A hypersurface Σ is a three-dimensional sub-manifold of a four-dimensional spacetime manifold M. The hypersurface may be time-like, space-like or null.

Let us consider that xμ (μ = 1; 2; 3; 4) are the coordinate of the four-dimensional spacetime, and ya(a = 1, 2, 3) are the inherent coordinates of the hypersurface.

The parametric equation of the hypersurface is

The three basis vectors provide the tangent vectors to Σ as

These basis vectors provide the induced metric or first fundamental form of the hypersurface by the following scalar product

[Actually, the elementary distance “ds” between two neighboring points in Σ (which are, therefore, also in M) is given by

where is the induced metric tensor in the hypersurface Σ.]

A unit normal nα can be defined if the hypersurface is not null.

Let us consider the surface Σ, defined by the Eq. (A1), which can be written in the form

Hence, the unit normal vector nμ is defined as

Here, and

Note that unit normal is defined when Σ is non-null as for null surface gμ𝜈 f,μf,𝜈 = 0.

The extrinsic curvature or second fundamental form of the hypersurface Σ is defined by

Extrinsic curvature is symmetric tensor, i.e., kab = kba.

Another form

Here, Ln stands for Lie Derivative.

trace of the extrinsic curvature.

Result

• (i) If k > 0, then the hypersurface is convex

• (ii) If k < 0, then the hypersurface is concave

hab is purely the inherent property of a hypersurface geometry, whereas kab is concerned with extrinsic aspects.

Remember

where hab is the inverse of the induced metric.

It is possible that any arbitrary tensor Tαβ can be projected down to the hypersurface with nonzero tangential components. The quantity that effects the projection is

One can make a 3 + 1 split of the spacetime M in a slightly different manner as follows (see Appendix C for more details):

Let us consider an arbitrary scalar field t(xα) such that t = constant describes a family of nonintersecting space-like hypersurfaces Σt.