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I've seem sometimes people saying that although uniform motion on a straight line cannot be detected and hence it is not absolute, acceleration is indeed absolute in Classical Mechanics (I don't know yet how this turns out to work in GR, so for this question I'm explicitly talking about CM).

Now, thinking about it for a while I think I got the idea and I want to know if it is this way it works. Suppose we consider one observer inside a train. With the train at rest, a ball is attached to the ceiling by a string. For this observer there are then two forces to be considered: the weight of the ball and the tension on the string.

Of course in this situation, the string is fully stretched and the ball sits making angle $\theta_0 = 0$ with the $y$ axis. Now, suppose the train moves with constant acceleration $a$. In that case, the ball will raise and now will have a new angle $\theta$.

The observer, however, still sees just the two forces which were acting before. In that case, it should be possible to compute a new acceleration being added, in order to change the state of motion of the ball so that it now makes an angle $\theta$ with the vertical.

So the idea, is that performing experiments himself, inside the train it is possible to detect the value of the acceleration? In that sense, acceleration is not something that has to be measured with respect to some outside frame? Is that the idea of absolute acceleration?

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    $\begingroup$ +1 Spot on. See my short answer discussing the same notion in GR. $\endgroup$ Mar 31 '15 at 4:43
  • $\begingroup$ IMO: it is not true: 'The observer, however, still sees just the two forces there were acting before' . In fact the observer will feel, and measure, the kick of the seat in his back side, or on any accelerometer (use the smartphone, if you wish). But your conclusion is correct. $\endgroup$ Mar 31 '15 at 5:02
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"So the idea, is that performing experiments himself, inside the train it is possible to detect the value of the acceleration?"

You're spot on. Couldn't write it better myself. BTW in GR we have the same situation, as long as you qualify what you mean. All observers will agree on the reading on an accelerometer fixed in a certain frame, and on the readings on an array of accelerometers positioned on a system of rods fixed relative to a given frame that thus lay down a local $XYZ$ Cartesian co-ordinate system. The latter will detect rotation. As with Newton's bucket, there is also an absolute notion of rotation in GTR. Look up the Lense-Thirring (frame dragging) effect, or see Mark Eichenlaub's wonderful exposition in his answer to the Physics SE question "Why do we say that the earth moves around the sun?". What's a little weird about this notion of absolute acceleration is that when you sit "stationary" on the surface of a nonrotating gravitating planet, your accelerometer reads the value corresponding to the Newtonian idea of local acceleration due to gravity. You can bring up an app on your smartphone or iPad to read its accelerometer. Do this and you'll see it reading $9.81{\rm m\,s^{-2}}$ when your phone is on the table. Now take it to your bedroom and drop it a meter or so onto a pillow. You'll see it register $0{\rm m\,s^{-2}}$ as it falls. The phone fleetingly followed its geodesic in spacetime and thus rode fleetingly in a truly inertial (in the GTR, not Newtonian sense) frame and it could tell you so. Pretty neat eh? Sometimes you will hear that acceleration can be relative in GTR: this means something different, i.e. that the Einstein Field Equations retain their form in all co-ordinate, including non inertial frames. In contrast, the equations of special relativity are different for accelerated observers in flat spacetime (see for example the difference between the metric on a Rindler chart and that for an inertial observer). But the trusty accelerometer - a "real" device - gives us an absolute notion of acceleration.

A highly famous man wrote your last paragraph down as a rather poetic allegory. See Galileo's famous Allegory of Salviati's Ship. In his day and culture, he was not free to write the kinds of things you have directly, so he had to hide them in allegories and plays.

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I do not believe your statement to be true, as what you're measuring now is not the absolute acceleration of the train but rather the acceleration of the train relative to the ball.

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