There are three simplest classifications of liquid crystals: (1) the nematic liquid crystals, (2) cholesteric liquid crystals, and (3) smectic liquid crystals. All are substances that flow like a liquid but have some orders in their molecular arrangements. Some of the mathematics of liquid crystals is described in the last chapter of Landau, et al. Theory of Elasticity, Volume 7, Course of Theoretical Physics, 3rd edition, 1986. Nematics have long molecules all aligned in the same direction, but otherwise randomly arranged. Cholesterics and smectics also have aligned molecules, which are arranged in distinct layers. For cholesterics the axes of the molecules are parallel to the plane of the layers, while for smectics they are perpendicular. The applications of liquid crystals as LCDs (liquid crystal displays) can be found in flat screen TVs, laptops, digital watches, calculators, and many similar electronic devices.

If the quantum vacuum of orderly arranged or aligned space-time charges is considered as a form of liquid crystals then its physical states of evolution can be described by the squares of energy, which is simply a transformation of the elasticity of the quantum vacuum given by ��²-��₀²��=0 where ��₀=��₀/��₀ and for non-equilibrium configurations, the squares of energy is given as ��²=��²��₀²-��₀²��² where ��₀ is of the order of Planck length and ��₀ is the thermal force given by ��₀=(����²/��⁴)Δ��/��² where �� is the universal constant of gravitation, �� is the Boltzmann’s constant, �� is the speed of light in vacuum, �� is the lattice distance between two space-time charges, Δ��=��₁��₂=(��₂-��₁)��₁ such that ��₁=1kelvin and ��₂>>��₁ if and only if both ��₁ and ��₂ are rational numbers. Clearly, the term ��²��₀² is practically zero unless the applied force �� is very large, while the term ��₀²��² is directly proportional to the difference of absolute temperature and the displacement �� of space-time charges.