Closed systems and mass/energy transference

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Hobold
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So I am reading a text written by my Thermodinamics professor and I find something that I still can't accept.

"(Sistema fechado) É aquele que não troca massa. (...) Um sistema fechado é dito isolado quando não troca energia. Entretanto, essa definição de sistema não é consistente com a teoria da relatividade, uma vez que quando energia atravessa a fronteira de um sistema, uma passa proporcional é também transferida segundo a equação de Einstein E=mc^2. (...)"

"(Closed system) does not transfer mass. (...) A closed system is said to be isolated when it doesn't transfer energy. However, this definition isn't consistent with relativity, as when energy crosses the boundary, there is a proportional mass being transferred as Einstein's E=mc^2 (...) However this amount of mass isn't anywhere close to detectable, as c^2 is sufficiently big."

I can't find this assertion to be true. I always thought Einstein's equation was only valid when releasing nucleus energy, such as through fission or fusion. I can't think how mass transference <-> energy transference would work, even if I consider mass as a parameter of the system (though it does not make sense to me).

Could anyone help me understand?
 
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Mass has energy, and energy has mass. The energy entering the system brings its mass with it.