What are some commonly known facts that are too bizarre for you to believe to be true?
For me it is the fact that our blood contains iron. I earlier used to believe the word stood for some 'organic element' since I couldn't accept we had metal flowing through our supposed carbon-based bodies, till I realized that is where the taste and smell of blood comes from.
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I wonder how long it would have taken for us to figure out time dilation in Einstein hadn't predicted it. I wonder if it would have taken until we observed it with satellites.
Without Einstein, I think that the discovery of time dilation would be delayed by only a few years. There were a lot of people working in theoretical physics already back then; someone else would inevitably dig through Lorentz' and Poincaré's papers, connect the dots, and say "waitaminute time might be relative". From that, time dilation is a consequence.
In special I wouldn't doubt that Max Planck would discover it.
I'm saying that because, in both science and engineering, often you see almost concurrent discoveries or developments of the same thing, because the "spirit of a time" makes people look at that aspect of reality or that challenge and work with it. The discovery of helium and the development of aeroplanes are examples of that.
IIRC the orbit of Mercure doesn't work with Newton Model, and astronomers were predicted the discovery of Vulcain a small planet between Mercure and the Sun. So a new model had to be invented since Vulcain couldn't be found.
We would have definitely figured it out once we built GPS, since you need to account for relativistic effects there.