Problem with equivalence principle.

[yuiop]

Einstien gives the example of an observer riding on the rim of a spinning disc. Using his length contracted rulers he measures the perimeter to be greater by a factor of gamma (y), than the $$2\pi R$$ he would have measured when the disc was stationary.

When he measures the radius with the same rulers he notices the radius is unchanged from when the disc was stationary. The observer concludes that the circumferance is not equal to $$2\pi R$$ and from here Einstien starts to introduce the concept of curved space that is non Euclidean.

The observer now has two contradictory measurements for the radius. The radius is R according to his rulers and R*y if he calculates the radius using $$circumference = 2\pi R$$. (Call that the geometrical or Euclidean radius?) The question, is which radius measurement method did Einstien use to base GTR on?

Lets assume for now that we assume space is curved in such a way that R > Circumference/$$2\pi$$.

Now we try another measurement. An omidirectional flash is reflected from a mirror at the centre of the disc and the time for the flash to go from the rim to the centre and back to the rim is measured. The clock on the rotating rim is slow due to time dilation relative to the clock of a non rotating observer standing outside the disc. The disk rider measurement of the time is less than that of an external observer and he might conclude that the radius as measured by light speed is less than the radius he measured with his rulers by a factor of gamma.

Using light to measure distance, space is curved in such a way that R < Circumference/$$2\pi$$.

Now we have 3 different measurements for the radius.

(a) Radius (Geometrical) = R*y
(b) Radius (Ruler) = R
(c) Radius (Light) = R/y

Which measurement should we trust? Which measurement is generally used in GTR when the measurement method is not specified? Measurements a and c suggest space is curved in two completely different ways. Should we trust the local ruler method which is the average of the other two measurements? After all when we state the local speed of light is always c, we use a local ruler and clock to determine c.

On the other hand, the international standard of units (SI) defines the metre in terms of the distance traveled by light in a given time. On that basic we should use measurement (c). Oddly AFAIK, Einstien never mentioned this obvious method to meaure the radius. There are other problems with defining distance in terms of light speed. If a series of mirrors are set up around the perimeter so that a light beam can be sent around the perimeter we notice that the time taken by the light depends on the direction the light is sent in. By the international definition, the length of the circumference is different depending upon whether the measurement is sent clockwise or anticlockwise. (This is the basis of the Sagnac effect)


So this leaves us with a dilemna. When a cosmologist states that some observation he has made agrees with predictions of GTR, how can we be sure if the R ,in gravitational redshift factor 1/sqrt(1-GM/c^2/R), has no firm definition. With such a choice of defining distance, GTR can be made to fit just about any observation.

 

[Dale]

With such a choice of defining distance, GTR can be made to fit just about any observation.

Actually, my understanding is that this is one of the strengths of GR. It only has one free parameter and that parameter is determined by the Netwonian limit. It cannot be made to fit arbitrary observations, but makes very specific predictions.

That said, I have no clue about your principal concern.

 

[pervect]

Einstien gives the example of an observer riding on the rim of a spinning disc. Using his length contracted rulers he measures the perimeter to be greater by a factor of gamma (y), than the $$2\pi R$$ he would have measured when the disc was stationary.

When he measures the radius with the same rulers he notices the radius is unchanged from when the disc was stationary. The observer concludes that the circumferance is not equal to $$2\pi R$$ and from here Einstien starts to introduce the concept of curved space that is non Euclidean.

The observer now has two contradictory measurements for the radius. The radius is R according to his rulers and R*y if he calculates the radius using $$circumference = 2\pi R$$. (Call that the geometrical or Euclidean radius?) The question, is which radius measurement method did Einstien use to base GTR on?

Einstein only used the rotating disk as a motivational tool. GR is not "based on" any such analysis. However, as far as the Ehrenfest paradox goes, Einstein was (one of?) the first to realize that the circumference was greater than 2 Pi. See for example Gron's historical paper online at http://www2.polito.it/ricerca/relgrav/solciclos/gron_d.pdf .

Einstein has here for the first time made it clear that the length of the periphery of a rotating disk in longer than not shorter as stated in Ehrenfest’s paradox. The reason for this difference is that Ehrenfest considered a hypothetical, but impossible situation where a disk had been put into rotational motion in a Born rigid way, while Einstein considered a situation in which the disk had been put into rotation in an arbitrary way, but the measuring rods were required to be Born rigid.

What Einstein (apparently) did not realize was that it is misleading to talk about the "space" on a rotating disk at all, because there is no surface of simultaneity in a rotating disk compatible with the Einstein synchronization convention. This point was left to other authors, and still causes a fair amount of confusion :-(.

See for example http://www2.polito.it/ricerca/relgrav/solciclos/tartaglia_d.pdf or http://arxiv.org/abs/gr-qc/9805089

I'll quote from the former:

Global geometric properties deduced for instance from light rays triangles
produce the same type of 'ambiguity' as the global measurements of time and
may be used only to deduce the presence of the rotational motion, not to assess
any unique 'curvature' of the space of the disk. Actually space has properties directly connected with 'simultaneity' surfaces for rotating observers. The
'synchrony' surface for a set of rotating observers is indeed a Riemann helicoid, whose 'global' curvature cannot be defined, the local curvature is of course
absent.

 

[J.F.]

Einstien gives the example of an observer riding on the rim of a spinning disc. Using his length contracted rulers he measures the perimeter to be greater by a factor of gamma (y), than the $$2\pi R$$ he would have measured when the disc was stationary.

When he measures the radius with the same rulers he notices the radius is unchanged from when the disc was stationary. The observer concludes that the circumferance is not equal to $$2\pi R$$ and from here Einstien starts to introduce the concept of curved space that is non Euclidean.

The observer now has two contradictory measurements for the radius. The radius is R according to his rulers and R*y if he calculates the radius using $$circumference = 2\pi R$$. (Call that the geometrical or Euclidean radius?) The question, is which radius measurement method did Einstien use to base GTR on?

In GR, by virtue of the general covariance, there is a problem of physical distance. As physical distance suggest to understand that basically it would be possible to measure using rays of light, system of mirrors and a chronometer.

Attempt to apply this hypothesis for calculation of height of a skyscraper yields different results. As the chronometer located on a ground floor goes more slowly a chronometer located on last floor such " measurement of height " a skyscraper by means of a ray of light, systems of mirrors and a chronometer will yield different results depending on that on what floor to arrange a chronometer

 

[yuiop]

In GR, by virtue of the general covariance, there is a problem of physical distance. As physical distance suggest to understand that basically it would be possible to measure using rays of light, system of mirrors and a chronometer.

Attempt to apply this hypothesis for calculation of height of a skyscraper yields different results. As the chronometer located on a ground floor goes more slowly a chronometer located on last floor such " measurement of height " a skyscraper by means of a ray of light, systems of mirrors and a chronometer will yield different results depending on that on what floor to arrange a chronometer

Hi, at the time I posted the OP I was trying to come to terms with the many notions of distance. (Still am ;)

Would you agree with the following? Imagine we have a ferris wheel of diameter R as measured by observer A on the ground and as measured by B on the perimeter of the wheel, (both using rulers). The perimeter of the wheel is 2*pi*R as measured by A and and 2*pi*R*gamma (greater) as measured by B. If the wheel breaks free and rolls along the ground it will roll a distance 2*pi*R*gamma along the ground in each complete revolution as measured by A (assume no slipping or resistance) and a distance of 2*pi*R as measured by observer B.

 

[J.F.]

The widely spread view that just this Principle underlies the Genera! Theory of Relativity is not quite correct. The basis of Einstein theory of gravity is another Equivalence Principle, having different and more deep content that metric field $$g_{{a} {b}}$$ of Riemannian space is proclaimed as the gravitational field. Just this is "the natural formulation of the Equivalence Principle" that Einstein had come to later.

 

[Ich]

What Einstein (apparently) did not realize was that it is misleading to talk about the "space" on a rotating disk at all, because there is no surface of simultaneity in a rotating disk compatible with the Einstein synchronization convention. This point was left to other authors, and still causes a fair amount of confusion :-(.

I don't think so. From his 1916 paper (p. 152 in the Collected Papers):

In classical mechanics, as well as in the special theory of relativity, the co-ordinates of space and time have a direct physical meaning.
...
But we shall now show that we must put it [=this view of space and time] aside and replace it by a more general view (...) if the special theory of relativity applies to the special case of the absence of a gravitational field.
...
Hence Euclidean geometry does not apply to K'. The notion of co-ordinates defined above, which presupposes the validity of Euclidean geometry, therefore breaks down in relation to the system K'. So, too, we are unable to introduce a time corresponding to physical requirements in K', indicated by clocks at rest relatively to K'.

He never talks about "space" on a rotating disk, he merely points out that the usual definition of coordinates (where dt or dx directly equal +-ds) breaks down, for both space and time coordinates.

I have some related question, pervect:
Your statement seems to imply that "space" is a volume orthogonal to time. If you don't talk about "space", but only about spacelike coordinates, where one of those coordinates is not orthogonal to time,
1. don't you arrive at Einstein's description?
2. isn't his explanation via non-Euclidean geometry in these coordinates eqally useful? Especially as one could tesselate the disk physically with suitable tiles of known area, arriving at A>pi r².
I's less abstract, anyway, but I agree one should also have seen the images of the spiraling four-velocity and plane of simultaneity you have in mind.

 

[pervect]

I'm not a historian, so I'm actually not particularly confident in making definite statements about what Einstein did or did not realize. I will point out, though, that for anyone actually interested in modern physics, it is a mistake to stop at Einstein's papers. Many good authors and physicists have contributed to the field in the last 100 years - focusing too much on Einstein seems to be a popular "cult phenomenon", where for whatever reason the focus is placed on one person (Einstein) rather than physics, and the modern understanding of it.

I would agree with the characterization that space is generally thought of as a volume orthogonal to time, and it is a key point that the spinning disk does not admit such a characterization, so there is no "space" on a spinning disk in this sense.

I would also agree that it is both fruitful and usual to adopt a coordinate system where space is not orthogonal to time to label the points on a spinning disk, or a spinning planet.

Most coordinate times used on the Earth, for example, use such a coordinate system.