WHAT YOU CAN DO
BOOSTING Use the scrollbar or the [LEFT-ARROW] and [RIGHT-ARROW] keys to adjust the boostVelocity (i.e. perform a Galilean or Lorentz transformation). The applet will display the spacetime diagram for an inertial observer traveling with that velocity. Note: -1 < boostVelocity < 1. DRAGGING Click-down and drag the mouse to slide the diagram around. ZOOMING ZOOM IN and OUT with the applet-buttons "Z" and "z" or the [UP-ARROW] and [DOWN-ARROW] keys. RESETTING To reset the display, hit [R]. |
What I want you to see.
"Time" runs upwards in the spacetime diagram.The applet tracks the Spacetime Diagram's (t,x) coordinates of the Mouse. What is a Boost?A boost (called a Lorentz transformation for Minkowski spacetime and a Galilean transformation for Galilean spacetime) is analogous to a (Euclidean) rotation in Euclidean space.Recall that in Euclidean space, points being pushed along by a rotation will trace out circles centered at the origin of the rotation. Each circle represents the set of points "equidistant" from the origin along different directions in Euclidean space. (Check "Equal intervals from O".) Lorentz-boosted clock-ticks travel on hyperbolas centered at the lower vertex. Each hyperbola represents the set of events "equidistant (in time)" from the origin for different inertial observers in Minkowski spacetime. For Galilean spacetime, the boosted clock-ticks travel along horizontal lines. Each horizontal line represents the set of events "equidistant (in time)" from the origin for different inertial observers in Galilean spacetime. Clock Effect.Suppose the "traveling" twin (traveling with boostVelocity 0.8) experiences 6 clock-ticks between the separation event O and the reunion event T.In "Galilean" spacetime, the "stay-at-home" twin also experiences 6 clock-ticks between the separation event O and the reunion event T. In "Lorentzian" spacetime, however, the "stay-at-home" twin experiences 10 clock-ticks between the separation event O and the reunion event T. Time Dilation.In Lorentzian spacetime, set the boostVelocity to 0.00. The applet displays the spacetime diagram of the "stay-at-home" twin. Looking at this diagram, the "stay-at-home" twin declares thatmy clock reads "5 ticks" when the "traveling" twin turns around at event Q. However, the "traveling" twin declares that my clock reads "3 ticks" when I turn around at event Q. It is sometimes said that the "stay-at-home" twin experiences time dilation when comparing the elapsed-times between a reference event O and two events: a distant-event Q he regards as simultaneous to a local-event: his clock-tick 5. The ratio 5/3 is called the gamma factor. (In Galilean spacetime, the analog of the gamma factor is always 1. Hence, there is no time dilation in Galilean spacetime.) Time Dilation is symmetrical.To set up a comparison, set the boostVelocity to 0.00. The "stay-at-home" twin declares thatmy clock reads "3 ticks" when the "traveling" twin's clock reads 3/gamma= 9/5 ticks. Now, set the boostVelocity to 0.80. The applet displays the spacetime diagram of the "traveling" twin. The "traveling" twin declares that my clock reads "3 ticks" when the "stay-at-home" twin's clock reads 3/gamma= 9/5 ticks.
Observe that "time-dilation" is symmetrical between the two observers until the traveling twin turns around and becomes a non-inertial observer. |
How do you resolve the Twin Paradox?The "stay-at-home" twin is an inertial observer.The outgoing "traveling" twin is an inertial observer. Although the incoming "traveling" twin is also an inertial observer, it's not the same as the "outgoing observer". In other words, the complete trip (outgoing and incoming taken together) is not an inertial trip. Recall what INERTIAL means. A particle traveling with the outgoing twin travels in a straight line with constant speed. If it is released before event Q, it will still follow the trajectory of the outgoing twin. However, at event Q, it will not "turn" to follow the trajectory of the incoming twin. Instead, it continues along the original straight line with the original constant speed. (Newton's First Law: The Law of Inertia). Realize that no choice of boostVelocity (i.e. no Lorentz or Galilean transformation) will ever straighten out the kink on the traveling twin's worldline. Hence, the two observers are not equivalent. Realize that there is no paradox if the spacetime is Galilean. But then in this case, the speed of light is not the same for all observers. (Check the "Light Cone of Q" and/or "Light Cone of O" and study the diagram in "Galilean" mode.) |