Question

Does time dilate for pedestrian relative velocities over very large distances?

Answer 1

I was looking at the lorentz transform for time when it occurred to me that if I wanted to create a time dilation factor of two, I could set the relative times and velocity arbitrarily, say 1sec and 2sec and 10m/s. When I solved for X I got an answer of almost exactly one light year! (1e16) Does this imply that objects in deep space are time dilated relative to us even at low relative velocities

Answer 2

Let K and K' be two inertial reference frames with relative velocity v. For any event, K records coordinates (x,t), while K' records (x′,t′). Assume E1 has coordinates (0,0) in both K and K'. E2 has coordinates (x,t) in K and coordinates (0,t′) in K'. The elapsed time between the two events are Δt=t in K and Δt′=t′ in K'.

The ratio of Δt′/Δt=√1−v2/c2 depends only on v, but the difference t′−t can be arbitrarily large, if x gets large. When we talk about time dilation, we care more about the ratio of elapsed time, rather than the absolute difference, which accumulates.

The term "time dilation" itself is very misleading.