test particles on the halo• Orbit of test particles in the disk• Confrontation of the model with observational data

• Conformastationary thick disk with halos• Interpretation of the EMT of the disk (halo) from the ZAMO• Stability and Interpretation of the EMT of the disk (halo)• Covariant thermodynamics of the conformastationary

disk-haloes.


Perspectives

• Orbit of test particles on the halo• Orbit of test particles in the disk• Confrontation of the model with observational data• Conformastationary thick disk with halos

• Interpretation of the EMT of the disk (halo) from the ZAMO• Stability and Interpretation of the EMT of the disk (halo)• Covariant thermodynamics of the conformastationary

disk-haloes.


Perspectives

• Orbit of test particles on the halo• Orbit of test particles in the disk• Confrontation of the model with observational data• Conformastationary thick disk with halos• Interpretation of the EMT of the disk (halo) from the ZAMO

• Stability and Interpretation of the EMT of the disk (halo)• Covariant thermodynamics of the conformastationary

disk-haloes.


Perspectives

• Orbit of test particles on the halo• Orbit of test particles in the disk• Confrontation of the model with observational data• Conformastationary thick disk with halos• Interpretation of the EMT of the disk (halo) from the ZAMO• Stability and Interpretation of the EMT of the disk (halo)

• Covariant thermodynamics of the conformastationarydisk-haloes.


Perspectives

• Orbit of test particles on the halo• Orbit of test particles in the disk• Confrontation of the model with observational data• Conformastationary thick disk with halos• Interpretation of the EMT of the disk (halo) from the ZAMO• Stability and Interpretation of the EMT of the disk (halo)• Covariant thermodynamics of the conformastationary

disk-haloes.


ReferencesG. A. González, A. C. Gutiérrez-Piñeres, and P. A. Ospina.Finite axisymmetric charged dust disks in conformastatic spacetimes.Phys. Rev. D, 78:064058, Sep 2008.

A. C. Gutiérrez-Piñeres, G. A. González, and H. Quevedo.Conformastatic disk-haloes in einstein-maxwell gravity.Phys. Rev. D, 87:044010, Feb 2013.

A. C. Gutiérrez-Piñeres.Conformastationary disk-haloes in einstein-maxwell gravity.arXiv preprint arXiv:1410.3148, 2014.

A. C. Gutiérrez-Piñeres.Conformastationary disk-haloes in einstein-maxwell gravity ii: the physical interpretation of the halo.Paper in preparation, 2014.

A. C. Gutiérrez-Piñeres, C. S. Lopez-Monsalvo, and H. Quevedo.Variational thermodynamics of relativistic thin disks.arXiv preprint arXiv:1306.6591, 2013.

H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt.Exact solutions of Einstein’s field equations.Cambridge University Press, 2003.

J. Synge.Relativity: the general theory.North-Holland Pub. Co.; Interscience Publishers, Amsterdam; New York, 1960.

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Disk-Haloes in Einstein-Maxwell gravity

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