Can an outside observer use electromagnets to solve the Barn-Pole Paradox?

[Kruse]

Please excuse me if I seem inexperienced (I am new to this forum), but lately I've been thinking a lot about a length contraction paradox called the Barn-Pole Paradox. The paradox:

A pole and a barn are both at rest relative to each other. When both are at rest, the pole is slightly longer than the barn. However, when somebody picks up the pole and runs toward the barn, length contraction occurs. In the frame of reference to an outside viewer, the runner and the pole both contract, and the pole appears to disappear entirely inside the barn before the runner emerges from the back door (we're assuming in this instance that there is a front door and back door, both of which are open). However, in the frame of reference of the runner, the barn seems to contract, making it impossible for the pole to entirely disappear inside the barn.

I've already read about closing doors and the runner stopping, but here is my question. Let's say that the outside viewer has a remote switch. When the viewer sees the runner and the pole disappear inside the barn, he or she quickly flips the switch on and off. The switch will trigger what has replaced the doors, which instead are electromagnets (we are also assuming that the pole is made of iroin). During this moment that the observer quickly switches the electromagnets on and off, what will happen in the frame of reference of the runner, where the pole is never entirely fit inside the barn?
Thank you for your time.

 

[JesseM]

The two electromagnets would turn on at different times in the frame of the runner, the front electromagnet would turn on when the front of the pole was still inside the barn, and the back electromagnet would turn on a bit later when the back of the pole had made it inside the barn.

 

[Kruse]

The two electromagnets would turn on at different times in the frame of the runner, the front electromagnet would turn on when the front of the pole was still inside the barn, and the back electromagnet would turn on a bit later when the back of the pole had made it inside the barn.

What causes this?

 

[JesseM]

What causes this?

The word "causes" doesn't really have a clear meaning in this context. On one level it's just a consequence of how different inertial frames are defined according to the Lorentz transformation. The transformation says that if you know the coordinates x,t of an event (like an electromagnet being turned on) in one inertial frame (say, the frame of the barn), then in the rest frame of an object moving at speed v along the x-axis of the first frame, the coordinates of this same event would be:

x' = gamma*(x - vt)
t' = gamma*(t - vx/c^2)

with gamma = 1/sqrt(1 - v^2/c^2). It's easy to check that if you have two events which occur at the same t-coordinate but different x-coordinates in the first frame (i.e. the events are 'simultaneous' in that frame), then the t' coordinates of the two events in the second frame will be different (they are non-simultaneous in the second frame). This feature is known as the relativity of simultaneity.

To understand why this transformation is the one that's used to define different inertial frames in relativity, it may help to understand that the Lorentz transformation is derived from two basic postulates: the first says that the laws of physics must work the same way in every inertial frame, the second says that light must have a speed of c in every frame. It's not hard to see why frames satisfying the second postulate must naturally disagree about simultaneity. For example, suppose I am riding on a rocket and I set off a flash at the exact midpoint of two light detectors at the front and back of the rocket. If I assume that both beams travel at the same speed in my frame, then this necessarily implies that both detectors must go off at the same time in my frame. But if the rocket is moving forward in your frame, then in your frame the detector at the front is moving away from the position where the flash was set off while the detector at the back is moving towards that position, so if both beams move at the same speed in your frame, in your frame the light must catch up with the detector at the back before it catches up with the detector at the front. If this isn't clear you might also take a look at this youtube video illustrating a slight different scenario (two flashes set off at the front and back of a moving vehicle with the light heading for the center, rather than vice versa as in my scenario) which is based on the thought-experiment Einstein described in sections 8 and 9 of his book Relativity: The Special and General Theory.

But as for the ultimate question of why the laws of physics have the property that it's possible to come up with a set of inertial frames satisfying both postulates, physics can't really answer that question. It so happens that all the fundamental laws physicists have found so far have the property that their equations are invariant under the Lorentz transformation above, but we could easily imagine a different set of laws that were instead invariant under some different transformation, like the Galilei transformation used in Newtonian physics:

x' = x - vt
t' = t

If the fundamental laws of physics were Galilei-invariant rather than Lorentz-invariant, it would be impossible to find a set of frames satisfying both postulates together, although you could find a set of frames satisfying either postulate individually (for example, you could still define a set of coordinate systems using the Lorentz transformation in a universe with Galilei-invariant laws, but then the equations of the basic laws of physics would have to be different in each frame so the first postulate would be violated, meaning for example that you couldn't use the same equations to predict how the electromagnet would influence the rod in the runner's frame as the ones you'd use in the Barn's frame). But all the evidence suggests that Lorentz-invariance is just one of the basic symmetries of nature.

 

[Kruse]

Thank you so much for this information, it certainly fills in a lot of gaps and explains a lot. I'll certainly look forward to learning more about simultaneity.