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?
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.
Meh, no string theory or singularity or unitology etc. Black Holes are accelerators. Every particle is broken down to the lowest level, accelerated to the highest speed and shot out. The path is non-vectorial, because the exitement of such magnitude makes the low-level particles act as "quantum binded particles". Which means, for every exited particle, there is another particle, it communicates. Through this binding efect, both particles and their respective "anti or dark matter" particles interact and lose energy. they slow down and under the rule of conservation of mass, stick together again. Far far away. the thing here is to think of binded particles as identical, reformable parts, rather than individual particles whizzing about.
Thats why general physics and traditional laws of nature break down in a black hole. Because the respective laws only go down as far as neutrons and protons.
Quarks, leptons, bosons, muons, tauonic particles etc.
Once we reach the bottom, we figure out the black holes. Until then, its all classical mechanics...on a cosmic scale :)