CHRONOMETRIC INVARIANCE AND STRING THEORY
Abstract
The Einstein–Hilbert Lagrangian R is expressed in terms of the chronometrically invariant quantities introduced by Zel'manov for an arbitrary four-dimensional metric gij. The chronometrically invariant three-space is the physical space γαβ = -gαβ+e2ϕ γαγβ, where e2ϕ = g00 and γα = g0α/g00, and whose determinant is h. The momentum canonically conjugate to γαβ is 
, where 
and ∂t≡e-ϕ∂0 is the chronometrically invariant derivative with respect to time. The Wheeler–DeWitt equation for the wave function Ψ is derived. For a stationary space-time, such as the Kerr metric, παβ vanishes, implying that there is then no dynamics. The most symmetric, chronometrically-invariant space, obtained after setting ϕ = γα = 0, is 
, where δαβ is constant and has curvature k. From the Friedmann and Raychaudhuri equations, we find that λ is constant only if k=1 and the source is a perfect fluid of energy-density ρ and pressure p=(γ-1)ρ, with adiabatic index γ=2/3, which is the value for a random ensemble of strings, thus yielding a three-dimensional de Sitter space embedded in four-dimensional space-time. Furthermore, Ψ is only invariant under the time-reversal operator 
if γ=2/(2n-1), where n is a positive integer, the first two values n=1,2 defining the high-temperature and low-temperature limits ρ ~ T±2, respectively, of the heterotic superstring theory, which are thus dual to one another in the sense T↔1/2π2α′T.
