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by Bruce Linnell |
This article explains reasons why I dissent from some of the the standard explanations given in General Relativity. It may not be relevent to someone unfamiliar with Special or General Relativity, though it could clear up some confusion to why my explanations are different from most of the literature..
For spherical objects like stars? Apparently not. For flat things like walls, YES!
The idea that you can't actually "see" Lorentz Contraction may be among the most major blunders in modern relativity. It may not be such a mathematical blunder, but a horrible blunder in teaching relativity. A 1959 article by James Terrell: Invisibility of theLorentz Contraction,supposedly "proved" that you cannot see Lorentz contraction. However. A careful reading of the paper will show that Terrell places the observer at one uniquely-determined angle in front of the object, where the object does indeed appear to be its normal uncontracted length. The animation here uses that angle as well.
This article remains uncontested to this day, as you can see in the wikipedia article, http://en.wikipedia.org/wiki/Terrell_rotation. Unfortunately, the other articles, which in their titles, at least, appear to support Terrell's conclusion, cost from five to fifty dollars to read. The wikipedia article, itself, acknowledges that "Thanks to the differential timelag effects in signals reaching the observer from the object's different parts, a receding object would appear contracted, an approaching object would appear elongated (even under special relativity) and the geometry of a passing object would appear skewed, as if rotated"
This is all corect up to the end of the sentence:"skewed, as if rotated," because "skewed" and "rotated" are different things.
Another point the Wikipedia article has correct is "A previously-popular description of special relativity's predictions, in which an observer sees a passing object to be contracted (for instance, from a sphere to a flattened ellipsoid), was wrong." But why was it wrong? It is not because the Lorentz Contraction is invisible but because of peculiar symmetries of the sphere itself. A sphere is a unique three dimensional shape that appears round no matter what angle you look at it. But take a circle, which is a two dimensional object, or any other two-dimensional surface,and it WILL apear tocontract as it goes by.
While Terrell's method of analysis was to usesomething called the aberration equation," another way to frame this problem is to ask where "what you can see" coincides with the path of the object.
What you can see is determined by all the light that is reaching you at this instant. It takes the form of a cone in space-time t = -√(x^2-y^2)
The path of a circle can be determined by taking avertical cylider of events in space-time and lorentz tranforming them.
The intersection of the cylinder with the cone is easiest found by first finding the set of intersection events, then lorentz transorming them... By the nature of the lorentz transformation, any events on the cone stay on the cone.
Resulting animations:
The side view (above) gives a good picture of the cone and the slanted cylinder, but to get a better idea of what the circle would look like to the observer, we switch to a top view:
Here we see the circle coming in from th right, where it appear elongated, then moving to the left, where it appears to be contracted. There is a particular angle of viewing, θ=ArcCot(v/c), during the circle's approach, where the Lorentz contraction "disappears" but that lasts only for a very short time. But that is the angle that Terrell chooses to use.
In Terrell's concluding paragraphs, he says "Any hopes of seeing the contraction in a rapidly moving space vehicle or astronomical body must be discarded." However, what we should notice is that while an object is approahing, it appears to move superluminally, and stretched out. As the object is departing, it appears to be contracted.
We should still look for the Lorentz contraction effect to be visible; just not at the angle where Terrell was looking for it. Instead of using θ=ArcCot(v/c), we should look at angle θ=0, where the object makes its closest approach to the observer. In many of the demontrations on this website, it is assumed that the object is passing, a long distance away, but at that approximate angle, θ=0, where the unmodified Lorentz Contracted length is most visible.
The further away the passing object is from the observer (tip of the light-cone) the bigger is the region where you can see an approximately "correct" length contraction effect.
IHere you can see the circle passing further away. When the image of the object makes its closest approach, you can see that it maintains a fairly constant shape, for at least a few frames of the animation.
Now, here's the question. At any given moment, draw two lines from the tip of the cone to both visible edges of the oval shape. Will the angle subtended by those two lines equal the height of the sphere? Sure. But does that in any way imply that Lorentz Contraction is "invisible" No. Because if we did the same animation with a cube going by, the face of the front face of the cube would be clearly Lorentz contracted, IF you viewed it at the right angle.
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by Bruce Linnell |