ConfusedRiou said:
OK OK i think i might have it. Is it the gravity that makes time go slower? The inertia from the acceleration of velocity.if this is true what comparison does gravtity have with time?
No, nothing to do with gravity. I think it's best to think in terms of a spacetime diagram with time on one axis and space on another, so you can see each twin's worldline as a path through spacetime. Here's one from the
twin paradox page, with time as the vertical axis and space as the horizontal one, drawn from the perspective of the frame where the Earth twin (Terence) is at rest so his position in space doesn't change over time in this frame:
You can see that the
length of the two paths is different--if these were ordinary paths in space, and you drove along each one with an odometer running, then even if the two cars had the same odometer reading when they departed from the point at the bottom of the diagram, they would have different odometer readings when they reunited at the top. In ordinary space, if a car travels in a straight line between two points with a separation \Delta x along the horizontal x-axis, and separation \Delta y along the vertical y-axis, then the distance traveled between the points (the amount the odometer reading will increase) is just given by the pythagorean theorem, it's \sqrt{\Delta x^2 + \Delta y^2}. In relativity clocks measure time elapsed along paths through spacetime in much the same way that odometers measure distance elapsed along paths through space, but the formula is slightly different--if a clock travels along a straight path between two points in spacetime (like Terence's path above, or like Stella's path between the 'start event' and the beginning of the 'turnaround'), and the two points in spacetime have a spatial separation of \Delta x and a time separation of \Delta t, then the amount the clock will increase is given by \sqrt{\Delta t^2 - (1/c^2)*\Delta x^2} (if you use a system of units where c=1, like light-years for distance and years for time, then this can just be written as \sqrt{\Delta t^2 - \Delta x^2}, which looks almost like the Pythagorean formula except for the minus sign). Because this formula is a little different than the Pythagorean formula, it actually works out that a straight-line path between two points in spacetime (like Terence's between the start event and the return event) will always correspond to a
greater amount of clock time elapsed than a non-straight path between the same two points (like Stella's path), unlike with spatial paths where a straight-line path between two points in space always has a
shorter distance than a non-straight path between the same points (because in Euclidean geometry a straight line is the shortest distance between points).
But aside from that difference, it's closely analogous. The reason the traveling twin ages less has to do with the overall shape of the paths, you can't pinpoint the moment where the Earth twin ages more, just like with spatial paths you can't pinpoint a particular section of the bent path that the extra distance is accumulated on the odometer. And just as different frames can disagree on which twin is aging more during a particular phase of the trip like the outbound leg, it's also true that different Cartesian coordinate systems in 2D space could disagree on which car had accumulated a greater odometer reading at a particular height along the y-axis. For example, suppose in the above diagram we have the y-axis oriented vertically and the x-axis oriented horizontally--in this case, if we pick the y-coordinate of the turnaround, Stella's car will have accumulated a greater odometer reading at that y-coordinate than Terence's car. But now suppose we orient the y-axis so it's parallel to the outbound leg of Stella's trip (which is equivalent to keeping the y-axis vertical and rotating the diagram so that the outbound leg of Stella's trip is vertical while Terence's is at an angle)--in this case, at the y-coordinate of the turnaround, it will be Terence who's accumulated a greater odometer reading than Stella. So until the two cars reunite at a common location, there is no objective frame-independent truth about which has accumulated a greater odometer reading at a given y-coordinate, just like in relativity there is no objective frame-independent truth about which clock has accumulated a greater amount of time at a given t-coordinate.
