Consistent answers in special relativity

Question

I was doing a question about if a train fits in a tunnel. From the train frame it does not and from the tunnel frame it does. Then the question asked are these two answers consistent? What does this mean? Does it mean thy agree on their relative velocity or something else? Please can you keep your answers general and if possible give a source.

Edit This question is ment to be about the definiton of consitent events within special relativity and what makes two points of view consistent. This Spefic example was just an aid to get my point across.

Answer 1

The site rules forbid us from giving the answers to homework problems, but this problem illustrates a fundamental issue in relativity so I think it's worth some general comments. Incidentally you may be interested to read the Wikipedia article on the ladder/barn paradox, though in it's efforts to be comprehensive I think the article gets a bit confusing.

There is an apparent paradox here, because the observer standing by the track measures the train and tunnel to have the same length while the observer on the train measures the train to be longer than the tunnel. Surely one of them must be wrong. When the question asks:

are these two answers consistent?

it really means:

are both observers correct or is one of them wrong?

Unless special relativity is broken both observers must be correct, so how can this be?

Suppose we measure the time $t_1$ when the front of the train exactly lines up with the far end of the tunnel, and the time $t_2$ when the rear of the train exactly lines up with the near end of the tunnel. When the track observer says the train is the same length as the tunnel they mean $t_1 = t_2$. When the train observer says the train is longer than the tunnel they mean $t_1 < t_2$.

An answer that would earn top marks if I were marking would be to use the Lorentz transformations to show that that starting in the track frame with $t_1 = t_2$, in the train frame $t_1$ really is less than $t_2$. In other words, events that are simultaneous in the track frame are not simultaneous in the train frame.

Answer 2

I was doing a question about if a train fits in a tunnel.

Did the question assignment include a specific consistent definition of what's meant there by "to fit", in the first place?

Presumably, in the setup which is typically considered,

• the ends of the tunnel (say participants $A$ and $B$) are supposed to be at rest to each other,

• the ends of the train (say participants $J$ and $K$) are supposed to be at rest to each other,

• the (ends of the) tunnel and the (ends of the) train are supposed to move at some particular non-zero speed $\beta := \frac{|\mathbf{ \vec v }|}{c}$ with respect to each other, such that

  • $A$ first met and passed $J$, then met and passed $K$,
  • $B$ first met and passed $J$, then met and passed $K$,
  • $J$ first met and passed $A$, then met and passed $B$, and
  • $K$ first met and passed $A$, then met and passed $B$.

It is then said that "the train fit in the tunnel" if:

• $B$'s indication of being passed by $J$ was before $B$'s indication simultaneous to $A$'s indication of being passed by $K$,
  and likewise (consistently) if:

• $A$'s indication of being passed by $K$ was after $A$'s indication simultaneous to $B$'s indication of being passed by $J$.

Moreover, it can similarly be defined whether "the tunnel fit in the train"; namely if:

• $K$'s indication of being passed by $A$ was before $K$'s indication simultaneous to $J$'s indication of being passed by $B$,
  and likewise (consistently) if:

• $J$'s indication of being passed by $B$ was after $J$'s indication simultaneous to $K$'s indication of being passed by $A$.

Finally, of particular interest are cases in which the ratio $R$ between the distance between the tunnel ends and the distance between the train ends satisfies

$$\sqrt{1 - \beta^2} \lt R \lt \frac{1}{\sqrt{1 - \beta^2}},$$

because in those cases,
related to the definition, within the theory of relativity, of (how to determine) "simultaneity", and
related to the definition, within the theory of relativity, of (how to determine) ratios of "distances", and
related to the definition, within the theory of relativity, of (how to determine) "mutual rest", and
related to the definition, within the theory of relativity, of (how to determine) "mutual speed", and
related to the definition of (how to determine) "fit" as described above (also based on the terminology and available definitions within the theory of relativity),
it follows that
"the train fit in the tunnel" and "the tunnel fit in the train".

From the train frame it does not and from the tunnel frame it does.

That's a very brief, superficial, arguably improper and seemingly paradoxial way of stating what I laid out in some detail above.

Then the question asked are these two answers consistent?

They are consistent based on the detailed definitions and setup description mentioned above.

Please [...] if possible give a source

At least there is a one well-known source for the definition, within the theory of relativity, of (how to determine) "simultaneity", namely
A. Einstein, Relativity: The Special and General Theory., chap. 8: On the Idea of Time in Physics.
The other relevant definitions are unfortunately only less widely documented.
Some related presentations on PSE on how to use these definition are Deriving formula for time dilation and What is the real meaning of length contraction? .