Gravitational Effect on (Relativistic) Equation of State

Transcription

1 Gravitational Effect on (Relativistic) Equation of State Hyeong-Chan Kim (KNUT) 51th workshop on Gravity and Numerical Relativity 2015, Seoul, Korea, Nov , Newtonian gravity: In preparation, H.K., Gungwon Kang (KISTI), - General relativity: In Preparation, H.K., Chueng Ji (NCSU).

2 Motivations Palatini-f(R) and Eddington-inspired Matters in a star? Born-Infeld gravity are plagued by the surface singularity problem:(sotiriou, Faraoni, 2010; Olomo, 2011). Singularity disappears if one allow extremely stronggravity 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?

3 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.

4 Basic principle: Statistics (Summary) The number of particles in unit phase volume is proportional to Partition function: Total energy and entropy: Heat Capacity: Number of ptls (normalized):

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

6 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 ener gy.

7 Ideal Gas in Constant Gravity Internal energy and entropy: U N /Nk B T X Gravitational potential energy:

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

9 Heat Capacities: Heat capacity for constant gravity: C V /Nk B 2.0 Monatomic gas X Heat capacity for constant temperature: G T /M X

10 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.

11 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.

12 Weak gravity limit: The new EoS 2: Limiting behaviors 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:

13 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.

14 Relativistic Case

15 Generalization: Ideal Gas in Constant Gravity N-particle system in a box (Rindler spacetime): One particle Hamiltonian: Rindler horizon N-particle Partition function:

16 Ideal gas law is satisfied locally (not globally) by local temperature. Continuity Equation: The Continuity Equation: The number density and momentum:

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

18 Total energy and Entropy

19 Gravitational potential: Gravitational potential Energy:

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

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

22 Newtonian Results: Partition function for:

23 Weak gravity case: Ultra-relativistic case:

24 Strong gravity regime: Parameterize the distance from the event horizon as Area proportionalit y

25 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

26 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 feature? 3

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

28 Conclusion Newtonian gravity: Locally Kept kept Macroscopically 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.

29 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

30 Thanks, All Participants.

31 Self-gravitating sphere

32 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.

33 Self-gravitating sphere in thermal equilibrium Statistics with a given the potential Density: Pressure:

34 Self-gravitating sphere in thermal equilibrium Determine the potential:

35 Re-inserting the potential to the partition function, Gravity part of of the partition fn: fn: Asymptotic forms: Total energy: Gravitational potential energy:

36 Entropy Entropy: The entropy increases monotonically. (No negative entropy problem)

37 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.

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

39 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 *

40 Conclusion Locally Kept kept Macroscopically 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.

41 Thanks, All Participants.