Equivalence principle and time dilation

[harrylin]

No, it doesn't. Wikipedia correctly defines proper acceleration as (http://en.wikipedia.org/wiki/Proper_acceleration#In_curved_spacetime) $$A^\lambda := \frac{DU^\lambda }{d\tau}$$ You appear to be missing the distinction between $=$ (equality) versus $:=$ (definition). From the definition you can derive the result in a momentarily co-moving inertial frame, but the definition is manifestly covariant.

That is according to Wikipedia proper acceleration "in the language of general relativity" under the header "curved spacetime"; I used the description of the intro for flat spacetime and its ref.1 (Taylor and Wheeler). Once more, I will try to avoid all such terms that could lead to quibbles over words.

 

[Dale]

So is it correct to say the following?

1. The number of "coordinate frames" that can be applied to any observer (point) in space-time is infinite
2. you can apply any number of those different coordinate frames simultaneously to the same "point". They are just imaginary "grids" that can be laid over space-time points to take measurements.
3. Things that may vary between points/observers in space-time are time and length, and all "observables" derived from those dimensions - including velocity, momentum and actual "simultaneity" meaning "when something happens".
4. If space-time is flat (all points in space-time can be considered mass-less) then the Minkowski Metric and the Lorentz Transformations based on it are used to relate measurements of variables taken in different coordinate frames. Often times this is done between one inertial coordinate frame and another different inertial frame (meaning one accelerated w/respect to the first) because that's a natural way to set up common problems in space-time.
   
5. Space-time diagrams allow you to overlay one coordinate frame and another, to read off variables according to the Minkowski metric and the Lorentz transforms.
6. There is only one "Proper coordinate frame" for any point, and that's the one special measuring grid defined inertially (meaning co-moving or unaccelerated) w/respect to that specific observer (point)

I'm going from memory here, so this would be progress if I can even get a B.

For 1 and 2, yes.

For 3 you have to be careful about what you mean by "observables". The actual number which is obtained from any measurement procedure (observation) is frame invariant. However, whether or not that measurement corresponds to a quantity like momentum or velocity depends on the reference frame.

For 4 and 5, yes.

For 6, I would probably just scrap that entirely. What you could say is that at any event on any particle's worldline you can construct a momentarily co-moving inertial frame (globally in flat spacetime or locally in curved spacetime).

 

[bcrowell]

This looks like a pop science book, not a scientific textbook or paper.

I'm currently using it as a textbook in a general education course.

Unfortunately, you can't learn science from pop science books, even if they're written by scientists.

This is an extreme generalization.

This statement is too vague to really tell you what the science says, and can easily be misunderstood to be saying something false.

The statement is taken out of context. Taken in context, I think Stannard's treatment is perfectly fine.

 

[PeterDonis]

This is an extreme generalization.

Perhaps. But a fair portion of the data on which I am basing it is threads here on PF which are started by someone misunderstanding a pop science explanation by a scientist. Brian Greene is the #1 source of these, but there are others.

I would love to have a better heuristic for when a book about science (to put it as neutrally as possible) is or is not a reasonable source for someone trying to actually learn the science, as opposed to just wanting to be told what "science says". But I have found it very hard to come up with one.

Taken in context, I think Stannard's treatment is perfectly fine.

I don't have the book so I don't know the context. If he explains what his statement means along lines similar to what I posted in post #2, then I would agree with you.

 

[harrylin]

Perhaps. But a fair portion of the data on which I am basing it is threads here on PF which are started by someone misunderstanding a pop science explanation by a scientist. Brian Greene is the #1 source of these, but there are others. [..].

There's also Barry Parker's book "relativity made relatively easy" - it is literally full of little mistakes and wrong explanations that may confuse the readers (I know this as my brother in law has it and he was confused by it). The author even gets MMX wrong. That's a pity because it's otherwise a very nice book with lots of history of science...
The only solution may be to take such a book and correct it. As a matter of fact, I already started with correcting Parker's book for my brother in law and other readers, but it's a big job and for the second part (GR + astrophysics) I would need help from an expert.