I had an idea, there should probably be a book called "A Primer of Natural Units"
Marcus, yes! You've already written a fair bit of it. Perhaps it should be illustrated? I'm not volunteering as an artist. My chickens would look like my Maid Marions.
Richard and Kea,
thank you both. Both your replies are very heartening.
If the project were to go ahead I would definitely not
use your posts without telling you, Richard,
and would only do so with prior permission.
Kea, what point or use do you see for a thin book about
this variant of the Planck units?
To get an audience would it have to discuss both conventional
(G=1) Planck units and this variant (that seems to appeal to
relavitivists and QG people and has 8piG = 1)?
Is there a point in a QG grad student's career when he or she
might enjoy such a book or get something out of it?
Or would "A Primer of Natural Units" inevitably be an eccentric adventure
of no real usefulness besides a few people's amusement?
(that last possibility does not mean I would refuse to write it
because i actually think the natural units are beautiful and i like
to look at nature thru them as lenses----but if it is apt to be
of very limited appeal to others I would like to know).
so if you don't mind giving an opinion, I'd appreciate it.
I'd actually prefer to avoid conventional Planck units and focus on these,
if that approach could be made to work
#243
Kea
859
0
a book
marcus said:
To get an audience would it have to discuss both conventional (G=1) Planck units and this variant (that seems to appeal to relavitivists and QG people and has 8piG = 1)?
Is there a point in a QG grad student's career when he or she might enjoy such a book or get something out of it?
Or would "A Primer of Natural Units" inevitably be an eccentric adventure of no real usefulness besides a few people's amusement?
I don't see why you would need to clutter up the book with 'conventional' Planck units. Personally, with a little editing, I think it would be an excellent first year level physics text. Physics should be taught in appropriate units, surely. To be honest I don't think a grad student would actually buy it. They don't have any money. But I have certainly enjoyed the entertainment. Much better than the illustrated wonderwoman knot book that recently appeared in my library.
I don't see why you would need to clutter up the book with 'conventional' Planck units. Personally, with a little editing, I think it would be an excellent first year level physics text. Physics should be taught in appropriate units, surely. To be honest I don't think a grad student would actually buy it. They don't have any money. But I have certainly enjoyed the entertainment. Much better than the illustrated wonderwoman knot book that recently appeared in my library.
Cheers
Kea
This is really encouraging. Thanks, Kea. Yes, physics should be taught in appropriate units! (what could be more appropriate?) I have been visualizing a small thin book, a "primer", and I like it more and more. Haven't got a picture of the table of contents yet
This summary needs to be brought forward periodically.
the force F = c4/(8piG) is the main constant in Gen Rel, the prevailing theory of gravity since 1915. The constant in the Einstein equation is not Newton's G, but rather F. In Quantum Gravity one often uses units in which |F| = 1
(this can come about by stipulating that |8piG|=1, since normally one already has adjusted the units so |c|=1)
the moment one sets
|F|= |c|=|hbar|=|k|=|e|=1
one has a fairly universal set of units and it is interesting to see what some familiar quantities come out to be.
Another way (suggested by a Kea post) to define the same units is to make the unit angular frequency be 3.7E42 per second. Call it Z, just to have a symbol. Adjusting the units to make the value |Z| = 1 gives the same set of units as setting |F| = 1.
I am trying out this version of natural units to see how they work. In order to try out the units one must keep a list of rough sizes of things handy----to use the units for practical purposes one must have a sense of scale. Here are some rough sizes of familiar things expressed in the units.
I periodically bring this list forward to keep it handy.
rough sizes:
Code:
q'ty expressed in nat. approximate size
E8 mass pound
E50 time units year
E33 length handbreadth (3.2 inch, 8.1 cm)
E34 length pace
E37 length half mile
E50 length lightyear
E-5 energy food Calorie
E-8 energy lab calorie
E-28 voltage quarter volt
E-28 energy quarter eV
10E-28 energy typical photon energy for green light
E-53 electmagn.field unit tesla
E-57 field unit gauss
E-29 temperatrure average Earth surface temp
E-9 speed 2/3 mph
E-7 speed 67 mph
E-6 speed speed of sound (cold air)
E-107 pressure conventional PSI on airgauge
14E-107 normal atmospheric pressure
E-39 (ang. format) frequency D on treble staff
E-50 acceleration one "gee"
E-40 force weight of 50 kg sack of cement, traditional "hundredweight"
E-49 power 144 watt bulb
some constants (approx.):
reciprocal proton mass 2.6E18
electron mass 2.1E-22
Hubble time 1.6E60
Lambda 0.85 E-120
rho-Lambda 0.85 E-120
rho-crit (critical density) 1.16 E-120
more exact Earth year 1.1676 E50
more exact lightyear 1.1676 E50
avg Earth orbit speed E-4
earth mass 1.38 E33
earth radius 7.86 E40
sun mass 4.6 E38
solar surface temp 2.0E-28
sun core temp 5E-25
solar constant 6.2E-117
CMB temperature 9.6E-32
earth surface air pressure 1.4E-106
earth surface gravity 0.88E-50
fuel energy released by one O2 17E-28
density of water 1.225 E8/E99
timescale:
3.700E42 rad per sec 1 (the unit frequency)
1/222 of a minute E42
4.5 minutes E45
As a handle on the natural timescale, imagine counting out loud rapidly at the rate of 222 counts a minute, each count is E42 natural time units. A thousand counts is 4 and 1/2 minutes. It just happens that one year is roughly E8 counts, or E50 natural.
So far we have been fairly casual about the names we call the natural units.
The natural units themselves, which is the system we are using, have simply been called, for instance, "natural length unit", "natural energy unit", or sometimes just "energy unit", or (when the type of quantity was understood already) just saying "natural", for short.
Early on, I was trying to use a lot of "named powers of ten" which were humanscale practical-size power-of-ten multiples of the natural units----like "pound" for E8 mass unit, and "food Calorie" for E-5 energy unit. Even that, I am doing less now, tho still do it some (to help provide a picture of rough size of things).
So the basic units themselves have not (so far at least) been dignified with a rigamarole of names and symbols, as they do in metric international system: for instance with Joule energy, Pascal pressure, Newton force, etc.
At least so far we don't have names and symbols. Instead, we've done things informally and just called them what they are: energy unit, force unit, power unit, voltage unit, etc.
All the same, I was thinking that, in some circumstances if not all the time, it might help to have a standard way to "spell out" the units we are using, and be very explicit-----
I don't know if this is a good idea, but I am reluctant to introduce a lot of new symbols and it would be possible to efficiently "spell out" all the units (any that one wanted to, when it seemed appropriate) using only ONE new symbol, plus familiar things like c and hbar and electron charge e.
but one would have to get used to "reading" the units in this type of spelling. I mean with the natural frequency Z.
frequency is a number of events per time
so 1/Z is a way of writing the natural time unit. We really don't have to have Z a bold letter so I will just make it ordinary.
of course c is the unit speed, and multiplying unit speed by unit time gives unit length: c/Z.
So c/Z is a way of denoting the unit length ("planck length" in this variant of the Planck units)
and cZ is a way writing the natural unit of acceleration, unit speed per unit time
and so on. (one has the option of reading Z as "per unit time" when interpreting units written this way)
hbar is a natural ratio of energy to freq. If you take any freq. and multiply by hbar you get an energy. one might even say that's what hbar is all about.
So hbarZ is a way of writing the natural unit of energy
and hbarZ/c2 is a way of writing the natural unit of mass
Maybe this would be useful. Have to try it out and see.
I guess the only thing to do is try and see if anyone understands and wants to reply.
My mass is 2E10 hbarZ/c2
does anybody want to reply and tell me (roughly approx. good enough), in more conventional terms, what that is? or describe some physical quantity at their end?
dont have to say your mass, can say the mass of something else not you, and there is linear dimension (c/Z is the unit and for convenience E34 c/Z is about 32 inches). Plenty of other types of quantity as well.
My wife and I just had supper---mostly it was rice and chicken breast with special orange sauce made with oranges off the tree.
Caloriewise supper, for me, was 1.5E-2 hbarZ.
thats how much food energy.
It would be nice if someone else would report a quantity or two in this notation.
BTW the point of the notation is that it is self-decoding. You only have to know the key which is that Z=3.7E42 per second and you can, if you want, find the metric equivalent of anything I say.
for instance as regards supper if you wanted to know what hbarZ is in joules (I am not proposing that you should want to know this, but if you did) you could just multiply the standard NIST value of hbar
which is 1.0545716E-34 by the metric value of Z which is 3.7E42
and that would give some number of metric Joules of energy----
and then 1.5E-2 of that, as I indicated, would be supper.
that is what i mean by the symbol being self-decoding.
#250
nightcleaner
Hi Marcus
I think you are on the right track here, and I have been working over the natural units also, from a dimensional analysis point of view. For example, force seems to me to work down to inverse time. I should do more calculation, instead of just dimensional analysis, so I will try to work your mass problem.
Personally I am not fond of named units either, or of the ones scaled to human conventions. I am still working on relating units using Z to units using F. I'll let you know if anything interesting comes up.
h-bar Z cE-2 is the mass unit which I have as .434 E-8 kg, so
2E10 .434 E-8 = 2 x .434 E2 = 86.8 kg = 190 lbs.
Have you lost some weight? I thought I remembered from an earlier post that you weighed about 200lb?
...
I'll edit this with a reply to the mass problem.
good! it is pretty dull having a onesided discussion about something like mass (which is already not very exciting I fear)
if you have a dog, estimate his/her mass if you don't wish to reveal your own.
I will demonstrate the self-decoding feature of the notation for mass:
the mass unit is hbarZ/c2
If one doesn't want metric equivalents then one simply leaves it at that.
But if one wants metric equivalents then
hbar = 1.0545716E-34 joule second
Z = 3.7E42 per second
one multiplies those together and gets 3.9E8 joule
then one divides by square of speed of light, which is about 9E16 (m/s)2
and gets around 0.434 E-8 kilogram.
so, like if you have a dog, and the dog's mass is 2E9 hbarZ/c2,
and you have to describe the dog to a Frenchman who insists being
told kilograms
then you just multiply 2E9 by 0.434E-8 and get 8.6 kilograms.
or you tell the dog to bite the Frenchman on the leg and ignore the question
(but you can convert if you want to, it is built into
the notation for the mass unit hbarZ/c2 itself)
h-bar Z cE-2 is the mass unit which I have as .434 E-8 kg, so
2E10 .434 E-8 = 2 x .434 E2 = 86.8 kg = 190 lbs.
Have you lost some weight? I thought I remembered from an earlier post that you weighed about 200lb?
I think you must be kidding. No, obviously I lie about my weight!
It really is 200 well just a bit over, traditional American pounds.
But I call it 2E10 hbarZ/c^2 for round numbers.
Am I supposed to carry it out to two decimal places when I am sensitive about it anyway?
Let's talk about your dog. Do you have a dog?
#253
nightcleaner
Yes, I do have a dog, or at least partial custody of a dog. His name is Tucker, a black lab, and he weighs in at very close to E10 mass units.
everybody please don't forget that the E notation is for powers of ten, as in a calculator.
if you want to write 1000 you can say E3
if you want to write c2, then don't say cE2, say c^2.
cE2 would, I guess, stand for 100c.
Chronos and i were talking about this just 3 or 4 days ago. it is easy to confuse nE2 and n^2
#255
nightcleaner
ok. noted. Probly one of the reasons I get confused easy. Thnks.
BTW, did you agree that F works down to dimensions of T^-1? and wouldn't these also be the dimensions of Z? Mass works down to dimensions of TL^-1, inverse velocity.
I have been trying to resolve all these units in terms of L, length, and T, time. Any ideas? Perhaps it is a useless endeavor.
Actually, I do find mass to be an interesting concept. I remember elsewhere working it down to T^-2 in dimensional units, and again elsewhere to inverse length, but I have too many notes and don't remember exactly how I got to those dimensions.
What do you think in general of the idea of using dimensional analysis to find the "meaning" of things like mass and energy? If I can get comfortable thinking of all physical quantities in terms of length and time, then I intend to use Einstein's spacetime equivalence to cancel out the universe. Sometimes I get tired of the universe. But then, I suppose, if I do find a way to cancel it all out, someone else will find a way to brew up a bunch of new ones. Oh well. And they say breaking mirrors is bad luck.
Oh, I was going to tell you, I read that the December 27 gamma burst was actually an expanding cloud of gamma, and the wavefront of the cloud was said to be moving at .3 c. So the Dec.24 event could have been a precurser after all. Perhaps a gravitational reaction to all those approaching gammas?
Yes, I do have a dog, or at least partial custody of a dog. His name is Tucker, a black lab, and he weighs in at very close to E10 mass units.
I myself am not svelte. I mass about 2.5E10M.
Great! this is my favorite PF post for the past couple of days, since whenever Kea wrote encouragements about the book. this is a great post.
I like black labs a lot. they are so beautiful and cute in a lumoxy way.
they can have enthusiasms too, like catching tennis balls in the air. they are enough dog that they have a distinct doggy smell. perhaps I should not generalize.
I am happy that Tucker mass is E10 mass units. that is like 43 kilogram so you do not pick that dog up lightly.
Yes I am not svelte either we don't have to be precise about these personal things.
like my daily food energy intake is 3E-2 hbarZ
that is 0.03 hbarZ.
(I know it is more congenial to be informal and say "0.03 energy unit" but I am trying out the notation)
does he eat dry food (kibble) or can food or other?
------
btw about collapsing units all down into one unit (or two), it can be done.
by setting stuff equal to one it is possible to reduce all units to one unit and all types of quantity down to one type. but this may actually be counterproductive. I like to keep the different types of physical quantity distinct----speed differnt from force, different from voltage, different from acceleration, different from density. But it is in some way a matter of taste. One tries to be consistent.
when I say |c| = 1 what I mean is that the value of c is one, expressed in these units. in other words that c = 1 length unit per time unit.
there is a subtle difference between this and saying baldly flat-out c = 1.
If you say c=1 it is like there are no units in the picture at all! c is just a number, the number 1. Or maybe it means that length and time are measured with the same unit! But that is too radical for me, so I say |c| = 1 meaning that there is a time unit and a length unit and they are distinct. but related so that c = 1 unit length per unit time.
that is the idea behind conventional Planck units too. Planck length and Planck time are two different things. And Planck mass is different, and so on. You can DO physics with all the units collapsed into one, and I believe somepeople like to do this. but I am very reluctant to take that step.
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#258
nightcleaner
marcus said:
let us estimate Tucker's energy intake per day
like my daily food energy intake is 3E-2 hbarZ
that is 0.03 hbarZ.
(I know it is more congenial to be informal and say "0.03 energy unit" but I am trying out the notation)
does he eat dry food (kibble) or can food or other?
He eats 3 cups of kibble each day, and numerous treats. I don't give him treats, just lots of pats and rubs and one-way conversation. My ex, now a very good friend, gives him p-nut butter in a Kong, sometimes five or more tbsp each day. And he gets table scraps, not from me of course. I take him for a run almost every day. He runs about a mile, then he has had enough. He is overweight, and gets scolded by the vet every time we take him in.
He smells very good. I even like to smell his paws. He is a very affectionate dog, but spoiled horribly, by my ex, of course. He doesn't like other dogs but he is crazy for attention from people. Any passer-by will do. He has to stay leashed when outdoors except when I take him out for a run. Then we go out in the deep woods where we never see other dogs or people.
I edited the post before this some. Maybe you didn't see it.
I was having some confusion about the mass unit. I have it variously as dimensions E c^-2, h-bar Z c-2, and sqrt (h-bar F c^-3.) Do you concur with these?
nc
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#259
nightcleaner
marcus said:
btw about collapsing units all down into one unit (or two), it can be done.
by setting stuff equal to one it is possible to reduce all units to one unit and all types of quantity down to one type. but this may actually be counterproductive. I like to keep the different types of physical quantity distinct----speed differnt from force, different from voltage, different from acceleration, different from density. But it is in some way a matter of taste. One tries to be consistent.
when I say |c| = 1 what I mean is that the value of c is one, expressed in these units. in other words that c = 1 length unit per time unit.
there is a subtle difference between this and saying baldly flat-out c = 1.
If you say c=1 it is like there are no units in the picture at all! c is just a number, the number 1. Or maybe it means that length and time are measured with the same unit! But that is too radical for me, so I say |c| = 1 meaning that there is a time unit and a length unit and they are distinct. but related so that c = 1 unit length per unit time.
that is the idea behind conventional Planck units too. Planck length and Planck time are two different things. And Planck mass is different, and so on. You can DO physics with all the units collapsed into one, and I believe somepeople like to do this. but I am very reluctant to take that step.
Yes, I understand your reluctance, but I have been working on this for some time and have actually a better reason in mind than the one about cancelling all appointments forever. It does seem that every time I get down to one or two base units, I lose track of the meaning. It is kind of a Tantulus myth. Just when I think I am making progress, the well dries up.
And after all, the base units of length and time are not absolutes anyway. There seems to be no Planck to hold onto in the swell of the sea of relitivity. Sometimes I think I should develop an interest in hockey instead. Things there at least are unequivocal.
He eats 3 cups of kibble each day...
I edited the post before this some. Maybe you didn't see it.
Let's eventually find out how to convert that 3 cups to Calories.
I just now saw the expanded edited version of the previous post.
I will provide some input to that process but I will not venture in that direction myself. It can get confusing when units are folded up into each other, one can lose track. So I personally refrain from it. but AFAIK anyone can do anything they want along those lines if they can manage to be consistent.
I was having some confusion about the mass unit. I have it variously as dimensions E c^-2, h-bar Z c-2, and sqrt (h-bar F c^-3.) Do you concur with these?
E c^-2, h-bar Z c-2, I agree. the other one I have to check. Yes, I concur with all three (the other one too)
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#261
nightcleaner
ok. He gets science diet lite, and say 1/4 cup pnut butter a day. Maybe about five dog biscuits. And the table scraps, which don't really amount to much. I can't tell you about the caloric value of the foods, because I am at my house and Tucker usually eats over at my ex. It is reduced fat pnut butter.
Marcus, I am tired tonight and have to retire early. Thanks for your conversation.
Yes, I understand your reluctance, but I have been working on this for some time and have actually a better reason in mind than the one about cancelling all appointments forever. It does seem that every time I get down to one or two base units, I lose track of the meaning. It is kind of a Tantulus myth. Just when I think I am making progress, the well dries up.
And after all, the base units of length and time are not absolutes anyway. There seems to be no Planck to hold onto in the swell of the sea of relitivity. Sometimes I think I should develop an interest in hockey instead. Things there at least are unequivocal.
I laughed out loud when I read that in icehockey
"things are at least unequivocal"
the tantalus picture is very descriptive of what i remember experiencing with possibly similar investigations which can be very exciting for a moment but in the end one clutches air
montaigne invented the essay, where one combines a thoughtful reflective attitude towards one's own life with just the right amount of philosophy balanced with humor.
selfAdjoint would know if it was really montaigne. if it was, then maybe I should read montaigne.
what you just wrote, the post about icehockey, was a bit like an essay.
where does wisdom come from except from shared reflection and honestly reported experience
Marcus, I am tired tonight and have to retire early. Thanks for your conversation.
thanks likewise. I won't bug you again about the Tucker calories (dumb idea). well enough. good night
the unit of electric current can be written Ze
(flow of electrons at rate of 3.7E42 electrons per second)
and the unit of voltage can be written as hbarZ/e
(unit potential energy per unit charge, i.e. per electron)
Ze current is 6E23 amps and hbarZ/e voltage is 2.4E27 conventional volts.
so when you multiply the current and voltage units together (which must give the power unit hbarZ2) you get
6E23 amps x 2.4E27 volts = 14.4E50 watts.
which I got earlier for the unit power. just checking.
I want to scope out what it might be like to use
Ze for current
hbarZ/e for voltage.
this lightbulb over my shoulder is drawing 120 volts and about 0.6 amps so what is it drawing in natural units?
E-24 current units = E-24 Ze
480 quartervolts (about) so 480E-28 = 4.8E-26 voltage units.
Let's round it to 5E-26 voltage units = 5E-26 hbarZ/e
that's what the lightbulb should have written where the specs are on the top: E-24 Ze and 5E-26 hbarZ/e.
and for the wattage it would be 5E-50 hbarZ2
Richard mentioned googlephobia the aversion to extreme numbers and I find it comes and goes in fits. sometimes quantities in terms of natural units seem too extremely small or large to assimilate.
(assimilate often means to find something similar)
the pressure unit can be denoted hbarZ4/c3
(which maybe deserves some explanation, why is that the unit force per unit area?)
and by coincidence
E-107 hbarZ4/c3 = about 1 PSI
(actually 1.06 PSI but still close enough to say that when I check the tires on the car I put in 30E-107 pressure)
we seem to be near the edge of what I can tolerate by way of extreme numbers, but anyway why is this how you spell the natural unit pressure?
because unit length is c/Z and area is (c/Z)2
and unit force (being energy/length) is hbarZ/(c/Z) = hbarZ2/c
so unit force over unit area is hbarZ2/c divided by (c/Z)2, and that is
hbarZ4/c3
If you work it out in metric (using 3.7E42 per second, and metric values of hbar and c) and by the way I don't see why you should think you need to, then it comes to 7.335E111 Pascal. Pascal is the sanctioned alias for "Newton per square meter"
so if you ride a bicycle and put 60 PSI in the tires then you are
riding on the pressure 60E-107 natural.
and regular atmospheric is 14E-107.
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#265
nightcleaner
I suppose this is why self-Adjoint cautioned me about reductionism. I have come this far to see the land where there is only unity and separation, and unity is removed in time as separation is removed in distance. The universe is very flat here, unmarked in all directions from where I stand to the horizon, and the sky is unmarked also from horizon to azimuth. The sky is a mirror, and so is the ground, so the horizon here is quite invisible. If I look straight down, I see only my own face stareing back at me. Why have I come to this place? To find myself. I would turn and go, but there is no turning here. Mach was right.
Well, I am strangely satisfied. Now that all the universe has come down to me and my feet, I remember, as an infant, the insane discovery of toes. That's me, out there, but how can that be? For this is me, in here. I can't be out there, and in here, both, can I? And if that is really me, what are these eyes?
Now there is the space between stacked mirrors. The Casimir force, and the degeneracy of protons in a neutrino beam. Why, exclusion is attractive, and neutrinos count the aeons in a proton life. I am amazed by these things and filled with a renewed sense of wonder.
I suppose this is why self-Adjoint cautioned me about reductionism. I have come this far to see the land where there is only unity and separation, and unity is removed in time as separation is removed in distance. The universe is very flat here, unmarked in all directions from where I stand to the horizon, and the sky is unmarked also from horizon to azimuth. The sky is a mirror, and so is the ground, so the horizon here is quite invisible. If I look straight down, I see only my own face stareing back at me. Why have I come to this place? To find myself. I would turn and go, but there is no turning here. Mach was right.
Well, I am strangely satisfied. Now that all the universe has come down to me and my feet, I remember, as an infant, the insane discovery of toes. That's me, out there, but how can that be? For this is me, in here. I can't be out there, and in here, both, can I? And if that is really me, what are these eyes?
Now there is the space between stacked mirrors. The Casimir force, and the degeneracy of protons in a neutrino beam. Why, exclusion is attractive, and neutrinos count the aeons in a proton life. I am amazed by these things and filled with a renewed sense of wonder.
intense essay. the brinkmanship of introspection.
I am becoming a fan. I will supply some outlandish looking Z-symbols for various types of physical quantity like the energy-density with appears to cause curvature (the unit of the stress-energy tensor)
I walk up the hill nearly every day. it is aerobic for about half hour and very important. Have you run the mile with Tucker today?
these Z symbols for the units are alienating and a bit scary, while the informal casual nomenclature is comfortable and easy.
it is very easy to say "E10 mass units" or "E10 natural" for the dog's mass.
it is off-putting to say E10 hbarZ/c2
but I think I should check it out anyway:
the natural unit of brightness (power radiating per unit area) is the
extreme brightness of
hbarZ4/c2
compare the surface of the sun, power radiating per unit area
it is roughly 3E-112 hbarZ4/c2
or compare the brightness of direct sunlight at Earth's distance from sun
6E-117 hbarZ4/c2
the point is the extreme difference between what I think of as very bright, like the brightest sunlight I ever see on earth, or like the face of the sun if you could look at it up close, and this universal unit brightness
they differ by over a hundred orders of magnitude, it is primordial
------------------
now what about curvature and energy density
the unit of area is (c/Z)2 and the unit of curvature is the reciprocal (Z/c)2
the force is hbarZ2/c
If you multiply an observed curvature by this force then you get the density of energy needed to produce that amount of curvature (main eqn. of GR)
try multiplying curvature x force units
(Z/c)2 x hbarZ2/c = hbarZ4/c3
and in fact what is on the righthand side is the natural unit of energy density------in other words one unit of energy density produces one unit of curvature and it is proportional
any actual energy or mass density one would encounter would be much much less than unit density, and any real observed curvature would be proportionally less than unit curvature too.
still just trying the notation out, poking around so to speak (btw pressure and energy density have the same unit---as in metric: Newton per sq. meter is algebraically the same thing as joule per cubic meter)
[edit: added on reflection. I'm thinking this Z notation is not reader-friendly enough.
it has the advantage of being explicit: definite and self-defining. but it is too cold and outlandish. maybe it could be used for sporadic marginalia and footnotes]
despite the downside outlandishness, to continue testdriving the Z notation, let's use it in calculating the number of CMB photons in a cubic mile of space.
the natural temperature unit is hbarZ/k
hbarZ is the energy unit and k is the Boltzmann constant which NIST lists
(if metric equiv needed hbarZ/k = 2.826E31 kelvin)
CMB temperature is 0.96E-31
first let's do it informally without Zees and calling pi2/15 =2/3
then the energy density of the microwave background is just 2/3 times the fourth power of the temp. let's be sloppy and call the temp E-31 so the fourth power is E-124.
so in natural units the energy density of the CMB is just (2/3) E-124
and the average photon energy is 2.7 times the temp, so 2.7E-31
we divide the total by the average to get the number of photons:
(1/2.7)(2/3)E-93
now a halfmile is E37 natural and a cubic halfmile is E111 volume units so the number of photons in a cubic halfmile is (1/2.7)(2/3)E18
and a cubic mile is 8 cubic halfmiles. so multiply by 8 and we get that the number of CMB photons in a cubic mile is 2E18
hey, nice. Two quintillion of these very old photons, that left home a few hundredthousand years after the Bang.
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now let's do it with the Zees, in a more uptight regular way.
the radiation is blackbody thermal, and the volumetric energy density of blackbody radiation depends on the temperature:
it is pi2/15 (times a constant which has value one and is just there to take care of the units) times the fourth power of the temp
the constant with value one is k4/(hbar3c3)
you can see it has value one because everything it is made of has value one (it wouldn't if you were working in metric so then you would need to multiply out or look up in book), because here
|k|=|hbar|=|c|=1
and in thermal radiation the average photon energy is 2.701kT
(1/2.701kT)(pi2/15)(k4/(hbar3c3)T4
and one of the Tees cancels, so it is only T3 now, and I will put in what the temperature of the CMB actually is: T=0.96E-31 hbarZ/k and get
I am jumping thru all this hoops and the number answer is going to be the same. this may diminish one's pleasure slightly. Anyway when you put all the constants in you find that almost all cancel, because they were there mainly to take care of (unnatural) units.
(1/2.701)(pi2/15)(0.96E-31)3 (Z/c)3
and that (Z/c)3 is the reciprocal of the unit volume(c/Z)3 means per unit volume
so it is a number per unit volume (a number of photons per unit volume) like we said in the first place.
the challenge is often just to get comfortable with the large numbers
which is mainly a business of practice with examples
so like E33 length units is a handbreadth (3.2 inch or 8.1 cm)
and a convenient area is E66 (roughly palmsize)
so suppose we want to know the Casimir attraction force on a palmsize E66 area where the separation is E30
the main thing is just raise sep to the fourth and get the reciprocal E-120
then we divide by 24 and that's the casimir force!
actually it is the force per area, so we multiply by our sample area E66
and get
(1/24)E-54
and that is the force felt by the palmsize area, expressed in natural force unit. we're done!
but then how do you imagine this delicate force? Well I always relate forces to what I feel on the sole of my foot when I am standing quietly in the garden. On each foot I feel E-40 force
(traditionally it would be called a "hundredweight")
and a hundredth of that is E-42 and it's the weight of a pound of butter.
But everybody probably has their own way of connecting to the force scale.
Anyway that E-54 force you see up there, for me it would be a trillionth of the weight of a pound of butter. that is: E-12 x E-42 = E-54
And it is astonishing to me that they can measure forces that delicate in the lab. but they do. the casimir force e.g. between two gold spheres has been measured and found to agree with formula. humbling thought.
