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Suppose you are inside a closed box which is moving with a constant speed with no acceleration whatsoever. You cannot see outside the box. If you are given all the instruments you need, what experiment could you perform to ascertain that you are, in fact, in motion?

This was a question asked in an interview for admission to a pure sciences research university.

The only thing I was able to think of was maybe if we somehow converted the mass of some object into energy it would be slightly more than what we would expect from $E=mc^2$ since it also has kinetic energy.

Is this even remotely correct? Does it even have an answer or was it simply a trick question?

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    $\begingroup$ It should be explicitly stated that no acceleration means no acceleration with respect to an inertial system, otherwise it is impossible to answer. $\endgroup$ Aug 16, 2020 at 10:28
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    $\begingroup$ If you feel inclined, notice that only "seeing outside" is not possible. This doesn't include other kinds of radiation you can't see... $\endgroup$
    – asdfex
    Aug 16, 2020 at 12:46
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    $\begingroup$ If by definition the box is moving, you don't need any instrument. But well... . The question you should have asked is "motion relative to what?". There is no such thing as motion without a referential. The question is flawed. Anyway, the answer is "none". $\endgroup$
    – Florian F
    Aug 16, 2020 at 13:53
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    $\begingroup$ @JKusin The point of the discussion is that you will find ejecting the same amount of material at the same speed relative to the box will always give the same "felt" (proper) acceleration (OK, you need to compensate for the reduced mass of the box!). A "low value" of something you cannot measure has no useful meaning. $\endgroup$
    – m4r35n357
    Aug 16, 2020 at 16:29
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    $\begingroup$ @JKusin Yes that is right, but it is special relativity, not general. You only need to use GR in curved spacetime. $\endgroup$
    – m4r35n357
    Aug 16, 2020 at 19:55

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You can't. This is a literal textbook question about Galileo's ship. I cannot understand why this seems not to be taught in schools, the idea is nearly 400 years old, and the description of it is only a couple of hundred words.

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This question is basically asking about the principle of relativity, which states:

Physical laws should be the same in every inertial frame of reference.

Since you're moving at constant speed, you are in an inertial frame of reference, and there is no detectable difference. That's why you are equally justified to say you are at rest and the rest of the universe is moving backwards at constant speed. This principle is also why we say there's no preferred frame of reference.

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  • $\begingroup$ Okay, that was what I was thinking but I wasn't sure if there was some caveat. Thanks $\endgroup$
    – Amadeus
    Aug 16, 2020 at 5:16
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First of all the question is a little ambiguous , but I assume that it meant "constant speed" relative to an inertial frame(Newtonian definition) if so , then there is no experiment you can do (done entirely in the box) to figure out whether your moving with constant speed (for example relative to earth) or other things are moving and your fixed or it's a combination of both .

Note:I assumed the non-relativistic case , but even considering SR my answer wouldn't change .

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The interview question is inherently flawed because the box must permit its contents one path of interaction with the exterior: gravity. The reason is that other masses within the box's light cone, however remote, will accelerate the box and must accelerate its contents as well. The contents is supposed to move with the box along the same geodesic. The box can not, even not within the realm of our thought experiment, shield its contents from gravity in the way we can assume it shields from radiation.

Now the gravitational field will be ever-so-slightly inhomogeneous; there will be tidal forces within the box which can be measured. The universe is supposedly isotropic when perceived from the cosmic rest frame. If you start moving relative to it, you'll have anisotropic inhomogeneities in the gravitational fields you measure. For example, if the box is large enough to build a Ligo in it you'll measure a red/blue shift of the gravitational wave frequencies from large mass collisions. Even if, to make it harder, we suspend such events for the duration of our thought experiment you still have "inhomogeneity events", just ones which are harder to measure (passing a far star etc.). These events will be anisotropic while you move relative to the cosmic rest frame.1

The only way to prevent gravity from betraying your movement is to imagine the box in an empty light cone. Then there are two questions left to ask the interviewer:

  1. What exactly was the box supposed to shield me from again?
  2. Which motion?

1 Of course, the gravity from far objects will be weak, their inhomogeneities minuscule, and unless the box is moving at cosmological speed their anisotropy will be so small that only the keenest and most patient experimenter will be able to obtain statistically significant data through the eons. But hey, it's a thought experiment. We are not putting people in boxes.
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As other's have said, there's no experiment you can do to determine your velocity.

maybe if we somehow converted the mass of some object into energy it would be slightly more than what we would expect from $E=mc^2$ since it also has kinetic energy.

No, that won't work. The object is (presumably) at rest in the frame of the box, so it has zero kinetic energy relative to that frame. There's no such thing as absolute kinetic energy or velocity, it's always measured relative to some frame.

But if you have, for example, an electron and a positron, which both have (rest) mass of $511\, keV/c^2$, and those particles have a total kinetic energy of $E$ relative to the box, when you annihilate the electron with the positron the gamma rays produced will have total energy of $E + 1022\,keV$.

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There always exists an inertial reference frame in which your velocity is not identically zero, so in some sense, you could say you are "in motion" in any situation. However, since the box is moving inertially, it would often be conventional to set its velocity to zero.

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More heat would be generated on different sides of the box from friction. So assuming they tell you the only thing that can be going on outside is movement or not, that would give an answer. Unless you are in a vacuum.

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  • $\begingroup$ Troll downvote? $\endgroup$
    – user272451
    Aug 16, 2020 at 15:04
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It can be done depending on the circumstances

We need to know (1) where you are relative to the nearest massive object (2) what the constant 'speed' is and its direction (3) how long you have to carry out the experiment (4) if the box is allowed to rotate.

For example

Suppose the box starts on the surface of the Earth, and moves in a straight line, tangent to the surface. You will leave Earth. As you do so, the apparent direction and force of gravity will change. A simple static pendulum will detect one and a spring balance will detect the other.

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    $\begingroup$ If you are in free fall (follow a geodesic) you cannot perceive the homogeneous part of gravity. If the box follows a straight line in a gravitational field it is not in free fall though but it is accelerated in GR which was not the situation presented in the interview question, presumably for the reason you have given. But the thought experiment does some hand-waving concerning gravity which invalidates the question. $\endgroup$ Aug 16, 2020 at 16:36
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    $\begingroup$ @Peter - Reinstate Monica - If you are in free fall then you are accelerating towards the Earth. This contradicts the requirement, "no acceleration whatsoever". $\endgroup$ Aug 16, 2020 at 16:38
  • $\begingroup$ @Peter - Reinstate Monica - P.S. The question, as you point out, is a mess. However, by pointing out the flaws, the OP could demonstrate their understanding of the situation to the interviewers. $\endgroup$ Aug 16, 2020 at 16:45
  • $\begingroup$ There is acceleration and proper acceleration (en.wikipedia.org/wiki/Proper_acceleration) ;-) which I was referring to: A force which nudges an object off its geodesic, vulgo free fall. Because the universe is full of gravity there is no "unacclerated" state in the naive sense; we are always pulled somewhere. (You can be unaccelerated relative to somebody else though, for example I'm unaccelerated relative to my keyboard, which is good. But "naively" as well as "properly" my keyboard and me are accelerated quite a bit each single moment, together.) $\endgroup$ Aug 16, 2020 at 16:45
  • $\begingroup$ @Peter - Reinstate Monica - The question, as you indicate, is a mess. However, by pointing out the flaws, the OP could demonstrate their understanding of the situation to the interviewers. $\endgroup$ Aug 16, 2020 at 16:46
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There are a few ways you could attempt to answer the trick question and thereby give the interviewer an idea of how you approach problems (after you've first said "under our current understanding of physics, there is no privileged inertial reference frame" and thereby answered the object-level question).

One of the instruments you need might be a drill or an X-ray-sensitive plate, for example, so that in a couple of different ways you can start to see outside the box. Maybe one of the instruments is an odometer of some sort (and discuss how you might be able to make an odometer for e.g. a car that is on a freely-moving conveyor belt).

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  • $\begingroup$ Certainly "is not useful" is in the eye of the beholder, but this is explicitly an interview question and there are standard interview techniques including "work around the implicit assumptions of the question". Do please enlighten me if I'm actually wrong. $\endgroup$ Aug 16, 2020 at 21:58