Imagine looking at a quasar millions of light years away. From your perspective, a photon emitted by the quasar has spent millions of years travelling through space, and its trajectory has been affected by the gravity of all the galaxies between the quasar and you. In non-Euclidean space-time, the path it has travelled along is a straight line.
However, the photon has no mass and travels at the speed of light. According to the theory of relativity, from the photon's perspective, no time passes between the moment it is emitted in the quasar and the moment it is detected in your eye. From the photon's perspective, there is no distance between the quasar and your eye. There is thus no path for it to have followed: for it, the curvature of space-time is not even a straight line; it is a single point.
Any particle which has mass can follow the same "path" but it sees a very different universe.
Is space-time a function of mass?
$E = mc^2$ expresses energy in terms of mass, time and space. Is there a way to reformulate this so that the observed value of the speed of light can be deduced from the existence of mass?