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Why would'nt this work?
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I think relativity demonstrates that light does have mass?
They might not have "rest mass" but they do have mass!
The eclipse experiment proved it, solar sails whilst hypothetical demonstrate it.
Relativistic mass is not helpful to our everyday understanding of mass, it's more helpful to discuss momentum, like the other commenter pointed out
Photons don't have mass, but they do have momentum.
How does that work?