Compact Stars in the Braneworld

Transcription

1 Compact Stars in the Braneworld Mike Georg Bernhardt Zentrum für Astronomie Heidelberg Landessternwarte 28 January 29

2 Outline Review: Relativistic Stars TOV equations Solutions of the TOV equations Neutron Stars and White Dwarfs Braneworlds Introduction Effective Field Equations on the Brane Compact Stars on the Brane Brane-TOV equations Analytical Exterior Solution Neutron Stars and White Dwarfs on the Brane Conclusions References

3 Outline Review: Relativistic Stars TOV equations Solutions of the TOV equations Neutron Stars and White Dwarfs Braneworlds Introduction Effective Field Equations on the Brane Compact Stars on the Brane Brane-TOV equations Analytical Exterior Solution Neutron Stars and White Dwarfs on the Brane Conclusions References

4 Tolman-Oppenheimer-Volkoff equations Our goal is to solve the Einstein equations for a static and spherical star. Ansatz for the line element G αβ := R αβ 1 2 Rg αβ = κt αβ ds 2 = e 2Φ(r) dt 2 + e 2Λ(r) dr 2 + r 2( dϑ 2 + sin 2 ϑ dϕ 2). Stellar matter modelled as a perfect fluid ( ϱ ) Tα β p = p. p

5 Tolman-Oppenheimer-Volkoff equations This yields the TOV equations of stellar structure m = 4πr 2 ϱ, p = (ϱ + p)φ, Φ = Gm + 4πGr3 p r(r 2Gm) with the line element ( ds 2 = e 2Φ(r) dt Gm(r) ) 1 dr 2 + r 2( dϑ 2 + sin 2 ϑ dϕ 2). r 3 equations, 4 unknown functions m(r), ϱ(r), p(r) and Φ(r). In order to close the system, we need an equation of state p = p(ϱ).

6 Solutions of the TOV equations Exterior: ϱ =, p = and m =, Φ Gm = r(r 2Gm), m = const =: M, e 2Φ = 1 2GM. r This leads to the Schwarzschild solution ( ds 2 = 1 2GM ) ( dt GM ) 1 dr 2 r r + r 2( dϑ 2 + sin 2 ϑ dϕ 2).

7 Solutions of the TOV equations Interior: An analytical solution exists only for the ideal case of a homogeneous density star with ϱ = const =: ϱ for all p(r). The TOV equations imply the internal Schwarzschild solution ( 3 ds 2 = 1 2GM 2 R 1 ) 2 1 2GMr2 2 R 3 dt 2 ) 1 + (1 2GMr2 dr 2 + r 2( dϑ 2 + sin 2 ϑ dϕ 2). R 3 Interesting property of this solution: GM R < 4 9, R max = ( ) 1/2 3πGϱ, Mmax = 4 ( 9 3πG 3 ) 1/2 ϱ.

8 Solutions of the TOV equations For realistic stars with inhomogenious density, there are no analytical solutions. We have to use numerical methods. The TOV equations are a system of first order, ordinary differential equations. They can be solved numerically using a Runge-Kutta-Algorithm. For the numerical calculations, I used the equation of state SLy for the neutron star matter.

9 log [g/cm 3 ] Neutron Stars trans 12 drip m/m 1..5 Kern innere Kruste äußere Kruste r [km]

10 Neutron Stars 15 R [km] M/M c/ nuc 15 R [km] M/M

11 White Dwarfs 6 log [g/cm 3 ] m/m r/r

12 White Dwarfs M/M log c [g/cm 3 ]

13 Outline Review: Relativistic Stars TOV equations Solutions of the TOV equations Neutron Stars and White Dwarfs Braneworlds Introduction Effective Field Equations on the Brane Compact Stars on the Brane Brane-TOV equations Analytical Exterior Solution Neutron Stars and White Dwarfs on the Brane Conclusions References

14 Introduction The braneworld paradigm is based on 3 assumptions: 1. 4D spacetime is a hypersurface of a 5D manifold, called bulk. 2. Gravity is a 5D phenomenom. The field equations in 5D have the same form as Einstein s equations in 4D: G AB = Λ 5 g AB + κ 5 T AB. 3. Matter and the fundamental interactions (except gravity) are confined to the hypersurface. Inspired by M-/string theory, the hypersurface is called a brane. The confinement of matter enters phenomenologically via T AB = δ(χ) [ λĝ AB + T AB ]. Due to the negative cosmological constant Λ 5, gravity is effectively localised near the brane. The 5 th dimension is not compactified, but can be infinitely large.

15 Effective Field Equations on the Brane On the brane an effective 4D gravity is induced by the 5D field equations: with G αβ = Λg αβ + κt αβ + 6κ λ S αβ E αβ Λ := 1 2( Λ κ2 5λ 2), κ := 1 6 κ2 5λ, In the limit S αβ := 1 12 TT αβ 1 4 T αµt µ β g αβ( 3Tµν T µν T 2). λ, κ 5, E αβ (with finite κ) gravity is localised on the brane and General Relativity is recovered.

16 Outline Review: Relativistic Stars TOV equations Solutions of the TOV equations Neutron Stars and White Dwarfs Braneworlds Introduction Effective Field Equations on the Brane Compact Stars on the Brane Brane-TOV equations Analytical Exterior Solution Neutron Stars and White Dwarfs on the Brane Conclusions References

17 Brane-TOV equations Now, we want to solve the modified Einstein equations G αβ = κt eff αβ, Teff αβ := T αβ + 6 λ S αβ 1 κ E αβ for a static and spherical star on the brane. Ansatz for the line element ds 2 = e 2Φ(r) dt 2 + e 2Λ(r) dr 2 + r 2( dϑ 2 + sin 2 ϑ dϕ 2). The brane metric, and also the Einstein tensor G αβ are the same as in 4D.

18 Brane-TOV equations Stellar matter modelled as a perfect fluid ) T β α = ( ϱ p p p This implies the high-energy correction S β α = 1 12 ϱ 2 (ϱ 2 +2ϱp) (ϱ 2 +2ϱp) (ϱ 2 +2ϱp) The static and spherically symmetric projected Weyl tensor can be written as 1 κ Eβ α = U 1 3 (U+2P) 1 3 (U P). 1 3 (U P)..

19 Brane-TOV equations We end up with the effective energy-momentum tensor ϱ eff Tα βeff p eff = p eff, p eff with ϱ eff := ϱ + ϱ2 2λ + U, p eff := p + pϱ λ + ϱ2 2λ + U + 2P, 3 p eff := p + pϱ λ + ϱ2 2λ + U P. 3

20 Brane-TOV equations The brane-tov equations follow from G αβ = κt eff αβ : m = 4πr 2( ϱ + ϱ2 ) 2λ + U, p = (ϱ + p)φ, Φ = G m + 4πGr3[ p + pϱ λ + ϱ2 2λ (U + 2P)], r(r 2G m) U = 2P 6P (4U + 2P)Φ 3 ϱ + p r λ ϱ with the line element ( ds 2 = e 2Φ(r) dt G m(r) ) 1 dr 2 + r 2( dϑ 2 + sin 2 ϑ dϕ 2). r

21 Brane-TOV equations 4 equations, 6 unknown functions m(r), ϱ(r), p(r), Φ(r) and the Weyl terms U and P. In order to close the system, we need an equation of state p = p(ϱ) plus an additional relation between U and/or P. I assume, that the Weyl terms obey an equation-of-state like relation P = wu. Then, the brane-tov equations read m = 4πr 2( ϱ + ϱ2 ) 2λ + U, p = (ϱ + p)φ, Φ = G m + 4πGr3[ p + pϱ λ + ϱ2 2λ + 1+2w 3 U ], r(r 2G m) U = 3 1+2w[ 2wUr (2 + w)uφ + 1 λ (ϱ + p)ϱ ].

22 Analytical Exterior Solution for w = 2 In the case w = 2 the exterior brane-tov equations can be solved analytically. Exterior: ϱ =, p = and m = 4πr 2 U, Φ = G m 4πGr3 U r(r 2G m) U = 4U r,, m(r) = m(r) + 4πU(R)R 4 ( 1 R 1 r e 2Φ = 1 2G m(r), r ( ) R 4 U(r) = U(R). r If U(R), then m(r) const in the exterior! )

23 Analytical Exterior Solution for w = 2 This leads to a Reissner-Nordström like solution ( ds 2 = 1 2G m + Q ) ( r r 2 dt G m + Q ) 1 r r 2 dr 2 with + r 2( dϑ 2 + sin 2 ϑ dϕ 2) m := m(r) + 4πU(R)R 3, Q := κ U(R)R 4. A numerical calculation shows, that for w = 2 we have in general U(R) <. That means, the effective mass tends to m < m(r).

24 Neutron Stars on the Brane As in the relativistic case, the brane-tov equations can be solved numerically. I used the same equation of state (SLy) as in the relativistic case, taking into account the inhomogeneous energy density inside the neutron star. The following pages show the results of the numerical integration for neutron stars.

25 Neutron Stars on the Brane w = 1 w = 1 λ [dyn/cm 2 ] = 1 35, 1 36, 1 37, 1 38 λ [dyn/cm 2 ] = 1 35, 1 36, 1 37, log [g/cm 3 ] m/m r [km] r [km]

26 Neutron Stars on the Brane w = 2 w = 2 λ [dyn/cm 2 ] = 1 35, 1 36, 1 37, 1 38 λ [dyn/cm 2 ] = 1 35, 1 36, 1 37, log [g/cm 3 ] m/m r [km] r [km]

27 Neutron Stars on the Brane w = 1 w = 1 λ [dyn/cm 2 ] = (dotted), λ [dyn/cm 2 ] = , 1 36, 1 37, , 1 37, log [g/cm 3 ] m/m r [km] r [km]

28 Neutron Stars on the Brane w =,6 w =,6 λ [dyn/cm 2 ] = , , λ [dyn/cm 2 ] = , 1 36, , 1 36, , 1 37, log [g/cm 3 ] m/m r [km] r [km]

29 Neutron Stars on the Brane w =,2 w =,2 λ [dyn/cm 2 ] = 1 35, 1 36, 1 37, 1 38 λ [dyn/cm 2 ] = 1 35, 1 36, 1 37, log [g/cm 3 ] m/m r [km] r [km]

30 Neutron Stars on the Brane w = 2 w = 2 λ [dyn/cm 2 ] = 1 35, 1 36, 1 37, 1 38 λ [dyn/cm 2 ] = 1 35, 1 36, 1 37, log [g/cm 3 ] m/m r [km] r [km]

31 Neutron Stars on the Brane w = 1 w = 1 λ [dyn/cm 2 ] = 1 35, 1 36, 1 37, 1 38 λ [dyn/cm 2 ] = 1 35, 1 36, 1 37, log [g/cm 3 ] m/m r [km] r [km]

32 Neutron Stars on the Brane R [km] w = 3 w = 1 w =,6 w =,2 w =,1 w =,2 w = 2 R [km] w = 3 w = 1 w =,6 w =,2 w =,1 w =,2 w = λ [dyn/cm 2 ] = 1 38 M/M c/ nuc 15 λ [dyn/cm 2 ] = M/M c/ nuc R [km] 11 R [km] M/M M/M

33 Neutron Stars on the Brane R [km] w = 3 w = 1 w =,6 w =,2 w =,1 w =,2 w = 2 R [km] w = 3 w = 1 w =,6 w =,2 w =,1 w =,2 w = λ [dyn/cm 2 ] = 1 37 M/M c/ nuc 15 λ [dyn/cm 2 ] = 1 36 M/M c/ nuc 13 2 R [km] 11 9 R [km] M/M M/M

34 Neutron Stars on the Brane 8 U [1 33 erg/cm 3 ] w = λ [dyn/cm 2 ] = 1 38 U [1 33 erg/cm 3 ] w = w = 1.2 w = r [km]

35 Neutron Stars on the Brane 2 w =,2 15 w =,2 15 w = w = 5 w =,2 w =, λ [dyn/cm 2 ] = 1 38 m/m w = w = w = 1 w = w = 1 λ [dyn/cm 2 ] = 1 37 m/m 1.8 w = w = w = 1 w = w = 1 w = 1 w = 1 w =,6 w =, w =,8 w = 3 w = w =,8 w = 3 w = 2 w = 1 w = r [km] r [km]

36 Neutron Stars on the Brane w =,2 15 w = 1 w =,2 5 λ [dyn/cm 2 ] = 1 36 m/m w = w = w = 1.7 w = 1.6 w = 1.5 w = w =,8 w = 2 w =,6 w = 3 w = r [km]

37 White Dwarfs on the Brane Basically, there is no difference between conventional and braneworld white dwarfs. Consider the effective density ϱ eff = ϱ + ϱ, ϱ := c2 ϱ 2 2λ + U c 2 and the parameters λ = 1 38 dyn/cm 2 and w = 2. For a white dwarf with ϱ c = 1 6 g/cm 3 the relative density correction ϱ/ϱ is 4, at the center and 4, at U min. For a neutron star with ϱ c = 1 15 g/cm 3 the relative density correction ϱ/ϱ is 4,5 1 3 at the center and 1% at U min.

38 Outline Review: Relativistic Stars TOV equations Solutions of the TOV equations Neutron Stars and White Dwarfs Braneworlds Introduction Effective Field Equations on the Brane Compact Stars on the Brane Brane-TOV equations Analytical Exterior Solution Neutron Stars and White Dwarfs on the Brane Conclusions References

39 Conclusions For white dwarfs there are basically no differences to general relativity. For neutron stars the relative corrections of mass and density are bigger than 1 3 if λ 1 39 dyn/cm 2. A brane tension λ < dyn/cm 2 is in contradiction with observed neutron star masses. Neutron stars on the brane are in general more compact: They have smaller radii (except for 1 w <,5) and are less massive (except for,5 < w,1 and ϱ c /ϱ nuc 4). A generic property of the brane-tov equations are exterior solutions with m const.

40 Outline Review: Relativistic Stars TOV equations Solutions of the TOV equations Neutron Stars and White Dwarfs Braneworlds Introduction Effective Field Equations on the Brane Compact Stars on the Brane Brane-TOV equations Analytical Exterior Solution Neutron Stars and White Dwarfs on the Brane Conclusions References

41 References M. G. Bernhardt Kompakte Sterne in der Branenwelt Diplomarbeit, Ruprecht-Karls-Universität Heidelberg (29) R. Maartens Brane-World Gravity Living Rev. Relativity, 7, (24), 7, C. Germani, R. Maartens Stars in the braneworld Phys. Rev. D 64, 1241 (21), arxiv:hep-th/1711v3 N. Deruelle Stars on branes: the view from the brane arxiv:gr-qc/11165v1 (21)