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I've just read this interesting Wikipedia article about time dilation and length contraction in special relativity.

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Derivation of time dilation

Applying the above postulates, consider the inside of any vehicle (usually exemplified by a train) moving with a velocity $v$ with respect to someone standing on the ground as the vehicle passes. Inside, a light is shone upwards to a mirror on the ceiling, where the light reflects back down. If the height of the mirror is $h$, and the speed of light $c$, then the time it takes for the light to go up and come back down is:

$t = \frac{2h}{c}$

However, to the observer on the ground, the situation is very different. Since the train is moving by the observer on the ground, the light beam appears to move diagonally instead of straight up and down. To visualize this, picture the light being emitted at one point, then having the vehicle move until the light hits the mirror at the top of the vehicle, and then having the train move still more until the light beam returns to the bottom of the vehicle. The light beam will have appeared to have moved diagonally upward with the train, and then diagonally downward. This path will help form two-right sided triangles, with the height as one of the sides, and the two straight parts of the path being the respective hypotenuses:

$c^2 \left(\frac{t'}{2}\right)^2 = h^2 + v^2 \left(\frac{t'}{2}\right)^2$

Rearranging to get $t'$:

$ (\frac{t'}{2})^2 = \frac{h^2}{c^2 - v^2}$

$\frac{t'}{2} = \frac{h}{\sqrt{c^2 - v^2}}$

$t' = \frac{2h}{\sqrt{c^2 - v^2}}$

Taking out a factor of $c$, and then plugging in for $t$, one finds:

$t' = \frac{2h}c \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} = \frac{t}{\sqrt{1 - \frac{v^2}{c^2}}}$

This is the formula for time dilation:

$t' = \gamma t$

In this example the time measured in the frame on the vehicle, $t$, is known as the proper time. The proper time between two events - such as the event of light being emitted on the vehicle and the event of light being received on the vehicle - is the time between the two events in a frame where the events occur at the same location. So, above, the emission and reception of the light both took place in the vehicle's frame, making the time that an observer in the vehicle's frame would measure the proper time.

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Derivation of length contraction

Consider a long train, moving with velocity $v$ with respect to the ground, and one observer on the train and one on the ground, standing next to a post. The observer on the train sees the front of the train pass the post, and then, some time $t'$ later, sees the end of the train pass the same post. He then calculates the train's length as follows:

$\ell' = v t'$

However, the observer on the ground, making the same measurement, comes to a different conclusion. This observer finds that time $t$ passed between the front of the train passing the post, and the back of the train passing the post. Because the two events - the passing of each end of the train by the post - occurred in the same place in the ground observer's frame, the time this observer measured is the proper time. So:

$\ell = v t = v \frac{t'}{\gamma} = \frac{\ell'}{\gamma}$

This is the formula for length contraction. As there existed a proper time for time dilation, there exists a proper length for length contraction, which in this case is $\ell$. The proper length of an object is the length of the object in the frame in which the object is at rest. Also, this contraction only affects the dimensions of the object which are parallel to the relative velocity between the object and observer. Thus, lengths perpendicular to the direction of motion are unaffected by length contraction.

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I understood all the calculations. But what I didn't understood was the proper time and the proper length.

They say that the proper time is the time measured in the moving train. But the proper length is the length measured by the observer on the ground. Why didn't they take the same frame for both proper time and proper length?

And they define the proper length as the length measured by the observer on the ground. But then they say this:

The proper length of an object is the length of the object in the frame in which the object is at rest.

But I don't understand this. To me this seems to be wrong, because the train is not at rest in the frame of the observer on the ground.

But the train would be at rest in the frame of the train itself. So in my opinion it would have been more logical to define both the proper time and the proper length to be the time and length measured in the frame of the train.

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    $\begingroup$ The proper time of a process P (that the object O, e.g. a train, undergoes) and the proper length of object O are defined in the same frame, namely the rest frame of O (in your case the train). $\endgroup$ – Luboš Motl Mar 29 '15 at 8:45
  • $\begingroup$ @LubošMotl i realise that this is an old post- not expecting a reply at all! But was just wondering, if the proper length and proper time are measured in the same frame for a particular measurement, does this not give a preferred frame which violates the principle of relativity? $\endgroup$ – 21joanna12 Mar 5 '17 at 19:30
  • $\begingroup$ Dear Joanna, there is a preferred frame for the definition of "proper" - it's the frame in which the given object is at rest - but there is no preferred frame in which the laws of physics would have a simpler form. If you have two trains on a collision course, each of them has its rest frame but none of these rest frames is privileged or special. The laws of physics have the same and equally simply/hard form in both frames. $\endgroup$ – Luboš Motl Mar 6 '17 at 6:29
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There is no contradiction. From your quote:

The proper length of an object is the length of the object in the frame in which the object is at rest.

and

The proper time between two events - such as the event of light being emitted on the vehicle and the event of light being received on the vehicle - is the time between the two events in a frame where the events occur at the same location.

Obviously, the rest frame of an object is a frame where all events involving it are happening at the same place.

Therefore, proper length and proper time are defined here in the same reference frame - that of the object at rest.

You should note that this is not the "best" definition of proper time though. Given the Minkowski metric $$ g_{\mu\nu} \mathrm{d}x^\mu\mathrm{d}x^\nu = \mathrm{d}s^2 = \mathrm{d}t^2 - \frac{1}{c^2}\mathrm{d}\vec x^2 $$ one defines the proper length of an observer travelling along a worldline $\gamma : [t_0,t_1]\to\mathbb{R}^4$ as the integral of the infinitesimal length element $\mathrm{d}s$ along it: $$ \tau = \int_\gamma \mathrm{d}s = \int_{t_0}^{t_1}\sqrt{\left(\frac{\mathrm{d}\gamma^0}{\mathrm{d}t}\right)^2 - \frac{1}{c^2} \sum_{i=1}^3 \left(\frac{\mathrm{d}\gamma^i}{\mathrm{d}t}\right)^2}\mathrm{d}t$$ which agrees with the former definition in the rest frame because the worldline of an observer in its own frame is just $\gamma(t) = (t,0,0,0)^T$, but has the advantage of being manifestly Lorentz invariant - this shows all observers will agree what the proper time for any other observer they see will be.

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  • $\begingroup$ ACuriousMind: Given your definition of the "coordinate curve $\gamma : [t_0, t_1] \rightarrow \mathbb R^4$" it seems questionable that the "dimensional" symbol "$c$" appears, as it does, in the square root expression which you wrote to express $\tau$. Wouldn't it be interesting and challenging, especially for the purpose of establishing notation, to consider instead a path "$\Gamma : [t_0, t_1] \rightarrow \mathcal S$", where "$\mathcal S$" is the set of events of a given region, and try to express $$\tau = \int_{\Gamma} {\rm d}s = \int_{t_0}^{t_1} \frac{{\rm d}}{{\rm d}t}s~ {\rm d}t =~... $$. $\endgroup$ – user12262 Mar 31 '15 at 0:12
  • $\begingroup$ How is the question's definition not manifestly Lorentz invariant? I think it's fairly obvious that all observers will agree on e.g. "that specific guy's biological time" or the "time measured by that specific watch", which is what the definition is saying. $\endgroup$ – Abhimanyu Pallavi Sudhir Jun 8 '18 at 5:00
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Proper time is a geometric property of spacetime: multipled by c it's simply "length" of a pice of a world line of the particle; "a geometric property" means independence on the coordinate system (as in the Euclidean space length of curve is independent on the coordinate system). Yet, spacetime geometry is different than Euclidean: the "length" in the spacetime means pseudo-length (i.e. it can be equal to zero for not identical points, as for the world line of a photon). ACuriousMind♦ explained it in really CuriousMindnes style!

In the spacetime of general relarivity (or in any other $n$-dimentional pseudo-riemannian manifold) the length of the curve is $$ L(\gamma )=\int_{\gamma} ds = \int \limits _{t_0}^{t_1}{\sqrt {{\Bigg |}\sum _{i,j=1}^{n}g_{ij}(\gamma (t)){\frac {d\gamma ^{i}(t)}{dt}}{\frac {d\gamma ^{j}(t)}{dt}}{\Bigg |}}}\,\, dt $$

where $ g_{ij}(\gamma (t))$ - metric tensor calculated at the point $\gamma (t), t\in \langle t_0, t_1\rangle $ . Finally, $$\tau=\frac{L(\gamma )}{c}$$ is a proper time of a particle moving along the curve (time can be measured by particle's clock, yet the lenght cannot be measured by a solid metric bar, as in the Euclidean space).

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A quantity or geometric relation being called "proper" can be understood as "referring to those participants who are thereby (directly, intrinsically) characterized".

Consequently we can consider for instance

  • the "proper length of a given train", as the distance of its two ends ("tip of the locomotive, $A$" and "ETD, $B$") between each other, provided they did indeed remain at rest to each other,

  • the "proper length of a given section of track", as the distance of its two ends (e.g. the two corresponding railway sleepers, $J$ and $K$) between each other, provided they did indeed remain at rest to each other,

  • the "proper time" (rather: duration) of locomotive tip $A$, from $A$'s indication of having been passed by $J$ until $A$'s indication of having been passed by $K$ (if $A$ had indeed been passed by railroad ties $J$ and $K$, in that order), and

  • the "proper time" (duration) of railroad tie $J$, from $J$'s indication of having been passed by $A$ until $J$'s indication simultaneous to railroad tie $K$'s indication of having been passed by $A$ (provided $J$ and $K$ they did indeed remain at rest to each other, so they could succeed in determining which indication of one had been simultaneous to which indication of the other).

A justification and occasion for emphasizing "proper" quantities and relations at all arises because in some presentations of the theory of relativity certain quantities and relations are indeed attributed "extrinsically", and thus "improperly"; for instance

  • by calling the distance between a certain pair of railroad ties (which is perfectly "proper" by itself, in this sense) also the "(improper) length of the moving train" (in distinction to its own proper length), or

  • by attributing some duration determined by railway ties improperly to (parts of) the trial; or simply by failing to carefully distinguish the duration of the track being occupied by a (moving) train, from the travelling duration of the train (itself).

p.s.
Some notes on the relation between durations "$\Delta \tau$" of participants and intervals "$s^2$" between pairs of events:

Considering two events which are "time-like" related to each other, i.e. such that there is at least one participant who took part first in one and then in the other event, then the magnitude of the interval between these two events, $\sqrt{-s^2}$ (i.e. using the sign convention of Wikipedia) is given by the duration of the (or any) participant who moved uniformly (unaccelerated, inertially) between these events; which (with the indicated convention) is the largest duration, from the indication of having participated in one of these two events until the indication of having participated in the other, of all applicable participants.

In turn, as far as the path of a participant can be piecewise approximated by sections of uniform motion, the duration of this participant from its indication of having participated in one event until its indication of having participated in another event can be expressed (approximatly) as a sum of interval magnituides over suitable pieces of the path; and, in the limit of describing the path by ever more additional events "on the way", with ever smaller pieces between successive events (in comparison to the interval magnituide between the two path ends), as an integral.

Finally it is to be emphasized that all these determinations are of course independent of any (possible, optional) assignments of coordinates to events and/or to participants and their indications.

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