Neutron Stars in the Braneworld

Transcription

1 Neutron Stars in the Braneworld Mike Georg Bernhardt Ruprecht-Karls-Universität Heidelberg Zentrum für Astronomie, Landessternwarte 24 April 29

2 Outline Introduction Why bother with Extra Dimensions? Braneworlds Why Neutron stars? Neutron Stars in 4D TOV equations Solutions of the TOV equations Neutron Stars in 5D Brane-TOV equations Interior Solutions Exterior Solutions Conclusions References

3 Outline Introduction Why bother with Extra Dimensions? Braneworlds Why Neutron stars? Neutron Stars in 4D TOV equations Solutions of the TOV equations Neutron Stars in 5D Brane-TOV equations Interior Solutions Exterior Solutions Conclusions References

4 Why bother with Extra Dimensions? The quest for unification Minkowski (199), Electromagnetism, 4D Einstein (19), General Relativity, 4D Kaluza (1919) and Klein (1926) Unification of Electromagnetism and Gravity in 5D? Supergravity, 11D or less String theory (only bosons), 26D 5 superstring theories (bosons and fermions), 1D M-Theory, 11D Solving the Hierarchy Problem Arkani-Hamed, Dimopoulos and Dvali (1998), 6D Randall and Sundrum (1999), 5D braneworld

5 Braneworlds The braneworld paradigm is based on 3 assumptions: 1. 4D spacetime is a hypersurface, called brane, of a 5D manifold. 2. Gravity is a 5D phenomenon, descibed by the field equations G AB = Λ 5 g AB + κ 5 T AB. 3. Matter and the fundamental interactions apart from gravity are confined to the brane, 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.

6 Braneworlds 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.

7 Why Neutron stars? Strong gravitational field Test bed for general relativity and alternative gravity theories Are observable properties of NS influenced by 5D effects? Observing NS observing extra dimensions? Can braneworld models be excluded by observing NS?

8 Outline Introduction Why bother with Extra Dimensions? Braneworlds Why Neutron stars? Neutron Stars in 4D TOV equations Solutions of the TOV equations Neutron Stars in 5D Brane-TOV equations Interior Solutions Exterior Solutions Conclusions References

9 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

10 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 ( ) 1dr ds 2 = e 2Φ(r) dt Gm(r) 2 r + r 2( dϑ 2 + sin 2 ϑ dϕ 2). 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).

11 Solutions of the TOV equations Analytical solutions of the TOV equations: 1. Schwarzschild solution for the exterior of a star 2. Interior Schwarzschild solution for a homogeneous density star For realistic stars with inhomogenious density, there are no analytical interior solutions. We have to use numerical methods. 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.

12 log [g/cm 3 ] Numerical Interior Solutions trans 12 drip m/m 1..5 Kern innere Kruste äußere Kruste r [km]

13 Numerical Interior Solutions R [km] M/M c/ nuc R [km] M/M

14 Outline Introduction Why bother with Extra Dimensions? Braneworlds Why Neutron stars? Neutron Stars in 4D TOV equations Solutions of the TOV equations Neutron Stars in 5D Brane-TOV equations Interior Solutions Exterior Solutions Conclusions References

15 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.

16 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)..

17 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

18 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

19 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 ].

20 Interior Solutions 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.

21 Numerical Interior Solutions 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]

22 Numerical Interior Solutions 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]

23 Numerical Interior Solutions 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]

24 Numerical Interior Solutions 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]

25 Numerical Interior Solutions 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]

26 Numerical Interior Solutions 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 Numerical Interior Solutions 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]

28 Numerical Interior Solutions 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 λ [dyn/cm 2 ] = M/M c/ nuc R [km] 11 R [km] M/M M/M

29 Numerical Interior Solutions R [km] w = 3 w = 1 w =,6 w =,2 w =,1 w =,2 w = 2 R [km] 2 1 w = 3 w = 1 w =,6 w =,2 w =,1 w =,2 w = λ [dyn/cm 2 ] = 1 37 M/M c/ nuc λ [dyn/cm 2 ] = 1 36 M/M c/ nuc 13 2 R [km] 11 9 R [km] M/M M/M

30 Exterior Solutions 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! )

31 Exterior Solutions 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).

32 Numerical Exterior Solutions 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]

33 Numerical Exterior Solutions 2 w =,2 w =,2 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]

34 Numerical Exterior Solutions w =,2 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]

35 Outline Introduction Why bother with Extra Dimensions? Braneworlds Why Neutron stars? Neutron Stars in 4D TOV equations Solutions of the TOV equations Neutron Stars in 5D Brane-TOV equations Interior Solutions Exterior Solutions Conclusions References

36 Conclusions 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). The relative corrections of mass and density of a neutron star 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. A generic property of the brane-tov equations are exterior solutions with m const.

37 Outline Introduction Why bother with Extra Dimensions? Braneworlds Why Neutron stars? Neutron Stars in 4D TOV equations Solutions of the TOV equations Neutron Stars in 5D Brane-TOV equations Interior Solutions Exterior Solutions Conclusions References

38 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)