I read a book on Einstein's Legacy (in Finnish). It was an interesting
tour through the predictions that Einstein made in physics. The author
is an astronomer who has lectured general relativity at a university
level. He starts each chapter by quoting an original article either by
Einstein or some other researcher using Einstein's ideas and theories.
This is the best overview of Einstein's scientific work I've read so far.
However, there are two claims in the book which surprised me (quotations
translated by me).
1) "Nowadays the Equivalence Principle has mainly historical meaning."
This is because it's just an approximation of GR. The evidence offered
is Einstein's paper in 1911 in which he calculated the bending of light
due to the Sun, and got an answer which is only half of the measured
value. The final prediction was made in his 1915 paper and this provided
a correct value.
Is it really so that the Equivalence Principle, if correctly applied,
fails to yield the correct value in the experiments carried out by
Eddington and others? Has the Equivalence Principle only pedagogical
value anymore, that is, is it virtually useless for research purposes?
2) "The speed of light is slowed down in a gravitational field."
The author cites Einstein's paper in 1911. In that paper, Einstein
derived a formula indicating that the speed of light is a function of
position in a gravitational field (that is, depends on the gravitational
potential). The empirical evidence is provided by the Shapiro time delay
experiments (https://en.wikipedia.org/wiki/Shapiro_time_delay). More
recently, the Cassini spacecraft has produced a very precise
confirmation of the time delay effect:
https://www.nature.com/articles/nature01997.
My question is the following: Does the modern interpretation of GR still
maintain that the speed of light depends on the gravitational potential
as stated by Einstein in 1911? If so, wouldn't this mean that when light
climbs from a gravitational well, it would experience both the
gravitational redshift
(https://astronomy.swin.edu.au/cosmos/g/Gravitational+Redshift) *and*
the decrease in speed?
A final note: Many sources seem to agree that "In GR the velocity of
light is only locally equal to c."