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Charge density zero? Yes. Mass density zero? No. Particles and
antiparticles have positive mass.

The mass density is indeed zero in a vacuum; that's the mass
density of virtual particles.

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?

Then you would claim that the mass density of a homogeneous
medium is anisotropic in the frame of someone moving with
respect to it. Please explain to me what observable quantity
is associated with an anisotropic mass density.

Leigh