Ulysees said:
While I don't normally reply to dishonest garrulousness that pretends not to understand things that are obvious to everybody else, I'll make one last exception.
whoo! i better go to etiquette school. is Miss Manners still around? (i honestly don't know, since i don't read a newspaper anymore.)
There are academics doing research today who notice many unexpected astronomical observations (like how fast a galaxy is rotating), and match this unexpected data by modifying the law of gravity in the long distance. Effectively G becomes a function of distance r. The observational data is then matched perfectly. Others match the data by imagining the ghostly influence of invisible matter - as in "The Invisible Man", this ghostly presence is imagined to exert forces and mess up the observations without anyone ever seeing it.
Either way, "invisible man" or "variable G(r)", someone had to bend the existing laws.
r is the distance from what? using cartesian coordinates, in a scenario with multiple bodies all contributing to the gravitational field, are you saying that at some fixed point (x
0, y
0, z
0), is the G there not some specific value? or does it change when the bodies around it that are contributing the the field, when those bodies move around? if
r means the distance to each body, then it's a change from the inverse-square law. so that's bending the physical law (changing the G
Mm/r2 to something else) , but it's not a changing G. it's a different issue.
if the existing law prevailed but was "bent" by changing G (say, w.r.t. time, but you could ask the same question as if G was different in the galaxy Andromeda from our G), if all dimensionless ratios of like-dimensioned quantity remained the same, no one will know the difference. if some dimensionless ratio (that we normally believe to be constant) has changed (which causes a bona fide different world that would be perceived as different by us and our measuring instruments), it's the dimensionless constant that's salient. that's the only thing that i have said from the beginning and it's the only thing I've not moved away from.
that was the original question of the thread.
" If the universal gravitational constant was changed from 6.67 ...". if G was changed from what we think it is from measurement using our meter sticks, atomic clocks, and kilogram standards,
if nothing else was changed (this was explicity specified by both you and by jackster), then the fact is that no mortal with instruments that cannot somehow step out of the context where they are subject to the laws of physics will know the difference. If your simulation program indicates an operational difference (i.e. a resulting simulation that isn't just like the normal one, with the current G, but with scaled time or length or mass), e.g. a result where some lengths get longer and other lengths get shorter (that's a
real difference), then the parameter(s) whose change was causing that is more fundamentally a dimensionless parameter. likely it is the masses of all of the particles, relative to the Planck Mass. i.e. if you said that this law:
F = G \frac{m_1 m_2}{r^2}
was changed to
F = (aG) \frac{m_1 m_2}{r^2}
where (
aG) is the new
G, i would say, think of it in terms of Planck Units, that the more fundamental representation is
F = G \frac{(\sqrt{a}m_1) (\sqrt{a}m_2)}{r^2}
and it's the masses that increased or decreased. Note that it's the same G, but we're postulating that all masses are getting scaled by the square root of whatever you're scaling G in your simulation program. But
if G remains constant, even if we change it, then those masses changed w.r.t. a unit mass that is defined in a system of units where G is fixed to a constant value. That is true for both Planck Units and Stoney Units. Assuming that all of the celestial bodies are made up of the same molecules with the same atom having the same sub-atomic particles, if we chose Planck Units to measure things, what happened is that the masses of all of these particles relative to the Planck Mass got scaled by \sqrt{a}. e.g. for the electron:
\frac{m_e}{m_P} \rightarrow \sqrt{a}\frac{m_e}{m_P}
or
\frac{m_e}{m_P} \rightarrow \sqrt{a} \frac{m_e}{\sqrt{\hbar c/G}}
See how that is the same as
\frac{m_e}{m_P} \rightarrow \frac{m_e}{\sqrt{\hbar c/(aG)}} ?
If you chose to measure things in, say, Atomic Units (here the unit mass is defined to be the mass of the electron and G is not normalized), then you would say G changed. But Nature doesn't give a rat's ass which system of units we use. If you keep in mind that when we are using a system of units to measure things, just like measuring a length of something with a ruler and counting tick marks (a dimensionless number), we are measuring that quantity against the unit definitions resulting in dimensionless numbers. When your measurement of G appears to change, it is the value of G
relative to the unit value of the kind of quantity, the dimension, that G also posseses. In SI,
G = 6.67428 × 10
-11 m
3 kg
-1 s
-2. In Atomic Units G is 2.4005 × 10
-43 L
A3 m
A-1 t
A-2 (where L
A, m
A, and t
A are the units length, mass, and time in Atomic Units; and if i didn't screw up the calculation). but there are expressions for those Atomic Units and then G comes out to be (making the substitutions)
G = 2.4005 \times 10^{-43}\left( \frac{\hbar^2 (4 \pi \epsilon_0)}{m_e e^2} \right)^3\left( m_e \right)^{-1}\left(\frac{\hbar^3 (4 \pi \epsilon_0)^2}{m_e e^4} \right)^{-2}
or
G = 2.4005 \times 10^{-43} \left( \frac{\hbar^2 (4 \pi \epsilon_0)}{m_e e^2} \right)^3 \left( \frac{1}{m_e} \right) \left( \frac{m_e e^4}{\hbar^3 (4 \pi \epsilon_0)^2} \right)^{2}
or
G = 2.4005 \times 10^{-43} \frac{e^2}{m_e^2 (4 \pi \epsilon_0)}
or
G = 2.4005 \times 10^{-43} \frac{e^2}{(4 \pi \epsilon_0) \hbar c} \frac{\hbar c}{m_e^2}
or
G = 2.4005 \times 10^{-43} \alpha \frac{\hbar c}{m_e^2} ( \alpha is the Fine-structure constant, a very important dimensionless and fundamental constant and it's about 137.035999
-1)
or
G = 2.4005 \times 10^{-43} \alpha \frac{(m_P^2 G) }{m_e^2}
or finally
\frac{m_e}{m_P} = \sqrt{2.4005 \times 10^{-43}} \sqrt{\alpha}
Expressing in Atomic Units what
G is, is precisely the same as saying what the ratio is of the mass of electron to Planck Mass is (with this factor of \sqrt{\alpha} tossed in. Now if you change that 2.4005 × 10
-43 value, you have done exactly the same as changing the mass of the electron to Planck Mass ratio. Or you changed the Fine-structure constant. Or a little of either. But the point is, no matter which way you look at it, no matter how you define your system of units, when you
think you've observed a change in a dimensionful constant like
G, what really changed, what fundamentally changed, the important thing that changed,
the quantity that really matters, is in the last analysis, a dimensionless parameter which can be, or usually is, a ratio of like-dimensioned quantities.
That, Ulysees and jackster, is the lesson. And I'm done presenting it in every which way, because, frankly, I'm running out of steam and if you don't get it, you just don't get it.
