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There's a rule in general relativity that says your 4-velocity is always normalized to the speed of light. If your 3-velocity is equal to zero, then your 1-velocity, which is the rate at which clocks tick in your local frame, accounts for all of the 4-velocity. If you increase your 3-velocity, then your 1-velocity has to decrease in order to make the total normalization of the 4-velocity hold. When your 1-velocity decreases, clocks tick slower in your local frame. It's pretty simple. The reason the 4-velocity is normalized is slightly more complicated. It has to do with special relativistic simultaneity.