Is this a way to send signals faster than c?
There is no such thing as a rigid body in relativity.
What happens is that the front end of the rod is compressed, and
this compressed form of the iron is transmitted along the rod until
it comes to the end near the Sun; the rate of transmission is (about)
the speed of sound in iron, much slower than c. When you think of iron
you think of something very rigid, but in this context, with very large
distances and when speeds are to be compared to c, think of the iron
rod as a foam rubber rod; the picture in your mind will be more accurate.
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