yes the kinematics of larger reference frame would be different if its observed from
smaller reference frame.
i am not sure at all how the dynamics would change. i have guess that larger reference frame has K² times more energy and 1/K times slower energy transfer speed, and 1/K times smaller energy density. this comes from energy of harmonic oscillator that has energy K²kx²/2 if the string constant is same.
how does different reference frames interact? is the larger reference frame "stiffer" since its time run slower or is it weaker and more fragile? i assumed above that slow time makes reference frame "stiffer".
about kinematics:
-if the K times larger reference frame is different in its natural unit length K times greater and time rate that is K times slower and there are no other changes, then
-all time dependent functions would have transformation f(t)' = f(t/K) and all path dependent functions would have transformation f(l)'=f(Kl)
-for example harmonic oscillator would oscillate K times slower and it would seem to
dampen K times slower in larger reference frame.
-all functions time derivatives if measured from normal reference frame, would be 1/K times slower
the constant velocity would be invariant or same measured in both reference frames
constant acceleration would be 1/K times slower in larger reference frame.
i may do math mistake here, but would all higher order time derivatives of any time dependent function be then always 1/K times slower.
i don't know how the higher order derivatives would seem to change, do they have some other rule than this. this could be if there are some other changes in larger reference frame than these two.
about the dynamics:
the different sized reference frame could have some other changes if these two changes arent enough to make the dynamics laws to work right.
i have a rough guess that K times larger reference frame has always K² times more energy than normal reference frame in all energy forms (including mass?). for example nonrelativistic kinetic energy would be K²mv²/2
and the energy transfer rate could be K times slower, the energy density E/V K times smaller... this comes from dimensional analysis of energy of harmonic oscillator kx²/2
- if the string constant k is same then energy of harmonic oscillator is K² times larger. i don't have to take slowness of time into account since energy transfer is also 1/K times slower, right?
odd thing about energy conservation is that if the energy is K² times larger and the reference frame does not exerience any forces during expansion, then this energy is free - or if small reference frame expands to larger one, it would have K² times more energy without actually doing any work.
-but what are the limitations how the process can happen, is another question... it may be property of space that for example if reference frame moves long distances it may change larger proportional to the distance relative to local reference frame. this can happen if every rest frame has variation in natural unit length but moving rest frame
does not change relative to another rest frame identically and therefore it seems to expand relative to local rest frame. -i mean if it is even possible to have inertial rest frames that have different unit length (and also time rate due to invariance of light velocity)
hi mfb, about the selection of units- i don't mean the definition of units by using different unit systems, my question is about that the all physics or most of the physics change in reference frame such way that every physical propertys size iow space dimension is K times larger (and also their time rate is K times slower due to light speed invariance principle) than they are in normal reference frame.the change between units between two different sized reference frame together in same place would be
t'=t/K time unit is K times longer
(x',y',z)= 1/K (x,y,z) every space dimension unit is K times larger.
and if the other frame is moving it also has relativistic changes.
and if natural unit length varies from place to place inside same inertial reference frame,
the change between units in different places would be
(x,y,z) = K (x,y,z) and t remains same. also if the frame is moving, it has relativistic effects