Final answer:
The difference between dτ_A and dt is due to the effects of gravitational potential on time intervals, a concept from general relativity. The 'universal' clock dt is measured by an observer outside the gravitational field, while dτ_A is the proper time experienced locally. The equality Δt_A=Δt_B indicates that periodic signals are similarly affected by gravity at both locations.
Step-by-step explanation:
The student is questioning why dτ_A differs from dt and why Δt_A=Δt_B leads to the Doppler frequency shift formula. This question delves into the realm of general relativity, where the concept of time is affected by gravity. The equation dτ_A = √(1−(2U_A/c^2))dt describes time dilation due to gravitational potential (U_A) at a point A.
The time interval dt is a coordinate time interval that an observer outside the gravitational field would measure. On the other hand, dτ_A is the proper time interval experienced by a clock at point A within the gravitational potential. It is called "proper time" because it's measured locally. The reason Δt_A=Δt_B (where Δt represents the coordinate time difference) is because, for signals sent periodically, the gravitational effects on time intervals should be identical if the potential is constant at both points A and B.
It's important to differentiate dt, as the 'universal' clock, seen from an outside observer’s perspective, from dτ, the proper time experienced by an observer in a frame at a specific gravitational potential. To reconcile this difference in time measurement due to gravity, the general theory of relativity posits that dτ will differ from dt depending on the gravitational potential.