Time Dilation and Proper Length

Why would time move slower for an astronaut traveling at deep space near relativistic speeds compared to an earthbound observer?

Because motion is relative, the spaceship would seem to be stationary and the Earth would appear to move. If an astronaut made a trip to deep space and traveled at a speed 0.8c, 6 years for him would be 10 years for someone on earth. Alternately, the astronaut would measure 10 years on his frame while he would measure 6 years to have elapsed on earth.

What did the analysis of simultaneity deduce about Einstein's postulates?

That time intervals have different values when they are measured in different inertial frames.

Length contraction

The decrease in the length of an object compared to its proper length when measuring a reference frame that is moving with respect to that observer.

What is Time Dilation?

The difference in time between two events, one being in an inertial frame and one being in a rest frame, between simultaneous events that occur at the same location.

What is proper length?

The distance between two points measured by an observer that is in rest relative to both inertial frames of reference. Someone sitting on a plane travelling at 0.8c will measure the proper length of a plane compared to someone stationary on the ground, who will see the plane to be shorter.

Given that one inertial frame of reference is accelerating past another at some velocity, what is one thing both observers in both frames will agree on?

The relative speed between them. Although both observers will measure different times and different lengths, they will both measure the same speed for the other reference frame.

What is Proper Time?

The time interval measured by the observer where both events occur at the same location. According to the earthbound observer, less time passes between events within the moving frame. The shortest elapsed time between events is in the inertial frame in which the observer sees the events.