Solving Paradox: Man Running with Pole Reaches Door Before Button is Pressed

[Jrs580]

TL;DR Summary

Special relativity paradox

I think I broke special relativity…not really but I am clearly over looking something. Imagin a man carrying a pole (like a pole vaulter) running at speed v. The length of the pole = L, so in the frame of the person watching this man run, the length of the pole is observed as L/γ. There is a button on the ground that opens a door some length away, let’s call that distance d=L/γ+ct+ε. The ct accounts for the time it takes for the button to open the door. So according the the person observing the man running…the door will always open just before the end of the pole gets there, no matter how fast he runs. Here’s where the paradox comes in…in the frame of the running man…the end of the pole gets to the door BEFORE he can push the button on the ground! To him, the distance between the button and door is length contracted…giving him the result of the pole hitting the door before it opens. What gives?

 

[DaveC426913]

The trick is - as it always is - in the relativity of simultaneity.

Read up on 'Ladder Paradox' or 'Barn Door Paradox'.

 

[FactChecker]

The ct accounts for the time it takes for the button to open the door.

I think this is a problem. Whose time is this and what does the other observer think of this time? From your description, it sounds like it is the stationary observer's time. To the runner, the distance to the door shrinks, but also the stationary observer's time is longer (by the runner's clock), so the button is farther away. The two effects cancel out. They all agree that the door opens in time, but for different reasons.

 

[Ibix]

TL;DR Summary: Special relativity paradox

in the frame of the running man…the end of the pole gets to the door BEFORE he can push the button on the ground! To him, the distance between the button and door is length contracted…giving him the result of the pole hitting the door before it opens. What gives?

It's difficult to see how you reached this conclusion because you waved your hands a lot instead of doing maths. For example, you could easily compute the distance you need between button and door in the two frames. If you do that I think you'll find it works - most likely you've forgotten that the signal doesn't need to cross the whole distance in the man's frame because the door is coming to him.

The other approach, the general one that works for all SR problems, is to write down the coordinates of events of interest in one frame, apply the Lorentz transforms (not just time dilation and length contraction, which are special cases with extra assumptions), and see what the results are. Drawing Minkowski diagrams afterwards is recommended if you want to learn to be able to solve these kinds of problem intuitively.

 

[Dale]

TL;DR Summary: Special relativity paradox

There is a button on the ground that opens a door some length away, let’s call that distance d=L/γ+ct+ε. The ct accounts for the time it takes for the button to open the door.

OK, so not even getting to relativity, the first thing to do is to fix your equation. Here $$d=L/\gamma+ct + \epsilon$$ is problematic, as mentioned by @Ibix and @FactChecker . The issue is that $t$ is a variable, so this $d$, as written, is a distance that expands at the speed of light. What you have as $ct$ needs to be a constant, and what you want is for the constant to be such that with the end of the rod at $L/\gamma + v t$ and with the flash of light from the button at $ct$ both reach the door at $d=L/\gamma + D$ at the same time. So, if we set $$L/\gamma + D = c t = L/\gamma + v t$$ then we can solve for $D$ and $t$ to get $$d=\frac{L}{\gamma}+\frac{L}{\gamma}\frac{v}{c-v}+\epsilon$$

Now, with $\epsilon=0$ the pole and the light from the button reach the door at the same time, with $\epsilon<0$ the light reaches the door first, and with $\epsilon>0$ the pole reaches the door first.

With that then use the Lorentz transform so that you account for the relativity of simultaneity, as @DaveC426913 mentioned, and all will work out.

 

[Jrs580]

t was intended to be the time it took for the button push signal to reach the door, assuming the signal travels at the speed of light. So, if we arbitrarily pick d, it fixes t.

 

[Jrs580]

I think the point is, I can boost the poles frame enough to make the distance between the button and the door effectively 0, which will be less than the length of the pole. So the pole will hit the door at relativistic speeds, but not at low speeds.

 

[PeroK]

I think the point is, I can boost the poles frame enough to make the distance between the button and the door effectively 0, which will be less than the length of the pole. So the pole will hit the door at relativistic speeds, but not at low speeds.

Have you heard of the relativity of simultaneity? Or, the Lorentz Transformation?

It's true that if you assume that SR entails only length contraction and/or time dilation, then it is self contradictory. But, SR isn't only these two things. That's what gives.

 

[Ibix]

I think the point is, I can boost the poles frame enough to make the distance between the button and the door effectively 0, which will be less than the length of the pole.

Then do the maths and show it instead of waving your hands and "thinking" it (which means you're guessing, really).

You will prove yourself wrong if you do the maths, as @Dale has already shown.

 

[PeterDonis]

t was intended to be the time it took for the button push signal to reach the door, assuming the signal travels at the speed of light. So, if we arbitrarily pick d, it fixes t.

That time is frame dependent, because the distance from the button to the door is frame dependent.

 

[Dale]

t was intended to be the time it took for the button push signal to reach the door, assuming the signal travels at the speed of light. So, if we arbitrarily pick d, it fixes t.

Yes, I understood that. But the notation is bad because $t$ is usually considered a variable, not a constant. And in any case we need a variable for time for the Lorentz transforms. So you either need to get rid of the $t$ or provide another variable to represent time. I showed how to get rid of your constant $t$.

I think the point is, I can boost the poles frame enough to make the distance between the button and the door effectively 0, which will be less than the length of the pole. So the pole will hit the door at relativistic speeds, but not at low speeds

No, you cannot. Use the expression I gave and actually use the Lorentz transform. For $\epsilon<0$ the pole will hit the door in both frames. For $\epsilon>0$ the pole will not hit the door in either frame. There is no way to make it hit in one and not in the other.

 

[Jrs580]

That time is frame dependent, because the distance from the button to the door is frame dependent.

ah ha, maybe this is where my problem is coming from, thanks!

 

[Jrs580]

OK, so not even getting to relativity, the first thing to do is to fix your equation. Here $$d=L/\gamma+ct + \epsilon$$ is problematic, as mentioned by @Ibix and @FactChecker . The issue is that $t$ is a variable, so this $d$, as written, is a distance that expands at the speed of light. What you have as $ct$ needs to be a constant, and what you want is for the constant to be such that with the end of the rod at $L/\gamma + v t$ and with the flash of light from the button at $ct$ both reach the door at $d=L/\gamma + D$ at the same time. So, if we set $$L/\gamma + D = c t = L/\gamma + v t$$ then we can solve for $D$ and $t$ to get $$d=\frac{L}{\gamma}+\frac{L}{\gamma}\frac{v}{c-v}+\epsilon$$

Now, with $\epsilon=0$ the pole and the light from the button reach the door at the same time, with $\epsilon<0$ the light reaches the door first, and with $\epsilon>0$ the pole reaches the door first.

With that then use the Lorentz transform so that you account for the relativity of simultaneity, as @DaveC426913 mentioned, and all will work out.

I see what you're saying now.

 

[Dragon27]

TL;DR Summary: Special relativity paradox

There is a button on the ground that opens a door some length away, let’s call that distance d=L/γ+ct+ε. The ct accounts for the time it takes for the button to open the door.

Just my two cents. From the surface reading of this quote it seems to me that you wanted the door to stay far enough from the runner with the pole, so that by the time the signal reaches the door (and it opens) the runner has not yet reached the door (or, more precisely, the front of the pole hasn't reached the door). So you took the length of the pole added the time (for the signal to reach the door) multiplied by the speed of light (the lowest upper bound of the runner's possible speed ;) ) plus some epsilon for good measure. Which means that the speed of signal will have to be $\frac{d}{t} = \frac{L}{\gamma t}+c+\frac{ε}{t}$ (i.e. more than the speed of light $c$) in your mental experiment. As a result, the opening of the door and the pressing of the button are causally unconnected (so the door can open both before or after the button is pressed in different frames of reference).

 

[Jrs580]

I figured out the problem. I was treating the “time it took for the button to open the door” as a constant and not considering that time changes when transforming to a new frame. All my “maths” add up now.

 

[Jrs580]

I see what you're saying now.

This fixed it…although I derived( L/γ c/c-v ) as the shifted distance. rather than v in the numerator

 

[PeroK]

This fixed it…although I derived( L/γ c/c-v ) as the shifted distance. rather than v in the numerator

There is a winning formula in all these problems:

1) Analyse the problem in one frame and note the coordinates of all relevant events (e.g. button pressed at $(t_1, x_1)$, door closed at $(t_2, x_2)$.

2) Transform all those events to the other frame using the Lorentz transformation.

The point is that nothing can possibly go wrong and no paradox (or apparent paradox) can possibly appear. The only way apparent paradoxes may appear is by short-circuiting this winning formula and making some flawed intuitive assessment or spurious calculation.

Note that SR, through an analysis of either its algebraic or its geometric structure, is seen to be mathematically consistent. Attempts to "break" it are, therefore, bound to fail. Although, of course, if you make a scenario complicated enough and do enough short-circuiting, then you may eventually introduce errors in your calculations that are difficult to detect.

 

[FactChecker]

I figured out the problem. I was treating the “time it took for the button to open the door” as a constant and not considering that time changes when transforming to a new frame. All my “maths” add up now.

That was at least one problem that you corrected to remove the obvious paradox. SR is treacherous any time you use the word "time", "distance", or "simultaneous". You have to always indicate who is observing that time, distance, or simultaneity and then convert it for other observers. It can become a bookkeeping exercise. I like the methodical approach that @PeroK advocates above. I don't have any practical experience with this myself.

 

[Dragon27]

I figured out the problem. I was treating the “time it took for the button to open the door” as a constant and not considering that time changes when transforming to a new frame. All my “maths” add up now.

Can you actually show the math? Because, from my point of view, the time it takes for the signal to reach the door being different in the moving frame of reference (in the frame of reference of the man with the pole) isn't really of any relevance, because you can make the distance between the button and the door smaller than the pole's length (which means that the end of the pole reaches the door before the legs of the runner reach the button). The solution is that, of course, you can't, because the distance to the door in the 'stationary' frame of reference has to be bigger than a certain minimum dependent on the pole's proper length and the pole's speed (owing to the fact that the signal's speed is limited by the speed of light), so that the distance in the 'moving' frame of reference turns out to be always bigger than the pole's length.

 

[Vanadium 50]

I don't see this converging. This recipe works 100% of the time.

1. Don't start with "there's a paradox!" or "I broke relativity!". I know you think you're being funny, but the time for that is after you have your answer,
2. Simplify, simplify, simplify. If you don't need 100 spaceships and 200 light beams to make your point, don't put them in.
3. Assign labels to every event in your scenario. A is when this sensor trips. B is when this door opens. Etc,
4. Assign coordinates to each of these labels.
5. Use the Lorentz Transformation as needed to compare events. (Be grateful for Step 2). To understand durations - differences in time - you must compare events at the same place. To understand distances - differences in lengths - you must compare events at the same time.

Then we can have an informed discussion, asking questions like "why did you assign G and H the same time coordiare if G always precedes H?"

 

[robphy]

If I may add to @Vanadium 50 ’s checklist,
6. Draw a position vs time diagram (aka spacetime diagram)… because a Spacetime diagram is worth a thousand words.
(Most relativity problems reduce to the analogues of trig-problems in Euclidean geometry. I think it’s probably hard to do a geometry problem using only words and formula.)

7. Use arithmetically-nice relative-velocities like (3/5)c, (4/5)c… not (1/2)c or 0.99c. This allows us to focus more on the physics (since the numbers are easier). Once the physics is understood, one can use a favorite velocity.

 

[PeterDonis]

Use arithmetically-nice relative-velocities like (3/5)c, (4/5)c…

Even better, don't assign specific numbers at all unless and until you absolutely have to. Use the general formulas. They're pretty simple and you can then be sure that your analysis isn't unintentionally restricting itself to particular values.

 

[Vanadium 50]

I said "simplify". I don't thinking arguing about whether "0.8c" is simpler than β is particularly fruitful.

I also think there is value in not making this "a simple 11,238 step procedure that anyone can do!"

 

[vanhees71]

Another simplification is to calculate everything first and draw the spacetime diagram then when forced by somebody to do so ;-)).

 

[PeroK]

6. Draw a position vs time diagram (aka spacetime diagram)… because a Spacetime diagram is worth a thousand words.

Sometimes an equation is worth a thousand (or even an infinite number) of diagrams. For example, can you draw a graph of the function?$$y = ax^2 + bx + c$$

 

[FactChecker]

Even better, don't assign specific numbers at all unless and until you absolutely have to. Use the general formulas. They're pretty simple and you can then be sure that your analysis isn't unintentionally restricting itself to particular values.

Good point. But doesn't that preclude the use of a Minkowski diagram?

 

[robphy]

Sometimes an equation is worth a thousand (or even an infinite number) of diagrams. For example, can you draw a graph of the function?$$y = ax^2 + bx + c$$

Of course… but I’m implicitly referring to the commonly seen
long verbal descriptions of some scenario, often using everyday-sounding words that really have technical definitions.

Having a diagram can help the reader parse the long verbal description and interpret it geometrically and physically.

 

[PeterDonis]

doesn't that preclude the use of a Minkowski diagram?

Not necessarily. You can still sketch a diagram and label events with coordinates. The basic geometry should be ok even if the line lengths and angles are not exactly correct, so the diagram can still be a huge aid to understanding a scenario.