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Mesoscopic Non-Equilibrium Thermodynamics Part 6

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Tham khảo tài liệu 'mesoscopic non-equilibrium thermodynamics part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Lorentzian Wormholes Thermodynamics 141 and V - p p e- r Ự1 - K r r -d d- 31 where we have taken into account that the components of an energy-momentum tensor which takes the form7 T v diag p pr in an orthonormal basis with the superscript 2 meaning the two-dimensional space normal to the spheres of symmetry are in this basis T e2 r p pr 4 and T - T e2 r p - pr 4. The Misner-Sharp energy in this spacetime reaches its limiting value E r 2 only at the wormhole throat r r0 which corresponds to the trapping horizon taking smaller values in the rest of the space which is untrapped. We want to emphasized that as in the case of the studies about phantom wormholes performed by Sushkov Sushkov 2005 and Lobo Lobo 2005 any information about the transverse components of the pressure becomes unnecessary. Deriving Eq. 29 and rising the index of Eq. 31 one can obtain d E 2nr2pe ỵ 1 - K r r 32 and t e r Ự1 - K r r p- 4pr. 33 Therefore we have all terms8 of Eq. 17 for the first law particularized to the Morris-Thorne case which vanish at the throat what could be suspected since we are considering a wormhole without dynamic evolution. Nevertheless the comparison of these terms in the case of Morris-Thorne wormholes with those which appear in the Schwarzschild black hole could provide us with a deeper understanding about the former spacetime based on the exotic properties of its matter content. Of course the Schwarzschild metric is a vacuum solution but it could be expected that it would be a good approximation when small matter quantities are considered which we will assume to be ordinary matter. So in the first place we want to point out that the variation of the gravitational energy Eq. 32 is positive negative in the outgoing ingoing direction in both cases9 since p 0 therefore this variation is positive for exotic and usual matter. In the second place the energy density w takes positive values no matter whether the null energy condition is violated or not. Considering the .