kknull said:
so... if we erase all Doppler effects, the twin who is traveling believes that his brother is younger, but when he returns home, he magically sees that his brother is older then himself??!
Huh?
To answer this question, I have to understand what you mean by "erasing all Doppler effects".
I don't understand what you mean by this question. In fact, it is the only part of your problem description that seems "magic" to me, because Doppler shifts are a part of physical law, and erasing them would take some sort of "magic" that I don't understand.
Offhand, it sounds like you're probablly mired with an incorrect notion of simultaneity - i.e. that you think "at the same time" has an unambiguous meaning. Unfortunately, "at the same time" means different things to different people according to relativity.
This can be seen with a space-time diagram.
<br />
\]<br />
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{\color{blue}\multiput(40,20)(0.12,0.36){167}{\line(0,1){0.36}}<br />
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\[<br />
Above is a space-time diagram. Arbitrary coordinates have been assigned to the set of events that comprise space-time. Every event in space-time is represented by one and only one location on the diagram.In this diagram the solid red line represents the path through space-time of observer #1.
The solid blue line represents the path through space-time of observer #2.
Observer #1 and observer #2 are moving with respect to each other. In fact, as you can see, we have adopted the coordinates of observer #1 to label the points on this diagram. This was an arbitrary choice on our part, and has no physical significance - coordinates never have any physical significance, they are the map, not the territory,.
For our convenience, however, we need to use some unique labels at every point to distinguish different points in space-time from each other.
We can see that on this diagram, for obsever #1, the "time coordinate" is the vertical position on this diagram, and the "space coordinate" is the horizontal position on this diagram.
The dotted red line represents the notion of simultaneity of obsever #1 - all points on this line are regarded as simultaneous by observer #1.
The dotted blue line represents the notion of simultaneity of observer #2.
Note that the two set of events that are regarded as simultaneous are not the same - they are different sets!
