Austin0 said:
I think the back would be contracted and the front would be relatively larger than the local metric or section of the middle CMIRF
yuiop said:
They both appear length contracted, because both the front section and the back section are both moving relative to the initial inertial RF, so they must both be measured to be length contracted in the initial IRF.
Agreed,,,, according to the initial inertial frame ( IIF) But I was talking about a CMIRF
yuiop said:
There is no "Middle CMIRF", just one CMIRF. In the initial inertial reference frame, the velocity at the back is greater than the velocity at the front, at any given simultaneous time (a horizontal line of the timespace diagram, with time on the vertical axis). In any given instantaneous CMIRF, the velocity at the front, back and middle is the same (zero) at CMIRF coordinate time zero. The velocities are all different in the initial inertial reference frame and all the same in any instantaneous CMIRF because the two observers have different opinons about what is simultaneous. It is counter-intuitive that the velocities at the front, middle and back of the rocket are all all the same in the CMIRF (at time zero) and it took me a while to convince myself that it was true when I first looked at the problem.
yuiop ---""One problem with the CMIRF measurement is that a CMIRF that is co-moving with the front of the ship is not co-moving with the back of the ship and vice verse so we would have to define a location for the CMIRF somewhere near the middle of the ship"".
Of course relative motion has no meaning " instantaneously" by definition. But as you yourself previously pointed out a single CMIRF does not apply to the system as a whole.
If the CMIRFs for a given moment are viewed as transparent coordinant grids then it would be seen that the metric of the CMIRF for the back as well as the natural ruler of the accelerating frame were smaller in the back relative to the middle CMIRF and conversely larger in the front , yes?
If the system is viewed as a composite of 3 CMIRFs the back 1/3 would be contracted and the front 1/3 would be expanded relative to the middle CMIRF yes?
yuiop said:
Again, I reiterate, that there is no distinction between the front, middle and back CMIRF. If a given CMIRF is moving at say 0.6c relative to the initial IRF, then the front, middle and back of the rocket are all momentarily stationary in that CMIRF.
Stationary yes but not having equivalent metrics
Austin0 said:
Why would length contraction be relevant if using calibrated clocks [all running same rate] and synchronization from a central master clock?
Wouldn't lengths derived from such clocks be equivalent throughout the system?
yuiop said:
They are equivalent throughout the accelerating system. They only appear length contracted in the CMIRF where the rocket appears momentarily stationary. This is slightly paradoxical. Why would a stationary rocket appear length contracted? The answer is that it is not an ordinary stationary rocket, but an accelerating stationary rocket.
I thought we were now,,,,
not talking about the accelerating rockets natural rulers,,,, but lengths derived from artificially calibrated clocks i.e. clocks equivalent to the middle CMIRF's clocks?WHy would those lengths not agree with the CMIRF?
Austin0 said:
Using the analogy of the back CMIRF wouldn't the local clock radar measurements agree with the ruler measurements if it is in fact equivalent to a locally instantaneous inertial frame? It would only be with radar measurements over greater distances in the system that the natural dilation of the rear clock would not agree with distance or am I missing again?
yuiop said:
Let us say we have two identical rulers such that when they are rest wrt each other and layed alongside each other they are clearly the same length. When one of these rulers is accelerating and the other is moving inertially, then in the rest frame of the inertially moving ruler, the accelerating ruler appears to be shorter, when both ends of the accelerating ruler are momentarily at rest with the inertial ruler. Again, this is counter-intuitive. It is commonly known/accepted that an accelerating clock ticks at the same rate as momentarily co-moving inertial clock and it might be natural to assume that an accelerating ruler measures the same as a momentarily co-moving ruler, but that is a mistake. I think I have probably made that mistaken assumption in the past.
Could you go over this a bit more?? It seems to me that an inertial ruler, instantaneously comoving with the back of the accelerating system would have the same length as that system's if the ruler is short. At some point of increased length the inertial ruler would actually be shorter than the natural accelerating ruler because the accelerating ruler increases in scale towards the front of the system. DO you see any error in this picture??
Here are some other points I am unsure of:
Austin0 said:
1) Born rigidity and Rindler coordinates are both based on a natural dilation differential between the front and the back. yes??
yuiop said:
No, Born rigidity does not really care about whether the clocks rates are differential or not. Born rigidity only claims that the radar length remains constant over time as measured in the accelerating system. It does not claim that the radar distance measured by a clock at the front is the same as the radar distance measured by a clock at the back. Radar distance only requires one clock, and using that one clock the radar distance remains constant over time in the accelerating system, so there is no requirement to refer to the relative rates or synchronicity of different clocks in Born rigid motion. Rindler coordinates on the other hand does take note of clocks at different locations, by definition of being a coordinate system. Rindler coordinates are highly artificial and are not what is measured by natural clocks clocks and rulers as in Schwarzschild coordinates.
Austin0 said:
This is taken to be equivalent to both a comparable gravitational dilation and a dilation equivalent to a difference in relative velocities??
yuiop said:
Coordinate systems like Rindler coordinates that are on a background of flat space, do not have tidal effects taken into account and have a proper acceleration proportional to 1/r rather than the 1/r^2 that we associate with gravity, on top of the complication of the artificial speed up of the clocks. It would be interesting to know if this 1/r^2 factor is recovered when we take the length contraction into account.
"""If all Rindler observers set their clocks to zero at T=0, then when defining a Rindler coordinate system we have a choice of which Rindler observer's proper time will be equal to the coordinate time t in Rindler coordinates, and this observer's proper acceleration defines the value of g above (for other Rindler observers at different distances from the Rindler horizon, the coordinate time will equal some constant multiple of their own proper time).[1] It is a common convention to define the Rindler coordinate system so that the Rindler observer whose proper time matches coordinate time is the one who has proper acceleration g=1, so that g can be eliminated from the equations (this means that even if we pick units where c=1, the magnitude of the proper acceleration g will depend on our choice of units"""":
_____________________________________________________________________________-
This seems to be talking about natural clocks showing proper time with coordinate time being calculated from the proper time. Is this right?
SO is what you and Passion Flower doing, artificially calibrating clocks to actually show Rindler coordinate time??
Are we agreed that natural clocks are dilated at the back and going faster toward the front?
Austin0 said:
2) This is assumed to be constant through time with a constant differential factor?
yuiop said:
Austin0 said:
3) A constant acceleration differential would seem to imply a linear increase in relative velocity between the various locations, eg. the front and the back Yes??
yuiop said:
This can not be answered if you do not specify from whose point of view you are talking about. "A constant acceleration differential would seem to imply a linear increase in relative velocity between the various locations" is true if you are talking about the velocity measured in the initial IRF after a finite period of time, but it is not true in the momentary CMIRF and it is not true in the accelerating reference frame. In the accelerating reference frame, the different locations are all considered to be stationary, just as you would not normally consider the the floor and ceiling of your house to have different velocities, even though they have different accelerations.
You seem to be disregarding physics here. Irregardless of internal interpretations within the system, constant proper acceleration as an application of force a =F/m must result in velocity , SO a constant acceleration of the back relative to the front must result in velocity relative to the front.Would yu disagree with this?
Now as measured in the Initial I Frame the acceleration is not exactly constant so the resulting relative velocity between the back and the front would not be completely linear but it would be continuously increasing yes??
WOuld this not have to mean a continuously increasing relative time dilation between the accelerating clocks at the back and the front?
Austin0 said:
4) How does a linear increase in relative velocities work out to be consistent with a constant dilation factor??
yuiop said:
It is a constant (artificial) dilation factor aplied to a single clock. Each clock has it own dilation factor proportional to the proper acceleration measured at the location of each clock, in the accelerating system.
This what I understand.
Each clock has a natural dilation factor due to the relative velocity resulting from the differential acceleration. I f you look at a Minkowski diagram of either Born acceleration or Rindler coordnate frames the picture is the same.
There is less proper time elapsed at the back of the system at any point of simultaneity [either of the initial frame or the accelerated frame] Are yu saying this is an artificial contrivance and does not represent the natural clocks and proper time within the systeM??
How do you apply a constant [artificial] dilation factor to a single clock?
What does this mean I am missing something here
Austin0 said:
5) The coordinate difference in acceleration and resulting relative velocities , as measured in the inertial frame where the accelerated system was initially at rest, must be consistent with the measured Lorentz contraction relative to that inertial frame, correct??
yuiop said:
Yes, the back of the rocket is length contracted to a greater extent than the front of the rocket exactly as we would expect from the Lorentz relations (in the co-moving inertial reference frames). When we transform to a different momentary CMIRF, everything still looks pretty much the same and this differential length contraction is observed in any CMIRF.
I am talking about overall contraction here.
Austin0 said:
How then could a linear increase in coordinate relative velocities between the front and the back be consistent with the non-linear increase in contraction??
Ditto I am talking about overall contraction here.
yuiop said:
Who said there the coordinate relative velocities in any frame have to have a linear increase?
OK then how could the increase in coordinate relative velocity between the back and the front match the asymptotic increase of the Lorentz contraction if the acceleration at the back is actually falling off faster than the front and could [maximally] only have a linear increase if there was no falloff at all?