- #36
Ibix
Science Advisor
- 12,319
- 14,262
Is the notation in the paper potentially slightly confusing? What it's calling ##d\tau## is the proper time of an infalling emitter between the emission of wave crests. What it's calling ##dt## is the proper time of an observer hovering at infinity between the reception of those crests. The latter is, of course, equal to the coordinate time differential between reception events. But it isn't equal to the coordinate time differential between the emission events.
Am I right so far? If so, observe that light pulses emitted radially outward ##\Delta t## apart at the same r arrive ##\Delta t## apart at infinity. Then draw a little triangle in the Schwarzschild t-r plane and start filling in results.
I admit I don't see immediately where the exponential comes from in this procedure. My guess would be the exact result is a truncated Taylor series for the paper's exponential. Or that I'm going about this the wrong way.
Am I right so far? If so, observe that light pulses emitted radially outward ##\Delta t## apart at the same r arrive ##\Delta t## apart at infinity. Then draw a little triangle in the Schwarzschild t-r plane and start filling in results.
I admit I don't see immediately where the exponential comes from in this procedure. My guess would be the exact result is a truncated Taylor series for the paper's exponential. Or that I'm going about this the wrong way.