Question about length contraction

[tom.capizzi]

reply to JesseM:

I looked at over a dozen sites that claimed to have an answer until I found one that reminded me of something I read in one of Einstein's own books - the train experiment. So don't believe me. Go to a real, physical library and lookup Einstein.

By the way, your comment:

"... (in fact moving objects don't always appear contracted visually, sometimes they appear expanded depending on the direction relative to the viewer) ..."

puzzled me. Maybe some math would help.

As to the issue of simultaneity, reread the post. I said "multiple stationary observers". Perhaps I should add "stationary relative to each other." So, there is no problem with simultaneity. In any case, it is Einstein's experiment not mine. Read his book.

I find it inconsistent that someone proficient in science would have difficulty understanding projective geometry references. Read "Flatland", or if you have already, read it again. A line segment only appears full length when viewed at 90 degrees. As the line is rotated (or the observer rotates) the line vanishes (contracts) to a single point. A 3-D observer off the plane can clearly see that the line segment never changes actual length, but merely rotates away from the axis that is visible towards the one that is not. The rotated line will also "fit" between two parallel lines that are separated by the contracted length, even though it is really not any shorter.
Maybe that isn't enough math for you, but you should know that you can't prove anything with any number of examples. Plus, a logical argument doesn't require specific numbers.

 

[JesseM]

reply to JesseM:

I looked at over a dozen sites that claimed to have an answer until I found one that reminded me of something I read in one of Einstein's own books - the train experiment. So don't believe me. Go to a real, physical library and lookup Einstein.

Can you be specific about the book/page number instead of just making these vague claims, and if you have the book in hand perhaps give a direct quote? For example, Einstein discusses a train thought-experiment in section 8 and section 9 of the book Relativity: The Special and General Theory which is available online, but first of all this thought-experiment is about the relativity of simultaneity rather than length contraction, and second I don't see him using words like "physical" or "real" or "actual" in his description, which was what I was asking about.

By the way, your comment:

"... (in fact moving objects don't always appear contracted visually, sometimes they appear expanded depending on the direction relative to the viewer) ..."

puzzled me. Maybe some math would help.

I was referring here to the the Terrell-Penrose effect which depends on the fact that the light rays from different points on a moving object which reach my eyes at the same instant were actually emitted at different time-coordinates in my frame. For example, one simple case of the Terrell-Penrose effect is that for an object traveling straight towards me, if a light ray from the back of the object reaches my eyes at the same moment of a light ray from the front, the light ray from the back must actually have been emitted at an earlier time-coordinate in my frame, which causes the object to visually appear stretched to a length greater than its coordinate length in my frame.

For example, suppose a ship with a rest length of 10 light-seconds is traveling towards me at 0.8c, traveling in the +x direction. If the front reaches my own position x=0 at t=100, then the front has x(t) = 0.8c*t - 80 (when you plug in t=100 you get x=0). Since the ship is length-contracted to a length of 6 light-seconds in my frame, that suggest that at t=100 the back must have been at position x=-6, so the back must have x(t) = 0.8c*t - 86. Now, suppose at t=50 the front emits a light signal from position x=0.8*50 - 80 = -40, which reaches my eyes 40 seconds later at t=90. Also suppose at t=20 the back end emits a light signal from position x=0.8*20 - 86 = 16 - 86 = -70, which reaches my eyes 70 seconds later, also at t=90. That means that at t=90, I am seeing the front end lined up with the x=-40 mark on my ruler, and also seeing the back end lined up with the x=-70 mark, so visually the object appears to be 30 light-seconds long at that moment, five times greater than its "true" length of 6 light-seconds in my frame.

See here for more on the difference between visual changes in length and "actual" length contraction of an object which is moving relative to your rest frame:

http://math.ucr.edu/home/baez/physics/Relativity/SR/penrose.html

And this page has a good discussion of differences in apparent visual length for objects moving towards you or away from you (showing how objects moving towards you appear stretched visually, not contracted, as shown by my example above):

http://www.spacetimetravel.org/bewegung/bewegung3.html

As to the issue of simultaneity, reread the post. I said "multiple stationary observers". Perhaps I should add "stationary relative to each other." So, there is no problem with simultaneity. In any case, it is Einstein's experiment not mine. Read his book.

What book? What page? The train thought-experiment in the book above is all about observers in relative motion with different definitions of simultaneity.

I find it inconsistent that someone proficient in science would have difficulty understanding projective geometry references.

I understand geometrical projections just fine, but you are being very vague about what specific thing (a worldline of some sort?) you want to project onto what surface (a surface of simultaneity? Something else?) so your comments are impossible to interpret. I don't want analogies to Euclidean geometry, I want to know what you are actually talking about in the context of spacetime, so that's why I'm asking for either math or a spacetime diagram.

Read "Flatland", or if you have already, read it again. A line segment only appears full length when viewed at 90 degrees.

Sure. This analogy doesn't help one whit in understanding what you are talking about with your comments about "actual distance" vs. "apparent distance" in spacetime though.

As the line is rotated (or the observer rotates) the line vanishes (contracts) to a single point. A 3-D observer off the plane can clearly see that the line segment never changes actual length, but merely rotates away from the axis that is visible towards the one that is not. The rotated line will also "fit" between two parallel lines that are separated by the contracted length, even though it is really not any shorter.

Again, completely obvious and also completely unhelpful. I'm beginning to think all you have is homespun analogies to Euclidean geometry, that you have no clear non-analogical idea about what your comments about actual vs. apparent distances and velocities actually mean in spacetime.

Maybe that isn't enough math for you, but you should know that you can't prove anything with any number of examples. Plus, a logical argument doesn't require specific numbers.

You sure are trying hard to rationalize not giving either any math or even a spacetime diagram. If you don't know how to describe your ideas in those terms, why not just admit that?

 

[tom.capizzi]

Einstein wrote more than one book. I read the paperback edition when you were in grade school and I no longer have my copy. It had two different train experiments in it, and at the time I was more interested in the one you refer to (relativity of simultaneity). The book is in an actual library. Online sources tend to be abridged versions anyway.

 

[JesseM]

I read the paperback edition when you were in grade school and I no longer have my copy.

If you read it decades ago, I doubt your memory of it is clear enough that you would specifically remember him using words like "physical" or "real" or "actual" to describe length contraction, which is the only thing my question was about. Just because a thought-experiment deals with physical objects like trains doesn't mean that the thing it's deriving is frame-independent (for example, he uses a train thought-experiment to show how simultaneity varies between frames), and personally I wouldn't use words like "physical" or "real" to refer to quantities that depend on one's choice of reference frame.

In any case, the question of whether you were justified in claiming "According to Einstein, length contraction of a moving body is a real, physical, actual phenomenon" is a bit of a sidetrack. Are you going to address my main question about whether you are able to what your notions of "actual" vs. "apparent" distance and velocity in terms of spacetime (preferably using a mathematical example or a spacetime diagram), not just using homey analogies to Euclidean geometry where it's completely unclear what specific elements in spacetime the geometrical objects are supposed to be analogous to?

 

[tom.capizzi]

Here's a URL for a demonstration of the train experiment in question. It is not the relativity of simultaneity experiment. It is designed for a college lecture, but contains very little math so that it can be understood by students of varying degrees of familiarity with the subject. Click on the link to visit the site.

http://www.fas.harvard.edu/~scdiroff/lds/QuantumRelativity/RelativityTrain/RelativityTrain.html

By the way, although I read the book many years ago, I aced relativity when I finished my college years in 2000.

 

[JesseM]

Here's a URL for a demonstration of the train experiment in question. It is not the relativity of simultaneity experiment. It is designed for a college lecture, but contains very little math so that it can be understood by students of varying degrees of familiarity with the subject. Click on the link to visit the site.

http://www.fas.harvard.edu/~scdiroff/lds/QuantumRelativity/RelativityTrain/RelativityTrain.html

Are you not reading what I write at all? I told you that my only question related to length contraction was what you meant when you said "According to Einstein, length contraction of a moving body is a real, physical, actual phenomenon". And this was following up on a previous discussion where you had said:

Can you define what you mean by "actually physical"?

I was referring to a statement made by ghwellsjr that "... those meter sticks ... will each contract physically along the direction of acceleration,..."

You may delete the word "actually" if it confuses the issue. From the rest of your reply, are you saying that meter sticks do not contract physically?

So, the focus here was on the question about what it means to say things like "contract physically". I don't understand what "physically" (or your synonyms 'real' and 'actual') is supposed to mean, since personally I only say something is "physical" if it is frame-independent, and clearly length-contraction is not. I don't have any problem understanding length contraction in general, I just don't understand your repeated questions and assertions about the ruler being "physically" contracted. So giving me some link on length contraction which doesn't say anything about the length contraction being "physical", "actual", or "real" is completely pointless, as I said I have no difficulty understanding how length contraction works in SR, I just don't understand all these questions and assertions about it being "physical". The only relevant reply would be to either give me a clear definition of what you mean by "physical", or show some example of another physicist using such language to describe it so we could see what they meant in context.

 

[Mentz114]

tom.capizzi,

here's a thought expriment that illustrates length contraction ( you may have come across this before of course.)

We have a railway carriage passing a station platform at contant speed. In the middle of the carriage, an operator sends a light signal from one source to both ends of the carriage. The light triggers paint spraycans located at the ends of the carriage which mark the platform. The operator has every reason to believe that the marks, measured by the station operator will be the same length apart as the ends of the carriage as measured by him. But the distance between the marks is not equal to the (rest) length of the carriage.

The reason is that the platform observer sees the spray cans going off one after the other, not at the same time, as the carriage observer does.

This 'length contraction' is an artifact of mixing measurements between inertial frames. Both carriage and platform remain the same length throughout.

 

[John232]

Length contraction shouldn't alter the velocity of the object. If it did it would cause a never ending loop of contraction caused by higher velocity because the amount of contraction is determined by the velocity. A beam of light sent perpendicular from an object with still maintain the same velocity in the direction of motion even if the length of that object was contracted.

An object traveling at a relative velocity will always measure everything in their own physics to be unaltered by an outside observerer traveling at a relative speed. For instance a spaceship would never be able to travel close to the speed of light and look down at Earth and alter our physics. I am sure this would always be the case since galaxies are out in space that travel close to the speed of light already, unless their relativistic effects are somehow void to traveling with the expansion of space itself.

If this was an experiment that was done I would think that the observer at rest saw in increase in momentum due to its relative mass.

 

[Grep]

Einstein wrote more than one book. I read the paperback edition when you were in grade school and I no longer have my copy. It had two different train experiments in it, and at the time I was more interested in the one you refer to (relativity of simultaneity). The book is in an actual library. Online sources tend to be abridged versions anyway.

That book is right next to me. Paperback, copyright 1961, called "Relativity: The Special and the General Theory". There is nothing in chapter 9, "The Relativity of Simultaneity" (with a train thought experiment) or chapter 10, "On the Relativity of the Conception of Distance" which even approaches what you're saying. I also went through any section where he dealt with this kind of thing.

I'd hesitate to call length contraction "physical" unless I had defined that word really unambiguously. I mean if I physically contract some object, that has implications. I've just applied energy to it, and it is changed as a result. Relativistic length contraction is nothing like that in any way. If I went at 0.9999c relative to Earth with a meter stick, it's still a meter stick. The fact that different observers in other inertial reference frames may give me all sorts of different lengths if they could measure it means nothing to me at that point, nor to the meter stick - it remains unaffected. Even they probably won't agree on how long it is unless they're in the same inertial frame of reference together.

When you use the word "physical" here, IMO, you stress the word past it's breaking point.

 

[tom.capizzi]

To address some of the objections that have been raised in this discussion, I have gone back to the source and re-read Einstein's derivation of Special Relativity. I have not yet located a copy of the revision which included the train experiment with multiple stationary observers. Nor have I seen his derivation which involved a series expansion of a function of two variables. I believe they were in the same printing, but I can't be specific yet. In any case, the proof presented in "Relativity: The Special and General Theory" is just as useful a starting point.
Einstein makes several references to "physical" in this text, although not in the context of length contraction. However, a clock which runs slow exhibits real, physical evidence in the angular position of the hands. To preserve the constancy of the speed of light, it is absolutely necessary for length contraction to be just as real. Otherwise, it would be possible to perform an experiment which would yield a different value for that velocity.
To derive the Lorentz transform (which specifies the amount of length contraction and time dilation) Einstein also assumes the principle of relativity. Essentially, this means if an object is moving at uniform speed in a straight line in one inertial frame of reference, then it will also be seen as moving in a straight line at uniform velocity in any other inertial frame regardless of the relative velocity of the two frames of reference. This limits the choice of transform to a linear combination of scaling and offset. By suitable location of the origins of the two coordinate systems, there is no offset, and the resulting transform is the Lorentz transform. This relationship can be expressed as a matrix. By appropriate rotation of the axes, all the relative velocity can be made parallel to a single axis. Since there is no effect perpendicular to the direction of motion, the most general matrix can be reduced to a 2x2 matrix with 4 elements, as yet to be determined.
By means of the constancy of the speed of light and the principle of relativity, enough independent equations are generated to solve for the 4 elements of the matrix, resulting in the familiar Lorentz transform. Although Einstein is said to have favored the use of world lines, they do not appear in his derivation of the Lorentz transform, so I don't think they are necessary for this part of the discussion.
It seems to me that the derivation involves a third assumption as well. The use of light rays to measure distance seems obvious, but I believe it may be too restrictive. It implicitly assumes that the distance between two points in a coordinate system is a simple straight line. I have shown at least one mathematical model for which this is not true. An infinite number of infinitesimal loxodrome spirals placed end-to-end looks just like a straight line. It has no thickness or cross-section, but the effective diameter of the string of spirals only agrees with the light ray measurement for a tilt angle of zero. With respect to relativity, if the sine of the tilt angle is v/c, then the length contraction factor is gamma. However, the "actual" diameter of the spiral parallel to the curve is unchanged by the tilt angle. The measured length, no matter how real or physical, is not the same as the "actual" length unless the velocity is zero.
This distinction can be expressed analytically by allowing measurements of length and time to be expressed with complex numbers instead of real ones. Then the assumption about measurement can be amended so that the light ray measures the real part of the complex number. The magnitude of the complex value is unchanged by the rotation, but the real part is contracted. Ironically, Einstein even derives the Lorentz transform from a 2x2 rotation matrix by allowing complex terms for both the rotation angle and one of the coordinates. To add to the irony, ordinary rotation is normally represented by an imaginary exponential, making the imaginary rotation of the Lorentz transform a real exponential. He then dismisses the importance of the complex representation in a single sentence with the remark "... on account of the relations of reality ...".
So, if we acknowledge that light rays measure only the real part of a complex length, the "actual" or total length is just the magnitude of the complex whole. After all, a stick which measures gamma meters in the frame of the moving observer is contracted to 1 meter when measured by an observer in the stationary frame. Similarly, if the real part of the path of a moving object is measured by the stationary observer to be 1 meter, then the magnitude of the complex length of the path is gamma meters. From the frame of the moving observer, the 1 meter path rushes by, contracted to 1/gamma meters. It would take a path of gamma meters to appear to be 1 meter to the moving observer, and this would be the real part of the complex path length from the moving point of view.
It seems to me that this explanation is consistent with the principle of relativity and the constancy of the speed of light, the two postulates upon which all of Special Relativity is based. Any disagreement stems from the assumption that we can understand a complex universe by looking at only the real parts.

 

[ghwellsjr]

Einstein makes several references to "physical" in this text, although not in the context of length contraction. However, a clock which runs slow exhibits real, physical evidence in the angular position of the hands. To preserve the constancy of the speed of light, it is absolutely necessary for length contraction to be just as real. Otherwise, it would be possible to perform an experiment which would yield a different value for that velocity.

The only things that we can say are true in an absolute sense are those that remain invariant when viewed from all possible inertial frames of reference. Time dilation and length contraction do not fit in that category but whenever a clock (or any object) accelerates, its length changes along the direction of acceleration and its tick rate (or aging rate) changes when viewed from any inertial frame of reference. Therefore, we can conclude that these changes are real, actual, and physical not just an artifact of the way we view them.

You have presented a very complicated argument which I haven't bothered to understand because I don't think it is necessary to answer your question. All you have to consider is the universally agreed upon fact that a clock that accelerates away from and back to a stationary clock has really, actually, and physically experienced a time dilation that the stationay clock did not experience. Now if that clock were a light clock, it must have also experienced a real, actual, and physical change in its length along the axis of acceleration or it would not produce the real, actual, and physical change in its time.

The reason why we hesitate to say that a clock experiences a physical length contraction when accelerated is that when viewed from a different inertial frame of reference that same acceleration may be a deceleration in which the clock experiences a length expansion, but there is a physical change one way or the other.