Hong and Kim studied (3+1) Morris-Thorne wormhole to investigate its higher dimensional embedding structures and thermodynamic properties. They showed that the wormhole is embedded in (5+2) global embedding Minkowski space. This embedding enables them to construct the wormhole entropy and wormhole temperature by exploiting Unruh effects.

There have been considerable discussions on the theoretical existence of wormhole geometry, since Morris and Thorne (MT) proposed a possibility of traversable wormhole, through which observers can pass travelling between two universes as a short cut. According to the Einstein field equations, the MT wormhole needs the exotic matter, which violates the weak energy condition.

On the other hand, it has been discovered the novel aspects that the thermodynamics of higher dimensional black holes can often be interpreted in terms of lower dimensional black hole solutions. In fact, a slightly modified solution of (2+1) dimensional Banados-Teitelboim-Zanelli black hole yields a solution to the string theory, so-called the black string. Since the thermal Hawking effects on a curved manifold were studied as Unruh effects in a higher flat dimensional space-time, following the global embedding Minkowski space (GEMS) approach several authors recently have shown that this approach could yield a unified derivation of temperature for various curved manifolds in (2+1) dimensions and in (3+1) dimensions. Moreover, the MT wormhole has been described in terms of its embedding profile surface geometry.

Hong and Kim, in their paper, analyzed the geometries of the MT wormhole manifolds to construct their higher dimensional flat embeddings, which they showed to be related with the embedding profile surface geometry of the wormhole. In these GEMS embeddings, they investigated the Hawking temperature and entropy via the Unruh effects to propose a possibility of “negative temperature” associated with the “exotic matter.”

They have thus showed that, on these flat embedding geometries, the wormhole temperature has negative (positive) values inside (outside) the exotic matter distribution accumulated mostly around the wormhole shape radius, and the wormhole entropy lower bound is twice the throat area of the wormhole.

**Reference***: Soon-Tae Hong, Sung-Won Kim, “Can wormholes have negative temperatures?”, Modern Physics Letters A, Vol. 21, No. 10, pp. 789-793 (2006). https://www.worldscientific.com/doi/10.1142/S0217732306019839*

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