Are 2 time intervals the same only for same speeds? I'm only amateur in modern phsyics having read some about relativity theory and time measurements depending on relative movement. Is it true that 2 clocks measure time the same only if the relative speed between the 2 clokcs is 0? and if the 2 clocks move in opposite directions then they will not measure the same time? Similar to the paradox that if one twin spends all life on spaceships travelling near the speed of light, then the travelling twin will be younger than the twin not travelling at fast speed? I wonder if the conclusion is correct that 2 clocks will measure exactly the same time intervals if and only if they are at exactly the same speed and therefore the relative movement between the clocks must be 0?
 A: You have to be careful about the difference between speed and velocity.  Saying that two clocks are moving at the same speed is different from saying that the relative speed between the two clocks is zero.  For example, as measured in some inertial frame of reference, two clocks can be moving at the same speed but in opposite directions, in which case their relative speed (the magnitude of their relative velocity) is not zero.
If the relative velocity between two identical clocks is zero, then the two clocks will measure time intervals the same.
It's also true that as measured in some "lab" inertial frame of reference, if two clocks are moving at the same speed, then they will measure time intervals the same as each other, even if the relative speed between the two clocks is not zero.  However, time intervals as measured by those two moving clocks will be different than time intervals as measured in the lab frame.  Also, just because the two clocks measure time intervals the same as each other as measured in the lab frame, that doesn't mean that it's true in general that the two clocks are measuring time intervals the same.  If two clocks have a non-zero relative speed, it's ambiguous as to whether the two clocks are measuring time intervals the same, or if not, which clock is running faster, unless you specify an inertial frame of reference in which the comparison is to be made.
The rule implied by time dilation is that in an inertial frame of reference in which one of two identical clocks is at rest, the other clock will be measured to be running slower than the clock that's at rest.  However, measurements made in an inertial frame of reference in which the other clock is at rest will come to the opposite conclusion as to which clock is running slow.
