Quantum Relativity

Physical Units

By Mark Lawrence

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We're used to measuring things with a lot of different types of units. In the US, we measure small things in inches, medium things in feet, and large things in miles. We weigh things in pounds. Time is measured in seconds, minutes, and hours. Some things are measured in combinations, like miles per hour. And, there's all sorts of units that don't play into our daily lives very often for things like forces, power, torque, etc. In physics, we deal with all of these units on a routine basis, and more.

Most of these units were chosen for historical reasons, many of which have since been forgotten. Feet were originally the length of a man's foot. Seconds were probably chosen by the average heart rate of a man. A minute is the time it takes for the sun or the moon to rise or set. Miles were originally 1000 paces of a Roman centurion. So, this is all very interesting for walking around Europe or watching sunsets, but it really doesn't have much to do with the fundamental laws of physics.

It's purely an accident of chemistry and biology that we see time and space as so different. Using the speed of light, c, as a conversion constant, we can measure distance in seconds or time in feet. One light second is about 186,000 miles. One foot of time is about one nanosecond. So, we agree to set c equal to 1, and we agree that we will use the same measure for time and space.

Einstein's special theory of relativity tells us that E = Mc2. But, we have just agreed that c is one, so energy and mass have the same units, pounds or kilograms or horsepower-hours, we can choose whatever we wish.

From quantum mechanics we learn that everything oscillates with a frequency η = E / h, where h is Planck's constant. This is completely fundamental - all energy oscillates, whether it's a photon or a bowling ball. It's just an accident of history and human perception that we choose to measure energy or mass with a different scale than time or distance. So, we'll agree to measure energy and mass using units of frequency, meaning "per second" or "per meter." Now, having made this agreement, Planck's constant is 1. We still haven't chosen a basic unit, but we have reduced most everything to this one unit.

Distance is now measured in meters. Velocity, distance per time, is meters per meter, so velocity has no dimensions. In our system of measurement, a velocity of one is the speed of light. The speed limit on most freeways is 65 miles per hour which equals about c / 10,000,000. If physicists were running the highways, apparently highway signs would say "Speed limit 10-7." That's it, no units.

Acceleration is velocity per second, so acceleration has dimensions of "per meter." Mass also has dimensions of "per meter," so F = Ma tells us that force has dimensions of "per meter2." Gauss' law of static electricity tells us that F = e2 / r2, so e, the electric charge, is dimensionless. The fine structure constant α = e 2/4π is also dimensionless.

Finally, we have one more fundamental unit in nature: G, Newton's gravitational constant. The units of G can be deduced from Newton's equation of gravity, F = GmM / r2. Since the force has dimensions of "per meter2," and the 1/r 2 term has dimensions of "per meter2," we see that GmM has no dimension. Therefore, since Mm has dimensions of "per meter2," G must have dimensions of "meter2." Now, our big leap: we'll set G to one, and therefore start using the same dimensions that the universe naturally uses - we'll call them "natural dimensions," sometimes referred to as "God's units."

In cgs (centimeter-gram-seconds) units, G = 2/3*10-7 cm3 / g-s2. Thus, we see that G / c3 = 1/4 * 10-38 s/g. Now, we multiply by h = 6.6*10-27 erg-sec = 6.6*10-27 g-cm2 /sec and we get 1.63 * 10-65 cm2. Finally, the square root of this number is √(hG/c3) = 4.04*10-33 cm. This will be our fundamental unit of length and time, which we will call the Planck, abbreviated as P. We can live with just one fundamental unit, but for convenience sake we will define one additional unit. Our mass and energy unit will be h / (c * Planck) = √(hc/G) = 5.45*10-5 g, which will call the Stone, abbreviated as E. Note that the Stone is simply 1 / Planck. Wherever we use Stone, we could write Planck-1.

Now we're left with just a few factors of 2π and such in various places. For example, our Lagrangian is now in units of mass, as we expect for an energy term, so the time integral of the Lagrangian, the action, is dimensionless, as we expect. We'll use the action to find the phase of a particle as phase = exp( i 2π S t ). We could have scaled our units to eliminate this 2π, but we prefer to leave it in as an explicit reminder of the difference between time and radians.

Below is a conversion table and a list of constants. This is enough in most cases to work real problems in natural units and get answers in MKS.

Conversion Factors from Natural Units
mMass1 stone5.45*10-5 grams5.45*10-8 kilograms
lLength1 planck4.037*10-33 centimeters4.037*10-35 meters
lLength1 planck4.037*10-25 Ångstrom4.265*10-51 light-years
tTime1 planck1.346*10-43 seconds3.74*10-47 hours
tTime1 planck1.558*10-48 days4.265*10-51 years
EEnergy1 stone4.9*1016 ergs4.9*109 joules
EEnergy1 stone3.06*1022 MeV3.55*1032 °K
VVolume1 planck36.58*10-98 cm36.58*10-104 meters3
vVelocity13*1010 cm/second3*108 meters/second
aAcceleration1 stone2.23*1053 cm/second22.23*1051 meters/second2
aAcceleration1 stone 2.27*1050 g
aTidal Acceleration1 stone25.52*1085 /second25.62*1084 g/meter
FForce1 stone24.9*1017 dynes8.22*10-45 Newtons
pPressure1 stone43.38*1079 dynes/cm23.38*1070 Newtons/meter2
dMass Density1 stone48.28*1092 gm/cm38.28*1083 kg/meter3

Physical Constants
ConstantMKS valueNatural value
G6.673*10-11 N m2 / kg21 stone2
e1.602*10-19 C.303
c3*1010 cm / sec1
h6.62607544*10-34 J s1
hc1.9856*10-23 kg m3 / s21
Boltzman's constant k1.38*10-16 ergs / °K2.82*10-33 stone / °K
me electron mass9.1096*10-31 kg = .511 MeV1.67*10-23 stone
mp proton mass1.6725*10-27 kg = 938.3 MeV3.066*10-20 stone
mn neutron mass1.6748*10-27 kg = 939.6 MeV3.07*10-20 stone
Compton wavelength h / 2π me c3.86*10-13 m = 3.86*10-3 Å9.56*1021 planck
Bohr radius h2 / 4π2 me e25.29*10-11 m = .529 Å1.31*1024 planck
Rydberg constant ½ me c2 α213.6 eV4.44*10-28 stone
mass of sun1.987*1030 kg3.646*1037 stone
mass of earth5.97*1024 kg1.095*1032 stone
mass of moon7.32*1022 kg1.343*1030 stone
radius of sun6.96*108 m1.724*1043 planck
radius of earth6371 km1.578*1041 planck
radius of moon1.7375*106 m4.304*1040 planck
mean orbital radius of earth1 AU = 1.495*1011 m3.703*1045 planck
mean orbital radius of moon3.844*108 m9.522*1042 planck
year3.156*107 s2.345*1050 planck
g earth9.8 m / s2 = G M / r24.35*10-51 stone
g sun273.4 m / s2 = G M / r21.23*10-49 stone
Schwarzchild radius of earth8.85 mm = 2GM / c22.19*1032 planck = 2*Mearth
Schwarzchild radius of sun2944 m = 2GM / c27.292*1037 planck

The following data is from NASA/JPL

Orbital and Physical Data for the Planets
PlanetMean radius
x1023 kg
Rotation Period
Sidereal h
m s-2
Escape Velocity
Orbit Period
Sidereal yr
Orbit Radius
Mercury 2440 3.30 5.4 1408 3.7 4.4 0.241 0.3871 0.2056
Venus 6052 48.7 5.2 -5832 8.87 10.4 0.615 0.7233 0.0068
Earth 6371 59.7 5.5 23.93 9.78 11.2 1.00 1.0000 0.0167
Mars 3390 6.42 3.9 24.6 3.69 5.03 1.88 1.5237 0.0934
Jupiter 69911 19000 1.3 9.92 23.1 59.5 11.86 5.2034 0.0484
Saturn 58232 5680 0.69 10.66 8.96 35.5 29.45 9.5371 0.0542
Uranus 25362 868 1.3 17.24 8.69 21.3 84.02 19.191 0.0472
Neptune 24624 1020 1.6 16.11 11.0 23.5 164.8 30.069 0.0086
Pluto 1151 .131 2.0 153 0.655 1.3 248 39.482 0.2488

Orbital And Physical Data for the Major Satellites
SatelliteMean Radius
x1020 kg
Orbit Radius
Moon 1737.5 732 3.34 27.322 384400. 0.0554
Phobos 11.1 0.107E-3 1.87 0.319 9380. 0.0151
Deimos 6.2 22.4E-6 2.25 1.262 23460. 0.0002
Io 1827 890 3.53 1.624 421800. 0.0041
Europa 1561 478 3.01 3.551 671100. 0.0094
Ganymede 2631 1476 1.94 7.155 1070400. 0.0011
Callisto 2410 1072 1.834 16.69 1882700. 0.0074
Mimas 198.6 0.381 1.17 0.942 185600. 0.0206
Enceladus 249.4 1.04 1.60 1.370 238100. 0.0001
Tethys 529.9 6.15 0.991 1.888 294700. 0.0001
Dione 559. 10.9 1.49 2.737 377400. 0.0002
Rhea 764. 23.1 1.24 4.518 527100. 0.0009
Titan 2575. 1340. 1.88 15.95 1221900. 0.0288
Iapetus 718. 19.4 1.25 79.33 3560800. 0.0284
Ariel 578.9 13.4 1.67 2.520 190900. 0.0012
Umbriel 584.7 11.6 1.40 4.144 266000. 0.0039
Titania 788.9 35.1 1.71 8.706 436300. 0.0011
Oberon 761.4 30.0 1.63 13.46 583500. 0.0014
Miranda 235.8 0.657 1.20 1.413 129900. 0.0013
Triton 1353 213 2.06 5.877 354800. 0.0000
Nereid 170. 0.308 1.5 360.14 5513400. 0.7512
Proteus 210. 0.502 1.3 1.122 117647. 0.0005
Charon 593. 16.1108. 1.853 6.387 19410. 0.0002