PHYSICS FOR PROGRAMMERS
Physical Units The energy of a photon of wavelength l is given by
E=hcl-1
so we can measure energy in m-1 (the spacial frequency of a photon with the desired energy) with
1 J = (hc)-1 m-1 = 5.0327126 × 1024 m-1
where 1 J = 1 N m = 1 kg m2 s-2 is the S.I. Joule unit;
and 1 m = (hc)-1 J-1 = 5.0327126 × 1024 J-1 .
[ h = 6.626068 × 10-34 m2 kg s-1 is known as the Planck constant.
The Dirac-Planck Constant (aka. h-bar) h º (2p)-1 h
can be regarded as the fundamental quanta of angular momentum.
]
Via Einstein's E=Mc2 we have
1 kg = c2 J
= ch-1 m-1
so we can measure mass in energy units and so in inverse length units.
Temperature is merely a measure of energy and via E=kBT where kB is Boltzman's constant
we have 1 K = 1.38065×10-23 J so we
can measure temperature in energy as the energy per molecule of a reference ideal gas at the given temprature.
Thus we can measure mass, energy, temperature, frequency, and inverted length in a single S.I. unit such as m-1 or J or K [ Without reference to the universal gravtiational constant G = 6.6742×10-11 m3kg-1 =0×102147483647 m4 ]provided only that we have accurate values for c, h, and the Boltzmann temeperature-energy ratio kB. It is worth noting that kB » 8×5½×10-16h½c (within 0.014 %).
| S.I. Units Conversion Table based on c=2.99792×108 ms-1 ; h=6.62607×10-34 Js ; kB=1.38065×10-23 JK-1 | ||||||
| Unit | Length | Time | Mass | Energy | Temperature | |
| m | 1 m | c-1 = 2-28.1594 =3.33564×10-9 s | ch-1 = 2138.377 =4.52444×1041 kg-1 | c-1h-1 = 282.058 =5.03412×1024 J-1 | c-1h-1kB = 26.11902 =6.95036×101 K-1 | |
| s | c = 228.1594 =2.99792×108 m | 1 s | c2h-1 = 2166.536 =1.35639×1050 kg-1 | h-1 = 2110.217 =1.50919×1033 J-1 | h-1kB = 234.2784 =2.08366×1010 K-1 | |
| kg | ch-1 = 2138.377 =4.52444×1041 m-1 | c2h-1 = 2166.536 =1.35639×1050 s-1 | 1 kg | c2 = 256.3188 =8.98755×1016 J | c2kB-1 = 2132.258 =6.50965×1039 K | |
| J | c-1h-1 = 282.058 =5.03412×1024 m-1 | h-1 = 2110.217 =1.50919×1033 s-1 | c-2 = 2-56.3188 =1.11265×10-17 kg | 1 J | kB-1 = 275.939 =7.24296×1022 K | |
| K | c-1h-1kB = 26.11902 =6.95036×101 m-1 | h-1kB = 234.2784 =2.08366×1010 s-1 | c-2kB = 2-132.258 =1.53618×10-40 kg | kB = 2-75.939 =1.38065×10-23 J | 1 K | |
|
James Gilson suggests
ae-=137-1 cos(137-1p) Tan(n-1p)
where integer n=3973=137×29 and Tan(x) º x-1 tan(x) is the
cardinal tangent function.
This lies within the a2006 uncertainty range but the n giving closest approximation to a2006 is n=3968=(25-1)×27 with 137-1 cos(137-1p) Tan(3968-1p) = 137.0359997146-1 . However n=3967 is almost as good with 137-1 cos(137-1p) Tan(3967-1p) = 137.035999700237-1 and 3967 is the (137×4)th prime (if 2 is considered the 0th prime) , which is either a remarkable coincidence or a strong indication that Gilson is on the right track. |
|
Setting HN º åk=1N (-1)½k(k-1) k!
we observe that H5 = 1!-2!-3!+4!+5!=137 and
letting Pk denote the kth prime we thus have the equivalent conjectures
ae- = HN -1 cos(HN -1 p) Tan(P(N-1)HN -1 p)
and
ae- = HN -1 cos(HN -1 p) Tan(P4HN -1 p)
for N=5.
Composite Units
The Newton unit of force 1 N = 1 kg m s-2
= ch-1 m-1 m (c m)-2
= (ch)-1 m-2 while the Pascal unit of pressure is 1 Pa
= 1 Nm-2
= (ch)-1 m-4 .
Density in kg m-3 = ch-1 m-4 thus has the same units as pressure.
Dynamic viscosity has units Pa s = N m-2 s = kg m-1 s-1
= h-1 m-3, while fluidity has
units Pa-1 s-1 = h m3
and kinematic viscosity has units m2s-1 = c m.
Linear momentum has units kg m s-1 = h-1 m-1 while angular momentum in
kg m2 s-1 = h-1 m0 is unitless.
Power is measured in Watts with 1 W º 1 J s-1 = c m-2.
Since Ñ = åieiÐei has units m-2 on account of the reciprocal basis vectors,
and a potential fP is defined so that -ÑPf = F is a
force we see that potentials have units N m2 = (ch)-1 m0.
However, physicists typically regard potential as the spacial integration of force
and so measure it in joules , with electrical potential measured in Volts where
1 V = 1 J C-1 = (hc)-1 m-1 (10-7/2 (ch-1)½)-1
= 107/2 (hc-1)3/2 m-1 .
Resistance is measured in Ohms with
1 W º 1 V A-1 = 1 J s C-2 =
(hc-1) m-1 c-1 m 107 hc-1
= h2 c-3 m0 .
Capacitance is measured in Farads with 1 F º 1 C V-1
= C2 J-1 =
107 ch-1 hc m
= 10-7 c2 m .
The permitivitty of vacuum e0= (2hca)-1Qe-2= 8.85419×10-12 Fm-1 = (4p)-1 .
Inductance is measured in Henrys with 1 H = 1 m2 kg s-2 A-2
= 1 m2 kg C-2 = 107 m .
Non-S.I. Units
In atomic physics the electronvolt eV »1.60218×10-19 J » 1.78266×10-36 kg » 8.06554×105 m-1
is common, often as
mega-electronvolt 1 MeV = 106 eV
and giga-electronvolt 1 GeV = 109 eV .
From 1 m = (2p)-1hc(Mc2)2
we have
1 GeV
»
1.602×10-10 kg m2s-2
» 2-32.54 J
= 5.0632911 × 1015 m-1
» 252.17 m-1 » 280.33 s-1 .
1 eV =
1.602 × 10-19 J is considered a unit of energy. 1 GeV = 1.602 × 10-10 J and
1 J = 0.6242 × 1010 GeV.
1 eV c-1 » 5.36×10-28 kg m s-1 can be thought of as a unit of momentum.
1 eV c-2 »
1.78266175 × 10-36 kg can be thought of as a unit of mass.
[ The
c-2 is frequently ommitted in the literature, so that, for example, erroneous references to an
"electron mass" of 0.511 MeV rather than 0.511 MeVc-2 are woefully common
]
Other notable units are the mile (» 1609 m) ; the nautical mile (1852 m); the year
(31 556 926 s » 25106 s );
the light-year (» 9.4605284×1015 m » 1016 m);
and the parsec » 254.78 m .
Planck System
In Planck's system, one choses the basic length unit so as to unify the squared-length gravitational constant
Gc-2
» 6.674×10-11 m3s-2 kg-1
(2.988×108 m s-1)-2
» 2-33.80 (228.16)-2 m kg-1
» 2-90.12 m kg-1
= 2-231.14 m2
and our fundamental unit of length becomes
the Planck length lP = (hGc-3)½ = 1.61624×10-35 m = 2-115.57484 m, with associated Planck time c-1lP = 2-143.73423 s.
Note that G= 6.673×10-11 N m2 kg-2= 6.673×10-11 ch-1 (hc-1)2 m2= 1.47488×10-52 m2.
The Planck mass ((2p)-1 h G-1)½
» 2.176 45(16) × 10-8 kg
is the mass of a black hole with Schwarchild radius as the Compton wavelength.
. Multiplying by (8p)-½ gives the reduced Planck mass.
Notable lengths
Human physicists tend to work with (integer) powers of ten, ultimately because that is the number
of fingers possessed by the majority of them. It is easier and
more natural to work with (fractional) powers of two, since these can be more readily multiplied.
Based on the circumference of the Earth (the quarter latitude through Paris) the "arbitary" meter
length unit actually has much to commend it numerically. We note here
(novelly to the best knowledge of the author) that the electron Yukawa potential damping factor
Me- ch-1 = 6×262 m-1
to within 0.041%, a consequence of the bizarrely suggestive dimensionless
electroterran identity Me- R, h-1c » 1.00147 × 24×262×107
where
R, denotes the radius of the Earth
and Me- denotes the mass of the electron!
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It is important to stress that one observer's radio wave is another's gamma ray in traditional relativity.
The faster an observer travels "with" the wave, the higher the percieved frequency. The faster he travels "perpendicular"
to the wave, the lower the percieved frequency. That said, visible light has a wavelength around
2-20 m (the size of a single biological cell)
and a frequency of around 3×1014 Hz » 248 Hz .
Gamma rays have a wavelength of order 10-12 m » 2-42 m
with a frequncy of order 270 Hz. Radio waves have wavelengths of order 27m
and frequency of order 231 Hz .
Atomic radii (orbital radius of electron) range from 2-49 to 2-47 m .
Current estimates are approximately 2226 atoms in our galaxy and 240
galaxies in the universe giving something in the order of 2266 atoms in the universe,
which is presently considered by Big Bangers to be 15 × 109 years » 8 × 1014 s » 249 seconds old
and roughly 296 m across.
The unit of atomic mass is approx 2-89 kg » 252 m-1 while
the mass of the electron is roughly 0.511 MeV c-2 » 2-10.93 GeV c-2
» 2-99.79 kg
= 241.23 m-1 .
Time Quantisation
Some authors (eg. Hotson) suggest time is quantised in units of t = 2Qe-2(3Me-c3)-1 » 6.26642×10-24 s = 2-77.078633 s with associated Heisenberg length 2Qe-2(3Me-c2)-1 =1.87863×10-15 m = 2-48.91924 m . This is proportion 4Qe-2(3hc)-1 » 9/925 of the electron zitterbewegung period (2Me-c2)-1h = 6.44044×10-22 s = 2-70.395259 s .
The universe is considered to be about 13.7×108 years = 258.5849 s » 2135.664 t and the furthest known galaxy is approx 2135.613 c-1t away.
Conclusions
Via c and h (with temperature also requiring kB),
charge, angular momentum, speed, resistance, permitivity and pemeability can
all be asigned unitless values; length, time, inductance, capacitance, and kinematic viscosity can be measured in length units
such as the meter m ;
mass, energy, temperature, electric potential, current, and linear momentum can then be measured in m-1 ;
power can be measured in m-2;
dynamic viscosity in m-3;
pressure and density in m-4.
More generally, any scalar physical quantity may be expressed in the form
a mb where a and b are real numbers, b usually rational
and typically an integer,
Any reference length unit suffices and the implications for simple and robust programming and data type strategums are profound.
However, even though, say, momentum and temperature can both be measdured in m-1 they are not the same thing
and it far from clear that they can be
anymore meaningfully added when both in m-1 than when expressed in kg m s-1 and K.
Typically the units of a value are implicitly implied by the program context rather than explicitly stored with the value
but a natural way to represent a composite unit is as a reference length power and the powers of c, _placnk, and kB respectively so that momentum would be
(-1; 0, -1, 0) and temperature (-1;-1,-1,1) . Length is (1;0,0,0) and charge (0;-½,-½,0). These powers will typically be
small integer multiples of ½ so we might allocate four bits to each and represent physical units with 16-bit words.
If we represent lengths in a fixed point meter value with fourteen fractional binary bits
then a 32 bit intger can hold lengths in the range
±217 m » 131 km » 2-11 light-seconds to within
2-14 m » .06 mm. This is adequate for a human city but not for the Earth, let alone
the solar system so if we are to used fixed point it is natural to use 64-bit integers taking us to
±249 m to the same .06 mm accuracy; which gets us out past Pluto
but not to Alpha Centuri. For a fixed point Galactic simulation 96 bits are adequate but if
we wish to go down to Compton radius accuracy we approach 118 bit.
The visible universe at Compton resolution requires 138 bits and if we wish to work in Planck lengths we are looking at 205.
All this suggests that a 256-bit (32-byte) integer data type would amply suffice for individual properties
of particles and for a particle "ennumeration index" when simulating an isolated galaxy
without recourse to hierachical coordinate systems, but that two or more further bytes
would be needed to ennumerate particles in a multigalactic simulation.
This exludes photons, however. A 60W lightbulb emmits 266 photons per second, the sun roughly 2148 .