Do The Math: Inferred Planck Unit Values Have No Uncertainties And One
Big G Value Forces Multiple Numerical Coincidences Among Constants
If ten cars in a pileup lose their wheels, remounting forty wheels
until all is right instead of the ten sets theory predicts is improper.
It is just as improper to trust the rules of algebra enough to infer
Planck mass [PM], Planck length [PL] and Planck time [PT] using velocity
of light [c], unit of action [hB] and gravitational constant [G] and to
then give uncertainties to the inferred. Uncertainty in experimentally
determined [G] means a choice among matched sets. In numerically listed
products of constants and maps of the networks of quantity calculations
interrelating them, only one matched set transforms a pervasive pattern
of close near misses into a pattern of exact numerical coincidences.
Prove these assertions by calculating three consistent sets with the
high, the low and the median CODATA 2022 [G] value of 6.67430(15)e-11,
by comparing the inconsistency among recommended Planck unit values to
the orders of magnitude less inconsistency among atomic unit values and
by examining the consequences of reality's one correct matched set.
CODATA 2022 recommended values of both [c] and [hB] are defined by
convention to be exact at 299792458 and 1.054571817e-34 and
[PM]^2 = [hB][c]/[G], [PL]^2 = [hB][G]/[c^3] and [PT]^2 = [hB][G]/[c^5].
So, {[PM]^2}[G] = unadjustable [hB][c], I
{[PL]^2}/[G] = unadjustable [hB]/[c^3] II
and {[PT]^2}/[G] = unadjustable [hB]/[c^5]. III
[I x II]^[.5] yields [PM][PL] = unadjustable [hB]/[c], IV
[I x III]^[.5] yields [PM][PT] = unadjustable [hB]/[c^2], V
[II x III]^[.5] yields [PL][PT]/[G] = unadjustable [hB]/[c^4], VI
[I / II]^[.5] yields [G][PM]/[PL] = unadjustable [c^2], VII
[I / III]^[.5] yields [G][PM]/[PT] = unadjustable [c^3] and VIII
[II / III]^[.5] yields [PL]/[PT] = unadjustable [c]. IX
Rules I - IX and CODATA 2022 medium [G] = 6.67430e-11 force a set of
Planck unit values consistent to the limits of the calculator employed:
[PM] = {[hB][c]/[G]}^[.5] = 2.1764343420511266686458767604584e-8,
[PL] = {[G][hB]/[c^3]}^[.5] = 1.6162550239285500506848148019963e-35,
[PT] = {[G][hB]/[c^5]}^[.5] = 5.3912464466619438794714935823914e-44,
[G] = {[PL]^3}/([PM]{[PT]^2}) = 6.6742999999999999999999999999993e-11,
[G] = [c^2][PL]/[PM] = 6.6742999999999999999999999999997e-11,
[G] = [c^3][PT]/[PM] = 6.6742999999999999999999999999999e-11,
[G] = [hB][c]/{[PM]^2} = 6.6742999999999999999999999999998e-11,
[c] = [G]{[PM]^2}/[hB] = 299792458.00000000000000000000001,
[c] = [PL]/[PT] = 299792457.99999999999999999999999,
[c] = {[PM][G]/[PL]}^[.5] = 299792458.00000000000000000000001,
[c] = [hB]/{[PM][PL]} = 299792458 and
[hB] = [G]{[PM]^2}/[c] = 1.054571817e-34,
[hB] = [PM]{[PL]^2}/[PT] = 1.054571817e-34,
[hB] = [c][PM][PL] = 1.054571817e-34 and
[hB] = [c^2][PM][PT] = 1.054571817e-34.
[G]'s maximum of 6.67445e-11 forces another precisely consistent set of
[PM] = {[hB][c]/[G]}^[.5] = 2.1764098855801400481262371758156e-8,
[PL] = {[G][hB]/[c^3]}^[.5] = 1.6162731858999514770990647336683e-35,
[PT] = {[G][hB]/[c^5]}^[.5] = 5.3913070284775191946258525745445e-44
and a third results with [G]'s uncertainty minimum at 6.67415e-11 where
[PM] = {[hB][c]/[G]}^[.5] = 2.1764587993465890732634518889652e-8,
[PL] = {[G][hB]/[c^3]}^[.5] = 1.6162368617530589806228983546288e-35 and
[PT] = {[G][hB]/[c^5]}^[.5] = 5.3911858641655987910906629766812e-44.
In both the CODATA 2018 and CODATA 2022 values, [c] = 299792458,
[hB] = 1.054571817e-34, [PM] = 2.176434e-8, [PL] = 1.616255e-35,
[PT] = 5.391247e-44, and [G] = 6.67430e-11 so that
[G] as {[PL]^3}/([PM]{[PT]^2}) = 6.674 299382451...e-11,
[G] as [c^2][PL]/[PM] = 6.674 300950128...e-11,
and [G] as [hB][c]/{[PM]^2} = 6.674 302097883...e-11
do approach [G] = 6.674 30e-11.
However, [c] as [G]{[PM]^2}/[hB] = 299792 363.768496... ,
[c] as [PL]/[PT] = 299792 422.791981... ,
[c] as {[PM][G]/[PL]}^.5 = 299792 436.661325...
and [c] as [hB]/{[PM][PL]} = 299792 509.554171...
for unadjustable [c] = 299792 458,
and [hB] as [G]{[PM]^2}/[c] = 1.054571 48552439...e-34,
[hB] as [PM]{[PL]^2}/[PT] = 1.054571 51179904...e-34,
[hB] as [c][PM][PL] = 1.054571 63564931...e-34
and [hB] as [c^2][PM][PT] = 1.054571 75949960...e-34
for unadjustable [hB] = 1.054571 817e-34.
Compare 2022 atomic unit consistency where AUN (atomic unit of) charge
[Q0] = 1.602176634e-19, AUN velocity [v0] = 2.18769126216e6, AUN mass
[Me] = 9.1093837139e-31, AUN length [a0] = 5.29177210544e-11, AUN time
[AT] = 2.4188843265864e-17, Hartree energy [He] = 4.3597447222060e-18,
fine structure constant [aa] = 7.2973525643e-3, vacuum permeability
[d] = 1.25663706127e-6 and vacuum permittivity [k] = 8.8541878188e-12:
[c] as [4Pi][hB][aa]/([d]{[Q0]^2}) = 29979245 7.814597...,
[c] as [hB]/{[Me][a0][aa]} = 29979245 7.817863...,
[c] as (1/{[d][k]})^.5 = 29979245 7.999821...,
[c] as ({[He]/[Me]}^.5)/[aa] = 29979245 8.001619...,
and [c] as [a0]/{[AT][aa]} = 29979245 8.001677...,
for unadjustable [c] = 29979245 8,
and [hB] as [He][AT] = 1.054571817 64618...e-34,
[hB] as {[a0]^2}[Me]/[AT] = 1.054571817 64659...e-34,
[hB] as [v0][a0][Me] = 1.054571817 64836...e-34
and [hB] as {[v0]^2}[Me][AT] = 1.054571817 65012...e-34
for unadjustable [hB] = 1.054571817 e-34.
Planck units exemplify the multiple numerical coincidences created in
an archive of quantity calculations involving fundamental constants:
If [G] matches [d]{[aa]^2} = 6.6917625561799575982689395955202e-11
& [aa][c][PL]/({[hB][c][k]}^.5) = 6.6917625561759644520352594583958e-11,
[PM] = {[hB][c]/[G]}^[.5] = 2.173592718946606101082353926929e-8,
[PL] = {[G][hB]/[c^3]}^[.5] = 1.6183680175812977582826487290039e-35 and
[PT] = {[G][hB]/[c^5]}^[.5] = 5.3982946348213261531837759874664e-44,
thus matching to {[PM]^2} = 4.7245053078576997811500159383496e-16
{[Q0]^2}[c^2]/([4Pi]{[aa]^3}) = 4.7245053107795088055899733604386e-16,
matching to {[PL]^2} = 2.6191150403300196900547781819157e-70
{[aa]^5}[Me][d][AT]/[v0] = 2.6191150418875452435025632093815e-70
and matching to {[PT]^2} = 2.9141584964340715087666256507899e-87
{[aa]^3}[a0][Me]/{[c^6][k]} = 2.9141584982010643608596121445625e-87.
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