Time dilation, time slowing down, as well as length contraction, mass becoming smaller and denser, come out of Einstein's equations because he wrote it into his equations by having his equations based on his theory that nothing of mass can travel faster than the speed of light and that the speed of light is the fastest possible speed (an upper speed limit) of the universe.  Einstein's equations, which he modified from the equations written for Relativity by Marcel Grossmann, were based on Einstein's theory of how the universe worked.  Therefore, simply solving the equations will give you no information about whether or not the results which they give are of any value.  Equations give you the results they were written to give, but those results are only of value if the theory on which the equations were based was correct.  Consequently, to know if there is time dilation and length contraction one does not solve the equations, as they were written to produce those results, one analyzes the theory from which the equations came; if the theory is wrong then the equations will provide wrong results.

The documents in my main paper: "Einstein's Theory of Relativity Misconceptions & Errors" and in four of my supporting papers, "Black Holes & Gravitational Lensing Misconceptions", "Einstein's Math Question", "Einstein's Doubts and Errors" and "Scinece's Questionable Information About Einstein and Relativity", provide ample proof that Relativity is wrong, that Einstein knew it was wrong, and that scientists around the world pretend that it is a valid theory because it pays them a lot of money to keep researching it.  As Relativity is wrong it is logical that time dilation and length contraction and the constancy and of the speed of light, and the speed of light being the unattainable (by objects of mass) speed limit of the universe, are also wrong.

At page 37, in the footnote, Einstein points out that because of the way he set up his equations nothing of mass can travel faster than the speed of light c.  That does not mean it is a true fact of the universe, it only means that when you use Einstein's equations you will never get a result showing something of mass going faster than c because the equations were written to prevent such a result from coming out of them.  Equations can be written to get the result you want to get, but equations are only lines on a piece of paper, they have no power to make the real world behave as they claim it will behave.

In my companion paper "Einstein's Doubts and Errors" using quotations from Einstein's Book, I show that his "Theory of Relativity" was based on his assumption that the universe is isotropic, which as explained in that companion paper, is a wrong assumption, which is why Einstein's equations give an answer for the mass of the universe that is 95% wrong.  That huge error which comes out of Einstein's "Theory of Relativity" is another very good reason to conclude that the time dilation and length contraction that comes from his equations are errors.
My 1976 paper "Useful Thoughts on Infinity" was studied by some of the best mathematicians in the world for over one year and they could find no errors in it; my paper proved that math's concept of infinity was wrong and that Einstein's Relativity was wrong.  My work was ignored so that scientists and mathematicians around the world could continue to get paid for studying relativity and could avoid having to change (at that time) over 75 years of the math world's concept of infinity.  I do not feel like giving out anything further for free, therefore, my work which can demonstrate that the Relativistic time dilation experiments which have been done for more than a hundred years are theoretically and experimentally faulty and consequently prove nothing, and my work which would disprove objects of mass having their momentum and consequently their mass become infinite at the speed of light, is not being given for free. However, I do provide a cryptic message, which anyone can see if they click here.  If science wants my work demonstrating that the time dilation experiments are theoretically and experimentally faulty and that an object's momentum / mass will not become infinite at the speed of light, then science will have to pay me for it, though I suspect that many good physcists and mathematicians already know the foregoing.

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Einstein Theory of Relativity Misconceptions and Errors paper.