Editing Ladder paradox (section)

==Resolution==
[[File:Ladder Paradox GarageScenario.svg|thumb|250px|left|Figure 4: Scenario in the garage frame: a length contracted ladder passing through the garage]]
[[File:Ladder Paradox LadderScenario.svg|thumb|250px|Figure 5: Scenario in the ladder frame: a length contracted garage passing over the ladder. Only one door is closed at any time]]
The solution to the apparent paradox lies in the [[relativity of simultaneity]]: what one observer (e.g. with the garage) considers to be two simultaneous events may not in fact be simultaneous to another observer (e.g. with the ladder). When we say the ladder "fits" inside the garage, what we mean precisely is that, at some specific time, the position of the back of the ladder and the position of the front of the ladder were both inside the garage; in other words, the front and back of the ladder were inside the garage simultaneously. As simultaneity is relative, then, two observers disagree on whether the ladder fits. To the observer with the garage, the back end of the ladder was in the garage at the same time that the front end of the ladder was, and so the ladder fit; but to the observer with the ladder, these two events were not simultaneous, and the ladder did not fit.

A clear way of seeing this is to consider the doors, which, in the frame of the garage, close for the brief period that the ladder is fully inside. We now look at these events in the frame of the ladder. The first event is the front of the ladder approaching the exit door of the garage. The door closes, and then opens again to let the front of the ladder pass through. At a later time, the back of the ladder passes through the entrance door, which closes and then opens. We see that, as simultaneity is relative, the two doors did not need to be shut at the same time, and the ladder did not need to fit inside the garage.

The situation can be further illustrated by the [[Minkowski diagram]] below. The diagram is in the rest frame of the garage. The vertical light-blue band shows the garage in spacetime, and the light-red band shows the ladder in spacetime.  The x and t axes are the garage space and time axes, respectively, and x{{prime}} and t{{prime}} are the ladder space and time axes, respectively.

In the frame of the garage, the ladder at any specific time is represented by a horizontal set of points, parallel to the x axis, in the red band. One example is the bold blue line segment, which lies inside the blue band representing the garage, and which represents the ladder at a time when it is fully inside the garage. In the frame of the ladder, however, sets of simultaneous events lie on lines parallel to the x' axis; the ladder at any specific time is therefore represented by a cross section of such a line with the red band. One such example is the bold red line segment. We see that such line segments never lie fully inside the blue band; that is, the ladder never lies fully inside the garage.

[[File:LadderParadox1 Minkowski.svg|center|thumb|325px|Figure 6: A Minkowski diagram of the ladder paradox. The garage is shown in light blue, the ladder in light red. The diagram is in the rest frame of the garage, with x and t being the garage space and time axes, respectively. The ladder frame is for a person sitting on the front of the ladder, with x{{prime}} and t{{prime}} being the ladder space and time axes respectively. The blue and red lines, AB and AC, depict the ladder at the time when its front end meets the garage's exit door, in the frame of reference of the garage and the ladder, respectively. Event D is the rear end of the ladder reaching the garage's entrance.]]