DW answered this.pmb_phy said:Actually Einstein held that the mass of a particle will be altered in a gravitational field. That notion of mass is identical to the notion of relativistic mass in definition..
You have answered your own question . The particle is at rest in it's own reference frame, but moving in others . It's (hypothetical) clocks and measuring rods must measure time and space differently than those in other reference frames.A particle is always at rest in its own reference frame so as such I don't understand what you mean by it seeing spacetime differently. Can you clarify please?.
Beauty may be in the (subjective) eye of the beholder, but mass is an objective property of the particle itself.Particle's don't make measurements. Observers do. Hence the observer dependant nature of mass etc.
Arithmeticly you may get the right answer, but that does not answer the original question, "where does the extra mass come from?" .Why? Its quite possible to do all calculations in special relativitywithout even knowing what proper velocity is. If you want to stick to 4-vectors and Lorentz invariants then you really shouldn't use the term "speed" or "velocity" etc. In such a case the magnitude of a particle's speed is the magnitude of the particle's 4-velocity and that is always c.
As you, yourself said, the particle is always at rest in its own frame. Therefore, it always has the same mass in that system.