can anyone derive gravitaional time dilation equation?

Question

i dont want the equation but i want the derivation.is there any einstein to derive it

can anyone give me the mathematical derivations

Answer 1

I recall a heuristic (nonrigorous) derivation that avoids messing with metrics. Conserve gravitational potential plus kinetic energy hf (Planck constant times frequency) of a photon as it travels between two locations. A photon's gravitational mass is E/c^2. Impose the condition that speed of light is c in both frames when the photon is at either location. The frequency of the photon in the frame of the initial location must remain the same to avoid the number of wavelengths between locations from diverging to infinity or going negative over time if a continuous beam is employed. The frequency, in the higher frame, though, must be less to conserve energy. Use the frequency of the photon as your "clock" in both frames and compare.