In a universe where space and time are two independent things. Where time is universal. Where time "passes at the same speed" in any point in space, and dilation never occurs, regardless of mass and speed. What would it imply and how would it affect the other laws of physics?
closed as too broad by Mołot, Andon, RonJohn, Logan R. Kearsley, JBH Apr 23 '18 at 4:53
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The answer to this depends heavily on how much of modern physics is not modified by the change. I recommend reading a good history of relativity to understand why time dilation is meaningful. If you can't find a good history, one of my answers may do in a pinch.
Basically, the most natural answer to your question is that your universe has an Aether, a medium in which light travels, which defines the frame in which the speed of light is calculated. Before experimenters like Michaelson and Moorley did experiments which disproved the Aether, this was actually the leading assumption. If there is an Aether, all of the equations of light (Maxwell's equations) can simply work without alteration.
If, however, your world has no time dilation and M&M's experiment yielded the same results that we see in our world, then this means that we have to play some more frustrating math games. The experiments which disproved the Aether showed that light exhibited what is called the Lorentz boost. This is a mathematical transform which takes the simple Maxwell's equations and turns them into equations which match the actual experimental results. However, they are nothing but mathematical equations. They did not have any physical significance until Einstein posited that they could be explained by stretching of space and time. This lead to the experiments which showed that not only light exhibited these behaviors, but physical objects did as well, showing that it was indeed a good idea to model these effects as space and time stretching.
If these experiments had gone differently, and found that physical objects did not act this way, the result would just be a frustratingly complex set of Maxwell's equations. There's a certain love of simplicity in the heart of physicists, and Maxwell's equations explain a lot while being very simple. It makes physicists happy. Physicists like to be happy, but they like the models that explain things more, so they'd have to accept these ugly equations.
As for effects on the universe itself, that's hard to say without pinning down precisely how physics works in your universe. However, a likely consequence would be that black holes cannot form. They depend on the ability of gravity to bend spacetime to come into being. However, whether they can exist or cannot is something that cannot be said without knowing your physics.