Appendix 2: 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.

In physics we have a lot of different types of units. We have joules for energy, watts for power, coulombs for charge, newtons for force, etc. However, there are only three fundamental units in classical physics: length, mass, and time. Normally, we measure these with meters, kilograms, and seconds (MKS) or centimeters, grams, and seconds (CGS). In the US, we also use feet, pounds, and seconds. All other units can be expressed as combinations of these. For example, the newton is a kilogram-meter per second2. The joule is a kilogram meter2 per second2.

Is this how the universe is actually structured? Are there actually fundamental distances, masses, and times? Are these units actually different?

It's purely an accident of chemistry and biology that we see time and space as so different. The chemical processes that our brains use to function work in milleseconds, so a natural time unit to a person is on the order of a second. But, if we had thought processes based on nuclear reactions, a natural time unit for a person would be a billion billion times smaller.

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. We have not yet decided on a unit for length, we've just decided at this point that we can use the same unit for spacial distance and for time.

Distance is 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."

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 or joules or horsepower-hours. We can choose whatever unit we wish to represent energy and mass.

From quantum mechanics we learn that everything oscillates with a frequency n = 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. We could choose meters for our basic unit. We could measure time in meters - a meter of time is about 3 nanoseconds. We could measure mass in inverse meters or inverse seconds. A kilogram has a frequency of about 1.5*1033 hertz (this number is just 1 / h), or a wavelength of about 2*10-42 meters (this is h/c).

Mass 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 a = e2/4p 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/r2 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 units," 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 Ö(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) = Ö(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 2p and such in various places. For example, our Lagrangian is now in units of mass or inverse meters, as we expect for an energy term. The time integral of the Lagrangian, the action, is dimensionless, as we expect. That is, since dt has units of length, mass * dt has units of length / length. We'll use the action to find the phase of a particle as phase = exp( i 2p ò L dt t ). We could have scaled our units to eliminate this 2p, 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 seconds1.346*10-43 seconds
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*108 cm/second3*1010 meters/second
aAcceleration1 stone2.23*1053 cm/second22.23*1051 meters/second2
FForce1 stone24.9*1017 dynes4.9*1012 Newtons
pPressure1 stone43.38*1079 dynes/cm23.38*1070 Newtons/meter2
dMass Density1 stone48.28*1092 gm/cm38.28*1083 kg/meter3

Conversion Factors from cgs to Natural Units
mMass1 gram1.835*104 stone
lLength1 centimeter2.477*1032 planck
lLength1 Ångstrom4.037*1012 planck
tTime1 second7.43*10-42 planck
EEnergy1 erg2.04*10-17 stone
EEnergy1 MeV3.27*10-23 stone
VVolume1 cm31.52*1097 planck3
vVelocity1 cm/second3.33*10-9
aAcceleration1 cm/second24.45*10-54 stone
FForce1 dyne2.04*10-18 stone2
pPressure1 dyne/cm22.96*10-80 stone4
dMass Density1 gm/cm31.21*10-92stone4

Conversion Factors from MKS to Natural Units
mMass1 kilogram1.835*107 stone
lLength1 meter2.477*1034 planck
lLength1 Ångstrom4.037*1012 planck
tTime1 second7.43*10-42 planck
EEnergy1 joule2.04*10-10 stone
EEnergy1 MeV3.27*10-23 stone
VVolume1 m31.52*10103 planck3
vVelocity1 m/second3.33*10-11
aAcceleration1 m/second24.45*10-52 stone
FForce1 newton2.04*10-13 stone2
pPressure1 newton/m22.96*10-71 stone4
dMass Density1 kg/m31.21*10-83stone4

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 / 2p me c3.86*10-13 m = 3.86*10-3 Å9.56*1021 planck
Bohr radius h2 / 4p2 me e25.29*10-11 m = .529 Å1.31*1024 planck
Rydberg constant ½ me c2 a213.6 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 earth6371 km1.578*1041 planck
radius of sun6.96*108 m1.724*1043 planck
mean orbital radius of earth1 AU = 1.495*1011 m3.703*1045 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

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Copyright © 2002 Mark Lawrence. All rights reserved. Reproduction is strictly prohibited.