The effect of Gravity on Equation of State Hyeong-Chan Kim (KNUT) FRP Workshop on String Theory and Cosmology 2015, Chungju, Korea, Nov , Newtonian gravity: In preparation, H.K., Gungwon Kang (KISTI), - General relativity: In Preparation, H.K., Chueng Ji (NCSU).

Motivations 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. Matters in a star ? 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) 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. General Covariance: Freely falling frame = locally flat EoS in freely falling frame = EoS in flat ST 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: Number of ptls (normalized): Heat Capacity:

Newtonian Stars and Equation of State A relation btw pressure and density is necessary. Spherically symmetric Newtonian star: EoS 1: (Ideal gas) This is not appropriate to integrate the EoS. An additional constraint: Adiabaticity With polytropic type EoS, one can integrate the EoM. EoS 2:

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: U N /Nk B T Gravitational potential energy: X

(X dep part of S N )/Nk B ✗ Ideal Gas in Constant Gravity Entropy: Ordering effect of gravity X

Heat Capacities: Heat capacity for constant gravity: Monatomic gas C V /Nk B X Heat capacity for constant temperature: G T /M X

EoS 1 of Ideal Gas in Constant Gravity However, local and the averaged values satisfy the ideal gas law. Distribution of particles is position dependent:

The new EoS 2 in the adiabatic case The energy is dependent on both of the temperature and gravity: First law: 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. Strong gravity (macroscopic system) limit: shows noticeable difference even in the non-relativistic, Newtonian regime: Therefore, for most astrophysical systems, the gravity effects on EoS can be ignored.

Then, when can we observe the gravity effect? A: Only when the macroscopic effects must be unavoidable. 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.) Macroscopic: size > kinetic energy/gravitational force System is being kept in thermal equilibrium compulsory. Near an event horizon where the gravity diverges.  Require general relativistic treatment. 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.

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: The total energy and entropy in Rindler frame: Ideal gas law is satisfied on the whole system if one define an average pressure for Rindler space.

Total energy and Entropy

Gravitational potential: Gravitational potential Energy:

Heat Capacities: Heat Capacities for constant volume, gravity and for constant volume, temperature:

Various Limits: Ultra-relativistic: Newtonian: Weak gravity: Strong gravity

Newtonian Results: Partition function for:

Weak gravity case: Ultra-relativistic case:

Strong gravity regime: Parameterize the distance from the event horizon as Area proportionality

Differentiating the definition of entropy: Thermodynamic first law From the functional form of the partition function: Combining the two, we get the first law: Gravitational potential energy

From the first law with dS=0, Equation of state for an adiabatic system From the definition of Heat capacities: Combining the two, we get: Fortunately, this is integrable: Universal f eature? 5 3

Newtonian gravity limit: Reproduce the Newtonian result. Strong gravity limit: Unruh temperature? At present, we cannot determine the dimensionless part.

LocallyMacroscopically Keptkept modified Conclusion There are some cases when the macroscopic effects can be observable. Strong gravity limit appears to determine the temperature of the system to be that of the Unruh temperature. Newtonian gravity: Rindler spacetime:

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 fn: Asymptotic forms: Total energy: Gravitational potential energy: Gravity part of the partition fn:

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: The system is unstable for X m X 1 <X< X M. The heat capacity for fixed T is negative for X< X M. Both decreases for 0<X< X M. The heat capacity for fixed T is positive for X M <X< X w. (implication?) Heat capacity X1X1 XwXw

Equation of State For small X: The correction term is order X 2 ~ For large X: where X should be determined from the relation, 2N k B T ≈ GM 2 /r *

LocallyMacroscopically Keptkept modified Conclusion 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.