Is time dilation related more to the speed of light or the distance away from earth the object is?
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I guess the speed of light, sort of. Basically, if I travel at twice the speed you do, I can cover twice the distance you do in the same time. Now think of "distance" as time itself. I'm traveling at twice the speed you do, I cover twice the time you do in the same time. We don't feel this affect because we are on Earth, which has mass, floating around Sun, which has mass, in the Milky Way, which has mass, and we have mass ourselves, and mass creates a dip in space-time. Comparatively, photons (particles of light) have no mass, which means they don't create a dip, hence they travel at the speed of light, and do not feel the passage of time, as they are not influencing time. The less you influence time, the slower time passes for you. If by any chance you're really into group theory, its easier to just describe space time through the Poincare group, its signature being (-,+,+,+).