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In Newtonian gravity (NG) it is known that the gravitational field anywhereinside a spherically symmetric distribution of mass is determined only by theenclosed mass. This is also widely believed to be true in general relativity(GR), and the Birkhoff theorem is often invoked to support this analogy betweenNG and GR. Here we show that such an understanding of the Birkhoff theorem isincorrect and leads to erroneous calculations of light deflection and delaytime through matter. The correct metric, matching continuously to the locationof an external observer, is determined both by the enclosed mass and massdistribution outside. The effect of the outside mass is to make the interiorclock run slower, i.e., a slower speed of light for external observer. We alsodiscuss the relations and differences between NG and GR, in light of theresults we obtained in this Lettework. Finally we discuss the GeneralizedShapiro delay, caused by the outside mass, and its possible laboratory test.

On a common misunderstanding of the Birkhoff theorem and light deflection calculation: generalized Shapiro delay and its possible laboratory test