First law of thermodynamics and emergence of cosmic space in a non-flat universe
The emergence of cosmic space as cosmic time progresses is an exciting idea advanced by Padmanabhan to explain the accelerated expansion of the universe. The generalization of Padmanabhan's conjecture to the non-flat universe has resulted in scepticism about the choice of volume such that the law of emergence can not be appropriately formulated if one uses proper invariant volume. The deep connection between the first law of thermodynamics and the law of emergence [1], motivate us to explore the status of the first law in a non-flat universe when one uses proper invariant volume. We have shown that the first law of thermodynamics, dE = TdS +WdV cannot be formulated properly for a non-flat universe using proper invariant volume. We have also investigated the status of the first law of the form −dĚ = ŤdS in a non-flat universe. We have shown that the energy change dĚv within the horizon and the outward energy flux are not equivalent to each other in a non-flat universe when we use the proper invariant volume. We further point out that the consistency between the above two forms of the first law claimed in ref. [2] will hold only with the use of the areal volume of the horizon. The failure in formulating the first law of thermodynamics with the use of invariant volume shows that the invariant volume will be a poor choice to describe any thermodynamic process in cosmology.