Is length contraction real or just a matter of perspective?

[JVNY]

I would appreciate views on the following understanding of length and distance in SR, restricted here to inertial movement. This is the beginning of a follow up to the thread: https://www.physicsforums.com/showthread.php?t=724025

1. The spatial length of an object is the spatial separation (distance) between its ends measured at the same time.

2. One can measure length (distance) using a number of methods, such as a specified rod, or coordinates set up in advance based on a specified rod.

3. Neither spatial separation nor simultaneity is invariant in SR. (see Taylor and Wheeler, Spacetime Physics, page 56: "not the same: space separations, time separations")

4. Because simultaneity and spatial separation are not invariant in SR, spatial length is not invariant in SR.

5. Any measure of the length of an object (or distance between two objects) in one reference frame is equally valid as a measure of the length of the object (or distance between the two objects) in any other reference frame.

Can you see why this . . . would yield a different length for the rod . . . because of the relative motion between O′ and O? O′ simply measures the rod to be a different (in fact shorter) length from its length as measured by O (the "rest" length since O is carrying the rod); however neither measurement is any more correct than the other i.e. they are both equally valid after being attributed to the respective observers.

WannabeNewton at https://www.physicsforums.com/showpost.php?p=4625309&postcount=6​

These seem to be fundamental to any further analysis. But I have read a number of pieces that seem to take a different view. For example, sometimes it seems that people think of proper length (spatial separation of ends of an object measured at the same time in the object's rest frame) as essentially real and invariant, and length contracted length as being coordinate dependent and not real. An example is:

Time dilation and length contraction are artifacts of remote observers. That is, in your own frame of reference, you are always the same size and your clock always ticks at one second per second. Some other observer, from a different frame of reference, SEES you as being length contracted and with a slow moving clock.

You, right now as you read this, are moving at almost the speed of light from the reference frame of an accelerated particle at CERN. Do you feel any different?

phinds at https://www.physicsforums.com/showpost.php?p=4625297&postcount=5​

But your length is also invariant in the other inertial reference frame -- it is just shorter there than in your frame. As WannabeNewton says, each length is equally valid. The muons in the classic experiment do not merely "see" the height of the mountain to be smaller than we do; the height of the mountain actually is smaller for the muons than it is for us. And neither measure of height is more valid or real than the other.

Another example imagines a construct of an odometer to determine invariant length or distance that is not dependent upon any coordinates:

The aim is not to involve frames per se, at all. Instead, generalize the idea of a 'road' going by. You reel out tape measure matching the speed of road as it passes. When done, you have a measure of how much road has gone by . . . The definition is coordinate and frame independent. The only thing specified is a congruence of world lines of 'reference objects'. This is much less than a coordinate system. Given this, only invariants are computed.

https://www.physicsforums.com/showpost.php?p=4621820&postcount=3​

But again, there is no invariant length or distance in SR.

Thanks in advance.

 

[WannabeNewton]

I agree with you on all your points. Measurements of "length" and "spatial distance" between events as made by one inertial observer can differ from similar measurements of "length" and "spatial distance" between the same events as made by another inertial observer but neither is any more correct than the other.

Even something like the "shape" of a rigid object has no observer independent meaning because of the relativity of simultaneity. If we have a rod parallel to the $x$ axis of $O$'s frame while accelerating uniformly along the $y$ axis then $O$ judges the rod to be a straight horizontal line at each "instant of time" (simultaneity line) relative to $O$ but an observer $O'$ moving uniformly along the $x$ axis will judge the rod to be parabolic because $O'$ has a simultaneity line that's tilted relative to that of $O$ and this slices the worldlines of all the points of the rod in a different way, yielding a parabolic shape.

That isn't to say however that one is not more preferred than the other. For example, often when dealing with general relativistic fluids we choose to work solely with properties of the fluid as measured by observers comoving with the fluid elements. The 4-velocity field of the fluid picks out a preferred family of observers and a preferred set of measurements of fluid properties. In particular, whenever defining quantities that depend on the "spatial distances" between fluid elements, such as the vorticity of the fluid, we use the spatial distances as measured by observers comoving with the fluid elements.

However "preferred" measurements should not be confused with "correct" measurements.

EDIT: as a side note, spatio-temporal intervals measured by an observer can easily be expressed in a coordinate-independent and frame-independent way even in the most general setting possible. See here: https://www.physicsforums.com/showpost.php?p=4559456&postcount=10

 

[jtbell]

But your length is also invariant in the other inertial reference frame -- it is just shorter there than in your frame.

I would not use the word "invariant" here. In the context of relativity theory, "invariant" has the specific meaning "has the same value in every inertial reference frame."

 

[Chestermiller]

Hi JVNY. I looked up your profile and see you are a non-scientist with a profession in finance. It is much easier to explain what you are asking about if we could use a little math. What is the extent of your math background? Any analytic geometry? Any calculus?

Chet

 

[Dale]

people think of proper length (spatial separation of ends of an object measured at the same time in the object's rest frame) as essentially real and invariant, and length contracted length as being coordinate dependent and not real.

I don't know about the label "real". I think it is a rather useless philosophical label. However, it is correct that length is frame variant while proper length in frame invariant.

If you have an inertial coordinate system and if you have two events on opposite ends of the object then the length is $\sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2}$ iff $\Delta t =0$. Since different frames will disagree about $\Delta t$ they will disagree about whether or not $\sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2}$ fulfills the requirement to be a length, and therefore length is frame variant.

If you have an inertial coordinate system and if you have two events on opposite ends of the object then the proper length is $\sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2 - c^2 \Delta t^2}$ iff $\Delta t =0$ in the rest frame. Since all frames will agree whether or not $\Delta t=0$ in the rest frame and since all frames will agree on the numerical value of $\sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2 - c^2 \Delta t^2}$ the proper length is invariant.

 

[JVNY]

Hi JVNY. I looked up your profile and see you are a non-scientist with a profession in finance. It is much easier to explain what you are asking about if we could use a little math. What is the extent of your math background? Any analytic geometry? Any calculus?

Chet

Chet, just the algebra used in texts like Taylor and Wheeler's Spacetime Physics; also I can use the computer to solve for the standard formulas for acceleration that are used for example in the relativistic rocket explanation at http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html

 

[JVNY]

It seems that there is a disagreement between DaleSpam on one hand and WannabeNewton and jtbell on the other. jtbell advises that the word "invariant" means "has the same value in every inertial reference frame." The length of an object does not have the same value in every inertial reference frame. And WannabeNewton agreed with all of the points in the original post. But DaleSpam considers the proper length to be "invariant."

Is this a real disagreement? Or is it just that the proper length is a "preferred" measurement that should not be confused with a "correct" measurement (in WannabeNewton's description)? I have studied the Taylor and Wheeler text, and it asserts that there is no measure of length that is invariant.

 

[PAllen]

It seems that there is a disagreement between DaleSpam on one hand and WannabeNewton and jtbell on the other. jtbell advises that the word "invariant" means "has the same value in every inertial reference frame." The length of an object does not have the same value in every inertial reference frame. And WannabeNewton agreed with all of the points in the original post. But DaleSpam considers the proper length to be "invariant."

Is this a real disagreement? Or is it just that the proper length is a "preferred" measurement that should not be confused with a "correct" measurement (in WannabeNewton's description)? I have studied the Taylor and Wheeler text, and it asserts that there is no measure of length that is invariant.

Proper length of an object is invariant, and is the same in any reference frame. Measured length of an object moving relative to you is differs from the length of the object measured in the rest frame. Proper length is only directly measured in the object's rest frame. In other frames it is computed (from several measurements).

Similarly, distance between two objects varies with the relative motion of who is doing the measurement. Proper distance between two specific events is invariant, and computed to be the same in any frame (but is only measured as a distance in a frame in which the events are simultaneous).

I don't think there are any real disagreements.

 

[Dale]

It seems that there is a disagreement between DaleSpam on one hand and WannabeNewton and jtbell on the other. jtbell advises that the word "invariant" means "has the same value in every inertial reference frame." The length of an object does not have the same value in every inertial reference frame. And WannabeNewton agreed with all of the points in the original post. But DaleSpam considers the proper length to be "invariant."

Is this a real disagreement? Or is it just that the proper length is a "preferred" measurement that should not be confused with a "correct" measurement (in WannabeNewton's description)? I have studied the Taylor and Wheeler text, and it asserts that there is no measure of length that is invariant.

There is no disagreement. I think you may be slightly confused. I agree with WBN that the points in the OP are correct and I agree with jtbell that invariant means has the same value in every inertial frame.

I am not sure why you think there is disagreement in the answers you have received.

"Proper length" is not "length". They are different quantities defined differently. "Proper length" is invariant, it has the same value in every frame. "Length" is variant, it has different values in different frames. I thought that I was very explicit on that point, even including formulas.

 

[JVNY]

OK, thanks, I may have misunderstood, so I will leave it to jtbell and WBN to reply if they have any disagreement. The next question involves radar distance. I think that I am misunderstanding something that WBN wrote on this point.

WBN gives an example of radar distance in the observer's rest frame as follows:

O attaches a mirror to the front end of the rod (he holds the back end of the rod). He then emits a beam of light towards the mirror which bounces off instantaneously from the front mirror and arrives back to him. He uses a clock to record the time for the round-trip and says it is Δt. Well if the speed of light in the forward direction is the same as the speed of light in the backwards direction, like we assumed, then surely the light beam reached the mirror when O's clock read Δt/2 and hence the distance traveled would obviously be L0=cΔt/2. This distance spans the length of the rod as measured by O.

https://www.physicsforums.com/showpost.php?p=4625309&postcount=6​

I understand this. If the rod has 100 proper length, then in the observer's rest frame the light takes proper time 100 out and 100 back, for a total of Δt = 200, and proper length of 200/2, or 100.

WBN then states:

Can you see why this "radar echo" method would yield a different length for the rod when a similar measurement is made by O′ because of the relative motion between O′ and O? O′ simply measures the rod to be a different (in fact shorter) length from its length as measured by O (the "rest" length since O is carrying the rod); however neither measurement is any more correct than the other i.e. they are both equally valid after being attributed to the respective observers. [same link as above]​

I cannot see how this method yields a shorter rod length for O'. For example, in an O' frame in relative 0.8c motion, the rod has length 60. The O' time for the forward flash to reach the mirror is 300 [being 60 / (1 - 0.8)], and for the return flash is 33.333 [being 60 / (1 + 0.8)], for a total Δt = 333.333, and Δt/2 = 166.67 (a value greater than 100). So if O' uses this method, he gets a radar method length greater (not lesser) than the proper length.

How should one interpret WBN's post? I have not seen this method used except in the rest frame. In other frames typically writers use other methods, like Einstein's of having lightning strike at both ends of the object at the same time in the O' frame (e.g., the platform frame in the train and platform example).

 

[WannabeNewton]

It's infinitely more cumbersome to do it explicitly using radar signals but you would have to have $O'$ send a light signal at some instant of his clock aimed at the farther end of the rod and then at a later instant send out another light signal now aimed at the closer end of the rod such that the two signals reach the respective ends of the rod simultaneously relative to $O'$. By calculating the radar distance to each of the two simultaneous events, $O'$ can then subtract the two to get the length of the rod in his rest frame.

If you draw a space-time diagram representing the rest frame of $O'$ then the two ends of the rod would have two parallel but slanted worldlines. The proper length of the rod is the length of the line segment formed by the intersection of the $t = 0$ simultaneity line of $O$ with the worldlines of the two ends of the rod. The length of the rod as measured by $O'$ is simply the intersection of the $t' = 0$ simultaneity line of $O'$ with the worldlines of the two ends of the rod which, in the space-time diagram, is simply the intersection of said worldlines with the $x'$ axis. Clearly $O'$ will measure a shorter length than $O$ will.

The reason I mentioned radar signals is that these $t = \text{const.}$ and $t' = \text{const.}$ simultaneity lines are established by the respective observers using radar sets that they each carry. In their respective rest frames they use their radar sets to establish what they consider to be lines of simultaneity by applying the convention $t_B = \frac{1}{2}(t_A + t'_A)$ alluded to earlier.

 

[JVNY]

OK, WBN, thanks. The radar method is certainly not cumbersome in the rest frame, so it seems useful there.

So, I think that I am down to the final measure, which is "coordinate." In inertial frames in SR, is there a distinct measure of "coordinate" length or "coordinate" distance? There does not seem to be any difference between (a) setting up coordinates covering a station platform and determining a passing train's length by reference to those coordinates, and (b) making marks on the platform at the train's front and rear at the same time in the platform's frame, then using a meter stick to measure the distance between the marks.

 

[Chestermiller]

OK, WBN, thanks. The radar method is certainly not cumbersome in the rest frame, so it seems useful there.

So, I think that I am down to the final measure, which is "coordinate." In inertial frames in SR, is there a distinct measure of "coordinate" length or "coordinate" distance? There does not seem to be any difference between (a) setting up coordinates covering a station platform and determining a passing train's length by reference to those coordinates, and (b) making marks on the platform at the train's front and rear at the same time in the platform's frame, then using a meter stick to measure the distance between the marks.

Yes, it's the same thing if you can make the two marks at the same time in the platform's frame.

 

[WannabeNewton]

In inertial frames in SR, is there a distinct measure of "coordinate" length or "coordinate" distance?

If it helps, imagine at first $O$ only has the clock he's carrying with him and a radar set. Furthermore imagine there's a clock at rest with respect to $O$ at each point in space. Right now $O$ can only label events with a time coordinate if they lie on his worldline but by using his radar set, $O$ can synchronize all the aforementioned clocks with his own and in doing so constructs what we call a global time coordinate. Whenever his clock reads some time $t$ so will all the other clocks in space because they have been synchronized with his so he can now assign a time coordinate to each point in space using his own clock.

By synchronizing each comoving clock in space with his own, $O$ has also defined what it means for events to be simultaneous because if two synchronized clocks $A$ and $B$ have their hands both positioned at $t$ at events $p_A$ and $p_B$ on their respective worldlines then $p_A$ and $p_B$ are deemed simultaneous. Einstein synchronization mathematically leads to the simultaneity planes $t = \text{const.}$

Imagine now a rigid rod connecting each such clock with another; this forms what one might call a "rigid coordinate lattice". By combining these rigid rods with his simultaneity planes, $O$ can now determine lengths of objects. This "rigid coordinate lattice" of rods and clocks is his coordinate system.

 

[JVNY]

Agreed. And this seems to be consistent with Chestermiller's response above: there is no separate "coordinate" length or distance in an inertial frame. The length of an object measured by using a ruler is exactly the same as the length measured by coordinates of the object's ends along the rigid lattice when noted at the same time (time being the time shown on the lattice clocks). Indeed, this is the same length as determined using the radar method as long as the object is at rest in the lattice frame (which will also equal the object's proper length). For an object at rest, proper length = length = radar length = coordinate length = ruler length. Or for the distance between two points at rest, proper distance = distance = radar distance = coordinate distance = ruler distance.

This takes us to back to Born rigid motion. If you have a set of points that are at rest with respect to each other then accelerate Born rigidly, they maintain their proper distance at all times. See http://www.mathpages.com/home/kmath422/kmath422.htm.

Put another way, they maintain their rest distance at all times. See Franklin, "Lorentz contraction, Bell’s spaceships, and rigid body motion in special relativity," page 8, at http://arxiv.org/abs/0906.1919 (referring to rest length of an object that is accelerated Born rigidly).

This makes sense, because they are at rest with respect to each other at all times.

They also maintain their ruler distance at all times. Observers riding along with them can slowly pass a ruler and measure the distance between the points throughout the acceleration, and they will agree that the points maintain the same ruler distance. See http://en.wikipedia.org/wiki/Rindler_coordinates

So, we have points that are at all times at rest with respect to each other and maintain their same proper, ruler and rest distances -- just like points that are at rest in an inertial frame.

However, the radar distance is not the same. Consider three ships, rear center and front, with equal proper distance between rear and center as between center and front. The ships begin at rest, then accelerate Born rigidly. Flashes of light sent simultaneously from the center toward the front and rear ships reflect off of those ships and return to the center. The forward flash returns before the rearward flash. This is true even though the proper distance between rear and center remains the same as the proper distance between center and front (and despite the fact that the rest distances and ruler distances remain the same also).

So, question 1: what is this radar distance? When we discussed the issue in the earlier thread, posters used examples of curves (such as the distance along a curve from one city to another along the surface of the earth). However, this cannot explain the radar distance here for two reasons. First, SR does not used curved spacetime; SR uses flat spacetime.

Second, the curved line along the surface of the Earth is the ruler distance between the cities. If you lay a series of rulers between the cities you will measure the longer curved distance (rather than the shorter straight line distance that you would get in a tunnel bored straight between the cities underground). But we have confirmed that the ruler distance between center and rear remains the same as the ruler distance between center and front, yet the forward light flash returns to the center before the rearward one does.

Question 2: is it clear what the coordinate distance is between rear and center (and between front and center)? The lattice of rods should remain unchanged in the Born rigid acceleration, and so the distances as measured by the lattice rods should remain the same. In order to ensure simultaneous measurements you have to adjust the clock rates so that each clock ticks at the same rate (choosing a single clock for the Rindler coordinate time rate). But, that does not seem to be enough. The Einstein method of synchronizing the clocks as WannabeNewton describes above depends on radar signals, and so there seems to be a circularity. You need to use radar to synchronize the clocks, but the radar distance is different from the ruler, rest and proper distance.

It is conceptually easier in GR, where I understand that the ruler distance between concentric rings around a black hole is longer than the difference in radii calculated using Euclidean geometry, due to curved spacetime.

 

[WannabeNewton]

This makes sense, because they are at rest with respect to each other at all times.

In the case of a rod being accelerated in a Born rigid fashion this is true but keep in mind that Born rigidity of a macroscopic object does not require all the microscopic constituent particles to be at rest with respect to one another because we can have Born rigid rotation wherein the constituent particles rotate relative to one another but spatial distances between neighboring constituent particles will remain constant as measured in the instantaneous rest frames of the neighboring constituent particles.

They also maintain their ruler distance at all times.

Indeed, so as long as this ruler distance is measured in the rest frames of the constituent particles making up the Born rigid object (see above).

However, the radar distance is not the same.

Indeed this is true in general and it is precisely due to the reason you stated: if we imagine a rod that is accelerated in a Born rigid fashion from back to front then the closer to the front we get the harder the constituent particles have to accelerate in order to maintain Born rigidity; relative to a background global inertial frame this means that the particles closer to the front of the rod have greater instantaneous velocities than do the particles closer to the back of the rod. However note that radar distance agrees with ruler distance if the constituent particles we are using for measurement are infinitesimally close to one another.

So, question 1: what is this radar distance?

Could you rephrase this question? I'm not sure if you're looking for some explicit calculation of radar distance, a conceptual explanation of radar distance, or something else entirely.

The Einstein method of synchronizing the clocks as WannabeNewton describes above depends on radar signals, and so there seems to be a circularity.

Einstein synchronization doesn't work for a rigid coordinate lattice carried by a uniformly accelerating observer. If a clock in the lattice is Einstein synchronized with the local clock of the observer at some initial local time then the clocks will become desynchronized immediately after. Because Einstein synchronization fails to be consistent for the rigid coordinate lattice carried by this observer, in the sense that initially synchronized clocks become desynchronized soon after, we cannot use Einstein synchronization to define simultaneity for this observer because a simultaneity convention derivative of a synchronization convention requires the synchronization convention to be consistent.

This is why radar distance doesn't agree to all orders with ruler distance for a uniformly accelerating observer: ruler distance automatically takes into account the clock desynchronization since ruler distance is calculated using the geometrical definition of simultaneity whereas radar distance relies on local and distant clocks being Einstein synchronized hence if we have the uniformly accelerating observer use radar distance we are assuming that simultaneous events have the same clock readings which will give us over-estimates and under-estimates of the ruler distance.

However for events infinitesimally close to events on the world line of the uniformly accelerating observer we will find that ruler distance and radar distance agree and this is because the clock desynchronization factor is negligible for clocks in the observer's rigid coordinate lattice that are infinitesimally close to the observer's own clock.

 

[pervect]

I would say that the invariant interval is independent of the observer, is defined between any two events in space-time regardless of whether or not they are part of an object, and is given in special relativity by the formula given by DaleSpam (the square root of dx^2 + dy^2 + dz^2 - dt^2 for a spacelike interval).

Proper distance is most rigorously understood as applying to a curve with endpoints. If you integrate the invariant integral (as defined above) along the specified curve between the specified endpoints you get the proper distance.

Often times it is assumed that given the endpoints, one "knows" the right curve to integrate along, in which case specifying the endpoints is _assumed_ to specify the curve. This may or may not cause confusion depending on how explicit the author was about the assumptions.

In the flat space-time of Special relativity, the length of an object (which may or may not be rigid) at some time t is equal to the proper distance (the integral defined above) between the ends of the object along a curve of constant time - which curve this is is observer dependent, it depends on the observer's notion of simultaneity.

As a consequence of the above definitions, the length of an object at time t in a frame in which it as rest in the flat space-time of SR is also equal to the invariant interval between the endpoints at time t.


In general relativity or in accelerated coordinate systems, one needs to introduce a metric. The invariant interval then becomes defined only for "nearby" points, and is equal to:

$$ds^2 = \Sum_{i=0..3, j=0..3} g_{ij} dx^i dx^j$$

where $g_{ij}$ are the metric coefficients. If g_00 = -1 and g_ii = +1 for i = 1..3, one recovers the usual SR formula.

Proper distance is still defined by the integral of the invariant interval using the new definition, and the length of an object is still defined by the proper distance between its endpoints. The potential confusion over which curve to integrate over is greater, however. Usually it's correct to assume a curve of constant coordinate time when trying to figure out which curve to integrate along.

It is in general no longer true that one can simply take the invariant interval between the endpoints to get the length of an object in GR, one has to use the procedure above.

 

[WannabeNewton]

The Einstein method of synchronizing the clocks as WannabeNewton describes above depends on radar signals, and so there seems to be a circularity. You need to use radar to synchronize the clocks, but the radar distance is different from the ruler, rest and proper distance.

It is conceptually easier in GR, where I understand that the ruler distance between concentric rings around a black hole is longer than the difference in radii calculated using Euclidean geometry, due to curved spacetime.

I missed the latter comment the first time around so let me address it now. While what you said about non-euclidean geometry in GR is true, keep in mind that the aforementioned issues regarding (and stemming from) clock synchronization exist in GR for the same reasons as in SR.

Imagine we have a rotating star (or rotating black hole if you prefer but it doesn't matter). Consider an observer outside the rotating star who is fixed with respect to the distant stars at some radius $R$ as well as a ring of clocks at $R$ (in the same plane as the observer) that are also fixed with respect to the distant stars. The clocks are therefore also all at rest with respect to one another and with respect to the observer; the ring formed by this observer+clocks system is Born rigid and all constituents of this system have the same proper acceleration. Can the observer Einstein synchronize all these clocks with his own?

Imagine further that at some given instant of the observer's local time he emits a light signals in the prograde direction around the ring and a light signal in the retrograde direction around the ring (each clock is equipped with a mirror angled appropriately in order to allow light beams to circulate around the ring). If the observer is to have any hope of Einstein synchronizing all the clocks in the ring then the retrograde signal and prograde signal should arrive back at the observer at the same time (as measured by his clock). It will turn out however that for the observer+clocks ring that we have constructed fixed with respect to the distant stars outside of a rotating star, the two signals will not arrive back at the same instant of local time. This is formally analogous to the failure of global Einstein clock synchronization for a rotating ring in flat space-time. This effect is called the Sagnac effect.

There is another method of clock synchronization that for a rigid lattice of inertial clocks is mathematically equivalent to Einstein synchronization but doesn't use light signals; this synchronization method is known as slow clock transport.* So you may think that we can instead try using slow clock transport for the ring described above because the Sagnac effect only manifests itself when we have counter-propagating signals (of any kind not necessarily electromagnetic) but it turns out that even using slow clock transport an observer in the ring fails to achieve synchronization with all the clocks in the ring.

*See section 7 of the following paper: http://arxiv.org/pdf/1002.0130v1.pdf

 

[stevendaryl]

I don't know about the label "real". I think it is a rather useless philosophical label.

Just a pet peeve, but it annoys me that physicists use the words "philosophy" or "philosophical" pejoratively in this way. A real philosopher is not going to use a useless label, any more than a physicist is. One of the main occupations of philosophy is to try to figure which labels are useful and meaningful, and which ones are not.

Almost all scientists engage in philosophy, but only use the word "philosophy" to describe other people's useless thoughts.

 

[stevendaryl]

It seems that there is a disagreement between DaleSpam on one hand and WannabeNewton and jtbell on the other. jtbell advises that the word "invariant" means "has the same value in every inertial reference frame." The length of an object does not have the same value in every inertial reference frame. And WannabeNewton agreed with all of the points in the original post. But DaleSpam considers the proper length to be "invariant."

That's not a disagreement. Proper length has the same value in every inertial reference frame.

 

[JVNY]

Thanks for a lot of good points to think about. Perhaps best to start with what seems most fundamental. Is spacetime different in an accelerating reference frame in SR than in an inertial reference frame in SR? I had learned that SR uses flat spacetime. But see the following from pervect:

. . . the length of an object at time t in a frame in which it as rest in the flat space-time of SR is also equal to the invariant interval between the endpoints at time t.

In general relativity or in accelerated coordinate systems, one needs to introduce a metric. The invariant interval then becomes defined only for "nearby" points . . .

This suggests that spacetime is different in SR in an accelerating frame (which is grouped with GR, and uses a metric) than in an inertial frame in SR (which is considered separately, and uses the interval).

And if it is different, is spacetime curved in an accelerating frame in SR?

 

[stevendaryl]

And if it is different, is spacetime curved in an accelerating frame in SR?

There are two different concepts: (1) A manifold (space, or spacetime) with curvature, and (2) curvilinear coordinates. If your manifold is curved, then you are forced to use curvilinear coordinates to describe a large enough region. You can use curvilinear coordinates in flat spacetime, but that doesn't make it curved.

An accelerating frame is just using curvilinear coordinates to make a set of accelerating observers "stationary" (that is, have zero spatial velocity).

 

[stevendaryl]

There are two different concepts: (1) A manifold (space, or spacetime) with curvature, and (2) curvilinear coordinates. If your manifold is curved, then you are forced to use curvilinear coordinates to describe a large enough region. You can use curvilinear coordinates in flat spacetime, but that doesn't make it curved.

An accelerating frame is just using curvilinear coordinates to make a set of accelerating observers "stationary" (that is, have zero spatial velocity).

Just another point: flat spacetime has a metric, just as curved spacetime does. The significance of an "inertial reference frame" is that, as expressed in those frames, the metric is so trivial that you can pretty much get away without ever mentioning it explicitly (although it is involved implicitly whenever you see expressions such as "$x^2 + y^2 + z^2 - c^2 t^2$" and "$E^2 - p^2 c^2$). In curved spacetime, or in noninertial, curvilinear coordinates, it becomes more important to explicitly talk about the metric.

 

[WannabeNewton]

Thanks for a lot of good points to think about. Perhaps best to start with what seems most fundamental. Is spacetime different in an accelerating reference frame in SR than in an inertial reference frame in SR?

Yes. For one, there can be global anisotropies in the speed of light. Take, for example, an observer riding on a rotating ring in flat space-time. As mentioned earlier if the observer emits prograde and retrograde light signals that circulate around the ring (by means of mirrors placed around the ring) then the observer will record different local times of reception of the counter-propagating signals. In this example this is directly related to the fact that the tangent vector field $\vec{\xi}$ to the world tube of the rotating ring has a non-vanishing curl $\vec{\nabla}\times \vec{\xi} \neq 0$ in the instantaneous rest frame of the observer. However it is important to keep in mind that this anisotropy in the speed of light relative to the observer riding on the rotating ring is global i.e. it only manifests itself after the prograde and retrograde signals complete an entire circuit. Locally the speed of light will still be isotropic regardless of the Sagnac effect.

EDIT: Also, aside from the obvious presence of inertial forces, accelerating frames of reference also have finite extents in space-time. Since you have access to MTW, see section 6.3 for a discussion of this.

And if it is different, is spacetime curved in an accelerating frame in SR?

No.

 

[WannabeNewton]

An accelerating frame is just using curvilinear coordinates to make a set of accelerating observers "stationary" (that is, have zero spatial velocity).

A frame is not the same thing as a coordinate system.

 

[stevendaryl]

A frame is not the same thing as a coordinate system.

That's true, but the vast majority of the uses of the word "frame" don't actually mean the technical definition. The distinction between "an accelerated frame" and "an accelerated (noninertial) coordinate system" isn't usually very important. Is it?

 

[stevendaryl]

What is the technical definition of a "frame"? It seems to me that it is most applicable in cases where spacetime can be factored into space + time, where the spatial part of the metric is time-independent. So is it an equivalence class of coordinate systems that agree on the factoring into space + time?

 

[WannabeNewton]

That's true, but the vast majority of the uses of the word "frame" don't actually mean the technical definition. The distinction between "an accelerated frame" and "an accelerated (noninertial) coordinate system" isn't usually very important. Is it?

I agree with you wholeheartedly that in most mundane cases it doesn't hurt to conflate the two terms (although MTW does this to the point where it just gets annoying) but there are situations even in SR wherein one needs to specifically talk about frames and not coordinate systems such as, for example, when speaking of gyroscopic precession. Also I do think it's good for anyone doesn't already know the difference between the two to have the distinction in mind in order to avoid potential future confusion(s).

In GR however the distinction between a frame and a coordinate system is crucial both mathematically and physically. I'm not saying you conflated these two in the setting of GR of course because you specifically referred to accelerating frames in SR. Besides you already know the subtleties involving the terminology so this is just a side note for the OP.

What is the technical definition of a "frame"?

Given an observer $O$ in an arbitrary space-time, a frame for $O$ is a smoothly varying choice of orthonormal basis $\{e_{\mu}\}$ for the tangent space to space-time at each event on $O$'s world line such that $e_0 = u$ where $u$ is $O$'s 4-velocity.

 

[stevendaryl]

Given an observer $O$ in an arbitrary space-time, a frame for $O$ is a smoothly varying choice of orthonormal basis $\{e_{\mu}\}$ for the tangent space to space-time at each event on $O$'s world line such that $e_0 = u$ where $u$ is $O$'s 4-velocity.

So, it's not part of the requirement for a "frame" that the metric components are independent of "time"? I've only seen people talk about frames in those cases (inertial and rindler frames in SR, "rest" frames in Schwarzschild spacetime).

 

[WannabeNewton]

So, it's not part of the requirement for a "frame" that the metric components are independent of "time"? I've only seen people talk about frames in those cases (inertial and rindler frames in SR, "rest" frames in Schwarzschild spacetime).

No that's certainly not a requirement. All the examples you listed are stationary space-times wherein the metric tensor is invariant under the flow generated by a time-translation symmetry, a property which is intrinsic to the space-time geometry and has no dependence whatsoever on what frame is used to make measurements using rods, clocks, and gyroscopes. This time-translation symmetry does however pick out a preferred frame field for which the metric components are independent of time and this is the natural frame field associated with the stationary observers in the space-time i.e. those observers who follow orbits of the time-translation symmetry. In Schwarzschild space-time, for example, these are the observers fixed with respect to the distant stars.

Another common term for frame in the context of SR and GR is "tetrad". See here for more: http://en.wikipedia.org/wiki/Frame_fields_in_general_relativity

 

[phyti]

I understand this. If the rod has 100 proper length, then in the observer's rest frame the light takes proper time 100 out and 100 back, for a total of Δt = 200, and proper length of 200/2, or 100.
I cannot see how this method yields a shorter rod length for O'. For example, in an O' frame in relative 0.8c motion, the rod has length 60. The O' time for the forward flash to reach the mirror is 300 [being 60 / (1 - 0.8)], and for the return flash is 33.333 [being 60 / (1 + 0.8)], for a total Δt = 333.333, and Δt/2 = 166.67 (a value greater than 100). So if O' uses this method, he gets a radar method length greater (not lesser) than the proper length.

If O' is measuring the rod located in the O frame, then with a = v/c, g = gamma,
and d = distance to reflection, total time t1 is

t1 = 2d/c(1+a) = 2t/(1+a) and

d' = ct' = ct/g(1+a) = d*sqrt[(1-a)/(1+a)].

d' = 100*sqrt[.2/1.8] = 100/3 = 33.3

In fig.1, applying inverse time dilation the O' clock shows 66.7 at x = 89. The perception of events as recorded by O' are overlaid for comparison. According to O' the reflection would have occurred at x' = 60.
In fig.2, the essential data for O' is transferred from fig.1 to show the perception from the O' frame.
With O' assuming a rest frame, and M moving past at -.8c, the distance at t' = 0 is
33.3*1.8 = 59.9 (60 without rounding errors).

Points of interest.
1. The reflection R occurs at Ox = 100. Since the signal is detected early according to the O' clock, O' concludes the universe has contracted by .6 along the x axis, i.e. an interpretation resulting from his own time dilation. Being an anaut and studying the SR manual, he would obviously be aware of this. He does have a choice.
2. R is the reflection event resulting from the signal path (magenta) as observed by O. R' is the reflection event as defined by the SR convention. With current knowledge there is no known method of verifying either case and so it is irrelevant since the round trips are equal. Considering science history that may change later.

https://www.physicsforums.com/attachments/65543

 

[JVNY]

Let's revert to WBN's earlier response:

I'm not sure if you're looking for some explicit calculation of radar distance, a conceptual explanation of radar distance, or something else entirely.

Perhaps it is best to start with explicit calculations. Let's restrict to straight line motion to simplify. Also, let's use three points (rear, center and front) and accelerate them Born rigidly. We will measure distance (rather than length) and not have to worry about accelerating the many particles within a single object. Since they are points, we can ignore the issues of how to accelerate all parts of an object simultaneously.

Start with the three points at rest in an inertial frame, rear at x=0.5, center at x=0.75, and front at x=1.0. The center is equidistant from the rear and the front: the distance is 0.25 from rear to center, and 0.25 from center to front, so a total distance of 0.5 from rear to front. The three points accelerate simultaneously Born rigidly: rear at proper rate a=2, center at proper rate a=1.33, and front at proper rate a=1. Simultaneously with starting to accelerate, the center flashes light forward and rearward. The light flashes reflect off of front and rear, then return to center.

As illustrated on the attached, the forward flash strikes the front at center time (Tc) 0.216. The rearward flash does not strike the rear until Tc=0.304. Forward flash returns to center at Tc=0.432. The rearward flash does not return to center until Tc=0.608. The hyperbolas show the world lines of the three points, and the dashed lines show the lines of simultaneity for the accelerating points.

1. Is the proper distance between rear and center always 0.25?
2. Is the proper distance between center and front always 0.25?
3. Is the ruler distance between rear and center always 0.25?
4. Is the ruler distance between center and front always 0.25?
5. Is the coordinate distance between rear and center always 0.25?
6. Is the coordinate distance between center and front always 0.25?
7. Is the radar distance between rear and center 0.304?
8. Is the radar distance between center and front 0.216?

The clock rates on each point are different, of course. This presumably affects the radar distance. Following the convention in http://en.wikipedia.org/wiki/Rindler_coordinates, clocks on the rear and center can be programmed to tick at a proportionately faster rate so that they tick at the same rate as the clock on the front, where g=1. Does one do this to calculate the radar distance?

 

[WannabeNewton]

The answers to your other questions are all "yes".

...clocks on the rear and center can be programmed to tick at a proportionately faster rate so that they tick at the same rate as the clock on the front, where g=1.

Where in the article is this stated? If the $g = 1$ world line describes the very front of the rod then the farther back we get from the front of the rod the slower comoving clocks tick relative to the clock comoving with the very front of the rod. It is ruler distance that takes this clock desynchronization into account when determining simultaneity lines for comoving observers.

The one-way speed of light depends on the global time coordinate $t$ which is also the proper time read by the clock comoving with the very front of the rod. Therefore radar distance will only give accurate results if all comoving clocks are synchronized with the comoving clock at the very front of the rod but we know this isn't true as stated above.

 

[JVNY]

Where in the article is this stated?

See the section "Relation to Cartesian chart," which states: "It is a common convention to define the Rindler coordinate system so that the Rindler observer whose proper time matches coordinate time is the one who has proper acceleration g=1, so that g can be eliminated from the equations"

I used the formulas for the relativistic rocket from http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html to generate the numbers on the Minkowski diagram. I am not sure upon further reflection whether the front ship is the correct clock to use. It has proper acceleration a=1, and the relativistic rocket explanation says that for years and light years, 1g is a=1.03. Is it right to use a=1 for the front ship, or do I have to change to a=1.03?

 

[WannabeNewton]

See the section "Relation to Cartesian chart," which states: "It is a common convention to define the Rindler coordinate system so that the Rindler observer whose proper time matches coordinate time is the one who has proper acceleration g=1, so that g can be eliminated from the equations"

Let me just address this quickly. This isn't the same thing as saying that all the comoving clocks have their rates adjusted so as to be synchronized with the clock comoving with the $g = 1$ world line. All this says is that the proper time read by the clock comoving with the $g = 1$ world line is the same as the global time coordinate $t$ of the Rindler chart. However from the relation $d\tau = x dt$ it is clear that all the other comoving clocks are desynchronized with the clock comoving with the $g = 1$ world line because their proper times disagree with the global coordinate time $t$.