Final answer:
The statement is correct; in flat spacetime, a timelike worldline intersecting with at least three distinct photon 2-surfaces suggests the worldline is a geodesic, and the photon 2-surfaces would be planes according to the postulates of relativity.
Step-by-step explanation:
The statement posed in the question touches on concepts from the theory of relativity and the geometry of spacetime. In flat spacetime, a timelike worldline represents the trajectory of a particle that travels slower than the speed of light. Such a worldline can be intersected by what are termed photon 2-surfaces, which are the surfaces in spacetime formed by the paths of photons emanating from a specific event.
If a timelike worldline belongs to at least three distinct photon 2-surfaces, the question suggests that this worldline is a geodesic segment, implying the simplest, straight-line path through spacetime that a particle can follow without any external forces acting upon it.
Moreover, the intersecting photon 2-surfaces are stated to be all planes. This part of the statement is specific to Minkowski (flat) spacetime where light cones, which contain all the possible world lines of photons emerging from an event, are represented by planes intersecting at 45° angles on a spacetime diagram. This is consistent with the postulates of relativity, which suggest that the speed of light is constant in all inertial frames. Hence, geodesic segments such as the worldline of a particle moving at constant velocity in flat spacetime would indeed appear as straight lines, and photon surfaces would be planes, in a spacetime diagram.