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Consider a uniform charge distribution occupying all of (flat) spacetime,

[tex]\rho(t,x,y,z) = \text{constant} \;\;\;\;\; ,\; (t,x,y,z) \in R^{1,3}[/tex]

Because this charge distribution is translationally invariant, it seems reasonable to expect that the electric field arising from the charge distribution is zero,

[tex]E(t,x,y,z) = 0 \;\;\;\;\;\;\;\;\; ,\; (t,x,y,z) \in R^{1,3}[/tex]

But then the electric field does not appear to satisfy Poisson's equation,

[tex]\nabla\cdot E = \rho/\epsilon_0[/tex]

Presumably this problem has a simple, well-known solution, but I have not encountered it before. Can anyone provide a reference or some insight?

[tex]\rho(t,x,y,z) = \text{constant} \;\;\;\;\; ,\; (t,x,y,z) \in R^{1,3}[/tex]

Because this charge distribution is translationally invariant, it seems reasonable to expect that the electric field arising from the charge distribution is zero,

[tex]E(t,x,y,z) = 0 \;\;\;\;\;\;\;\;\; ,\; (t,x,y,z) \in R^{1,3}[/tex]

But then the electric field does not appear to satisfy Poisson's equation,

[tex]\nabla\cdot E = \rho/\epsilon_0[/tex]

Presumably this problem has a simple, well-known solution, but I have not encountered it before. Can anyone provide a reference or some insight?

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