Short question about length contraction

[HALON]

I've started reading Mermin's It's About Time. Damn though...he doesn't cover rotations, which is what the ordinary clocks on the book's cover do. I suspect this may be my last post for a while.

Meanwhile, Brian Greene's book (which I haven't read in years) put this idea into my head "an object's motion is shared between the dimension of time and the dimension of space". So the diagonal path of light in the Michelson Morley experiment "dilates" by a factor of $γ$ in the dimension of time. It "dilates" because $γ=1/(1-v^2/{c^2})^{1/2}$ Therefore the photon's path in the dimension of space must "contract" by the reciprocal factor $1/γ.$ Well, knock me out. Because when I simplified the equation for the double diagonal path as $2/(c^2-v^2)^{1/2}$ (from Greene's end notes for the "mathematically inclined reader" LOL) and called it a radius, I found the "clock face" obeyed the same rules, because its circumference divided by the area equalled the length contraction factor $1/γ$, and the area divided by the circumference equalled the "time dilation" factor $γ$. Try it out, it's fun.

Why then is the slowing of clocks called "time dilation"? Maybe because the clock, um, expands in time while its energy remains invariant? (Yep, the big hand can't go faster but the circumference is longer). So its mass expands in time too? Didn't I read that old fashioned relativistic mass increases ("dilates"?) by $mγ$ with increasing speed? Isn't that...hmmmm... coincidental to the slowing of clocks when a body's mass increases in space? It's kind of "equivalent". What does that teach us about mass? For me, this is the missing chapter in every relativity book I've read.

 

[PeterDonis]

Brian Greene's book (which I haven't read in years) put this idea into my head "an object's motion is shared between the dimension of time and the dimension of space".

This is another of those Brian Greeneisms that, while it isn't wrong, causes a *lot* of misunderstandings. From the rest of your post, I think it is leading you in a direction that is going to increase your confusion, not decrease it.

(Btw, here's a suggested heuristic for spotting pop science things that, while not wrong, are likely to cause misunderstandings if you're actually trying to understand the science: does the author actually use the model he's presenting in the pop science book, in his research papers--the ones that get peer reviewed? If the answer is "no", which, AFAIK, it is for the Brian Greene model you refer to, that's a big red flag, IMO, that the pop science presentation is going to cause confusion. If the author himself isn't using it in his own research, why is he trying to pawn it off on lay readers?)

 

[HALON]

This is another of those Brian Greeneisms that, while it isn't wrong, causes a *lot* of misunderstandings.

In fairness to Brian Greene, the sharing of motion between dimensions was his "heuristic". The rest of the model was my take. The mathematics agreed neatly, so actually I'm not confused at all. It seems others are. So I'll retire from this thread to reflect.

 

[ghwellsjr]

I think you have come to a wrong conclusion. Are you thinking that time stops for a photon, so that the time of each tick of a clock (so to speak) propagates outward in all directions at the speed of light like an expanding soap bubble? Is that what you're thinking?

I'll put it in terms of the Doppler effect. When the distance between the signal emitter and the receiver widens, more wavelengths will fill the wider distance which causes the receiver to perceive them as a lower frequency as the signal is "stretched" over a longer distance. And the converse is true when the distance shortens.

It sounds to me like you're trying to describe how conventional non-relativistic Doppler works where the waves are traveling in a medium at a relatively slow speed and we can ignore any Time Dilation for the moving receiver.

But when it comes to light signals, in addition to the frequency shift due to the receiver encountering a differing number of waves, there is also an additional frequency shift due to Time Dilation that is dependent only on the speed of the receiver according to the rest frame of the signal emitter. It doesn't matter what direction the receiver is moving, it has the same Time Dilation independent of direction or any acceleration that causes just a change in direction.

But in the very different circumstance of one body rotating about another body, the radial distance between them is constant. Now the central body receives a lower frequency, while the rotating body receives a higher frequency. This is called the transverse Doppler effect.

Yes, in this case, there is no conventional Doppler but the rotating body is subject to Time Dilation according to Special Relativity. The slowing down of the clock of the rotating body is the reason why it detects a Doppler shift of a higher frequency and why it emits a lower frequency that is detected by the central body as a lower frequency. It's not any more complicated than that.

The rest of your post has nothing to do with Transverse Doppler.

So rather than the signal filling a longer or shorter distance, in this situation the signals fill a larger or smaller time volume. I don't know how else to picture it. (You can draw it quite simply using each clock "tick" as the radius of the time bubble, and the Lorentz transformation is the same). Anyway, Einstein said a body's acceleration is equivalent to a gravitational field- which is associated with mass. So if you use $e=γmc^2$ you can describe the bubble's increase in relativistic mass and come to an analogous conclusion regarding a body's relative increase in mass. Relative and relativistic mean different things. To illustrate the idea, see this image from Wikipedia

http://en.wikipedia.org/wiki/Gravitational_redshift#mediaviewer/File:Gravitational_red-shifting2.png

A body with greater relative mass receives a higher frequency of wavelengths, just as a body with greater relativistic mass does. You can re-imagine the larger sphere representing the rotating body compared to the central smaller sphere. But if you do so, the sphere's shown are now filled with relative time, not relative mass. (Or you can say each is composed of different relativistic mass, but not necessarily of different relative masses.).

Einstein introduced the equivalence principle. But equivalence does not mean identical. As I'm trying to convey, relativistic mass is not identical to relative mass. But as long as this distinction is clear, then it's OK to say a body's relativistic mass increases while its relative length contracts, although it's kind of using mixed terminology. Maybe that's why the term relativistic mass is avoided nowadays. In my early readings it was difficult to understand why writers would say the relativistic mass would increase, while elsewhere they would say it's length contracted. The distinction between relativistic and relative were not clear in the textbooks I read.