# Knights spears

1. Nov 23, 2013

### gatusso

Hello everyone,

If two knights, one with a 5m long lance (A) and the other one with a 3m long lances (B) horizontally hold their lances in the direction of each other and move towards each other with a speed close to speed of light, who will win? because if the speed is really high, for each of them the other one has a shorter lance and will predict he will be the winner. Actually B with a shorter lance at rest than A will claim he will win. I read the ladder paradox but i cant relate it to this problem. can someone help me.(this is not hw question tho, my friend and I were debating)

2. Nov 23, 2013

### Staff: Mentor

If by "win", you mean "knocks the other one off his horse, so that he can't do the same to me", then there is a certain closing speed (try calculating it - it's fun!) below which the guy with the 5m lance wins, above which neither win because they both get hit by the opposing lancepoint.

In any case, all observers everywhere, regardless of their speed, will agree about the result. There will never be a paradox where some observers see an unhorsed knight and other don't.

3. Nov 23, 2013

### gatusso

But then how can u reconcile this with one observer's view who sees his lance to be longer than his rival's? can u explain the speed after which, they both get hit

4. Nov 23, 2013

### Staff: Mentor

You might try googling for the "bug-rivet paradox", which is closely related.

There are two important things going on here:

First, what happens at one end of a lance does not instantaneously affect what happens at the other end, so when my lancepoint hits the other guy in the chest and knocks him off his horse, his lancepoint is not immediately deflected away from my chest. As I view events, it is possible for his lancepoint, already very near to my chest, to hit my chest after my lancepoint has hit his chest (but before I've seen my lancepoint hit his chest, because I can't see that until light from that event has traveled to my eyes).

Second, there's the relativity of simultaneity. If the closing speed is high enough (I'll repeat myself - try calculating it - it's fun!) some observers will see my lance hitting him to happen first, other observers will see his lance hitting me happening first.

You can also try drawing a space-time diagram tracking the four interesting objects through space-time: The two lancepoints and the two targets. (Ghwellsjr has software for drawing these, and if he's watching this thread he's probably working on it already).

Last edited: Nov 23, 2013
5. Nov 23, 2013

### ghwellsjr

Yes, I have started but can't continue right now, maybe I'll have something by tomorrow.

6. Nov 23, 2013

### Meir Achuz

Sir Longlance defeats Sir Shortlance in any coordinate system.
This is seen most easily in the earth system where they approach each other at the same speed,
so there is no preference for either.
The situation is a bit tricky in the rest system of Shortlance, especially if he tries to keep all times the same.
What happens is that he gets unhorsed before he knows it. He gets fooled by trying o do things at equal time in his rest system. This takes a bit of algebra to show.
Take the positions where Longlance's spear strikes Shortlance's armour (to use the knightly spelling).
to be x_1=0 and y_1=0. x_1 is the position of the tip of Longlance's lance,
and y_1 is the position of Shortlance's body.
x_2=-L/gamma is the position of Longlance's body,
and y_2=-S/gamma is the position of the tip of Shortlance's lance.
All of this is at the same time, which can be taken as t=0.
Then a Lorentz transformation to Shortlances's rest system gives t'_2L=-vL and t'_2S=-vS,
while t'_1L=t'_1S=0. These times are not equal, so the usual Lorentz contraction is not applicable.

7. Nov 23, 2013

### yuiop

What about when the velocity of Longlance relative to Shortlance is such that the the length contraction factor is 0.6 and the length of Longlance's spear is the same length as the Shortlance's spear as measured in Shortlance's rest frame? In this reference frame they would both strike each other simultaneously and that would count as a draw. In another reference frame moving in the opposite direction to Longlance relative to Shortlance's rest frame, Shortlance strikes first. As Nugatory pointed out, you have to careful how you define 'win'.

8. Nov 24, 2013

### ghwellsjr

As promised, here is a set of spacetime diagrams for the speed of each knight traveling at 0.6c in the "earth" frame. I have used the unit of light-meter to refer to the length of time that light travels in one meter. The green knight is traveling to the right with the red tip of his lance extended out 5 meters in front of him (as will be apparent later in his rest frame). The black knight is traveling to the left with the blue tip of his lance extended out 3 meters in from of him.

Here is the diagram for the "earth" frame:

Note that the red tip of the longer lance contacts the black knight before the blue tip of the shorter lance contacts the green knight, as expected. However, the shock wave of the impact cannot travel through each lance to the respective knight nearly as fast as the speed of light and I have drawn in two signals to show this limit for each lance. Thus, even though the green knight "wins" in this reference frame, he still gets impaled by the blue tip of the black knight's lance.

If we look at the green knight's rest frame, we see that there is even more time between the red tip of his lance impacting the black knight but the maximum shock wave still takes the same amount of the black knight's time to reach his blue tip and again, it's too late:

Now for the rest frame of the black knight:

In this frame, the length of the green knight's lance is contracted to about 2.5 meters, shorter than the black knight's 3-meter lance, and so the black knight "wins" but since it takes a long time for the shock wave to travel down the green knight's lance back to the red tip, the black knight still gets knocked off his horse.

#### Attached Files:

File size:
11.7 KB
Views:
137
File size:
28.4 KB
Views:
144
• ###### Lances3.PNG
File size:
31.6 KB
Views:
140
Last edited: Nov 24, 2013
9. Nov 24, 2013

### Meir Achuz

Who gets knocked off cannot depend on coordinate system no matter what space-time diagrams are drawn.
The four times (for the front and back of each lance) can be the same in only one Lorentz system.
Trying to set them all equal in a different Lorentz system leads to the errors above.

10. Nov 24, 2013

### TumblingDice

What errors do you see? George's diagrams and commentary seem valid, without any "who gets knocked off" issues.

The space-time diagrams illustrate how the concept of simultaneity in different reference frames and the maximum speed of a wave in a rigid object can influence the perception of how the event 'transpires'. (Which tip touches first, and how each knight suffers the consequences of the collisions.)

Last edited: Nov 24, 2013
11. Nov 24, 2013

### Staff: Mentor

We aren't getting different results by varying the coordinate system here, we're getting different results by varying the physical situation. For any relative velocity of the two knights, there is exactly one possible physical outcome and all observers, regardless of their velocity and choice of coordinates will observe that outcome - no one doubts that. But different relative velocities of the knights will produce different outcomes, just as the relative velocity between a bullet and its target affects the coordinate-independent damage done to the target by the bullet.

12. Nov 24, 2013

### TumblingDice

From your post #4, I thought that the 'sizzle' behind this thought experiment involved the calculation of simultaneity when combined with the maximum signal speed in a rigid object. So at .6 c, different reference frames will see different tip strike first, however speed of rigid object resolves any paradoxes and same knights end up on the ground.

Have I been thinking too quickly or just plain wrong?

13. Nov 24, 2013

### Meir Achuz

The space-time diagrams are all wrong, because (in the earth diagram) the red lines both stop when the red knight is dehorsed. The red lines must therefor also stop in all the other diagrams at the time given by the Lorentz transformation to those frames.

14. Nov 24, 2013

### Meir Achuz

Another way to see that only Sir Shortlance is dehorsed is to agree that this happens in the earth frame.
The other Lorentz frames correspond to observing the joust from a moving anachronistic car. Whichever direction the car is traveling, you will see Sir Shortlance dehorsed. Your driving by the scene cannot save him.
You may observe some funny time sequences, but he will be dehorsed.

15. Nov 24, 2013

### PAllen

Nugatory is correct on this and his explanation was good so I chose not to add to this thread till now. Note that their is no frame dependence on what happens: either long lance wins, or both lose. The correct outcome is produce in any frame with correct analysis.

As an intuition for such situations, I have long felt SR books should have a section on the interaction of matter at high relative speeds. The key point is that even 'unobtainium' with a speed of sound and bonding influence equal to the speed of light behaves, to first order, as a collection of independent dust particles when the relative speed is high enough. The modern concept (relative to when I first learned this stuff) of congruence of world line makes this precise. Model an object (lance) as a congruence of world lines. Posit an interaction at one world line (the initial contact between a lance and and a knight). Consider the future directed light cone from this event. The other world lines in the congruence must behave as if this collision event did not happen until they cross that light cone.

Understanding this, the resolution is simple. Above a certain relative speed, the two events: Knight A getting skewered, knight B getting skewered, have spacelike separation. This, of course, means there cannot be any preferred time ordering of these events.

This gets to the core intuition which this 'paradox' forces you to reconsider: "If I get hit first, I don't get to knock off the other knight". This has a whole bunch of caution words for SR: 'first' only has invariant meaning for causally connected events. Then, you are talking about events in two different places, which implies simultaneity (invariantly: spacelike separation) is going to be key to analysis. This 'obvious' statement becomes false in principle above a certain relative speed.

Last edited: Nov 24, 2013
16. Nov 24, 2013

### yuiop

I get the critical minimum relative speed of the 2 knights to be (8/17)c for a draw (both unseated). Is that what you get?

This relative speed is determined by whether or not the 'potential event' (e2) of Longlance being hit by Shortlance is in the future light cone of the event of Shortlance being hit by Longlance (e1) and calculated by solving $v >= c(1-3/5 \sqrt{1-v^2/c^2})$ for v (if I have done it right).

I am assuming that the deflection shock wave travels along the spear of Shortlance at the speed of light, but in reality the shock wave is slower than c, so the minimum speed will be a bit higher.

Last edited: Nov 24, 2013
17. Nov 24, 2013

### PAllen

I get this as well, by a different (but equivalent) calculation.

18. Nov 24, 2013

### Staff: Mentor

I assume you mean "the red line and the green line both stop when the green knight is dehorsed". That is not correct; the *green* line stops (actually, it changes trajectory) when the green knight is dehorsed, but the *red* line (the worldline of the tip of the green knight's lance) does *not*, because it takes time for the change in the green knight's trajectory to propagate down the green knight's lance to its tip (the red line), since the change cannot propagate faster than the speed of light. Until that happens, the green knight's lance tip (the red line) will remain on its original trajectory, and will therefore still unhorse the black knight if the relative velocities of the two knights are large enough that the two unhorsing events (blue line crosses green line, and red line crosses black line) are spacelike separated. That's what ghwellsjr's spacetime diagrams (correctly) show.

19. Nov 24, 2013

### Staff: Mentor

Wouldn't slower shock wave propagation (for real materials, a *lot* slower--typical sound speeds in solids are about $10^{-5} c$) make the minimum speed for a draw *lower*, not higher?

20. Nov 24, 2013

### WannabeNewton

OP, work out the following exercise from Schutz "A First Course in General Relativity" and then move on to your scenario; the main idea behind the two is the same (finite propagation speed of sound).

File size:
89.6 KB
Views:
93