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If the Higgs field permeates all space, why some claim, that total universe energy equals (or is very close to) zero?

Qmechanic
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Dukunocil
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Because space is very, very flat. Even if we don't understand how the fields and masses balance out, simple observation tells us in the long haul there's not much gravitational curvature -- and that means that overall, the universe averages out to zero.

Terry Bollinger
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