On a common misunderstanding of the Birkhoff theorem and light
  deflection calculation: generalized Shapiro delay and its possible laboratory
  test
  In Newtonian gravity (NG) it is known that the gravitational field anywhere
inside a spherically symmetric distribution of mass is determined only by the
enclosed mass. This is also widely believed to be true in general relativity
(GR), and the Birkhoff theorem is often invoked to support this analogy between
NG and GR. Here we show that such an understanding of the Birkhoff theorem is
incorrect and leads to erroneous calculations of light deflection and delay
time through matter. The correct metric, matching continuously to the location
of an external observer, is determined both by the enclosed mass and mass
distribution outside. The effect of the outside mass is to make the interior
clock run slower, i.e., a slower speed of light for external observer. We also
discuss the relations and differences between NG and GR, in light of the
results we obtained in this Lettework. Finally we discuss the Generalized
Shapiro delay, caused by the outside mass, and its possible laboratory test.
