By Baumgarte T.W., Shapiro S.L.

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Stellar models constructed from such an equation of state are called polytropes. 74)) that ρ0 = P/(Γ − 1) and ρ = ρ0 + P/(Γ − 1). There are a number of physically interesting stars that can be modeled as polytropes in a first approximation. For example, stars supported against collapse by the pressure of noninteracting, nonrelativistic, degenerate fermions can be modeled as n = 3/2 polytropes, while stars supported by noninteracting, ultrarelativistic, degenerate fermions can be modeled as n = 3 polytropes.

5. 3. 1: A foliation of the spacetime M . The hypersurfaces Σ are level surfaces of the coordinate time t, Ωa = ∇a t. The normal vector na is orthogonal to these t = constant spatial hypersurfaces. The 4-metric gab allows us to compute the norm of Ω, which we call −α−2 Ω 2 = g ab ∇a t∇b t ≡ − 1 . 21) As we will see more clearly later, α measures how much proper time elapses between neighboring time slices along the normal vector Ωa to the slice, and is therefore called the lapse function. We assume that α > 0, so that Ωa is timelike and the hypersurface Σ is spacelike everywhere.

95) According to the above equation, χ0 must lie in the range 0 ≤ χ0 ≤ π/2. 27). 96) where a 1 = (1 + cos η) . 97) am 2 In our synchronous coordinate system the star thus remains homogeneous throughout the collapse. 93). At this time a central singularity forms at the center of the star. It is both instructive and straightforward to probe the spacetime geometry of OS collapse. 3 shows the worldlines of infalling Lagrangian fluid elements as well as the location of the black hole event horizon. The event horizon first forms at the center and grows monotonically outward to encompass the entire star.

### Numerical relativity: solving Einstein's equations on the computer by Baumgarte T.W., Shapiro S.L.

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