Static spacetime

A static spacetime has a timelike Killing vector that is hypersurface orthogonal. Coordinates adapted to it can be chosen with no time-space cross terms.

Static Killing field is a Ricci eigenvector ( $R_{ab} k^{b} ∥ k_{a}$ )

k = \partial_{t}

and

d s^{2} = N (x)^{2} d t^{2} - h_{ij} (x) d x^{i} d x^{j}

. The only connection coefficients with time indices that can be nonzero are

Γ_{0 i}^{0} = \partial_{i} lo g N

and

Γ_{00}^{i} = N h^{ij} \partial_{j} N

. Substitution into the Ricci tensor gives

R_{0 i} = 0

. Thus

R_{ab} k^{b}

has only its time component and is proportional to

k_{a}

. Equivalently

k_{d} R^{d}_{[a} k_{b]} = 0

. This local statement does not require the Vacuum Einstein equations and is independent of the overall curvature-sign convention.

Rescaling the spatial metric by the squared static lapse function gives a useful three-dimensional quotient metric

γ

R_{ij} (γ) = 2 \partial_{i} U \partial_{j} U

, while

U

On a complete nonsingular spatial quotient with no inner boundary and standard asymptotic flatness, a bounded harmonic function tending to zero on all ends is zero. The static vacuum conformal spatial metric is then Ricci flat; three-dimensional curvature from the Ricci tensor makes it flat. Completeness and an ordinary Euclidean asymptotic end exclude nontrivial flat quotients, yielding Minkowski spacetime with the standard global time coordinate.

The stress-energy current from a Killing vector associated with the static time translation gives a positive conserved energy for a massive real scalar. Its hypersurface independence follows from current conservation and the divergence theorem.

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Static spacetime is a concept in general relativity that refers to a type of spacetime geometry that is both time-independent (static) and has a specific symmetry. More formally, a static spacetime is one where the gravitational field does not change over time and exhibits certain symmetries, particularly time translation symmetry and spatial symmetry. Key characteristics of static spacetimes include: 1. **Time Independence**: The metric tensor, which describes the geometry of spacetime, does not vary with time.