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Register a new userExtensivity, entropy current, area law and Unruh effect
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Added by
Francesco Becattini
Publication date
2019
fields
Physics
and research's language is
English
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We present a general method to determine the entropy current of relativistic matter at local thermodynamic equilibrium in quantum statistical mechanics. Provided that the local equilibrium operator is bounded from below and its lowest lying eigenvector is non-degenerate, it is proved that, in general, the logarithm of the partition function is extensive, meaning that it can be expressed as the integral over a 3D space-like hypersurface of a vector current, and that an entropy current exists. We work out a specific calculation for a non-trivial case of global thermodynamic equilibrium, namely a system with constant comoving acceleration, whose limiting temperature is the Unruh temperature. We show that the integral of the entropy current in the right Rindler wedge is the entanglement entropy.
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Total entropy generated by the Unruh effect is calculated within the framework of information theory. In contrast to previous studies, here the calculations are done for the finite time of existence of the non-inertial reference frame. In this case only the finite number of particles is produced. Dependence on mass of the emitted particles is taken into account. Analytic expression for the entropy of radiated boson and fermion spectra is derived. We study also its asymptotics corresponding to limiting cases of low and high acceleration. The obtained results can be further generalized to other intrinsic degrees of freedom of the emitted particles, such as spin and electric charge.
Using a graphical analysis, we show that for the horizon radius $r_hgtrsim 4.8sqrttheta$, the standard semiclassical Bekenstein-Hawking area law for noncommutative Schwarzschild black hole exactly holds for all orders of $theta$. We also give the corrections to the area law to get the exact nature of the Bekenstein-Hawking entropy when $r_h<4.8sqrttheta$ till the extremal point $r_h=3.0sqrt{theta}$.
Applying Clausius relation, $delta Q=TdS$, to apparent horizon of a FRW universe with any spatial curvature, and assuming that the apparent horizon has temperature $T=1/(2pi tilde {r}_A)$, and a quantum corrected entropy-area relation, $S=A/4G +alpha ln A/4G$, where $tilde {r}_A$ and $A$ are the apparent horizon radius and area, respectively, and $alpha$ is a dimensionless constant, we derive modified Friedmann equations, which does not contain a bounce solution. On the other hand, loop quantum cosmology leads to a modified Friedmann equation $H^2 =frac{8pi G}{3}rho (1-rho/rho_{rm crit})$. We obtain an entropy expression of apparent horizon of FRW universe described by the modified Friedmann equation. In the limit of large horizon area, resulting entropy expression gives the above corrected entropy-area relation, however, the prefactor $alpha$ in the logarithmic term is positive, which seems not consistent with most of results in the literature that quantum geometry leads to a negative contribution to the area formula of black hole entropy.
The area of a cross-sectional cut $Sigma$ of future null infinity ($mathcal{I}^+$) is infinite. We define a finite, renormalized area by subtracting the area of the same cut in any one of the infinite number of BMS-degenerate classical vacua. The renormalized area acquires an anomalous dependence on the choice of vacuum. We relate it to the modular energy, including a soft graviton contribution, of the region of $mathcal{I}^+$ to the future of $Sigma$. Under supertranslations, the renormalized area shifts by the supertranslation charge of $Sigma$. In quantum gravity, we conjecture a bound relating the renormalized area to the entanglement entropy across $Sigma$ of the outgoing quantum state on $mathcal{I}^+$.
We propose a test for the circular Unruh effect using certain atoms - fluorine and oxygen. For these atoms the centripetal acceleration of the outer shell electrons implies an effective Unruh temperature in the range 1000 - 2000 K. This range of Unruh temperatures is large enough to shift the expected occupancy of the lowest energy level and nearby energy levels. In effect the Unruh temperature changes the expected pure ground state, with all the electrons in the lowest energy level, to a mixed state with some larger than expected occupancy of states near to the lowest energy level. Examining these atoms at low background temperatures and finding a larger than expected number of electrons in low lying excited levels, beyond what is expected due to the background thermal excitation, would provide experimental evidence for the Unruh effect.
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