Charged Boson Stars and Black Holes in Horndeski Gravity

Transcription

1 Yosef Verbin The Open University of Israel Charged Boson Stars and Black Holes in Horndeski Gravity Work in collaboration with Yves Brihaye, University of Mons, Belgium arxiv:

2 Boson Stars Self-gravitating localized solutions of the Einstein-Klein-Gordon (complex) field equations. Stabilized by conserved U(1) charge = particle number Similar to Heisenberg uncertainty in QM Kaup, PR 172, 1331 (1968) Ruffini & Bonazzola, PR 187, 1767 (1969) & Tuesday Talk by Ulrich Sperhake 2

3 General Field Equations: m G T 0, T * * g m Conserved U(1) current and charge i J * * 2 J 0 N d x det( g) J 3 0 One may add = particle number 4 /4 self-interaction M. Colpi, S.L. Shapiro and I. Wasserman, Phys. Rev. Lett. 57 (1986)

4 Spherically-Symmetric Stationary Solutions: f () r e it g dx dx A( r) 2 dt 2 B( r) 2 dr 2 r 2 d 2 sin 2 d 2 Localized asymptotically flat solutions: For r : f ( r) 0, A( r) 1, B( r) 1. Actually, asymptotically Schwarzschild: 2 2 A( r) 1 2 GM / r, 1/ B( r) 1 2 GM / r Mass and particle number: Solutions are determined by f (0). 4

5 Profiles for f(0)=0.1,0.25,0.5,0.75,1.00, Rescaling to dimensionless variables: r mr, f f

6 Masses: in terms of M 2 Pl / m Particle numbers: in terms of MPl 1/ G GeV gr M M / m 2 2 Pl kg Pl / m m/1ev M m/1ev Earth Stability against fusion : up to crossing point where M ( f (0)) N( f (0)) at about f(0)= and M=N= Stability against small perturbation- more restrictive: up to maximal M,N : M 15.91, N at about f(0)= Sometime written as: M 6 max max Binding energy per particle: (Nm-M)/N max / , N / max

7 Adding Electric Charge 2 Field Equations: D D m G F T e J * * 2 2 T D D D D F F g D m F Conserved local U(1) current, particle number and charge: i J * D * D J 0 N d x det( g) J, Q en 0 7 charged boson stars exist [Jetzer, Phys Rep 220, 163 (1992) ]

8 e=0.2 further rescaling: 2 e e / m, a0 a0 Profiles for f(0)=0.1,0.25,0.5,0.75,1.00, Solution asymptotically RN. Very similar to e=0 except a () r 0. Densities too.

9 copy of e=0 profiles Profiles for f(0)=0.1,0.25,0.5,0.75,1.00,1.25 a ( r) 0 0 9

10 e=0.6 Profiles for f(0)=0.1,0.25,0.5,0.75,1.00,

11 e=0.2 copy of e=0.2 profiles Profiles for f(0)=0.1,0.25,0.5,0.75,1.00,

12 Partial Summary Boson Stars Depth of Gravitational Potential = A(0) or rather 1-A(0): Dependence on f(0) and e: Gravitational potential becomes deeper with: larger f(0) larger e 12

13 Charged Boson Stars - general behavior M solid ; N dashed electrostatic repulsion increases mass. f(0) value of maximal mass decreases with e Critical coupling constant appears: ecrit m Binding energy per particle decreases with e 2 /2 13

14 Horndeski Scalar-Tensor Theory 1) A small cosmological constant explains very well the current universe acceleration, but its origin is obscure. 2) Dynamical explanation involve in most cases scalar fields. 3) What is the most general scalar-tensor field theory whose field equations are second order (no Ostrogradsky ghosts)? 4) Answer:, 14

15 Special Horndeski: Fab Four, where Special sector- screening property: The most general subclass of Horndeski theory in which flat space is a solution, despite the presence of. 15

16 Horndeski- John Boson Stars Take only John term with a constant coefficient: VJ. Complex scalar field. General Lagrangian with possible electromagnetic coupling: Ordinary boson stars: e0, 0. Field equations: A Einstein: recall: 16

17 The additional Horndeski (only John ) terms: all higher order derivatives in Y and Z cancel by explicit expansion. 17

18 Spherical Symmetry + stationary scalar f : a 0 : G 0 : 0 r G r : 18

19 mass, particle number, charge: define mass function by: Total mass: scalar + vector + Horndeski Particle number density: Total number: charge: Q en Note also field dependent gravitational constant : gravitation stronger with. 19

20 Horndeski- John Boson Stars-neutral and charged solutions are asymptotically RN: neutral boson stars solutions exist only for a finite interval of f(0) if 0: 0 f(0) f(0) max. interval shrinks in both directions, but faster for 0. 20

21 Mass, Particle Number and Binding Energy effect of on neutral boson stars M solid ; N dashed Larger, stronger gravity, less particles form bound states. For the same N, larger are more stable (slightly). f decreases with. Technical reason for f (0) max : equations cannot be decoupled. The inversion matrix becomes singular and its determinant () r vanishes. 21 (0) max

22 Explicit expression:, 22

23 e=0.7 solutions exist only for a finite interval of f(0) if 0 : 0 f(0) f(0) max Charged Boson Stars Note: minimal A(0) on 0 plots non-zero, But very small. For 1, f(0) 0.691, A(0) 0.02 Solution seems close to RN with the scalar field almost confined to the interior. 23

24 Profiles for e 0.7, 1, f(0)=0.05, 0.1, 0.25, 0.5, 0.6, Solutions near maximal f(0) look very different from ones The 0 end-point solutions become close to RN as increases.

25 Mass, Particle Number and Binding Energy e=0.7; several values of M solid ; N dashed Much higher M and N for charged BS than the neutral. Much smaller binding energy per particle. 25

26 Charged Boson Stars - branch 2 Second branch exists for 0 and above a threshold. BS with higher masses and much higher B.E./N. End points are extremal RN solutions. 26

27 Mass and Binding Energy per Particle lower: branch 1 ; upper branch 2 27

28 Mass and Binding Energy per Particle lower: branch 1 ; upper branch 2 P1 P2 P 1 P 2 28

29 Profiles for f(0)=0.0963,0.15,0.2,0.26,0.2967, moving along the branch2 curve from P1 to P2. 29 The first and last are almost ERN identical outside the horizon.

30 30 The two branch structure

31 31 Summary and Conclusions

32 32 Summary and Conclusions

33 Thank you for your attention 33