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22 Nov. 2013 ;
The 47th Workshop on Gravitation and Numerical Relativity
Physics at the surface of a star in
Eddington-inspired Born-Infeld gravity
Matters in
a star
?
*Hyeong-Chan Kim
Korea National University of Transportation
Brief survey on EiBI gravity
Eddington gravity from Palatini formalism:
Eddington (1924):
The equation of motion:
Solution:
This can be rewritten as GR after equating quv in terms of guv.
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)
Born-Infeld Generalization: Inequivalent to GR:
EiBI gravity(Banados, Ferreira; 2010)
•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
.
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.
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.
4.
5.
6.
Tensor perturbations (Escamilla-Rivera,Banados,Ferreira, 2012)
Precursor of Inflation (PRL; arXiv:1305.2020; Cho, K, Moon)
A nongravitating scalar field (PRD; arXiv:1302.3341;Cho,K,2013)
Cosmological and astrophysical constraints are
satisfied (Felice, Gumjudpai,Jhingan;Avelino, 2012)
7. | |<3 x 105 m5s-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.)
Reducing EOM
Define auxiliary metric:
Now, the variation of the action w.r.t.
where,
.
The metric compatibility gives,
Now, the metric variation of action w.r.t.
gives,
gives,
The surface singularity problem
We are interested in the asymptotically flat space-time: l = 0.
Perfect fluid + spherically symmetrical star in EiBI gravity
Perfect Fluid:
Metric and Auxiliary metric for spherically symmetric star:
Rewriting the energy density:
=
Solving equation of motions, (Harko et.al. 2013)
EoM1::
1
EoM2::
A main result: the stellar objects becomes more
massive by 22%~26% depending on the equation
of state. (Harko et.al. 2013)
Mass function:
TOV equation:
2
The equation of state for polytropic fluid, Surface singularity of curvature
Continuity eq:
Equation of state:
Integrating,
The star surface
is at
Scalar curvature,
1
Possible divergent contributions comes from the discontinuity of
the derivatives of
analytic functions.
Because,
contributions come from
, most singular
Surface singularity
From ,
Differentiating once more,
Near the surface of the star, we may use the surface value,
and using the limit
To get,
Mending the equation of state
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,
Pressure= total momentum transfer per unit area, unit time
Neutron stars.
: Number density
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 .
Validity check for the nonrelativistic degenerate Fermi gas
One reason to suspect the validity:
• 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.
If the star is very cold? We need other reason.
Let’s check the geodesic deviation equation.
Using
, and taking
Similar expression can be obtained for angular directions too.
Hooke’s Law
with frequency
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!
Pressure due to the
geodesic deviation:
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
Surface singularity happens again!
Awkward situation
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.
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
,
Modified curvature at the surface
Calculate the curvature once more:
For
The curvature scalar at the surface of the star becomes,
This is an acceptable value.
It does not contain .
Importance of the result
• A high spacetime curvature (geodesic deviation) may
modify the effective equation of state of matters.
• 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/R2.
• The EiBI gravity can be saved from the flaw.
Further studies
Two main obstacles of the theory:
– Re-examination on surface singularity  Removed
– The initial growth of tensor perturbation  Almost removed
Now, physics remains:
–
–
–
–
–
–
–
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?
Thank you for Listening!