In Newton–Cartan theory, one starts with a smooth four-dimensional manifold M and defines two (degenerate) metrics. A temporal metric t a b with signature ( 1 , 0 , 0 , 0 ) , used to assign temporal lengths to vectors on M and a spatial metric h a b with signature ( 0 , 1 , 1 , 1 ) . One also requires that these two metrics satisfy a transversality (or "orthogonality") condition, h a b t b c = 0 . Thus, one defines a classical spacetime as an ordered quadruple ( M , t a b , h a b , ∇ ∇ ) , where t a b and h a b are as described, ∇ ∇ is a metrics-compatible covariant derivative operator; and the metrics satisfy the orthogonality condition. One might say that a classical spacetime is the analog of a relativistic spacetime ( M , g a b ) , where g a b is a smooth Lorentzian metric on the manifold M .