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The effect of Gravity on
Equation of State
Hyeong-Chan Kim
(KNUT)
FRP Workshop on String Theory and Cosmology
2015, Chungju, Korea,
Nov. 27-28, 2015
- Newtonian gravity:
In preparation, H.K., Gungwon Kang (KISTI),
- General relativity:
In Preparation, H.K., Chueng Ji (NCSU).
Motivations
Matters in a
star
Palatini-f(R) and Eddington-inspired
Born-Infeld gravity are plagued by the
surface singularity problem:(Sotiriou, Faraoni, 2010;
Olomo, 2011).
Singularity disappears if one allow extremely strong
gravity modifies the matter EoS (H.K., 2014)
 Save the theory.
Q: It appears genuine that strong gravity affects on EoS.
Are there similar effects in GR and Newtonian?
?
Equation of State in General Relativity (Ordinary Wisdom)
General Covariance:
Freely falling frame = locally flat
EoS in freely falling frame = EoS in flat ST
Scalar quantity
Density, pressure, temperature are scalar quantities.
Therefore, their values in other frames must be the same as
those in the freely falling frame.
Extremely strong curvature?
A statistical system has a size.
When curvature or gravity is extremely strong, is it possible to set
up a locally flat coordinates for a given system?
 We need to check it.
Statistics (Summary)
Basic principle:
The number of particles in unit phase volume is proportional to
Partition function:
Total energy and entropy:
Heat Capacity:
Number of ptls (normalized):
Newtonian Stars and Equation of State
Spherically symmetric Newtonian star:
A relation btw pressure and density is necessary.
This is not appropriate to integrate the EoS.
(Ideal gas)
EoS 1:
An additional constraint: Adiabaticity
EoS 2:
With polytropic type EoS, one can integrate the EoM.
Generalization: Ideal Gas in Constant Gravity
N-particle system in a box:
One particle Hamitonian:
One particle partition function:
Landsberg, et. al. (1994).
Order parameter for gravity:
Ratio btw the grav. Potential energy to the thermal kinetic energy.
Ideal Gas in Constant Gravity
Internal energy and entropy:
UN /NkBT
X
Gravitational potential energy:
Ideal Gas in Constant Gravity
✗
Entropy:
(X dep part of SN)/NkB
Ordering effect of
gravity
X
Heat Capacities:
Heat capacity for constant gravity:
2.4
2.2
CV/NkB
2.0
Monatomic gas
1.8
1.6
0
1
2
3
4
X
Heat capacity for constant temperature:
X
GT/M
EoS 1 of Ideal Gas in Constant Gravity
Distribution of particles is position dependent:
However, local and the averaged values satisfy the ideal gas law.
The new EoS 2 in the adiabatic case
First law:
The energy is dependent on both of the temperature and gravity:
Adiabaticity:
We get,
which can be integrated to give a new EoS,
X
contains thermodynamic variables.
This gives difference from the old EoS.
The new EoS 2: Limiting behaviors
Weak gravity limit:
The correction is second order.
Therefore, one can ignore this correction in the small size limit
of the system.
Therefore, for most astrophysical systems, the
gravity effects on EoS can be ignored.
Strong gravity (macroscopic system) limit:
shows noticeable difference even in the non-relativistic,
Newtonian regime:
Then, when can we observe the gravity effect?
A: Only when the macroscopic effects must be unavoidable.
Macroscopic: size > kinetic energy/gravitational force
System is being kept in thermal equilibrium compulsory.
The (self) gravity (or curvature) increases equally or faster than the
inverse of system size. (e.g., Palatini f(R) gravity near the star surface.
This is impossible in GR.)
The size of the system is forced to be macroscopic.
Ex) The de Broglie wavelength of the particles is very large
(light, slowly moving particles e.g., the scalar dark matter).
 Require quantum mechanical treatment.
Near an event horizon where the gravity diverges.
 Require general relativistic treatment.
Relativistic Case
Generalization: Ideal Gas in Constant Gravity
N-particle system in a box (Rindler spacetime):
One particle Hamiltonian:
N-particle Partition function:
Rindler horizon
Continuity Equation:
The Continuity Equation:
The number density and momentum:
Ideal gas law is satisfied locally
(not globally) by local temperature.
Total energy and Entropy:
Define pressure in Rindler space:
Ideal gas law is satisfied on the whole system if one
define an average pressure for Rindler space.
The total energy and entropy in Rindler frame:
Total energy and Entropy
Gravitational potential:
Gravitational potential Energy:
Heat Capacities:
Heat Capacities for constant volume, gravity and
for constant volume, temperature:
Various Limits:
Newtonian:
Strong gravity
Weak gravity:
Ultra-relativistic:
Newtonian Results:
Partition function for:
Weak gravity case:
Ultra-relativistic case:
Strong gravity regime:
Parameterize the distance from the event horizon as
Area
proportionality
Thermodynamic first law
Differentiating the definition of entropy:
From the functional form of the partition function:
Combining the two, we get the first law:
Gravitational
potential energy
Equation of state for an adiabatic system
From the first law with dS=0,
From the definition of Heat capacities:
Combining the two, we get:
Fortunately, this is integrable:
5
Universal f
eature?
3
Newtonian gravity limit:
Reproduce the Newtonian result.
Strong gravity limit:
Unruh temperature?
At present, we cannot determine the dimensionless part.
Conclusion
Newtonian gravity:
Locally
Macroscopically
Kept
kept
kept
modified
There are some cases when the macroscopic effects can be observable.
Rindler spacetime:
Strong gravity limit appears to determine the temperature of the system
to be that of the Unruh temperature.
Future plan
1) System around a blackhole horizon
2) Quantum mechanical effect?
3) Self gravitating system?
4) Relation to blackhole thermodynamics?
5) Dynamical system?
5) Etc…
Thanks, All Participants.
Self-gravitating sphere
Self-gravitating sphere in thermal equilibrium
1.
Self-gravitating ideal gas in a box of radius
2.
Assume that the whole system is at the same temperature.
(isothermal system in the presence of gravity)
Strategy
1.
Do statistics as if the gravitational potential is given.
 relation between the density and pressure = EoS
2.
Solve gravitational EoM:
 get the gravitational potential.
3.
Insert the obtained potential back to the statistics:
 get physical quantities.
Self-gravitating sphere in thermal equilibrium
Statistics with a given the potential
Density:
Pressure:
Self-gravitating sphere in thermal equilibrium
Determine the potential:
Re-inserting the potential to the partition function,
Gravity
part
of the
partition
Gravity
part
of the
partition
fn: fn:
Asymptotic forms:
Total energy:
Gravitational potential energy:
Entropy
Entropy:
The entropy increases monotonically. (No negative entropy problem)
Constraint from self-gravitating condition
X is not an independent parameter but dependent on the temperature and size.
X is uniquely determined only when
No static spherically symmetric
(stable or not) configuration exist when the system is too dense:
Therefore, low temperature, extremely dense stars do not exist in this theory.
Heat capacity
Heat capacity:
The system is unstable for
X1<X< XM.
The heat capacity for fixed T is
negative for X< XM.
Xm
X1
Both decreases for 0<X< XM.
The heat capacity for fixed T is positive for XM<X< Xw. (implication?)
Xw
Equation of State
For small X:
The correction term is order X2 ~
For large X:
where X should be determined from the relation,
2N kB T ≈ GM2/r*
Conclusion
Locally
Macroscopically
Kept
kept
kept
modified
However, there are some cases when the macroscopic effects can be
observable.
As an example, we deal a self-gravitating sphere in thermal equilibrium.
Thanks, All Participants.