Originally Posted by rpj
Imagine the object moving from a to b to c. From a to b it would take less time measured withing the object than measured outside the movement from a stationary point.
|
For differences in time due only to velocity, the measurement doesn't depend on where you are in space. It depends on what spacetime coordinate system (reference frame) you're using. Each coordinate system exists throughout space and time. Each coordinate system applies both inside and outside of all the objects being described, regardless of how they're moving relative to each other.
Consider a reference frame that keeps pace with the object (moves with the object = has the same velocity as the object). We call this the object's rest frame because the object is at rest in such a coordinate system. Measured according to this frame, a certain time passes between the event of the object being at A and the event of the object being at B. But we can't use this time to calculate the velocity of the object, because in the object's rest frame - by definition - it has no velocity: it's standing still.
It's only when measured according to a reference frame (spacetime coordinate system) moving relative to the object, that the object has any velocity at all. For example, we could set up a coordinate system in which points A, B and C are at rest. Now it's the object that moves past these points, rather than the points that move past the object.
From the perspective of the object's rest frame, time goes slow according to the rest frame of the moving points. From the perspective of the points' rest frame, time goes slow according to the rest frame of the moving object. But in each case, it goes slower by the same amount! From this, we see that the velocity of the moving points is the same when measured according to the object's rest frame as the velocity of the moving object when measured according to the points' rest frame. So there is no paradox.
Originally Posted by rpj
From b to c it would take even less time measured from within the object and thus it would be accelerating as time goes slower and slower but the distance inbetween a and c remains constant.
|
If the distance from A to B is the same as the distance from B to C, according to a reference frame where the object is moving, such as the rest frame of the points, then it will cover the same distance in the same time - this time being always
the time according to the reference frame in which the object's position is changing. We can't use the smaller amount of time that passes according to the object's rest frame, because according to that frame, the object isn't actually moving.
And it certainly wouldn't make sense to start measuring velocity using the time as measured according to one frame, then switch to using the time according to another frame, then treating the reduced time as if it was the yet to be reduced time, reducing it, and finding the limit as this recursive process is performed infinitely many times.
Originally Posted by rpj
My paradox is how can the object be accelerating if you measure time from withing the object and be at constant speed when time is measured from a stationary point.
Relativity aproves that they can be multiple places where time goes at different speeds. To divide a certain space into multiple areas each area needs to take up a certain space.
When the object moves, all around the object time is going slower.
|
Again, for considering only the sort of time differences between coordinate systems due to their movement at a constant velocity relative to each other, the amount of time that passes between two events doesn't depend on
where in space you are, only on what spacetime coordinate system you use to locate events. When we say "in a reference frame", we mean "according to a coordinate system", rather than inside an object or within some particular region of space. The coordinate system applies to all of the spacetime under discussion.
