QG units

[Martin]

the force F = c⁴/(8piG) is the main constant in Gen Rel. The constant in the Einstein equation is actually 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)

 

in keeping with the natural units idea we can also set Boltzmann k (the temperature energy ratio) and the electron charge equal one. 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.

 

As a convenience in trying out the units and getting some practice with them, here's a list of rough sizes of familiar things.

 

rough sizes:

q'ty expressed in QG     approximate size
E8 mass units            pound
E50 time units           year
E45 time units           4.5 minutes
E33 length               handbreadth (3.2 inch, 8.1 cm)
E34 length               pace
E37 length               half mile
E50 length               lightyear
E-9 speed               2/3 mph
E-7 speed               67 mph
E-6 speed               speed of sound (cold air)
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-29 temperature        average earth surface temp
E-31 temperature       cosmic microwave background
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, "hundredweight"
E-49 power             144 watt bulb
E-53 electmagn.field unit    tesla 
E-57 field unit               gauss

[J.C.MacSwell]

Let's call an E-31 black hole "Olber's Mass". If you look far enough into space, eventually you'll see one!

[Martin]

Let's call an E-31 black hole "Olber's Mass". If you look far enough into space, eventually you'll see one!

 

Hi JC, such a "olber" hole would certainly blend in with the background.

 

same temp as empty space so perfect camouflage

 

I have an organization problem. trying to think in what order to present things

[Martin]

the basic good thing about using any kind of natural units is clean formulas

here is a sampling of formulas.

in each case i am assuming that the calculation is done in natural units terms, so that I don't have to specify the units each time I say something.

 

1. for a satellite in circular orbit

mass = 4 x period x speed³

 

e.g. a planet's year is E50 and its speed is E-4 (both very like Earth's)

how massive is its star?

 

answer:

4 E(50-12) = 4E38 natural mass units

 

(this approximately what the sun's mass is. I often interpret natural mass units by thinking of E8 units (which comes to 434 grams) as a "pound-like" mass, so the mass of the sun calculated here is 4E30 "pounds" but that is just a way to get a handle on it that works for me)

 

 

2. for black hole radius, area, temperature, evaporation time

 

radius = (1/4pi) mass

area = (1/4pi) mass²

temp = mass^-1

evaporation time = (80/pi) mass³

 

3. radiant energy density and brightness

(energy per unit volume, power per unit area)

 

energy density = (pi²/15) temp⁴

brightness = (pi²/60) temp⁴

 

4. average photon energy

the famous Riemann zeta function makes a cameo appearance here.

3zeta(4)/zeta(3) = 2.701 tells the average thermal photon energy at some temp. Multiply the temperature by 2.701.

Once you have 2.701, can forget about Riemann zeta function unless math number theory interests.

 

avg photon energy = 2.701 temp

 

Since sun surface temp is 2E-28, the average sunlight photon has energy 5.402E-28.

 

Room temperature is 1.04E-29, so the average energy of a photon in the room with you right now is 2.8E-29

 

(In metric units the formula for the average photon energy at temp involves the same number 2.701 but also the Boltzmann constant k, which you probably have to look up.

metric formula:

avg. photon energy = 2.701 kT.

 

5. critical density of universe

(just multiply the square of the hubble parameter by 3)

H = (5/8)E-60

H² = (25/64)E-120

critical density = 3(25/64)E-120 = (75/64)E-120

It's the overall concentration of energy needed in the universe so that it can be spatially flat---too little makes negative curvature and too much makes positive curvature, either way triangles dont add up to 180 degrees--- and since it looks flat, folks think the actual density is at or close to critical.

This is where "0.83 joules per cubic km" comes from. It is just a metric translation of 1.2E-120

 

 

6. radian time in low orbit.

(time to go one radian, that is 1/2pi of full circle, lowest possible orbit)

radiantime² = 6/density

 

e.g. if density of planet is E-91 (similar to water) then square of radiantime is 6E91 = 60E90, so radiantime roughly 8E45 = 8 x 4.5 minutes.

 

e.g. if density of planet is 6E-91 (similar to Earth) then square of radiantime is E91 = 10E90, so radiantime roughly 3E45 = 3 x 4.5 minutes.

 

7. the heat capacity of water, per molecule

For the liquid, it is 9

So making some liquid water's temperature increase by E-30 takes an amount of energy equal to (the number of molecules) x 9E-30. The latent heat of vaporization is 1.7E-28 per molecule.

 

for a metallic solid, heat capacity is about 3 per atom

for a biatomic gas like air, 5/2 per molecule at constant volume, 7/2 per molecule at constant pressure

 

8. some 1/137 stuff

 

1/137 (more exactly 1/137.036...) is the coulomb constant. it tells the force between two charges separated by a distance. just multiply the charges by 1/137 and divide by the square of the distance.

 

1/137 also tells the force between parallel currents (measured on a test segment with length equal half the separation). just multiply the currents by 1/137

 

it also tells the strength of the magnetic field inside an empty coil where N is the turns per unit length and I is the current

field = (4pi/137)NI = (4pi) x current in coil x turns per unit length

 

(1/137)² tells the energy needed to ionize a hydrogen atom. multiply the rest energy of an electron (2.1E-22) by it and you get a quantity of energy called the Hartree----which is twice the ionization energy (so you still need to divide by two)

[Kygron]

Why would I want to use these units when "my" units work just as well?

[Martin]

Why would I want to use these units when "my" units work just as well?

 

Hi Kygron, i was hoping you would visit this thread. I like your sig----we could think that more.

 

I have no reason to argue about preferences of that sort. If you want to only use one system that is fine.

For me personally it has been helpful to be able to look at nature thru several sets of units (both the more manmade kind and the more natural or intrinsic kind)

 

I havent gotten started with this thread. i need to figure out how to begin it. or to organize it---but maybe no organization is best just getting started with some examples

 

I dont want to seem to dismiss your question. You understand I can only give my person opinion. I am very interested in QG (I think it is the leading field in theoretical physics where the most is happening) and I see QG research papers using these natural units just based on the main fundamental constants (no arbitrary chosen human or earth stuff). And they do it for practical reasons. It is really sweet when c, hbar, 8piG, k are all one. the usual arithmetic goes away and equations get very clean and it is easier to see relationships.

 

So a set of units is like lenses to see nature with and natural units feel more like contact lenses, or maybe seeing 20-20 not needing any lenses.

 

or a set of units is like the clothes that nature wears and natural units is more like very light casual clothes or no clothes. instead of heavy woolen tweed

 

but that is just how it turns out to be for me personally. You might find using these units very awkward. I want to find some examples for you to try so you can see how they feel to you.

[Martin]

BTW Kygron do you have some favorite topics to think about like

 

celestial mechanics (planets, orbits, masses...)

temperature (radiant heat, heat capacity, speed of sound, convection...)

electricity

colors of light

black holes (temperature, evaporation time, ...)

[Kygron]

If you've just gotten started then it may not be a good idea to jump into personal info

 

My question was a direct answer to my sig, as you may have guessed. I personally had no clue as to what all that math was that you posted, I just tryed to look through it and assume it was usefull to someone who knows those topics.

[Martin]

If you've just gotten started then it may not be a good idea to jump into personal info

...

 

I was just about to turn in for the night and got an idea of something to tell you. It is personal' date=' expect that is just how I think (at some level)

 

You and I live on sunlight, tomorrow I will go out into my garden and all this really bright sunlight will be pouring down and it is because the sun is a certain temperature. hot things glow.

 

and the stefanboltzmann constant is the deep universal proportion in nature that connects their temperature to the POWER PER UNIT AREA or brightness, of their glow.

 

Now in clean units, the stefanboltzmann is just pi^2/60

 

but in metric it is a complicated thing which you can find at official websites

it looks like

oh heck I dont want to type it all out. wait. I will

 

[math']\text{stefboltz sigma} = \frac{\pi^2}{60}\frac{k^4}{\hbar^3 c^2}[/math]

 

and in the metric system you plug in all these values which you may well have to look up in some book and raise to powers and multiply out and you get some number (which I forget) of watts per square meter per kelvin-fourth

 

the pi^2/60 is what nature put in there, and the rest is constants, conversion factors, that the metric system needs to connect temp to energy

and energy to power and light

 

so what. so I want to look at the sun and be cognizant of its temperature (by the color of its light) and have an immediate grasp of how that temperature is translated into brightness.

 

so I look at nature in a set of units where speed of light is one, and hbar is one, and the stefboltz sigma is what nature made it namely

 

[math]\text{stefboltz sigma} = \frac{\pi^2}{60}[/math]

 

well that is one reason, it is kind of stripped down, it streamlines things

[Martin]

Kygron, what I am telling you about is an aesthetic prejudice

you dont have to share it but I suspect you understand it.

 

You and I are made of atoms

the most basic thing about an atom, or a molecule, is Coulomb constant which tells how the attraction between two charges depends on distance

 

in metric coulomb is made of the elementary charge e and the EM coupling constant alpha (about 1/137) which is the basic QED constant

 

 

[math]\text{coulomb const} = \frac{1}{137.036..}\frac{\hbar c}{e^2} = \frac{1}{4\pi \epsilon_0}[/math]

 

none of that matters, it is just metric system conventions, except for the alpha, except for the 1/137. if you look up metric values of hbar and c and electron charge e and multiply out you will get the conventional metric value of the coulomb constant. but it is a damned cluttered nuisance

 

If you do natural units then the value of the coulomb constant is just alpha namely about 1/137 because of course the e and hbar and c value one, and dont enter in the arithmetic.

 

[math]\text{natural units coulomb constant} = \frac{1}{137.036..}[/math]

[Martin]

so I like using natural units for several reasons

one is that i learn things about nature by using them

and one is it puts my mind into closer, less obstructed, contact

with basic things like what are we made of

what we live on

why the wind just came up, in the trees outside

what our planet is doing

 

my chorus gave a concert today! we sang brahms songs (liebeslieder waltzes, do you happen to know any choral music you like?) must turn in for the night now

[Kygron]

Well, I agree that it looks pretty. Easier to remember too. But are you leaving out info? Is the fact that the constant is based on c important?

 

Why have a "physics" type formula that doesn't use "physics" type values? If all constants are left out and all you're left with is a number, why not just set that number to 1 as well and call it a natural unit?

[Martin]

I dont have names for the natural units. so they got left out in the previous. here is what it would look like with placeholder symbols for the units. I am trying to say 1/137 "force unit lengthsquared per chargesquared unit"

 

[math]\text{natural units coulomb constant} = \frac{1}{137.036..}\mathbb{F L}^2 /\mathbb{C}^2}[/math]

 

Same thing for Stefan-Boltzmann sigma: "natural power per area per temp-fourth unit"

[math]\text{natural units stefboltz sigma} = \frac{\pi^2}{60}\mathbb{P L}^{-2}\mathcal{T}^{-4}[/math]

 

when i put in the units, then nothing is left out

the natural units of time, length, force, mass are determined uniquely as the system in which:

 

c (speed of light in vacuum) is the unit of speed

 

hbar is the unit of angular momentum

 

c^4/(8p G), the coefficient in the einstein eqn of Gen Rel, is the unit of force

 

once one has adjusted the units to make these basic things have a value of one, then there is no more room to adjust the units. So one cannot "set anything" further equal to one.

 

having defined the units, by setting the values of these universal constants equal to one, we just need to explore the sizes of the units and learn how to say ordinary things in those terms. these are close cousins of "Planck" units like "Planck length", "Planck time", mass, energy etc.

so the length unit is very small, even as Planck length is small.

 

there is a tradeoff: calculating can be very easy because many of the most often used constants have value one

but getting used to the sizes of the units can take a while because many of them are extreme small or large

 

that is why I have listed some sizes, in that codebox at the beginning:

a half mile is E37 length units!

 

The half mile is a lot of length units because the length unit is very small!

 

It is small (by our human standards) because there is only one thing it can be if you adjust the units so that this bunch of nature's main constants all have value one, and that happens to be small compared to human scale.

[Martin]

the force F = c⁴/(8piG) is the main constant in Gen Rel. The constant in the Einstein equation is actually 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)

 

in keeping with the natural units idea we can also set Boltzmann k (the temperature energy ratio) and the electron charge equal one. 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...

 

might as well get started. my attitude is if it isnt fun for you dont bother, just use whatever units you are used to. we can start with temperature

the natural scale essentially says zero is absolute zero and one is the temperature of the big bang, the maximum imaginable temperature

 

going down from there

big bang (putative) 1

core of sun 5E-25

sun surface 2E-28

tungsten filament in 100watt litebulb E-28

body temp 1.1E-29

common outdoors (49Fahrenheit) temp E-29

 

the temp scale is not arbitrary. the temp is immediately informative in the sense that it tells you right away something about the color of the light that a hot thing glows, the temperature IS in a certain sense the energy of a typical photon in the thermal glow at that temperature

 

the blackbody spectrum is a skewed distribution so we have to say what we mean but roughly speaking since the sun surface temp is 2E-28

a fairly typical energy for a photon in sunlight to have is 2E-28 energy units.

 

Actually the visible range of photon energies is 7E-28 (red) up to 13E-28(violet) and a typical energy for a green photon is right in the middle of the visible range 10E-28

 

that means that most of the photons in sunlight are infrared warmth that we dont see. for our eyes to see them they have to be up in the high end of the distribution. but anyway the surface temp 2E-28 tells you right away something about the light

 

and the photon energy tells you right away about the wavelength

a green photon, for instance, energy 10E-28 = E-27, must have wavelength E27!

 

Now the breadth of my hand is E33 natural units. So I have a "handle" on the light of the sun.

I go out this morning, brilliant sunlight, I can see by its color (more or less) that the temperature of the thing in the sky is 2E-28

and all around me there is this color green 10E-28 = E-27

and I hold out my hand E33

and think about the size of a wavelength being E27 which is a millionth of E33, my handwidth.

 

And I want to understand the brightness of the thing in the sky so I square its temperature twice to get the fourth power of temp = 16E-112

(everybody all right that E-28 to the fourth is E-112?)

and multiply by pi-square/60, which is about 1/6 (since pi-square about 10)

so the brightness of sun surface, the natural units of power radiating per unit area is (16/6) times E-112

So I can understand the brightness of the sunlight around me on the ground by how much that has been spread out.

Our distance from the sun making a ball with so much more area than the surface of the sun.

 

Actually doing the arithmetic with the area is maybe not so important. We could do it and find out the brightness of sunlight (natural power units per unit area) here at earth. main thing is we have a direct handle on it.