1. Matters in a star ? Physics at the surface of a star in Eddington-inspired Born-Infeld gravity *Hyeong-Chan Kim Korea National University of Transportation 22 Nov. 2013 ; The 47th Workshop on Gravitation and Numerical Relativity

2. Brief survey on EiBI gravity

3. Eddington (1924): The equation of motion: Solution: Eddington gravity from Palatini formalism: This can be rewritten as GR after equating q uv in terms of g uv. Eddington’s action is an alternative starting point to GR. However, it is incomplete in that it does not include matter. Couplings the matter with the connection were further studied. It eventually was shown to be equivalent to the Palatini version of GR. (N.Poplawski,2009)

4. EiBI gravity (Banados, Ferreira; 2010) Born-Infeld Generalization: Inequivalent to GR: Small = GR limit; Large = Eddington limit. For vacuum, it is the same as GR. Inside matters, it deviates from GR. Non singular initial state for radiation filled universe. Eddington-inspired Born-Infeld gravity Palatini formulation of gravity is dependent on the connection only. denotes the determinant of the metric. is a dimensionless constant related with the cosmological constant. The matter field couples only with the metric.

5. Progress 1.Singularity free solutions for stars composed of dust, polytropic fluids ( Pani,Cardoso, Delsate, 2011 ) 2.Non singular initial state for perfect fluid with positive equation of state(EoS; w>0). de Sitter state for w=0. (PRD, Cho,K,Moon 2012) 3.Tensor perturbations ( Escamilla-Rivera,Banados,Ferreira, 2012 ) 4.Precursor of Inflation (PRL; arXiv:1305.2020; Cho, K, Moon) 5.A nongravitating scalar field (PRD; arXiv:1302.3341;Cho,K,2013) 6.Cosmological and astrophysical constraints are satisfied ( Felice, Gumjudpai,Jhingan;Avelino, 2012 ) 7.| |<3 x 10 5 m 5 s -2 /kg ( Casanellas,Pani,Lopes,Cardoso,2011) 8.Surface singularity for compact star, No additional DoF problem other than GR (PRL; Pani, Sotiriou, 2012) 9.Neutron star etc. (PRD; arXiv:1305.6770;Harko et.al.)

6. Reducing EOM Define auxiliary metric: Now, the variation of the action w.r.t. gives, where,. The metric compatibility gives, Now, the metric variation of action w.r.t. gives,

7. The surface singularity problem We are interested in the asymptotically flat space-time: 

8. Perfect fluid + spherically symmetrical star in EiBI gravity Perfect Fluid: Metric and Auxiliary metric for spherically symmetric star: = Rewriting the energy density:

9. Solving equation of motions, (Harko et.al. 2013) EoM1:: EoM2:: Mass function: TOV equation: 1 2 A main result: the stellar objects becomes more massive by 22%~26% depending on the equation of state. (Harko et.al. 2013)

10. The equation of state for polytropic fluid, Surface singularity of curvature Continuity eq: Equation of state: Integrating, Scalar curvature, 1 Possible divergent contributions comes from the discontinuity of the derivatives of analytic functions. Because,, most singular contributions come from The star surface is at

11. Surface singularity From,Differentiating once more,Near the surface of the star, we may use the surface value, and using the limit To get,

12. Mending the equation of state

13. Non-relativistic degenerated Fermi gas For an ideal fluid, the singularity is happening. Important example of this type: non-relativistic degenerate Fermi gas. Electron gas in metals and in white dwarfs, Neutron stars. Dense Fermi particles when the Fermi energy exceeds by far the temperature. (High density and low temperature; e.g. two electrons per unit phase space volume). Rough estimation: Fermi momentum: is proportional to the energy density. : Number density Pressure= total momentum transfer per unit area, unit time

14. Validity check for the nonrelativistic degenerate Fermi gas The approximation holds when the temperature is smaller than the Fermi energy. Therefore, for low number density it fails to hold. At the surface of the star, the energy density goes to zero. One reason to suspect the validity: Let’s check the geodesic deviation equation. Using, and taking Similar expression can be obtained for angular directions too. Hooke’s Law with frequency If the star is very cold? We need other reason.

15. Pressure by the geodesic deviation Following the interpretation of Hooke’s law, any two nearby geodesics will cross each other times irrespective of the distance. Oscillation of geometry! Oscillation is free from scale over all the surface of the star. The characteristic scale is nothing but the radius of the star ? After setting, and We get, The pressure = In the low density, and the surface singularity disappears because However, the curvature becomes too small so that term is subdominant. Therefore, there is no reason for the existence of Pressure due to the geodesic deviation: Awkward situation Surface singularity happens again!

16. Characteristic scale and the pressure by geodesic deviation To avoid this awkward situation, we need a delicate balance between the diverging curvature effect and the modified of the equation of state due to geodesic deviation. The curvature must be not too small and not too large. Since the curvature does not diverge, the characteristic scale must decrease! Balance between the gravity and the Fermi liquid  the ratio of their correlation scales must be a good measure.

17. Modified equation of state for the polytropic fluid We propose a modified equation of state near the surface: decreases with the mass of the particle and size of the star. increases with the mass of the star. goes to zero in the GR limit. For dominates the pressure when In that regions, the equation of state takes after that with, where no singularity exists at the star surface. Higher curvature Geodesic deviation Pressure increase to modify the equation of state Suppress curvature

18. For Modified curvature at the surface Calculate the curvature once more: The curvature scalar at the surface of the star becomes, This is an acceptable value. It does not contain.

19. A high spacetime curvature (geodesic deviation) may modify the effective equation of state of matters. Importance of the result The (usual) equation of state is defined in a flat spacetime and is ported to the curved spacetime in a locally inertial coordinates.  Therefore, geodesic deviation is not taken into account. Similar singularities will be modulated. e.g., 1) singularity happening phase transition [Sham.et.al.2012] 2) surface singularity in the Palatini f(R) gravity … Newtonian limit will be recovered because it goes as 1/R 2. The EiBI gravity can be saved from the flaw.

20. Further studies Two main obstacles of the theory: –Re-examination on surface singularity  Removed –The initial growth of tensor perturbation  Almost removed –Any observable effects (Most Urgent) ? –Neutron and quark stars, blackholes –Density perturbations [ arXiv:1307.2969,Yang,Du,Liu; 1311.3828, Lagos et.al. ] –Cosmological anisotropy? –Vector fields and other higher spin fields with EiBI? –Effects on standard model physics? –High energy regime physics? Now, physics remains:

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