# Effect of Generalized Uncertainty Principle on Main-Sequence Stars and White Dwarfs

Authors: Mohamed Moussa
Journal: Advances in High Energy Physics
 (2015)

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## Abstract

This paper addresses the effect of generalized uncertainty principle, emerged from different approaches of quantum gravity within Planck scale, on thermodynamic properties of photon, nonrelativistic ideal gases, and degenerate fermions. A modification in pressure, particle number, and energy density are calculated. Astrophysical objects such as main-sequence stars and white dwarfs are examined and discussed as an application. A modification in Lane-Emden equation due to a change in a polytropic relation caused by the presence of quantum gravity is investigated. The applicable range of quantum gravity parameters is estimated. The bounds in the perturbed parameters are relatively large but they may be considered reasonable values in the astrophysical regime.


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## Body

## 1. Introduction

Different approaches for the quantum gravity have been proposed in string theory and black hole physics to provide a set of predictions for a minimum measurable length and an essential modification of the Heisenberg uncertainty principle (GUP) [1–11] and a modification in the fundamental commutation relation [xi,pj]. According to string theory, it has been found that strings cannot interact at distances smaller than their size, which yields generalized uncertainty principle [1]. From black hole physics [2, 3], the uncertainty principle, Δx~ħ/Δp, is modified at the Planck energy scale when the corresponding Schwarzschild radius is approximately equal to Compton wavelength at the Planck scale. Higher energies result in a further increase of the Schwarzschild radius, to yield Δx≈lPl2Δp/ħ. The above approaches along with a combination of thought experiments and rigorous derivations suggest that the (GUP) holds at all scales and is represented by the following [1–11]:(1)ΔxiΔpi≥ħ21+βΔp2+p2+2βΔpi2+pi2,where p2=∑jpjpj, β=β0/(Mpc)2=β0(lp2/ħ2), Mp is Planck mass, and Mpc2 is Planck energy. It was shown in [9] that inequality (1) is equivalent to the following modified Heisenberg algebra:(2)xi,pj=iħδij+βδijp2+2βpipj.This form ensures, via the Jacobi identity, that [xi,xj]=0=[pi,pj] [10]. Different minimal length scale scenarios inspired by various approaches to the quantum gravity have been reviewed in [12]. Apparently, this suggests a modification of the physical momentum [9–11](3)pi=p0i1+βp02,while xi=x0i with x0i; p0j satisfies the canonical commutation relations [x0i,p0j]=iħδij, such that p0i=-iħ∂/∂x0i, where p is the momentum satisfying (2). The upper bounds on the parameter β have been derived in [13, 14] and it was found that it could predict an intermediate length scale between Planck scale and electroweak scale. It was suggested that these bounds can be measured using quantum optics techniques and gravitational wave techniques in [15, 16] which is considered as milestone in the quantum gravity phenomenology. It is noteworthy that the thermodynamical phenomenological implications of GUP in quantum gases are numerous, for example, [17, 18]. In [19] the quantum gravity influences on the statistic properties classical nonrelativistic, ultrarelativistic, and photon gases were addressed. Small corrections to the energy and entropy at low temperatures were found and some modifications to the equations of state are employed. The authors considered two configurations. The first is the white dwarf which is composed of a nonrelativistic cold nuclei; the mass of the star was calculated. It is found that there is a quantum gravity correction that depends on the number density of the star that leads to an increase in the mass of the star and the quantum gravity tends to resist the collapse of the star. The other configuration is that the star is almost composed of ultrarelativistic particles. They found that the increase in the star energy leads to monotonical blowup of Fermi pressure which resists the gravitational collapse.We will use this form of GUP (1) to investigate the implications of the quantum gravity on statistical properties of quantum gases. After redefining the new phase space we will put the partition function in a form that is consistent with GUP and then rewrite the thermodynamic properties for quantum gases, namely, photons, nonrelativistic ideal gases, and fermions in degenerate state. Using these results we will follow the proposal which was investigated in [20] in studying the stability of main-sequence stars and white dwarf. Also a simple method is used to study the effect of quantum gravity in a different stage of stellar evolution and a modified mass-radius relation for the white dwarf is calculated and boundaries in the quantum gravity perturbative terms will be determined.


## 2. Statistical Mechanics and GUP

In this section we investigate the GUP modification in the formalism of the grand-canonical ensemble. If we considerN particles with energy states Ei, for each state there is ni particles in that state. The grand-canonical partition function is obtained from the sum over all of the states:(4)Z=∑ni∏ieμ-Ei/KBTni,where T is the temperature, KB is the Boltzmann constant, and μ is the chemical potential. The connection to thermodynamics is obtained after introducing the partition function defined as [21](5)ln⁡⁡Z=∑iln⁡⁡1+aze-Ei/KBT,where z=exp⁡⁡(μ/KBT) and the parameter a takes value depending on the considered particles. If they are fermions, a takes +1 value, and if the considered particles are bosons, a takes -1 value.By considering large volume, the summation turns to be∑i‍→∫d3xd3p/((2πħ)31+βp23) where we are considering the invariant phase space in the existence of the GUP that has been derived in [22]. We should put the grand canonical partition function for these systems in a form consistent with GUP framework. The GUP can be considered in the phase space analysis by two equivalent pictures: (a) considering deformed commutation relations (i.e., deformed the measure of integration) simultaneously with the nondeformed Hamiltonian function or (b) calculating canonical variables on the GUP-corrected phase space which satisfy the standard commutative algebra (i.e., nondeformed standard measure of integration), but the Hamiltonian function now gets deformed. These two pictures are related to each other by the Darboux theorem in which it is quite possible to find canonical coordinates on the symplectic manifold which satisfy standard Heisenberg algebra [23, 24]. In this paper we are going to consider deformed measure of integration with nondeformed Hamiltonian function. Based on the new phase space the partition function is(6)ln⁡⁡Z=V2πħ3ga∫ln⁡⁡1+aze-E/kBTd3p1+βp23with E=p2c2+m2c41/2. Pressure, number of particles, and internal energy will be determined by the relations(7)P=KBT∂∂Vln⁡⁡Z,n=KBT∂∂μln⁡⁡ZT,V,U=KBT2∂∂Tln⁡⁡Zz,V.We will expand all the terms that containβ and we will keep the terms that are proportional to ~β only. This approximation breaks down around maximum measurable energies such that βp2≈1. In this case we need an exact solution. But one can trust the perturbative solution where βp2≪1. Now we will use the modified partition function to calculate the thermodynamics of photons, nonrelativistic ideal gases, and degenerate fermions.


### 2.1. Photon Gas

For a gas of photonsg=2, a=-1, and μ=0. The energy density is given, using (7), by(8)u=UV=1π2ħ3∫EeE/KBT-1p2dp1+βp23,where E=pc. Use the approximation 1/(1+βp2)3≃1-3βp2; one gets(9)u=cπ2ħ3∫dpecp/KBT-1p3-3βp5=4σcT4-β821π4KB6ħ3c5T6,where σ=π2k4/60ħ3c2 is the Stefan-Boltzmann constant. For β=0 one can recover the usual Stefan-Boltzmann law.And the pressure of system (7) is given by(10)P=cπ2ħ3∫dpecp/KT-113p3-35βp5=4σ3cT4-β8105π4KB6ħ3c5T6.It is clear that the modification in phase space decreases the available number of microstates, which holds the thermodynamic properties and hence decreases the internal energy and pressure of the system. In order to obtain the pressure-energy density relation divide (10) by (9):(11)Pu=13+β1621KBπc2T2.If we remove the quantum gravity effect we can recover the usual relationu=3P. In order to obtain the equation of state (EOS) P=P(u) we first solve (9) for the temperature to get(12)T=c4σ1/4u1/4+β107πKBc2c4σ3/4u3/4.Substitute (12) into (11) to get the first order correction to the EOS:(13)P=u3+β162115π2ħ3c1/2u3/2.Equation (13) represents a modified equation of state due to generalized uncertainty principle.


### 2.2.  Nonrelativistic Ideal Gases

Let us now consider the effect of quantum gravity on nonrelativistic ideal gases. In order to do that the partition function for one particle with the modified phase space is given by(14)z1=V2π2ħ3∫e-p2/2mKBTp2dp1+βp23.The total partition function is given byZ=(1/N!)z1N by which we can derive the free energy for the system with the equation F=KBTln⁡⁡Z. Using Stirling’s formula ln⁡⁡N!=Nln⁡⁡N-N we can determine the pressure(15)P=-∂F∂VT,N=NKBTV.So we recover the usual EOS for nonrelativistic ideal gas P=nKBT, where n=N/V is the number density which shows that the quantum gravity does not play here. The same behavior is established in [23]. Suffice calculation of the pressure at this point, which we need, and the other thermodynamic properties of the nonrelativistic ideal gases can be obtained easily by solving the integration in (14) and then calculating the free energy.


### 2.3. Degenerate Fermion Gas

Fermions have a property that at most one can occupy each quantum state. AtT=0 all states with energy less than Fermi energy level are occupied and all states with greater energy are empty. So the total energy of the system is the lowest possible consistent with the exclusion principle. The energy of the highest occupied state at absolute zero is the Fermi energy and corresponding momentum is referred to as the Fermi momentum pf. In this case the distribution function goes to step function. According to these assumptions and using (7) the particle number density is given by(16)n=1π2ħ3∫0Pfp2dp1+βp23=1π2ħ313pf3-β35pf5.Solving this equation for Fermi momentum, one gets(17)pf=3π2ħ3n1/3+β353π2ħ3n.The second term ensures that the Fermi momentum is increased due to the presence of quantum gravity and hence Fermi energy. Also the energy density for fermions in degenerate state is given by(18)u=1π2ħ3∫0Pfp2c2+m2c41/2p2dp1+βp23.Using the substitutional(19)p=mcsinh⁡⁡xthe energy density integration will be(20)u=m4c5π2ħ3∫0xfsinh⁡2⁡x+1-3βm2c2sinh⁡4⁡x-3βm2c2sinh⁡6⁡xdx=m4c5π2ħ3F1y+1-3βm2c2F2y-3βm2c2F3y,where(21)y=pfmc=sinh⁡⁡xfwith(22)F1y=-12ln⁡⁡y+1+y2+12y1+y2F2y=38ln⁡⁡y+1+y2+14y1+y2y2-32.F3y=-516ln⁡⁡y+1+y2+y1+y2516+y2y26-524By the same way, we can derive the pressure from (7); after making use of the integration by parts and function expansion, one finds that(23)P=c2π2ħ3∫0pfdpp2c2+m2c41/213p3-35βp5=m4c5π2ħ313F2y-35βm2c2F3y.Equation (23) represents the ground state pressure. There is a considerable change due to the presence of the function F3(y) and y itself depends on modified Fermi momentum. We should notice that the value of y is increased due to the increasing in Fermi momentum, this leads to a considerable growing in the pressure (23). The change in pressure of the radiation and degenerate fermions will affect the polytropic relations for astroobjects. In the next section the modified pressure effect in the some stars stability and some other applications will be investigated exactly.


## 2.1. Photon Gas

For a gas of photonsg=2, a=-1, and μ=0. The energy density is given, using (7), by(8)u=UV=1π2ħ3∫EeE/KBT-1p2dp1+βp23,where E=pc. Use the approximation 1/(1+βp2)3≃1-3βp2; one gets(9)u=cπ2ħ3∫dpecp/KBT-1p3-3βp5=4σcT4-β821π4KB6ħ3c5T6,where σ=π2k4/60ħ3c2 is the Stefan-Boltzmann constant. For β=0 one can recover the usual Stefan-Boltzmann law.And the pressure of system (7) is given by(10)P=cπ2ħ3∫dpecp/KT-113p3-35βp5=4σ3cT4-β8105π4KB6ħ3c5T6.It is clear that the modification in phase space decreases the available number of microstates, which holds the thermodynamic properties and hence decreases the internal energy and pressure of the system. In order to obtain the pressure-energy density relation divide (10) by (9):(11)Pu=13+β1621KBπc2T2.If we remove the quantum gravity effect we can recover the usual relationu=3P. In order to obtain the equation of state (EOS) P=P(u) we first solve (9) for the temperature to get(12)T=c4σ1/4u1/4+β107πKBc2c4σ3/4u3/4.Substitute (12) into (11) to get the first order correction to the EOS:(13)P=u3+β162115π2ħ3c1/2u3/2.Equation (13) represents a modified equation of state due to generalized uncertainty principle.


## 2.2.  Nonrelativistic Ideal Gases

Let us now consider the effect of quantum gravity on nonrelativistic ideal gases. In order to do that the partition function for one particle with the modified phase space is given by(14)z1=V2π2ħ3∫e-p2/2mKBTp2dp1+βp23.The total partition function is given byZ=(1/N!)z1N by which we can derive the free energy for the system with the equation F=KBTln⁡⁡Z. Using Stirling’s formula ln⁡⁡N!=Nln⁡⁡N-N we can determine the pressure(15)P=-∂F∂VT,N=NKBTV.So we recover the usual EOS for nonrelativistic ideal gas P=nKBT, where n=N/V is the number density which shows that the quantum gravity does not play here. The same behavior is established in [23]. Suffice calculation of the pressure at this point, which we need, and the other thermodynamic properties of the nonrelativistic ideal gases can be obtained easily by solving the integration in (14) and then calculating the free energy.


## 2.3. Degenerate Fermion Gas

Fermions have a property that at most one can occupy each quantum state. AtT=0 all states with energy less than Fermi energy level are occupied and all states with greater energy are empty. So the total energy of the system is the lowest possible consistent with the exclusion principle. The energy of the highest occupied state at absolute zero is the Fermi energy and corresponding momentum is referred to as the Fermi momentum pf. In this case the distribution function goes to step function. According to these assumptions and using (7) the particle number density is given by(16)n=1π2ħ3∫0Pfp2dp1+βp23=1π2ħ313pf3-β35pf5.Solving this equation for Fermi momentum, one gets(17)pf=3π2ħ3n1/3+β353π2ħ3n.The second term ensures that the Fermi momentum is increased due to the presence of quantum gravity and hence Fermi energy. Also the energy density for fermions in degenerate state is given by(18)u=1π2ħ3∫0Pfp2c2+m2c41/2p2dp1+βp23.Using the substitutional(19)p=mcsinh⁡⁡xthe energy density integration will be(20)u=m4c5π2ħ3∫0xfsinh⁡2⁡x+1-3βm2c2sinh⁡4⁡x-3βm2c2sinh⁡6⁡xdx=m4c5π2ħ3F1y+1-3βm2c2F2y-3βm2c2F3y,where(21)y=pfmc=sinh⁡⁡xfwith(22)F1y=-12ln⁡⁡y+1+y2+12y1+y2F2y=38ln⁡⁡y+1+y2+14y1+y2y2-32.F3y=-516ln⁡⁡y+1+y2+y1+y2516+y2y26-524By the same way, we can derive the pressure from (7); after making use of the integration by parts and function expansion, one finds that(23)P=c2π2ħ3∫0pfdpp2c2+m2c41/213p3-35βp5=m4c5π2ħ313F2y-35βm2c2F3y.Equation (23) represents the ground state pressure. There is a considerable change due to the presence of the function F3(y) and y itself depends on modified Fermi momentum. We should notice that the value of y is increased due to the increasing in Fermi momentum, this leads to a considerable growing in the pressure (23). The change in pressure of the radiation and degenerate fermions will affect the polytropic relations for astroobjects. In the next section the modified pressure effect in the some stars stability and some other applications will be investigated exactly.


## 3. Applications to Astrophysical Objects

### 3.1. Main-Sequence Stars

Most main-sequence stars have a mass in the range about10 to 50 solar mass. They are composed by a mixture of nonrelativistic gas and radiation that are stood in hydrostatic equilibrium by gravity [25]. The total pressure is Pc=Pg+Pr, where Pc, Pg, and Pr are center star, gas, and radiation pressure, respectively. We can write βc=Pg/Pc and 1-βc=Pr/Pc. By definition (1-βc) and βc are the fractional contribution of radiation and gas to the central pressure of the star and both are less than one; then(24)Pgβc=Pr1-βc.The classical gas pressure can be expressed, using (15), in the form(25)Pg=ρcKBTm¯.We used the relationn=ρc/m¯; ρc is the central density of the star and m¯ is the average mass of the gas particle that constructs the star defined as m¯=uNmN where uN is the mean molecular weight and mN is the nucleon mass. Using (10), (25) into (24), one finds that(26)ρcKBβcm¯Tc=4σ3c1-βcTc4-8105βπ4KB6ħ3c51-βcTc6,where Tc is the central temperature of the star. We can find the temperature by solving the above equation:(27)Tc=1-βcβcm¯3cKB4σ1/3ρc1/3+β1701-βcβcm¯π4KB7ħ3c3σ2ρc.Using this expression to calculate the star pressure from the relation Pc=Pg/βc=(ρcKB/βcm¯)Tc one gets(28)Pc=K1ρc4/3+βK2ρc2=PsUP+psGUP,(29)K1=1-βc1/3βc4/33cKB44σm¯41/3,K2=1701-βcβc2π4KB8ħ3c3σ2m¯2.The first part of (28) is the ordinary polytropic relation with a a polytropic exponent Γ=4/3 (polytropic index np=3) [26] perturbed by a polytrope Γ=2 (polytropic index np=1) where np=1/(Γ-1).The best example may be used to estimate GUP perturbation is our sun. If we assume thatβc is constant overall the stars and its chemical composition is not changed so uN is constant. For our sun 1-βc=10-3 and uN=0.829 [27] and central density ρc=1.53×105 kg/m3; then(30)psGUPPsUP=1.8×10-49β0.The perturbed pressure should not exceed the original one such thatpsGUP/PsUP≪1, so this equation suggests that the upper value of β0 should be β0<1048. This bound is far weaker than that set by electroweak measurements.According to (28) the quantum gravity modifies the polytropic relation by adding a new term proportional to ρ2. This suggests a new modification in Lane-Emden equation [28]. The modified Lane-Emden equation is established in the appendix. Studying modified Lane-Emden equation and its solution is not the scope of this paper so results that are investigated by the this equation will be postponed to another research.Main-Sequence Stars Stability. In order to discuss the stability concept, the energy of a Newtonian polytrope systems with pressure P can be expressed in terms of the internal energy and gravitational potential such that(31)E=k1PV-k2GM2R,where k1 and k2 are constants, G is the gravitational constant, and M and R are the mass and radius of the star. At hight densities it is important to take into account the general relativistic effects in star stability. To deal with it a general relativistic correction term should be added to (31), in lowest order approximation; namely, Ecorr=-k3G/c2M7/3ρc2/3, where k3 is constant that depends on the actual distribution of matter. According to [27], for nP=3 polytrope, the condition of stability is Γ>4/3+2.25(GM/c2R). We can see that addition of a relativistic correction contracts the region of stability or increases the critical value of Γ. In our case GM/c2R~10-6 so this correction can be neglected. So we expect that the new term in a polytropic equation (28) with np=1, which has a positive contribution, will not affect the problem of stability since Γ>2.Using (28) into (31) the energy of main-sequence stars will be(32)E=C1Mρc1/3+βC2Mρc-k2M5/3ρc1/3,where C1 and C2 are constants. The mass M can be obtained by using the equilibrium condition ∂E/∂ρc=0; the result is(33)M=3k313C1+βC2ρc2/33/2.Then(34)dln⁡⁡Mdln⁡⁡ρc=β3C2C1ρc2/3.Equation (34) ensures that dln⁡⁡M/dln⁡⁡ρc>0 which is the stability condition [27] for main-sequence object. These results show that the generalization process for the uncertainty principle does not change the stability of the main-sequence objects.Minimum and Maximum Masses for Main-Sequence Stars. In this part we will discuss the evolution stage of main-sequence stars in the presence of quantum gravity to derive the minimum and maximum masses of these objects. Initially, in the stage of star forming, the star must has mass enough to generate a central temperature which is high enough for thermonuclear fusion. At this stage the central pressure comes by the electrons and ions which are forming an ideal classical gas; their pressure can be expressed as in (25). The star will be made if this pressure is close to the pressure needed to support the system. It is known that the upper limit relation between the pressure and density at the center of the star is given by [29](35)Pc≤π61/3Gρc4/3M2/3.This relation is valid for any homogeneous star in which the mass is concentrated towards the center. Equating (25) and (35) one can get the initial central temperature in this stage:(36)KBTc≤π61/3m¯Gρc1/3M2/3.We can see that the quantum gravity does not appear in the picture. As shown in this equation the central temperature will rise as the central density increases with star contraction. The contracting temperature continues to rise until the electrons at the center become fully degenerate and occupy the lowest energy states in accordance with exclusion principle. At this stage, at the center, the pressure comes from the electrons that are in degenerate state plus the pressure from the classical ions. Quantum gravity will not change the classical ions pressure, according to (15), but a nonrelativistic degenerate electrons pressure will change according to (23). In nonrelativistic limit, where y≪1, the degenerate pressure (23) will be(37)P0non-relat=115me4c4π2ħ3y5-335me6c7π2ħ3y7.Use (17), (21) in above equation and add the result to (15); finally we can write the central pressure at this stage as(38)Pc=a5mene5/3+β12a235mene7/3+niKBTc,where a=(31/3π2/3ħ)2. The number density of the electrons ne and ions ni can be expressed in terms of central density, simply ne=ni=ρc/mH, where mH is the mass of hydrogen atom. A hydrostatic pressure is achieved if this pressure equals the pressure needed to support the mass (35). This leads to the central temperature being as follows:(39)KBTc≤π61/3GmNM2/3ρc1/3-a5meρcmH2/3-β12a235meρcmH4/3.The first term is associated with classical ions and second and third terms with degenerate electrons. The last two terms become important and are increasing with central density until the temperature will cease to rise as the mass contracts. From this equation we can find the maximum value of temperature:(40)KBTcmax⁡≤zM4/31-607βmezM4/3,(41)z=54π62/3meamH8/3G2.Equation (40) proves that the maximum central temperature decreases due to quantum gravity. The contracting mass achieves stardom if that maximum central temperature reaches ignition temperature for the thermonuclear fusion of hydrogen. Denote this ignition temperature by Tign; then the minimum mass for the star will be given by(42)Mmin⁡≥KBTignz3/41+457βmeKBTign.This relation shows that the quantum gravity effect increases the minimum mass of the main-sequence stars. The approximation has a meaning if the second term in (42) is less than unity. If we take the ignition temperature for hydrogen to be about 1.5×106 K, this suggests that the upper value of β0<1047.The situation is changed if the radiation pressure takes a part in this stage of revolution. The star will be disrupted if radiation becomes the dominant source for the internal pressure. This property will set a limit to the mass of a main-sequence star. The pressure at the center of hot massive stars is due to electrons and ions (as a classical gases) and radiation; the pressure that is produced by these species is derived in (28). Comparing this with the pressure needed to support the star (35) to get the maximum mass, one finds that(43)π61/3GMmax⁡2/3≥K1+βK2ρc2/3,which means that the maximum mass is increased due to the presence of quantum gravity. This increase depends on the central density. Again the second term in the right-hand side of inequality (43) should be less than the first; this suggests that β0<1048.


### 3.2. White Dwarfs

The white dwarfs consist of Helium atoms that are almost in an ionization state. The temperature of the star is about107 K, so the electrons are in a relativistic energy regime and can be considered in a complete degenerate state and the Helium nuclei do not play a part, in first order approximation, in the dynamic of the system. Now we will determine the effect of GUP on white dwarf radius according to pressure modification expressed in (23). If the number of Helium nuclei is N/2 then there are N electrons. Suppose that M=2Nmp is the mass of the star, so the particle density of the electron is(44)ns=3M8πmp1R3.Using (17), (44) into (21), one gets(45)y=M¯1/3R¯+35βme2c2M¯R¯3,(46)M¯=9π8Mmp,R¯=mecħR,where R is the radius of the star and mp is the proton mass. The equilibrium configuration of white dwarf is caused by the pressure of the degenerate electrons against Newtonian gravitational collapse which is due to the nuclei, which means that(47)P0=κ4πGM2R4=K′M¯2R¯4,K′=κG4π8mp9π2mecħ4.We have two extreme cases as follows.(i) When the electron gas is in a low density, nonrelativistic dynamics may be used such thaty≪1, and (23) will lead to(48)P0=45Ky5-3635βme2c2Ky7,K=mec212π2mecħ3.Equating (47) and (48), we got the equation(49)R¯3-45KK′M¯-1/3R¯2-4835βme2c2K′KM¯1/3=0.The solution of that equation is(50)R¯≃45KK′M¯-1/3+157βme2c2K′KM¯.This equation shows that the radius of the star is increasing; the elevation depends on the mass of the star. The used approximation is true if the second term in the right-hand side of (50) is much less than the first one; this will put a limit in the upper value of β0 such that it should be β0<1044; we used a white dwarf mass M=0.41M⊙ [30].(ii) When the electron gas is in a high density, that relativistic effect comes to play strongly such thaty≫1. We can write(51)P0=Ky4-y2-β185Kme2c2y63-y44.Using (51) into (47) and simplifying, then(52)R¯4+K′KM¯4/3-1-310βme2c2M¯2/3R¯2-65βme2c2M¯4/3=0.The solution of that equation, keeping only the terms that proportional to ~β, is(53)R¯≃R¯01+β310me2c2R¯022M¯4/3R¯02-12M¯2/3,where(54)R¯02=M¯2/3-K′KM¯4/3.Using (46) into (54) and solving for R and letting R→RCh, one gets(55)RCh=9π1/32ħmecMmp1/31-MMCh2/31/2,MCh=93π1/264κħcGmp23/2mp.R
Ch is the unmodified white dwarf radius or Chandrasekhar radius limit and MCh is the Chandrasekhar limit mass, of order 1.44M⊙. First part of (53) is the Chandrasekhar radius and the second term is the radius correction due to GUP affect. These analyses show that the increasing in star radius depends on 1-M/MCh2/3-1 which formally goes to infinity as M→MCh. It is clear that the quantum gravity effect becomes significant when mass of the white dwarf gets very close to the Chandrasekhar limit in high density stars. Equations (23), (50), and (53) attract attention to a noticeable point that the growing in star radius ensures that quantum gravity resists gravitational collapse by increasing the electrons degeneracy pressure. These results are consistent with that established in [19]. Our approximation will have a meaning if the second term in the right-hand side in (53) is less than one; this puts an upper limit for β0 such that β0<1043. We have used M=0.59M⊙ [30].We should also notice that, in the limity≫1, in ultrarelativistic regime, the function F3(y) behaves like ~y6 which almost goes like ρ2. This result is consistent with polytrope behavior in which Γ=2, that is, corresponding to stability range of the star. So we can conclude that this approach of quantum gravity does not produce any instability to white dwarfs.


## 3.1. Main-Sequence Stars

Most main-sequence stars have a mass in the range about10 to 50 solar mass. They are composed by a mixture of nonrelativistic gas and radiation that are stood in hydrostatic equilibrium by gravity [25]. The total pressure is Pc=Pg+Pr, where Pc, Pg, and Pr are center star, gas, and radiation pressure, respectively. We can write βc=Pg/Pc and 1-βc=Pr/Pc. By definition (1-βc) and βc are the fractional contribution of radiation and gas to the central pressure of the star and both are less than one; then(24)Pgβc=Pr1-βc.The classical gas pressure can be expressed, using (15), in the form(25)Pg=ρcKBTm¯.We used the relationn=ρc/m¯; ρc is the central density of the star and m¯ is the average mass of the gas particle that constructs the star defined as m¯=uNmN where uN is the mean molecular weight and mN is the nucleon mass. Using (10), (25) into (24), one finds that(26)ρcKBβcm¯Tc=4σ3c1-βcTc4-8105βπ4KB6ħ3c51-βcTc6,where Tc is the central temperature of the star. We can find the temperature by solving the above equation:(27)Tc=1-βcβcm¯3cKB4σ1/3ρc1/3+β1701-βcβcm¯π4KB7ħ3c3σ2ρc.Using this expression to calculate the star pressure from the relation Pc=Pg/βc=(ρcKB/βcm¯)Tc one gets(28)Pc=K1ρc4/3+βK2ρc2=PsUP+psGUP,(29)K1=1-βc1/3βc4/33cKB44σm¯41/3,K2=1701-βcβc2π4KB8ħ3c3σ2m¯2.The first part of (28) is the ordinary polytropic relation with a a polytropic exponent Γ=4/3 (polytropic index np=3) [26] perturbed by a polytrope Γ=2 (polytropic index np=1) where np=1/(Γ-1).The best example may be used to estimate GUP perturbation is our sun. If we assume thatβc is constant overall the stars and its chemical composition is not changed so uN is constant. For our sun 1-βc=10-3 and uN=0.829 [27] and central density ρc=1.53×105 kg/m3; then(30)psGUPPsUP=1.8×10-49β0.The perturbed pressure should not exceed the original one such thatpsGUP/PsUP≪1, so this equation suggests that the upper value of β0 should be β0<1048. This bound is far weaker than that set by electroweak measurements.According to (28) the quantum gravity modifies the polytropic relation by adding a new term proportional to ρ2. This suggests a new modification in Lane-Emden equation [28]. The modified Lane-Emden equation is established in the appendix. Studying modified Lane-Emden equation and its solution is not the scope of this paper so results that are investigated by the this equation will be postponed to another research.Main-Sequence Stars Stability. In order to discuss the stability concept, the energy of a Newtonian polytrope systems with pressure P can be expressed in terms of the internal energy and gravitational potential such that(31)E=k1PV-k2GM2R,where k1 and k2 are constants, G is the gravitational constant, and M and R are the mass and radius of the star. At hight densities it is important to take into account the general relativistic effects in star stability. To deal with it a general relativistic correction term should be added to (31), in lowest order approximation; namely, Ecorr=-k3G/c2M7/3ρc2/3, where k3 is constant that depends on the actual distribution of matter. According to [27], for nP=3 polytrope, the condition of stability is Γ>4/3+2.25(GM/c2R). We can see that addition of a relativistic correction contracts the region of stability or increases the critical value of Γ. In our case GM/c2R~10-6 so this correction can be neglected. So we expect that the new term in a polytropic equation (28) with np=1, which has a positive contribution, will not affect the problem of stability since Γ>2.Using (28) into (31) the energy of main-sequence stars will be(32)E=C1Mρc1/3+βC2Mρc-k2M5/3ρc1/3,where C1 and C2 are constants. The mass M can be obtained by using the equilibrium condition ∂E/∂ρc=0; the result is(33)M=3k313C1+βC2ρc2/33/2.Then(34)dln⁡⁡Mdln⁡⁡ρc=β3C2C1ρc2/3.Equation (34) ensures that dln⁡⁡M/dln⁡⁡ρc>0 which is the stability condition [27] for main-sequence object. These results show that the generalization process for the uncertainty principle does not change the stability of the main-sequence objects.Minimum and Maximum Masses for Main-Sequence Stars. In this part we will discuss the evolution stage of main-sequence stars in the presence of quantum gravity to derive the minimum and maximum masses of these objects. Initially, in the stage of star forming, the star must has mass enough to generate a central temperature which is high enough for thermonuclear fusion. At this stage the central pressure comes by the electrons and ions which are forming an ideal classical gas; their pressure can be expressed as in (25). The star will be made if this pressure is close to the pressure needed to support the system. It is known that the upper limit relation between the pressure and density at the center of the star is given by [29](35)Pc≤π61/3Gρc4/3M2/3.This relation is valid for any homogeneous star in which the mass is concentrated towards the center. Equating (25) and (35) one can get the initial central temperature in this stage:(36)KBTc≤π61/3m¯Gρc1/3M2/3.We can see that the quantum gravity does not appear in the picture. As shown in this equation the central temperature will rise as the central density increases with star contraction. The contracting temperature continues to rise until the electrons at the center become fully degenerate and occupy the lowest energy states in accordance with exclusion principle. At this stage, at the center, the pressure comes from the electrons that are in degenerate state plus the pressure from the classical ions. Quantum gravity will not change the classical ions pressure, according to (15), but a nonrelativistic degenerate electrons pressure will change according to (23). In nonrelativistic limit, where y≪1, the degenerate pressure (23) will be(37)P0non-relat=115me4c4π2ħ3y5-335me6c7π2ħ3y7.Use (17), (21) in above equation and add the result to (15); finally we can write the central pressure at this stage as(38)Pc=a5mene5/3+β12a235mene7/3+niKBTc,where a=(31/3π2/3ħ)2. The number density of the electrons ne and ions ni can be expressed in terms of central density, simply ne=ni=ρc/mH, where mH is the mass of hydrogen atom. A hydrostatic pressure is achieved if this pressure equals the pressure needed to support the mass (35). This leads to the central temperature being as follows:(39)KBTc≤π61/3GmNM2/3ρc1/3-a5meρcmH2/3-β12a235meρcmH4/3.The first term is associated with classical ions and second and third terms with degenerate electrons. The last two terms become important and are increasing with central density until the temperature will cease to rise as the mass contracts. From this equation we can find the maximum value of temperature:(40)KBTcmax⁡≤zM4/31-607βmezM4/3,(41)z=54π62/3meamH8/3G2.Equation (40) proves that the maximum central temperature decreases due to quantum gravity. The contracting mass achieves stardom if that maximum central temperature reaches ignition temperature for the thermonuclear fusion of hydrogen. Denote this ignition temperature by Tign; then the minimum mass for the star will be given by(42)Mmin⁡≥KBTignz3/41+457βmeKBTign.This relation shows that the quantum gravity effect increases the minimum mass of the main-sequence stars. The approximation has a meaning if the second term in (42) is less than unity. If we take the ignition temperature for hydrogen to be about 1.5×106 K, this suggests that the upper value of β0<1047.The situation is changed if the radiation pressure takes a part in this stage of revolution. The star will be disrupted if radiation becomes the dominant source for the internal pressure. This property will set a limit to the mass of a main-sequence star. The pressure at the center of hot massive stars is due to electrons and ions (as a classical gases) and radiation; the pressure that is produced by these species is derived in (28). Comparing this with the pressure needed to support the star (35) to get the maximum mass, one finds that(43)π61/3GMmax⁡2/3≥K1+βK2ρc2/3,which means that the maximum mass is increased due to the presence of quantum gravity. This increase depends on the central density. Again the second term in the right-hand side of inequality (43) should be less than the first; this suggests that β0<1048.


## 3.2. White Dwarfs

The white dwarfs consist of Helium atoms that are almost in an ionization state. The temperature of the star is about107 K, so the electrons are in a relativistic energy regime and can be considered in a complete degenerate state and the Helium nuclei do not play a part, in first order approximation, in the dynamic of the system. Now we will determine the effect of GUP on white dwarf radius according to pressure modification expressed in (23). If the number of Helium nuclei is N/2 then there are N electrons. Suppose that M=2Nmp is the mass of the star, so the particle density of the electron is(44)ns=3M8πmp1R3.Using (17), (44) into (21), one gets(45)y=M¯1/3R¯+35βme2c2M¯R¯3,(46)M¯=9π8Mmp,R¯=mecħR,where R is the radius of the star and mp is the proton mass. The equilibrium configuration of white dwarf is caused by the pressure of the degenerate electrons against Newtonian gravitational collapse which is due to the nuclei, which means that(47)P0=κ4πGM2R4=K′M¯2R¯4,K′=κG4π8mp9π2mecħ4.We have two extreme cases as follows.(i) When the electron gas is in a low density, nonrelativistic dynamics may be used such thaty≪1, and (23) will lead to(48)P0=45Ky5-3635βme2c2Ky7,K=mec212π2mecħ3.Equating (47) and (48), we got the equation(49)R¯3-45KK′M¯-1/3R¯2-4835βme2c2K′KM¯1/3=0.The solution of that equation is(50)R¯≃45KK′M¯-1/3+157βme2c2K′KM¯.This equation shows that the radius of the star is increasing; the elevation depends on the mass of the star. The used approximation is true if the second term in the right-hand side of (50) is much less than the first one; this will put a limit in the upper value of β0 such that it should be β0<1044; we used a white dwarf mass M=0.41M⊙ [30].(ii) When the electron gas is in a high density, that relativistic effect comes to play strongly such thaty≫1. We can write(51)P0=Ky4-y2-β185Kme2c2y63-y44.Using (51) into (47) and simplifying, then(52)R¯4+K′KM¯4/3-1-310βme2c2M¯2/3R¯2-65βme2c2M¯4/3=0.The solution of that equation, keeping only the terms that proportional to ~β, is(53)R¯≃R¯01+β310me2c2R¯022M¯4/3R¯02-12M¯2/3,where(54)R¯02=M¯2/3-K′KM¯4/3.Using (46) into (54) and solving for R and letting R→RCh, one gets(55)RCh=9π1/32ħmecMmp1/31-MMCh2/31/2,MCh=93π1/264κħcGmp23/2mp.R
Ch is the unmodified white dwarf radius or Chandrasekhar radius limit and MCh is the Chandrasekhar limit mass, of order 1.44M⊙. First part of (53) is the Chandrasekhar radius and the second term is the radius correction due to GUP affect. These analyses show that the increasing in star radius depends on 1-M/MCh2/3-1 which formally goes to infinity as M→MCh. It is clear that the quantum gravity effect becomes significant when mass of the white dwarf gets very close to the Chandrasekhar limit in high density stars. Equations (23), (50), and (53) attract attention to a noticeable point that the growing in star radius ensures that quantum gravity resists gravitational collapse by increasing the electrons degeneracy pressure. These results are consistent with that established in [19]. Our approximation will have a meaning if the second term in the right-hand side in (53) is less than one; this puts an upper limit for β0 such that β0<1043. We have used M=0.59M⊙ [30].We should also notice that, in the limity≫1, in ultrarelativistic regime, the function F3(y) behaves like ~y6 which almost goes like ρ2. This result is consistent with polytrope behavior in which Γ=2, that is, corresponding to stability range of the star. So we can conclude that this approach of quantum gravity does not produce any instability to white dwarfs.


## 4. Discussion

Various approaches to quantum gravity such as string theory, black hole physics, and doubly special relativity predict a considerable modification in Heisenberg uncertainty principle to be a generalized uncertainty principle. This modification leads to a change in the energy-momentum dispersion relation and in the physical phase space. In this paper we studied the modification due to a generalized uncertainty principle (1) in the statistical and thermodynamic dynamic properties for photons, nonrelativistic ideal gases and degenerate fermions. There is a considerable decrease in the number of accessible microstates in quantum gravity background. This approach leads to a decrease in the internal energy and pressure of photon gas. Pressure and internal energy of fermions in degenerate state are increased and pressure of nonrelativistic ideal gases is not changed. These results are used in studying the stability of the main-sequence stars and white dwarfs. It is found that this approach of quantum gravity does not produce any changes in stars stability.We follow a very simple way to study the effect of quantum gravity in a different stage of stellar evolution. It is found that there is no change in pressure or temperature in the initial stage where the pressure is coming from classical ions. In the second stage, after stardom contracting, where degenerate electrons control the central pressure, the quantum gravity leads to a decrease in the central maximum temperature and an increase in the minimum and maximum masses of the main-sequence stars. This result shows that the quantum gravity has an effect on the building process of the universe and it will have a more strong effect in highly dense objects.A mass-radius relation for white dwarf is also investigated. The results show that there is an elevation in the star radius that is proportional to the mass of the star. For the case of high dense stars the elevation in the radius goes to infinity as the mass of the star reaches Chandrasekhar limit. This means that quantum gravity resists gravitational collapse. This result goes in parallel with that found in [19]. Unfortunately, quantum gravity behaves in a different way from the prediction of the usual model for white dwarf. The current observation indicates that white dwarfs have smaller radii than theoretical predictions [30–32]. This is not consistent with our results. But this result may be reasonable if we add this perturbed term, which comes from quantum gravity, to the other perturbed terms that are coming from other physical aspects such as Coulomb corrections, lattice energy, correction due to rotation, and magnetic field (which may be much larger than the correction due to GUP) and more realistic density distribution for electronic gas. The sum of all these terms may be consistent with the current data.On the other hand, implications of the GUP in various fields, such as high energy physics, cosmology, and black holes, have been studied. In [13] potential experimental signatures in some familiar quantum systems are examined. β0 is a numerical parameter that quantifies the modification strength if we assume that parameter is of the order of unity. This assumption renders quantum gravity effects too small to be measured [13]. But the β0 dependent terms are important when energies (momenta) are comparable to Planck energy (momenta) and length is comparable to the Planck length. If we cancel this choice, the current experiments predict large upper bounds on it, which are compatible with current observations. It is agreed that such an intermediate length scale, linter≃lPlβ0, cannot exceed the electroweak length scale ~1017lPl [13]. This means that β0≤1034. The authors in [13, 14] determined a bound in β0 parameter which proves that GUP could be measured in a low energy system like Landau levels, Lamb shift, potential barrier, and potential step. They calculated the intermediate length scales such that linter≃1018lPl,1025lPl, and 1010lPl of which the first two are far bigger than the electroweak scale and the last is smaller but may get further constrained with increased accuracies. In this paper the boundary values of β0 parameter, in cases of main-sequence stars, are 1048, 1047, and 1048 and those from white dwarf are 1044 and 1043. These can be converted to intermediate length scales, linter≈1024lPl for main-sequence stars and linter≈1022lPl for white dwarf. They are far bigger than the electroweak scale but it is approximately compatible with the results addressed by [14] in one case, and it can be considered reasonable values in the astrophysical regime.



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*Source: 343284-2015-05-18.xml*