- #36
mysearch
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Response to #29: Part 2 of 2
Section-3:
It is probably easier to replicate the 2 definitions you modified:
You change the words `time` and `length` to `clocks` and `rulers`, therefore I wanted to clarify whether there was an important physical implication in this change rather than semantics. In cosmological expansion, the overall volume of space expands, but atoms are unaffected by this expansion, i.e. atoms do not get bigger as cosmological space expands. However, I have always interpreted length/ruler contraction in a different way, i.e. conceptually the stationary observer would perceive everything in the moving frame of reference to contract in the direction of motion, even atoms. This is why [x`] is measured locally by a ruler as 1 metre, while the stationary observer perceives both the ruler and object being measured to both be contracted.
Again, we have a similar change, but the term ` length expands` is changed to `rulers contract`. Based on similar assumption as outlined previously, I assumed that the local observer’s ruler would expand in-line with any expansion in space and as such, the local observer would not perceive any stretching of the ruler as its was orientated along the radial direction. However, a distant observer would perceive both the ruler and object being measured to both be expanded.
As such, this still seems to be an open issue that I would like to clarify.
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Section-4:
Yes, I agree. By definition, the calculated radius or coordinate-r corresponds to that measured in flat spacetime. Wasn’t too sure about the next bit, as I would have thought we could simply say that the circumference was measured without reference to any velocity. However, I agree that horizontal/tangential circumference would not be affected by gravity. At this point, I would like to introduce two definitions, although you will probably disagree with the second:
[tex]Coordinate-r = circumference/2\pi[/tex]
[tex]Spatial-r = \gamma(coordinate-r)[/tex]
The implication of the second definition suggests that the radial distance expands, although it is probably not measurable by the local observer for the following reasons starting with your words:
While you begin with `Yes`, I don’t think you will agree with my statements at the end of section-3. I was assuming space expands in the radial direction due to the increasing curvature of space. However, I was also assuming that the local ruler would also be affected and therefore the increase in the spatial-radius is not measurable locally, only observed at a distance. So, to conclude, let me try to apply my assumptions to the 2 observers under consideration, i.e. the distant observer (x, t) and the shell observer (x`, t`) within the gravity well using my assumptions:
x` > x, i.e. [tex]x’ = \gamma(x) [/tex]
t` < t, i.e. [tex]t'= (t)/ \gamma [/tex]
The velocity [v] is a function of distance over time:
v' = x'/t' [tex]= \gamma(x)/ (t)/ \gamma = \gamma^2 (x/t) [/tex]
As such, [v’] is greater than [v] by [tex]\gamma^2[/tex] as required by the very first equation. Again, apologises for such a long posting, but as stated, I wanted to try and address all the open issue from my current understanding.
Section-3:
It is probably easier to replicate the 2 definitions you modified:
o As velocity [v] approaches the speed of light [c], time slows and length contracts in the direction of motion, at least, with respect to a `stationary` observer.
You change the words `time` and `length` to `clocks` and `rulers`, therefore I wanted to clarify whether there was an important physical implication in this change rather than semantics. In cosmological expansion, the overall volume of space expands, but atoms are unaffected by this expansion, i.e. atoms do not get bigger as cosmological space expands. However, I have always interpreted length/ruler contraction in a different way, i.e. conceptually the stationary observer would perceive everything in the moving frame of reference to contract in the direction of motion, even atoms. This is why [x`] is measured locally by a ruler as 1 metre, while the stationary observer perceives both the ruler and object being measured to both be contracted.
o On approaching a gravitational mass [M], time slows and length expands in the direction of gravitational pull, as a function of radius [r], at least, with respect to a `distant` observer.
Again, we have a similar change, but the term ` length expands` is changed to `rulers contract`. Based on similar assumption as outlined previously, I assumed that the local observer’s ruler would expand in-line with any expansion in space and as such, the local observer would not perceive any stretching of the ruler as its was orientated along the radial direction. However, a distant observer would perceive both the ruler and object being measured to both be expanded.
As such, this still seems to be an open issue that I would like to clarify.
--------------------------------------------------------------
Section-4:
Jorrie and I were of the opinion that the radius calculated by a local observer by measuring circumference would agree with radius measured by the observer at infinity.
Yes, I agree. By definition, the calculated radius or coordinate-r corresponds to that measured in flat spacetime. Wasn’t too sure about the next bit, as I would have thought we could simply say that the circumference was measured without reference to any velocity. However, I agree that horizontal/tangential circumference would not be affected by gravity. At this point, I would like to introduce two definitions, although you will probably disagree with the second:
[tex]Coordinate-r = circumference/2\pi[/tex]
[tex]Spatial-r = \gamma(coordinate-r)[/tex]
The implication of the second definition suggests that the radial distance expands, although it is probably not measurable by the local observer for the following reasons starting with your words:
Yes, the local observer can never measure the change in the proper length of his ruler. I prefer to think that the observer at infinity sees the ruler as length contracted when orientated vertically.
While you begin with `Yes`, I don’t think you will agree with my statements at the end of section-3. I was assuming space expands in the radial direction due to the increasing curvature of space. However, I was also assuming that the local ruler would also be affected and therefore the increase in the spatial-radius is not measurable locally, only observed at a distance. So, to conclude, let me try to apply my assumptions to the 2 observers under consideration, i.e. the distant observer (x, t) and the shell observer (x`, t`) within the gravity well using my assumptions:
x` > x, i.e. [tex]x’ = \gamma(x) [/tex]
t` < t, i.e. [tex]t'= (t)/ \gamma [/tex]
The velocity [v] is a function of distance over time:
v' = x'/t' [tex]= \gamma(x)/ (t)/ \gamma = \gamma^2 (x/t) [/tex]
As such, [v’] is greater than [v] by [tex]\gamma^2[/tex] as required by the very first equation. Again, apologises for such a long posting, but as stated, I wanted to try and address all the open issue from my current understanding.