The Initial Value Problem in Special Relativity

Abstract

We consider the relativistic equations for a perfect fluid

$$ divT = 0, $$

((2.1.1))

in flat Minkowski spacetime,

$$ ds^2 = \eta _{ij} dx^i dx^j = - d\left( {ct} \right)^2 + d\left( {x^1 } \right)^2 + d\left( {x^2 } \right)^2 + d\left( {x^3 } \right)^2 , $$

((2.1.2))

where

$$ T^{ij} = \left( {p + \rho c^2 } \right)w^i w^j + p\eta i^j , $$

((2.1.3))

denotes the stress-energy tensor for the fluid. Recall that in Minkowski spacetime,

$$ divT \equiv T_{j,i}^i $$

((2.1.4))

where again we assume summation over repeated up-down indices, “,i” denotes differentiation with respect to the variable xⁱ, and in general all indices run from 0 to 3 with x⁰ = ct. In (2.1.3), c denotes the speed of light, (we take c = 1 when convenient), p the pressure, w = (w⁰, ...,w³) the 4—velocity of the fluid particle, ρ the mass-energy density, and η^ij ≡ η_ij ≡ diag(−1, 1, 1, 1).