Author(s)

Christopher Pilot

Physics Department, Gonzaga University, Spokane, WA, USA.

Using a space filled with black-body radiation, we derive a generalization for the Clausius-Clapeyron relation to account for a phase transition, which in-volves a change in spatial dimension. We consider phase transitions from dimension of space, n, to dimension of space, (n - 1), and vice versa, from (n - 1) to n -dimensional space. For the former we can calculate a specific release of latent heat, a decrease in entropy, and a change in volume. For the latter, we derive an expression for the absorption of heat, the increase in entropy, and the difference in volume. Total energy is conserved in this transformation process. We apply this model to black-body radiation in the early universe and find that for a transition from n = 4 to (n - 1) = 3, there is an immense decrease in entropy accompanied by a tremendous change in volume, much like condensation. However, unlike condensation, the volume change is not three-dimensional. The volume changes from V₄, a four-dimensional construct, to V₃, a three-dimensional entity, which can be considered a subspace of V₄. As a specific example of how the equation works, we consider a transition temperature of 3 × 10²⁷ Kelvin, and assume, furthermore, that the latent heat release in three-dimensional space is 1.8 × 10⁹⁴ Joules. We find that for this transition, the internal energy densities, the entropy densities, and the volumes assume the following values (photons only). In four-dimensional space, we obtain, u₄ = 1.15×10¹²⁵ J⋅ m^-4, s₄ = 4.81×10⁹⁷ J⋅ m^-4⋅ K^-1, and V₄ = 2.14×10^-31 m⁴. In three-dimensional space, we have u₃ = 6.13×10⁹⁴ J⋅ m^-3, s₃ = 2.72×10⁶⁷ J⋅ m^-3⋅ K^-1, and V₃ = 0.267 m³. The subscripts 3 and 4 refer to three-dimensional and four-dimensional quantities, respectively. We speculate, based on the tremendous change in volume, the explosive release of latent heat, and the magnitudes of the other quantities calculated, that this type of transition might have a connection to inflation. With this work, we prove that space, in and of itself, has an inherent energy content. This is so because giving up space releases latent heat, and buying space costs latent heat, which we can quantify. This is in addition to the energy contained within that space due to radiation. We can determine the specific amount of heat exchanged in transitioning between different spatial dimensions with our generalized Clausius-Clapeyron equation.

Clausius-Clapeyron Relation, Spatial Dimension, Phase Transition, Inflation

Pilot, C. (2019) A New Type of Phase Transition Based on the Clausius-Clapeyron Relation Involving a Change in Spatial Dimension. Journal of High Energy Physics, Gravitation and Cosmology, 5, 291-309. doi: 10.4236/jhepgc.2019.52016.