# First Law of Thermodynamics and Emergence of Cosmic Space in a Non-Flat Universe

## TL;DR

This paper investigates the applicability of the first law of thermodynamics in a non-flat universe within the emergent cosmic space framework, finding that only the areal volume allows a consistent thermodynamic description.

## Contribution

It demonstrates that the first law of thermodynamics cannot be properly formulated using proper invariant volume in a non-flat universe, but can be consistent when using areal volume.

## Key findings

- First law cannot be formulated with proper invariant volume in non-flat universe.
- Energy change and flux are not equivalent in non-flat universe with proper invariant volume.
- Consistency is achieved only with areal volume of the horizon.

## Abstract

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 \cite{mahith}, 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 $-dE = TdS$ in a non-flat universe. We have shown that the energy change dE 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 have further shown that the consistency between the above two forms of the first law claimed in Ref. \cite{caiakb} will hold only with the use of the areal volume of the horizon. Thus, a consistent formulation of the above two forms of the first law of thermodynamics demands the use of areal volume.