The time delay of a solar grazing photon

Moving clocks and clocks in a gravitational field slow down, not due to the effects of special relativity nor to the space-time curvature of general relativity, but due only to the principle of equivalence and the conservation of energy. However, some might argue that there has been a further "test" of the effect of gravity on time, namely, the measurement of the time-delay of a round-trip, solar-grazing radar beacon performed by Shapiro in the 1960s. In this test, Shapiro bounced a radar pulse off Mars at superior conjunction (a feat in itself for the time), and compared the measured round-trip travel time of this pulse with the expected round-trip time of a signal traveling at c for the entire trip, as determined from highly accurate planetary ephemerides. Shapiro had predicted this time-delay long before being technologically able to make such a measurement. While general relativity can be used to correctly obtain the magnitude of this delay, it is not the only explanation. As earlier with the analysis of clocks in motion and of clocks in a gravitational field, this paper derives the same result without invoking the space-time curvature of general relativity.