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Charge density zero? Yes. Mass density zero? No. Particles and
antiparticles have positive mass.
A spherical (isotropic) distribution of *real* particles, as observed
by one stationary with respect to the particles' center of mass, will
be contracted (anisotropic) along the direction of relative motion, as
observed by one moving with respect to the particles' center of mass.
Why would it be different for a distribution of virtual particles?