Kim and Lee studied the thermodynamic equilibrium of matter in a wormhole ‘station’ which connects various distinct asymptotic regions of a spacetime.

An example of the ‘station’ is a traversable wormhole connecting two distant regions. We all know that, a wormhole is a physical object which connects two distinct regions of a spacetime. We can imagine a wormhole as a three-dimensional space with two spherical holes (‘mouths’) in it. These holes are connected to each other by means of a ‘handle’ having a ‘throat’ in it. Usually, it is assumed that the length of the ‘handle’ does not depend on the distance between the ‘mouths’ in an external space. The wormhole can be used to travel through the space if it is stable or lives long enough.

In the presence of a traversable wormhole, various interesting phenomena happen which are beyond ordinary physical sense. Kim and Lee illustrated a few of them. First, the geometry is not symmetrical under the reflection with respect to the ‘throat’. Therefore, the size of the ‘mouths’ may not be the same in general. Secondly, the Arnowitt-Deser-Misner (ADM) mass of the wormhole measured by an asymptotic observer reside in the upper half is not the same as that in the other half. If you are an observer who is travelling through a wormhole, the proper time may not flow with the same speed in the upper/lower half. A time travel was shown to be possible when the two asymptotic regions are adjoined. Now, Kim and Lee, focused on a ‘static’ (traversable) wormhole based on general relativity in the absence of cosmological constant and studied the thermal equilibrium of self-gravitating matters in the wormhole spacetime.

In order to understand whether a wormhole station remains in thermodynamic equilibrium or not. We have to consider a space-like section of the wormhole station which are connected to 4 asymptotic regions (or you can consider any number of asymptotic regions but yes, it must be more than 2). Just refer fig 2 given below:

As you can see in figure above, the ‘station’ consists of a ‘core’, N-‘mouths/outsides’, and N-‘branches’ which connects a ‘mouth’ with the ‘core’, where each ‘mouth’ plays the role of a gate to/from an outside. The station may consist of combinations of ordinary+exotic selfgravitating matters in thermal equilibrium. A typical example of N = 2 case is a traversable wormhole in which two outsides are connected by a wormhole ‘throat’ (an example of ‘core’). The wormhole ‘handle’ is made of two ‘branches’+‘core’. A ‘mouth’ Mn, which is a border between the station and the outside An, is assumed to be hard enough to prevent matter inside from spilling out (where, n =1, 2, 3…N).

Now, lets consider two regions A1 and A2, which are disjointed. Well, when they are disjointed the two regions have unequal temperature, and authors showed that, this difference does not cause any problem. However, when they are adjoined, the difference of the asymptotic temperatures seems to cause a trouble mainly because of the transitive relation between the objects in thermal equilibrium, called “zeroth law of thermodynamics”, which states that if two systems have the same T, then the two are in thermal equilibrium.

“This pathology actually comes from the incompleteness of the previous mentioned coordinates which originates from the multiply connected property of the space.”

“Still there remain a few important questions. First, what is the first law of the thermal system in the wormhole? Specifically, what is the entropy of the system consisting of matter plus wormhole? Second, does the equilibrium is stable or not? In other words, what happens when a mass falls in a wormhole in thermal equilibrium. Further studies are required to answer these questions.”— concluded authors of the study

**For more:*** Hyeong-Chan Kim, Youngone Lee, “Thermodynamic Equilibrium of a Wormhole Station”, Arxiv, pp. 1-13, 2019. **https://arxiv.org/abs/1902.02957*

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