Assume a photon enters the event horizon of a black hole. The gravity of the black hole will draw the photon into the singularity eventually. Doesn't the photon come to rest and therefore lose it's mass?
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We don't know what will happen when a photon or any other particle hits a singularity of a black hole. The singularity is a phenomenon of classical general relativity and the singularity is really is an indication that classical general relativity breaks down there. To really understand what happens near a singularity we need a full quantum mechanical version of general relativity. String theory is the best quantum mechanical version of general relativity that we currently have but string theory is not developed enough to give a definitive answer to your question.