Looking at Both Sides of Einstein’s Gravitational Field Equation

The letter by Alex Harvey (Physics Today, February 2002, page 73 ) claims that the term $\Lambda g_{\mu \nu }$ is necessarily an ingredient of the geometrical, or left-hand, side of Einstein’s gravitational field equation. The corresponding matter-free ( $T_{\mu \nu }=0$ ) solution for Λ > 0 is the de Sitter space, which contains an embedded repulsive force driving particles apart. Accelerated cosmic expansion therefore appears as a natural consequence of the geometry of space-time. This view conflicts with the “majority opinion” by which the Λ term is identified with some kind of “dark energy.”

Whichever side of Einstein’s equation has Λ, the equations are still solved by the same curved de Sitter space-time, which contrasts with the flat Minkowski space-time obtained when Λ vanishes.

People nevertheless tend to distinguish the two cases because, on the right-hand side, the Λ term is another source of gravity, expressed by a constant times the stress-energy tensor ( $\kappa T_{\mu \nu }$ ). Adopting $T_{\mu \nu }=\left(p+\rho \right)u_{\mu }{u}_{\nu }+p{g}_{\mu \nu }$ , the form valid for a perfect fluid with energy density ρ and pressure p, one can define a Λ fluid by setting Λ = κρ. Then the two formulations agree for an equation of state p = −ρ.

Because Λ is a constant, ρ is constant as well, and such a fluid has negative pressure for positive ρ. The relabeling does not change the physics fixed by general relativity: The reformulated cases are equivalent. One should nevertheless be on the alert if ρ is not constant—even a slowly fluctuating field differs from the Λ case.

However, as long as ρ is constant, the difference between the two formulations lies only in the language used to describe physically identical situations. General relativists are well acquainted with similar apparent interpretational ambiguities when different coordinate systems are used to describe the same space-time geometry.