Velocity vs Distance - Clarifying a Moronic Question

[pervect]

Pervect,

I read the link about Born acceleration.

While much of the math went over my head, I accepted his calculations as correct, and concentrated on the logical flow of his statements.

Unfortunately, what the article means to me is that this desribes a phenomenon an inertial frame would observe, but I don't see that the accelerating object itself actually experiences anything unusual.

If I had two clocks in my rocket, one at the fron, one at the rear, you, an inertial observer, would read a difference, not just in their time, but also in their physical location (I see the "bending" being talked about). In fact, the time and material distortions sound like you'd be challenged to even identify that it was my Rocket you were looking at :)

BUT, I do not see how this article argues that I would observe anything odd at all.

Sorry. That was an awesone attempt you made, but I still don't see any effects for my frame of reference.

The particular effect I'm talking about, which is also described in the textbook I referenced (MTW) using more advanced math, is that in order for the head and the tail to keep a constant distance from each other, the head must accelerate at a different rate than the tail.

This effect should be obvious just by inspecting the space-time graph, actually. In particular, the particle at the pivot point requires infintie accelleration, and particle not at the pivot point only require a finite acceleration. This should be a strong clue that the accelerations are not equal, as the finite acclerations for particles at any other point than the pivot point are not equal to the infinite acceleration of the pivot point.

Even if you can't follow the math (basic calculus in this case), the following quote from the link I cited tells you this directly and explicitly.

Trailing sections of the rod must undergo a greater acceleration in order to maintain Born rigidity with the leading end, and the required acceleration is inversely proportional to the distance from the pivot event.

 

[JesseM]

You just said "as long as there is no gravity involved". That appears to contradict "Equivalence".

In that instance I wasn't even talking about using the equivalence principle, just making the point that to figure out what things will look like in a non-inertial frame in flat spacetime, it's easiest to first calculate what they'll look like in an inertial frame using the rules of SR, then do a coordinate transformation; obviously this wouldn't work in curved spacetime, where you couldn't figure out anything using only the rules of SR. Also, just to be clear, by "no gravity" I meant "no spacetime curvature", and the equivalence principle only deals with arbitrarily small regions of spacetime where curvature can be neglected. In flat spacetime physicists do sometimes talk about a "uniform gravitational field", but other physicists would say the name is a bit of a misnomer, since unlike gravitational fields involving spacetime curvature this sort of uniform field would not involve tidal forces, and it wouldn't diminish with distance from some central mass. See The "General Relativity" Explanation of the Twin Paradox, which talks about issues surrounding "uniform gravitational fields".

So, there is no need to consider any non-inertial coordinate systems to answer the question of how three accelerating clocks would behave, as seen in an inertial frame,

BINGO! I thought I made it clear my difficulty was with the assertions that time in my frame (on the Rocket, which is accelerating, which is therefore not inertial)

But a non-inertial coordinate system is not a "frame" in the same sense as an inertial frame where the laws of SR hold. You can't assume that in a non-inertial coordinate system the laws of physics will work in anything like the way they do in inertial frames; for example, the coordinate speed of light will not in general be c in non-inertial coordinate systems. And more relevant to this example, you can't assume that if two clocks are at rest in a non-inertial coordinate system, like the clocks at the front and back of the rocket, they will be ticking at the same rate, as they would be in an inertial frame. The rate a clock ticks is not in general a function of its velocity in non-inertial coordinate systems.

Also, you didn't respond to my comment about how, once you have figured out how the clocks will behave in an inertial frame, it is simply a matter of doing a coordinate transformation to find how they will behave in any given non-inertial coordinate system (note that this has nothing whatsoever to do with the equivalence principle, you could use exactly the same procedure to figure out how things would look in a non-inertial coordinate system in Newtonian physics). Do you disagree with this?

Perhaps I was not clear. Points A and B, North and South poles (of axis of rotation)

I don't understand what you mean by "North and South poles (of axis of rotation)" here. Are you talking about the Earth's axis of rotation so you have one point at the Earth's north pole and one at the south pole in Antarctica, or are you saying the lab itself is rotating?

A & B are accelerating. They share the acceleration component due to the Earth's orbit. Their average acceleration over one full orbit will be the same.

If A & B are at the north and south pole, do you mean the rotation of the Earth on its axis (once per day), or the orbit of the Earth around the sun (once per year)? And if they are so far apart, what does this have to do with the equivalence principle, which deals with only with measurements in a small laboratory where the curvature of spacetime is too small to register on your measuring equipment? What you said earlier was:

I'm being told that equivalence means gravity and acceleration are treated the same, yet apparently all these weird things happen on my rocket (which could be accelerating at 1g), but they aren't happening here on the planet, despite being accelerated at 1g also.

The equivalence principle in this case is about the clock at the front and back of the rocket accelerating at 1 G functioning the same as a clock at the top and bottom of a similar-sized rocket standing at rest in a 1 G gravitational field. It wouldn't say anything about an equivalence between two clocks near each other in an accelerating rocket and two clocks thousands of miles apart at either pole of the earth, where any region of spacetime that includes both must necessarily include a great deal of spacetime curvature due to the Earth's gravity.