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 [tex]2\pi R[/tex] 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 [tex]2\pi R[/tex] 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 [tex]circumference = 2\pi R[/tex]. (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/[tex]2\pi[/tex]. 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/[tex]2\pi[/tex]. 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 travelled 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.