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- W32430860 abstract "Geometrically speaking the fundamental problem of GTR is that of finding 4-dimensional manifolds M4 endowed with a metric g ab of signature (+ − − −), satisfying Einstein’s equation (see Part I, §§37.4, 39) $${R_{ab}} - frac{1}{2}R{g_b} = frac{{8pi G}}{{{c^4}}}{T_{ab,}}$$(1)where Tab is the energy-momentum tensor of the medium. From the most general point of view there is only one restriction on the tensor Tab, namely that the “energy density” be non-negative, which comes down to the condition that for every time-like vector ξ = (ξa) (i.e. such that gab ξ,a ξb > 0) at each point of M4, we have Tabξa ξb ≥ 0. However in the present chapter we shall for the most part concern ourselves with the so-called “hydrodynamical energy-momentum tensor” of an isotropic dense medium (forming a “perfect fluid”), already introduced in Part I, §39.4, important in the investigation of the gravitational fields of macroscopic bodies: $${T_{ab}} = (p + varepsilon ){u_a}{u_b} - p{g_{ab}},$$(2)where u = (u0) is the 4-velocity ( u, u) = 1), p is the (isotropic) pressure, and ɛ is the energy density (see loc. cit.)." @default.
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- W32430860 date "1985-01-01" @default.
- W32430860 modified "2023-09-27" @default.
- W32430860 title "The Global Structure of Solutions of Higher-Dimensional Variational Problems" @default.
- W32430860 doi "https://doi.org/10.1007/978-1-4612-1100-6_8" @default.
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