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You have to start where Lorentz, Larmor, Poincare, and Einstein started: the problem with the concancy of the speed of light. Start by talking about the number of experiments done to try to find the speed of Earth relative to the ether in the 19th century, including the famous one by Michelson and Morely. All such experiments, no matter how accurate, always showed the speed of light to be the same no matter what season the Earth was in (no matter what direction it was moving relative to the supposed ether). These experiments convinced Lorentz to come up with the Lorentz transformations as a replacement to the Galilean transformations. Also, Maxwell's equations for electricity and magnetism suggest that the speed of light should be the same for all observers, which is what prompted Poincare and Einstein to look for a replacement to Newtonian dynamics. The combination of Maxwell's theories and the experimental evidence that light appears to move at the same speed for all observers is what started each of these men on a search for a new theory, and Einstein was the one who took it the furthest. Ask your friend how light can always appear to move at the same speed no matter who observes it or what the source of the light is. This is very different from Newtonian physics, which asserts that different observers will disagree with the speed of a baseball if they are moving at different speeds. But the speed of light is always the same, no matter how fast or how slow or what direction you're moving. This obviously requires a dramatic reshaping of classical ideas. The solution found, in the special theory of relativity, was that lengths must contract and time must dilate as speed increases. Follow up your argument by stating that this theory has been sufficiently proven by experiment and many of our modern conveniences wouldn't work without an understanding of the theory of relativity. For instance, GPS satellites have to account for time dilation in order to perform accurately.
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