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By Harrison D.M.

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In classical dynamics, the evolution of a system is uniquely determined by the initial positions and velocities of its constituents. By analogy, the evolution of a general relativistic gravitational field is determined by specifying the metric quantities gab and ∂t gab at a given (initial) instant of time t. In particular, we need to specify the metric field components and their first time derivatives everywhere on some 3-dimensional spacelike hypersurface labeled by coordinate x 0 = t = constant.

23 When nuclear interactions are included, high-mass neutron stars are better represented by a “stiffer” equation of state, but the resulting models are often crudely modeled as n = 1 relativistic polytropes. 24 When using a polytropic equation of state to construct stellar equilibrium models, it is always possible to scale out the constant K . In gravitational units K n/2 has units of length, so that we can introduce a new set of nondimensional quantities, often denoted by a bar: r¯ ≡ K −n/2r, ρ¯0 ≡ K n ρ0 , P¯ ≡ K n P, ρ¯ ≡ K n ρ, M¯ ≡ K −n/2 M, M¯ 0 ≡ K −n/2 M0 .

But before we describe these modifications, we will derive the standard set of 3 + 1 equations in this chapter. To introduce the subject, we shall begin by discussing the initial value problem in electrodynamics, a simpler field theory than general relativity, but often a good place to 3 Darmois (1927); Lichnerowicz (1944); Four`es-Bruhat (1956); see also York, Jr. (1979) and references therein. 26 Chapter 2 The 3+1 decompostion of Einstein’s equations gain intuition. Specifically, we will cast Maxwell’s equations into 3 + 1 form in Minkowski spacetime.

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