What spacetime is the maximally symmetric vacuum solution with a positive cosmological constant?

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The spacetime that is the maximally symmetric vacuum solution with a positive cosmological constant is de Sitter space.

In general relativity, “vacuum” means that ordinary matter and radiation are absent, while the cosmological constant may still contribute to spacetime curvature. A positive cosmological constant produces constant positive curvature, making de Sitter space the Lorentzian analogue of a sphere. “Maximally symmetric” means it has the greatest possible number of spacetime symmetries for its dimension.

The geometry is named after Dutch astronomer Willem de Sitter, who studied it in the early twentieth century. De Sitter worked with Albert Einstein on cosmological models, and the solution became an important example of an expanding universe driven by vacuum energy. Modern cosmology uses de Sitter-like behavior to describe accelerated expansion associated with dark energy.

A common mix-up is anti-de Sitter space: it is also maximally symmetric and vacuum, but it has a negative cosmological constant and negative curvature. Minkowski space instead has zero cosmological constant and zero curvature, while Rindler space is a coordinate description of part of flat spacetime for uniformly accelerated observers.

Source: Wikipedia · fact-checked Sept. 2026

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