de Sitter and anti-de Sitter geometries

The vacuum equation requires the Ricci tensor to equal Lambda times the metric. Maximal symmetry further fixes the full Riemann tensor to the constant-curvature form, a stronger statement than the vacuum equation alone. Schwarzschild with Lambda, for example, need not be maximally symmetric. Define a positive length L through the absolute magnitude of Lambda. Scalar curvature changes sign between the two maximally symmetric cases, while the contracted curvature square has the same positive value for equal L.

Neither the sign of scalar curvature nor the word vacuum determines whether a particular chart has a horizon. A horizon is a causal feature, whereas the displayed invariants describe local curvature. Changing L rescales curvature continuously: doubling L divides scalar-curvature magnitude by four and the curvature square by sixteen. The positive-Lambda effective fluid has positive energy density and negative pressure as discussed earlier, violating the strong energy condition without contradicting the field equation. Negative Lambda cannot simply be described by the same positive-density fluid.

In a static de Sitter chart the metric factor is one minus radius squared over L squared. Static observers exist for radius less than L; the chart’s horizon at L is not a curvature singularity. It depends on the observer’s causal patch rather than enclosing a localized central mass. A different family of comoving observers yields a spatially flat FLRW patch whose scale factor grows exponentially with constant Hubble rate c divided by L. This flat spatial slicing does not make the four-dimensional spacetime flat.

The static and expanding descriptions cover different patches and use different observer families. They are not competing predictions about whether the same invariant geometry changes with time. The flat FLRW patch does not cover all of global de Sitter, and its exponential scale factor should not be generalized to a universe containing arbitrary matter, radiation, or spatial curvature. The exact vacuum relation between H and L follows from the specified solution. Perturbations and realistic cosmic history require additional equations and initial data.

The global static anti-de Sitter metric instead has factor one plus radius squared over L squared. It stays positive at every nonnegative radius, so this chart contains no de Sitter-like zero of its static factor. The universal covering spacetime unwraps the periodic time associated with the simplest hyperboloid embedding, avoiding that embedding’s closed timelike circles. Statements about the global geometry should identify this cover rather than silently treating periodic and unwrapped time as the same spacetime.

Radial light can reach the conformal boundary in finite global time. That boundary is timelike, so specifying data on a bulk spacelike slice alone does not determine later incoming influence. A well-posed field evolution must include suitable boundary conditions, such as an explicitly chosen reflecting prescription; different matter and gravitational problems need compatible choices. Other anti-de Sitter patches can have coordinate horizons, so the absence of a zero in this global static factor is not a claim about every chart. The quadratic instrument below contrasts signed scalar labels only, not these global causal properties.