There is no such thing as a solid rod. Everything acts like a spring at some level. So when you push on the end of the rod it will compress and it takes some amount of time for this pressure to move along the rod. The movement is the same as sound which means that the movement goes at the speed of sound through the material. For steel the speed of sound is about 3-5km/s. About 10-20 times faster then the speed of sound in air. But still orders of magnitude slower then the speed of light. So even at the fastest end of the spectrum it would take about 60.000 years for the movement to go through the entire light year long rod.
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