# A paradox of length contraction

Suppose the proper length of a train is longer than that of a bridge, and the bridge can't bear the total weight of the train but can bear it partially. As the train goes very fast, it becomes shorter than the bridge and it can be entirely on the bridge. As a result, the bridge will collapse. However, in the train's frame, the train is even longer than the bridge, and it is only partially on the bridge; so the bridge will not collapse. What is the solution to this paradox?

• I'm not an expert but something must be wrong in your argument. It's against the first posulate of relativity becauae different events are being ovserved from different frames. From the bridge frame, there comes a time when the whole train is on the bridge but from the train frame no such event ever happens. So, I don't know what but something is wrong in your argument
– Dove
Jan 30 '17 at 3:23
– Paul
Jan 30 '17 at 4:47
• A perfectly rigid bridge whose total stability at any moment is determined by the total weight it carries is a nonlocal object that does not exist in relativistic mechanics. Perfect rigidness implies faster-than-light. The resolution to this paradox comes from considering the bridge as a flexible object through which signals travel slowly, as described here. Jan 30 '17 at 4:54
• I suppose that you are assuming that the force and stress experienced by the bridge is instantaneous when in the real world it takes time for strain to happen. And by the time the stress forces are 'felt' by the train, it's come and gone. In the real world that will save the bridge. But what about the relativistic increase in mass? And will that be 'felt' in the bridge's reference frame? Jan 30 '17 at 5:02
• @lelouch: The relativistic mass cannot be relevant here --- the relativistic increase in mass is only in the direction of motion, and so is irrelevant to forces in orthogonal directions --- such as, in this case, downward. Feb 6 '17 at 5:31

The "paradox", i.e. genuinely contradictory, frame dependent accounts of what will happen, is avoided through relativity of simultaneity. That is, from the train's standpoint, it straddles the bridge when an observer riding on the train can say that the two events of (1) its forward end has crossed the bridge and (2) its hinder end has yet to begin to cross the bridge are both simultaneous. This simultaneity is of course wholly possible.

Likewise, we imagine that we can identify a whole time sequence of simultaneous events defined by the train's ends and conclude that at no time will the amount of train on the bridge fell the latter.

However, the pairs of events are not simultaneous from the standpoint of an observer at rest relative to the bridge. If they were, then we would genuinely have a paradox as you describe - deduced from the symmetry of the Lorentz contraction. Different pairs of events are simultaneous in this bridge frame, such that we would still conclude that at no time is there enough train on the bridge to fell the latter.

There are two aspects of this problem, the kinematic and the dynamic. The kinematic part is precisely the same as the ladder paradox as it will resolve the "conflicts" between different accounts of what section of the train is "on the bridge". The dynamic part is a great deal more subtle, as pointed out in tparker's answer but I'm not sure that Supplee's paradox is applicable. The problem is highly dynamic: a train travelling at a significant fraction of $c$ relative to the bridge will be off the bridge before any significant dynamics play out. Acoustic pulses are launched into the bridge by the train as it passes and will then travel into the bridge - the question - now quite different from what the OP originally thought - is whether these pulses will knock the bridge down in the train's wake.

User Shen says:

Don't talk so much about simultaneity. It doesn't help. The question is very simple: Will the bridge collapse or not? You simply didn't answer this question.

I've been meaning to come back to this answer so I'll make these further comments. The short answer is that one cannot know just from the span of the bridge and its static failure load and the speed, length and weight of the train alone. Indeed, I think the span of the bridge turns out to be irrelevant in a first approximation, whereas its elastic properties and detailed shape will be highly relevant. An intuitive guess is that the bridge will not fall down to a first approximation, because the impulse imparted to the bridge per unit length is independent of the train's speed, as I discuss in more detail below.

The OP, as might be a reasonable first guess, tries to approximate the condition for the bridge collapse as a condition on whether there is a certain weight on the bridge at a certain time. As in the ladder paradox, simultaneity concepts show why this proposition can't have a sound truth value. So simultaneity is certainly relevant, but, I agree with you, will not answer the question.

There are delays in the propagation of stress throughout the structure, so this problem, ultimately, is one of relativistic acoustics. But probably a good approximation to the correct, fully relativistic approach would be as follows. As the train passes over the bridge negligible acoustic propagation happens unless the bridge is unbelievably long -I should think even the Malmö-København whopper (anyone know of a longer bridge?) could be handled by this approach. After the train has passed, mechanically you have a bridge into whose surface a uniform impulse per unit length has been transferred; this per unit length impulse is approximated by $\frac{m\,g}{L}\,\frac{L}{v}=\frac{m\,g}{v}$ with $m$ the train's weight, $v$ its speed relative to the bridge and $L = L(v) = L_0/\gamma(v)$ the train's length as measured from the bridge frame and $L_0$ the train's length from a train-comoving inertial frame. The bridge's span is probably irrelevant to a first approximation. Alternatively, this is roughly equivalent to a force per unit length of $\frac{m\,g}{L}$ imparted uniformly along the whole bridge's span for a time period $\frac{L}{v}$. You would then solve the wonted acoustic propagation equations for these initial conditions and check whether the structure would withstand the passage of these waves, given the bridge's mechanical properties.

Let's now look at where relativity comes into this simplified analysis, and sketch how the conclusion is frame-independent (which, I believe, conceptually is much more important than whether a particular bridge falls down).

The easiest frame to do the calculation from is from the bridge frame, where the contracted length $L(v) = L_0 / \gamma(v)$ of the train determines the force per unit length on the Earth's surface (be it bridge or not - the pressure is constant). It is indeed higher than the normal force per unit length that would be exerted if the train were stationary. However, the effect is counteracted in the approximate model, because the higher force per unit length is present for a shorter time, since $L(v)/v < L_0/v$, and the impulse per unit length remains the same as it would be in the case where the stationary train were on the bridge for time $L_0/v$. So the impulse is independent of the speed. In this simplified model, this impulse is the whole story. From the train's frame, it sees a lower normal force per unit length on the bridge because the bridge is contracted from its frame and the train's weight is more spread. The contact time for each bridge point with the train is $L_0/v$, so the impulse is less in the train frame. But we have to transform the equations of acoustics. The material of the bridge becomes contracted along the train's motion such that its elastic properties are anisotropic and the properly transformed acoustic equations will foretell the same results in response to this lower impulse as in the bridge frame. We must get the same answer in both cases with the properly transformed acoustic equations and there is no genuine paradox.

The use of $m\,g$ for the train's weight is not strictly correct; as noted elsewhere, there will be gravitodynamic effects. But I think these will be very small: the train will not alter the gravitational field appreciably as its mass is negligible compared to that of the Earth, and I can't see any reason why the force on it upwards from the ground would in principle be much different from the static case (so all the elements of the stress energy tensor arising from the train and its interaction with the Earth will be vanishingly small). To a good first approximation, because the train is moving along a gravitational equipotential, it is, just as it would be were it stationary with respect to Earth's surface, still being uniformly accelerated upwards at $g$ relative to its locally comoving inertial frame.

Still another aspect of this problem to consider is that the idealized thought experiment would need to take place in a uniform gravitational field $g$ on the surface of some body which is considerably less curved that the Earth's surface, so that "uniformly accelerated upwards at $g$ relative to its locally comoving inertial frame" in the last paragraph holds true. A train moving at relativistic speeds in a vacuum tube would need to hold itself down on the track, as it is clearly moving at greater than low Earth orbit speed.

• Don't talk so much about simultaneity. It doesn't help. The question is very simple: Will the bridge collapse or not? You simply didn't answer this question.
– Shen
Apr 29 '17 at 1:19
• @Shen I think I have a conclusive answer to the "paradox", i.e. I now discuss in more detail how the acoustics are frame-independent, but I don't think one can answer whether the bridge will fall. As a first approximation, it will not fall, because the impulse exerted on the bridge is independent of speed. Apr 29 '17 at 7:12

I actually don't think this situation is precisely equivalent to the ladder paradox. The resolution to the ladder paradox is the usual "both the ladder's and the barn's perspectives are equally correct, and neither one of them is 'objectively' correct about whether the ladder fits in the barn."

But this paradox is different, because the bridge either objectively does or does not collapse, and both the train and the bridge will agree on whether it does. So we can't just say "oh, they're both correct in their own way" and sing Kumbaya.

I think the key insight is that the force that the train exerts on the bridge is due to gravity, which means that we can't consider the problem solely from the perspective of special relativity - we need to go to general relativity. I think that this paradox is therefore actually more closely analogous to Supplee's paradox. I'm not sure what the answer is, but I think that the bridge will collapse, solely by analogy with the GR analysis of Supplee's paradox. But I am quite confident that the resolution to this paradox is actually significantly more subtle than the resolution to the ladder paradox.

– Shen
Feb 20 '17 at 4:41
• You are correct that there are two aspects of this problem, the kinematic and the dynamic. The kinematic part is precisely the same as the ladder paradox as it will resolve the "conflicts" between different accounts of what section of the train is "on the bridge". The dynamic part is a great deal more subtle, but I'm not sure that Supplee's paradox is applicable. The problem is highly dynamic: the train is off the bridge before any significant dynamics play out and acoustic pulses then travel into the bridge after the train passes - the question is now whether the pulses knock the bridge down Feb 21 '17 at 22:37

Probably it is not necessary to consider gravitation and weight of the train. Well, the bridge can fall because the train is too heavy. If the bridge is fated to fall, it will.

Moments when different parts of the train start falling will be frame dependent though.

The bridge can fall because of another reason, that doesn’t matter. Let's say that proper length of the train and the bridge is the same. The train has ten cars, the bridge has ten columns. If the both train and the bridge are at rest (train stays on the bridge) each column is straight under every car.

Let’s train moves in the bridge’s frame. We explode columns Nr. 1 and Nr. 10 (or all 10 together) under the bridge simultaneously in the bridge’s frame, when center of the bridge and center of the train coincide. Beams from flashes (explosions) travel towards the observers in the center of the train and center of the bridge. Obviously, train will fall; all his cars go down simultaneously in the bridge’s frame. However, these events – explosions were not simultaneous in the train’s frame, and when the first car was already falling down, the last one was still on the track.

We can put a bomb into the every car and launch explosions simultaneously in the train’s frame. These explosions will destroy the train, the bridge and some buildings close to the track to the left and to the right of the bridge. However, these explosions will be not simultaneous in the bridge’s frame. Observer on the bridge will decide that the first column was destroyed earlier than the last one.

Gravito-magnetic effect increases the train's weight, but that is not a problem, because different frames can agree about that.

If two important parts of the bridge stay together because they are oppositely electrically charged, then in the train's frame the bridge breaks because there is a magnetic repelling force between those parts, like there always is between charged objects that are moving.

And in the bridge's frame the bridge breaks because the train is heavy and short.

We have now explained how train's weight can break steel rods that experience forces that are transverse to the motion of the rods as seen from the train. The explanation is the magnetic forces inside the rods.

But the first thing to break may be a longitudinal rod just as well, that depends on the design of the bridge.

If a longitudinal rod breaks, the explanation for that in the train's frame is that because of the length contraction of the bridge the geometry of the bridge is such that weight on the bridge causes extra large stresses on the longitudinal rods.

Just saw this. I am writing an answer because this is too long for a comment. Paradoxes appear in physics thought experiments when one is using two different frameworks of physics in one problem. In this case the thought experiment confuses special relativity and Newtonian gravity.

Length contraction happens in the direction of relativistic motion. Direction perpendicular to that motion are not affected. It has to be in an inertial frame, i.e. no accelerations.

In empty flat space the reciprocity of gap and train in contraction creates no paradox, as the whole description is within the special relativity physics problem. The train sees the bridge contracting and the bridge sees the train contracting.

A transverse vector is introduced by gravity, and I think people who comment that general relativity has to be employed are correct. This makes the system described not inertial, since a general relativity curvature is introduced (an acceleration).

Gravity that makes the train fall or not fall in the thought experiment is classical Newtonian. Special relativity and Newtonian physics are two different frameworks, valid in different regimes. Lorenz transformations do not work in the framework of Newtonian mechanics. For velocities approaching c they are mathematically incompatible. General relativity includes special relativity for flat spaces and no paradoxes can appear :

The train will follow a curvature which will redefine the direction of motion at each point, thus having an accelerating , not an inertial frame. A simple length contraction is the effect of instantaneous velocity of order c in an inertial frame not an accelerating one.

As the train goes very fast, it becomes shorter than the bridge and it can be entirely on the bridge. special relativity

As a result, the bridge will collapse. Newtonian gravity

However, in the train's frame, the train is even longer than the bridge, and it is only partially on the bridge; special relativity

so the bridge will not collapse. Newtonian gravity

The limiting case of flat space must be reachable in a thought experiment that introduces a diminishing gravitational curvature; a train hurtling at relativistic velocities has a strong contribution to the energy momentum GR tensor, and the gravitational interaction cannot be ignored.

I do not have the tools to write up the equations for the curvature followed by such a train in the gravitational field of the earth, but having seen other paradoxes resolved by clearly stating the boundary conditions and potentials I am confident that once the mathematics is put down one will see that "there is no symmetry between the spacetime paths of the bridge and the train" because the bridge does not see the gravitational field that the train sees, and so there is no paradox.

I hope that the discussion makes clear that I expect that the bridge will collapse in both frames, or not collapse in both frames: the curvature path that the observer on the bridge sees is different than the curvature path that the observer on the train sees.

The problem cannot be stated in correct mathematics using special relativity's inertial frames and Newtonian gravity, thus a paradox appears.

• You discussed a lot but didn't come to the point. The question is: Will the bridge collapse or not? You simply didn't answer it.
– Shen
Apr 29 '17 at 12:20
• You discussed a lot about the symmetry but didn't come to the point of this issue. The paradox is: in one frame the bridge will collapse but in another frame it will not, whereas in fact only one can be true. The question is: Will the bridge collapse or not? You simply didn't get it and didn't answer it.
– Shen
Apr 29 '17 at 12:30
• I edited. If the dynamics, discussed by others, are such that the bridge collapses in the train rest frame, it will collapse in the bridge rest frame too. The mathematical solutions will be different, taking into account the distortion of paths due to general relativity. Apr 29 '17 at 12:34

Your argument amounts to reductio ad absurdum - it disproves Einstein's relativity. Here is a similar disproof: