Fermion Stars with an Extra Dimension

7 Pages·0.11 MB·

by  Nahomi Kan

Fermion Stars with an Extra Dimension
Nahomi Kan1 and Kiyoshi Shiraishi2
Graduate School of Science and Engineering, Yamaguchi University,
Yoshida,Yamaguchi-shi,Yamaguchi 753-8512, Japan
0
Abstract
0
0 Many efforts have been devoted to the studies of the phenomenology in particle
2 physics with extra dimensions. We propose the degenerate fermion star in the five
dimensions,andstudywhateffectscausedbythegeometryofextradimensionsshould
n
a appearinitsstructure. WenotethatKaluza-Kleinexcitedmodeshaveeffectsforthe
J larger scale ofextradimensionsandexaminetheconditions onwhich differentlayers
3 should be caused in theinside of thestars. We expoundhow the effects of theextra
1 dimensions appears on physical quantities.

2 1 Introduction
v
7
2 Recently, physics dealing with higher dimensions has been noted [1, 2]. Considering extra dimensions
0 in GUT, one points out four forces could be unified at low energy scale. And these are closely related
1 with superstring theories, 3 supergravity, and the others. There are many works on extra dimensions.
0
One of them is “Neutron stars and extra dimensions” by Liddle, Moorhouse and Henriques [5]. They
0
consideredthe neutronstarsinfivedimensionsofwhichthefifthoneiscompactifiedintoS1,andshowed
0
/ that the maximum mass of the stars decreases under the background. However there are some problems
c
in their result. First, they did not consider Kaluza-Klein (K-K) excited modes caused by the geometry
q
of extra dimensions. Secondly, they put some hypotheses both on the equation of state and on that
-
r of conservation, so arbitrariness is left. Whereas, in this paper, taking account of the K-K modes, we
g
: propose the star in the five dimensions, which is made of degenerate Dirac fermions, and indicated that
v
the excited modes have effects to the inside of star. After asking for the maximum mass and the radius
i
X by use of numerical calculation, we will make it clear what the interior structure of star should be.
r
a
2 Matter
2.1 (3+1) dimensions
Inthe four dimensionaltheory,the thermodynamic potentialof afermiongas withmass m andhalfspin
is
Ω = 21V d3p ln 1+e−β(√p2+m2−µ) +(µ µ) , (1)
4 − β (2π)3 ↔−
Z h (cid:16) (cid:17) i
where we put subscript to emphasize that it stands for a four dimensional quantity. We will restrict
our system to degenerate fermion gas and take the zero temperature limit. Then, the thermodynamic
potential becomes
V m4 µ µ2 1 2µ2 5 +3ln µ + µ2 1 (m<µ)
Ω4(m)≡0−, 24π2 (cid:20)mqm2 − (cid:16) m2 − (cid:17) (cid:12)(cid:12)m qm2 − (cid:12)(cid:12)(cid:21) (m≧µ) . (2)
 (cid:12) (cid:12)
(cid:12) (cid:12)
Thermodynamical quantities follow from Eq.(2).

1E-mail:[email protected]
2E-mail:[email protected]
3ThepossibilityoflargeextradimensionswasoriginallydiscussedbyAntoniadis[3],withtherelationtostringtheories.
Thefirststringrealizationoflowscalegravitymodelswasgivenin[4].
2.2 (4+1) dimensions
We willextend our argumentinto the five dimensionaltheory. Here we suppose that the fifith dimension
is compactified into S1 with a radius b, b is not so large. We impose the periodic boundary condition on
a wave function in the fifth dimension:
ψ(χ) eip5·χ , (3)
∼
ψ(χ+2πb) ψ(χ) , (4)
∼
with the fifth coordinate χ and momentum p , then
5
n
p = (n: integer) . (5)
5
b
Therefore the relativistic energy involving the fifth dimension is
n
E = p2+( )2+m2 (6)
5
b
r
= p2+M2 , (7)
p
where
n 2
M2 +m2 , (8)
≡ b
(cid:16) (cid:17)
(nisinteger. Unlessn=0,theK-Kmodesbecomeeffective.) UsingEq.(7),thethermodynamicpotential
of the fermion gas with mass m and half spin in the (4+1) dimensional space-time is
Ω = 21V d4p ln 1+e−β(√p2+M2−µ) +(µ µ) , (9)
5 − β 4 (2π)4 ↔−
Z h (cid:16) (cid:17) i
If we turn the integralover the fifth dimensional momentum into the sum over quantum number n, that
is :
1
dp , (10)
5
→ b
Z n
X
and in addition,
V =2πbV , (11)
4
then the thermodynamic potential is
Ω5 = −2β1V (d23πp)3 ln 1+e−β(qp2+nb22+m2−µ) +(µ↔−µ)
n Z (cid:20) (cid:18) (cid:19) (cid:21)
X
n2
= Ω m2+ = Ω (M) . (12)
4 b2 4
n r ! n
X X
As well as the previous section, we will deal with the degenerate fermion gas and low temperature near
to zero. The thermodynamic potential, therefore,
Vb−4f(µb,mb), (µ>M)
Ω5 = 0−, (µ≦M) , (13)
(cid:26)
where
∞ (Mb)4 µb µb 2 µb 2 µb 2 µb
f 2 5 1+3ln 1+ (µ>M) .
≡n=X−∞ 24π2 Mb (cid:18)Mb(cid:19) − !s(cid:18)Mb(cid:19) − (cid:12)(cid:12)(cid:12)s(cid:18)Mb(cid:19) − Mb(cid:12)(cid:12)(cid:12)
(cid:12) (cid:12) (14)
From Eq.(13), thermodynamical quantities are as follows: (cid:12)(cid:12) (cid:12)(cid:12)
1 ∂Ω 1 Ω 1 1
5 5
P = = = f , (15)
−2πb ∂V −2πb V 2πbb4
1 ∂Ω 1 1 ∂f ∂f
5
P = = x +y 4f , (16)
5 −2πV ∂b 2πbb4 ∂x ∂y −
(cid:18) (cid:19)
U 1 1 ∂f
ρ= = x f , (17)
V 2πbb4 ∂x −
4 (cid:18) (cid:19)
where P is the pressure in the fifth dimension, and we put x=µb, y =mb.
5
3 Space-Time
We take the line element to be of the form
dr2
ds2 =e−Φ e−2δ∆dt2+ +r2(dθ2+sin2θdϕ2) +b2e2Φdχ2 , (18)
− ∆ 0
(cid:20) (cid:21)
where we regard ∆, δ, and Φ (b=b eΦ) as functions depending only on r, the distance from the origin.
0
Next, the energy-momentum tensor is
Tµ =diag.( ρ,P,P,P,P ) , (19)
ν − 5
inwhichwesupposeisotropicpressureinthespaceofthreedimensions,andrepresentthefifthdimensional
pressure as P . The equation of conservation is
5
Tµr =0 . (20)
µ
∇
From this, we find the chemical potential µ satisfies that
e23Φ+δ
µb= µ b , (21)
0 0
√∆
where µ depends on r and µ and b are each the value of µ and b when r is close to zero.
0 0
4 Equations
Now, we are just deriving Einstein equations. Before that, we will rewrite Eq.(14). After putting
x=µb, y =mb,
Eq.(14) reduces to
∞ ′
(y2+n2)2 x x2 x2
f(x,y) = 1 2 5
n=−∞ 24π2 " y2+n2sy2+n2 − (cid:18) y2+n2 − (cid:19)
X
x p x2
+3ln + 1
(cid:12) y2+n2 sy2+n2 − (cid:12)#
(cid:12) (cid:12)
∞ ′(cid:12) (cid:12)
(cid:12)p (cid:12)
= (cid:12)f˜(x,y) ( x2 y2 >n2(cid:12)) , (22)
−
n=−∞
X p
where the sum over n is done unless n2 exceeds x2 y2 . (To remark this, we put prime on a sum
−
symbol.) In addition, we put
p
Y = y2+n2 . (23)
Using the leading formulae, Einstein equations arpe
M˜′ 3 1 ′ ∂f˜
⋆ ∆˜(Φ′)2 =4π e−6Φ x f˜ , (24)
r˜2 − 8 m4b40 ∂x − !
X
1 3 1 e−6Φ ′ ∂f˜
δ′+ (Φ′)2 = 4π x , (25)
r˜ 4 − m4b40 ∆˜ ∂x!
X
∆˜′ 2 8π 1 e−6Φ ′ 3Y2 2y2 ∂f˜
′′ ′ ′
Φ + δ + Φ = − x
∆˜ − r˜! − 3 m4b40 ∆˜ Y2 ∂x
X
12Y2 8y2
− f˜ , (26)
− Y2
(cid:19)
where
M˜ = G(4)m2G(4)mM , (27)
⋆ 0 0 ⋆
q
r˜= G(4)m4r , (28)
0
q
2M˜
∆˜ =1 ⋆ . (29)
− r˜
In the above equations, we define a usual Newtonian constant G(4) as
0
G(5)
G(4) 0 , (30)
0 ≡ 2πb
0
in which G(5) stands for the Newtonian constant in the fifth dimension. And M is the mass of star
0 ⋆
depending on r. A prime means derivative with respect to r˜. We can solve these equations numerically.
In the next section, we will show the result.
5 Results
We will exhibit the relationship between the mass and the central density in fermion stars in Fig.1, and
the mass and the radius in Fig.2. It turns out that the larger the extra dimension grows, the smaller
the maximum mass becomes. Furthermore, it is remarkable that two maximum points appeared when
mb =3,4.
0
0.3
0.25
0.2
M 0.15
M¯
0.1
0.05
-3 -2 -1 0 1 2
log (ρ /m4)
10 0(4)
Fig 1: Plots of the mass M of the fermion stars versus its central density ρ for the various scales
0(4)
of the extra dimension. The dashed line corresponds to mb 0. The dot-dashed line corresponds to
0
≈
mb =1, the two dot-dashed line to mb =2, the three to mb =3, the four to mb =4.
0 0 0 0
0.3
0.25
0.2
M
M¯
0.15
0.1
1 2 3 4 5 6 7
R
R¯
Fig 2: Plots of the mass M of the fermion stars versus its radius R for the various scales of the extra
dimension. The correspondence between the dashed lines and the fifth dimension is the same as Fig.1.
Next, in Fig.3, we draw the interior structure of the stars having the maximum mass, which is based
on Fig.2. We can read off it from Fig.3 that the excited modes have effects in the core of the stars with
increase in the extra dimension. We have two solutions in the case of mb = 3,4. For mb = 3, one of
0 0
them is a larger star than that for mb = 2 and its central density is lower, which is shown in Fig.1.
0
While the other gets to be a smallerone and its centraldensity is higher. Inthis solution, higher excited
mode (n=2) is caused mainly in the center of stars and lower mode (n=1) in the vast region including
the core. Proceeding to mb =4, this inclination appears more remarkably.
0
6 6
4 4
2 2
0 0
-2 -2
-4 -4
-6 -4 -2 0 2 4 6 mb =4 -6 -4 -2 0 2 4 6
0
6 6
4 4
2 2
0 0
-2 -2
-4 -4
-6 -4 -2 0 2 4 6 mb =3 -6 -4 -2 0 2 4 6
0
6
4
2
0
-2
-4
-6 -4 -2 0 2 4 6 mb =2
0
6
4
2
0
-2
-4
-6 -4 -2 0 2 4 6 mb =1
0
Fig3: Theevolutionsofthefermionstarswiththeincreaseoftheextradimension. Inthedarkestcircle,
n=2modeismosteffectiveandn=1modeinthemiddle one. Thepalecirclestandsfornoexcitedmode.
6 Conclusion
WehaveshownthatconsideringK-Kmodeinthefivedimensionaltheory,astheextradimensionbecomes
larger, the maximum mass more and more decreases. And also we have obtained two series of solutions.
One is that the fermion star is getting larger and its central density lower with the increase of the extra
dimension, that is, the larger and lighter star is formed. In these solutions, only n=1 mode is caused in
thecoreofthestars. Theotheristhatastheextradimensiongrows,thoughthestaroncebecomeslarger
and its central density lower for mb =2, it is getting smaller and smaller and its central density higher
0
for the much larger dimension, namely, the smaller and heavier star is created. In these stars, n=2,1
modes is caused, the former is in the center of the stars and the latter is in the vast regionincluding the
core. As far as we examine, in this time, we have provedthat the structure of the fermion stars depends
on the scale of the fifth dimension, that is, the excited modes have effects to the inside of stars. Taking
the scale of the fifth dimension much larger,however,we need to analyze the stability of stars explicitly.
And, in this paper, we have imposed the periodic boundary condition on a wave function in the fifth
dimension,while we canadoptthe generalone,thatis: ψ(χ+2πb) eiϕψ(χ). For the anti-periodicone,
∼
ψ(χ+2πb) ψ(χ), the effective mass of the fermion M is M = m2+(n±12)2, on which we are
∼− (h) (h) b
working,so, the details willbe explained elsewhere. Meanwhile,we canqapply ourtopics to cosmologyin
which we will suppose the time dependence of the fifth dimension and think over how the stars should
be created in the time-dependent process. Although we took the zero temperature limit in this paper,
we can treat the finite one too. Furthermore we will try to extend our argument into higher dimensional
theories (the six dimensions, the ten dimensions and the others),andconsider whatthe star made ofthe
bulk matter in the brane world should be, in which we are going to researchas the next theme.
References
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