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Lets say that at each end of a train car there is a rocket engine and that the train car is presently at rest on the tracks. Also, the two engines are pointing in opposite directions. If someone positioned at the middle of the car flips two switches simultaneously, both rocket engines will fire at the same time. In turn the car does not move since the power output from both opposing engines cancels out. Even if the train car is in motion at a fixed velocity, again this has no effect, so the passenger feels no change at all in the train cars velocity.

However, from someones point of view who is at rest relative to the train tracks, it appears that one rocket engine fires before the other, but this observer too notices no change in the trains velocity.

How can this be?

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    $\begingroup$ Near duplicate of this question from earlier today: physics.stackexchange.com/q/470546 $\endgroup$ – Duncan Harris Apr 5 at 0:10
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    $\begingroup$ "In turn the car does not move...". But each end of the car moves. If the rocket at the right end of the train fires outward, then the right end of the train moves leftward; simultaneously the left end moves rightward and the train is compressed. (Remember that the right end can't possibly "know" instantaneously that the engine at the left end has fired.) In a frame where the train is moving, one compression takes place before the other. $\endgroup$ – WillO Apr 5 at 0:20
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    $\begingroup$ relativity does not allow absolute rigidity. $\endgroup$ – JEB May 25 at 14:45
  • $\begingroup$ @WillO, you said, "In a frame where the train is moving, one compression takes place before the other." Do you mean, "when one frame is moving in relationship to another frame, one compression "appears" to take place before the other (because of the extra distance light from the far-limb must travel)? $\endgroup$ – Thomas Lee Abshier ND May 29 at 23:47
  • $\begingroup$ @ThomasLeeAbshierND : I don't understand your question at all. What does it mean for a frame to be "moving in relationship to another frame"? I mean exactly what I said. $\endgroup$ – WillO May 30 at 0:26
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Sequence of events, seen from the train's frame.

  1. Here's the train, sitting on the track.

  2. Rockets just went off at both ends. The ends of the train have been propelled inward. The train is getting shorter now. Maybe it will soon be crushed completely, or maybe some internal forces will slow down the shrinkage and the train will be permanently shorter.

Sequence of events, seen from the track frame:

  1. Here's the train, moving rightward the track.

  2. The left rocket just went off. The left side of the train is now moving faster than the right side. The train is shrinking. Maybe it will continue to shrink until the train is crushed out of existence, or maybe some internal forces will slow the shrinkage.

  3. The right rocket just went off. The right side is now moving slower than before. This increases the rate of shrinkage. Once again, the train might end up permanently smaller, or crushed out of existence.

Edited to correct: Wait, wait, wait. I'd missed the part where the train starts out at rest relative to the track. Given that, there's only one frame of interest here --- namely the frame in which both the train and the track are initially stationary. So there's no need to bring in the track observer at all. The above "sequence of events, seen from the train's frame" remains correct. The "sequence of events, seen from the track frame" is correct if the train is moving along the track, but superfluous otherwise.

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  • $\begingroup$ It's still non-obvious how the central car never moves, i.e. why the two compression shock waves reach the center at the same time, despite one of them having had longer time to travel. $\endgroup$ – Emilio Pisanty May 26 at 10:10
  • $\begingroup$ @EmilioPisanty: I don't see why you find this non-obvious in the train's frame, given the symmetry. $\endgroup$ – WillO May 26 at 13:20
  • $\begingroup$ In the train's frame it's obvious. But it bears explaining in the external frame - how the relativistic addition of velocities makes the explanation work there. For OP's sake =). $\endgroup$ – Emilio Pisanty May 26 at 13:42
  • $\begingroup$ @EmilioPisanty : I see what you're saying, but I tend to think the more valuable lesson for the OP is that the best way to get the answer is to do your calculations in a well chosen frame. $\endgroup$ – WillO May 26 at 14:36
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A train car has opposing rockets mounted to each end. Igniting them simultaneously, from the perspective of the middle of the car, produces no net force/acceleration on the car. An observer standing on the track a distance from the car sees one rocket engine ignite before the other.
* The distant observer expects that the car will move, given the indication of unequal application of force.
* Why doesn't the distant observer see the car move?

enter image description here

Short Answer:

  • Background: The forces generated by the simultaneously ignited rocket engines compress both ends of the train-car. That force is transmitted between atoms via repulsive EM fields at the speed of light, but inertial and interatomic forces slow the force transmission, resulting in pressure waves traveling through the mass of the train-car at phonon speed.
  • The observer has no effect on the forces acting on the mass of the train-car. Thus, for the remote observer's measurements to yield the same experimental results, the position and momentum measured in the distant frame must correspond exactly to that measured in the local frame.
  • Thus, the distant observer must correct the measurement times he/she records (by the time delay introduced into the distant observer's measurements by the extra distance to the far limb) to reflect the forces and positions of the atoms in the local train-car frame at every moment.
  • Answer: The time recorded by the distant observer for the times of ignitions (and positions of the pressure wave) must be corrected for the differential in transit time between the near-limb and the far-limb (so the measurement time of events will correspond exactly in both the distant and local frames).
  • Therefore: a) Adjust the distant observer's measurement of the far-engine ignition time, and b) Adjust the distant observer's measurement time for each position of the far-limb's pressure wave.
  • Adjusting the distant observer's measurement time by the time delay (due to the speed of light transit time of the extra distance to the far limb) corrects the measured time to reflect the actual sequence of events local to the train-car.
  • When this correction is applied, the remote observer obtains the same measurements as the local observer, namely that:
  • 1) The near-engine and far-engine fired simultaneously,
  • 2) The center of mass did not move,
  • 3) The pressure wave proceeded from both ends toward the center of mass symmetrically.

More Detailed Analysis:

Observer at the Center of Mass:

  • Compressive force from Rocket Engines: The local (Center of Mass) observer sees two opposing equal, opposite, and simultaneous forces compress the train-car mass. The pressure compresses/shortens the mass equally on both ends, but the Center of Mass does not displace.

  • Transient Pressure Wave: At the local COM observer, a) the leading edges of both pressure waves are measured to simultaneously initiate, b) both pressure waves cross simultaneously at the Center of Mass, c) the pressure waves both continue to the ends and simultaneously reflect off opposite ends, d) both pressure waves repeatedly reflect off the ends until the impulse is fully damped by inelastic collisions (e.g., converting some of the work of compression into thermal energy, irreversible/plastic bond deformation, radiation...), d) the balance of work done on the mass is stored as potential energy (repulsion of orbitals between atoms) as elastic compression. At steady-state after the transients dissipate, the train-car is compressed.

Distant Observer - down the tracks:

  • The extra distance traveled by light from the far-limb vs. the near limb is: $\Delta d = d_{far} - d_{near}$
  • The near-engine is measured to ignite at time $t_{near}$, and the far-engine is measured to ignite later at time $t_{far}$. Thus, the distant observer, without correcting for the speed of light delay in the signal from the far-engine, may (incorrectly/naively) assume that the center of mass will move, given the appearance of the unopposed force for the period of time $\Delta t = t_{far}-t_{near} = \frac {\Delta d}{c} $.
  • NB: the distant observer's perspective has no influence on the application or propagation of force on the mass of the train-car. There are no a) entanglement phenomena, b) no forces acting at distance between the distant observer and the local train-car frame, and c) no relativistic speeds, hence no time dilation and length contraction considerations in this problem.
  • To determine how the body actually responds in the local frame, we must transform/adjust the time measurements taken by the distant observer, and convert them into the local train-car time frame where the forces actually act.
  • Again, the far-engine, due to its greater distance to the observer and the signal propagation rate of $c$, requires an extra increment of time to travel that extra increment of distance at the speed of light.
  • The extra distance traveled by light from the far-engine is: $\Delta d = d_{far} - d_{near}$
  • The measurement times recorded for the signals at the distant observer are: $t_{far}$ and $t_{near}$
  • The extra time elapsed before the distant observer detects the far-engine ignition is: $\Delta t = t_{far}-t_{near} = \frac {\Delta d}{c}$

Resolution of the Expectation of Movement
* (Note: There is actually no observed movement of the center of mass, and we must resolve this with our (initially incorrect/naive) expectation.)

  • Adjusting the time of the distant observer's measurement of the far-engine signal, compensates for the delay in the time of arrival of the light signal.
  • Expectation Bias: We live in a world of ordinary/daily human body sensory perception where we do not need to consider the speed of light to qualify the data we perceive.
  • This problem illuminates our habit, our bias, and our expectations. We expect that measuring a sooner/earlier time of ignition (i.e., near-limb vs. far-limb) means that there is a time of unopposed force, which implies acceleration/movement of the center of mass.
  • Local Reality: In fact, the far-engine and near-engine ignition times are identical in the frame where the force is mediated by electrostatic fields, where atomic orbitals are compressed and exert repulsive force. The force between layers of metal atomic-lattice is unaffected by a distant observer's perspective and measurement of those locally acting EM fields.
  • In reality, all inertial frame, far-limb measurements are delayed compared to near-limb measurements. Thus, the local mass upon which force is applied is always the proper/real frame to consider as the frame where forces are actually/effectively applied and propagated. Thus, the speed of light delay must always be considered and used to adjust the measured time when comparing the light signals from an object with a near-limb and a far-limb.
  • Thus, we must take the local frame - the perspective of the Center of Mass - as the primary frame, the unchanging standard, as it is the frame in which electrostatic forces act and compress the train-car.
  • Note: This is not an entanglement problem where remote detection influences a local measurement. Rather, this is a problem of adjusting the measurement times of events recorded by the distant observer so that the distant observer correctly identifies the times of force initiation at the local object.

Regarding Rigidity, Delayed Measurement, and Center of Mass Displacement:

  • Is the near-limb pressure wave seen to reach the COM before the far-limb pressure wave? If it did, it seems that the near-limb pressure wave would unopposed at the COM. It seems like this may be the case because comparing the near-limb and far-limb pressure wave up until that point, the near-limb pressure wave is always perceived before its corresponding far-limb pressure wave. If this was true then there would be an unopposed force acting at the COM.
  • It's not true. The COM is a point in space in the body of the mass; at that point, the near-limb and far-limb pressure waves arrive at exactly the same time as measured by the Distant Observer and at the COM. No adjustment of the Distant Observer's measurement is necessary to show simultaneity as the time of measurement of the pressure wave at the COM is identical. Thus, there is no net force measured at the COM.
  • This must be so because the pressure wave from the near and far limb are initiated at the same time in the local train-car frame. And, the pressure wave from the near-engine reaches the COM at the same time as the pressure wave from the Far-Engine. Thus, at the moment of observation of the pressure wave at the COM, $t_{COM}$, both the near and far limb pressure waves are measured as acting on the COM. Thus, the sum of forces acting at the COM is zero, and neither the distant nor local observer see or expect COM displacement.
  • All contradictory expectations of displacement between the COM and the distant observer are resolved by correcting every moment of time measurement, converting his/her measurements into "local train-car time". The local frame experiences exactly equal and opposite compression forces at every moment and the corrected distant measurements reflect the same symmetric cancellation effects.

Summary: the local-frame, where force is applied to a mass, is the primary frame to which we must refer in our evaluation of the actual local acceleration and displacement of a mass when observed at a distance. Thus, we must correct the time measurements of distant observations to transform the distant-frame measured times into local-frame time to correct our measurements to correctly predict the actual movement of the mass and pressure waves.

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Observations are frame dependent. That means, if the observer is closer to one engine, they see that engine fire first. Observations are not chronologically accurate. Just because we see something first, doesn't mean it occured first. Now, causal events are always perceived in chronological order. You can't see a building explode before seeing a rocket strike it. It's only unrelated events which occur in different locations which can be observed in reverse chronological order.

Imagine 2 stars. A is 10 light years away and B is 5 light years away. If A goes supernovae right now we'll observe the supernova in 2029. If B goes supernovae next year, we'll observe the supernova 5 years later, in 2025. We observe the later event, 4 years before the prior event. Distant from an even determines how soon we detect it. We then need to calculate when it actually occured.

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