The Impo ance o New onian KE and P on Special Rela i i y
F ancesco R. Rugge i Hanwell, N.B. Oc . 6, 2025
Special ela i i y is o en hough o as a ising as a ma h ans o m linked o Maxwell’s
elec omagne ic equa ions, o h ough Eins ein’s hough expe imen s using ligh and he
in a iance o ames mo ing a cons an speeds ela i e o each o he .
He e, we a gue ha e en hough New on’s equa ions seem o be la gely ocused on he
no ion o momen um, which is a ec o , he second law dp/d = F leads o a scala (unde
o a ion), namely p do p /2m which is no simply p do p. In o he wo ds, we a gue ha
New on’s no ion o bo h a p ( ec o ) and p do p /2m (scala unde o a ions) as desc ibing a ee
pa icle s a e ac ually leads o special ela i i y. In o he wo ds, a single objec , ei he p o p do
p /2m is no su icien o desc ibing a ee pa icle. We no e ha an objec a es is also
desc ibed by a numbe mo and p=0. Fo a pa icle mo ing a a e y low , he s a e is s ill mo o
i s app oxima ion e en hough he e is mo and .5mo . As a esul , we sugges ha New on’s
idea o a ec o p and scala .5mo desc ibing a s a e ac ually be ex ended o a pa icle a es ,
and ha one mus conside mo ( /b)bb and mo h( /b) whe e and h a e unknown unc ions.
One may see ha a cons an b mus be in oduced and b should ha e he same alue as seen
om any cons an ly mo ing ame, i.e. should ep esen a pa icle wi h mo=0. We use he idea
o a scala unde o a ion and a ec o as desc ibing a s a e as cha ac e izing a ee pa icle
s a e. We no e ha o he es s a e p=0 and so one only mobb. This means ha one may
in oduce a linea , i.e. ma ix ans o ma ion, because mobb, mo and .5mo a e all linea in
he ac o mo. We sugges ha gi en he p esence o b, which has eloci y uni s, i mus be a
cons an and ha e he same alue as seen in all ames. This, we a gue, implies ha a clock in
a es ame canno ha e he same pe iod as one in a mo ing ame.
As a esul , we sugges ha special ela i i y ollow om New on’s idea o cha ac e izing a
ee pa icle s a e by a dual, mo, .5mo and p ec o and ha one may ob ain special ela i i y
om his idea alone because i equi es he in oduc ion o a speed b which is he same as seen
in all ames mo ing a cons ain .
Special Rela i i y
New onian mechanics was de eloped in he 1600s, bu special ela i i y did no eally appea
un il he la e 1800s wi h s udies o p ope ies o ans o ma ions o Maxwell’s elec omagne ic
equa ions. In 1905. Eins ein in oduced hough expe imen s using ligh (c being conside ed he
same as seen in all ames and a maximum speed) which ea ed ames mo ing a cons an
speeds wi h espec o each o he as being in a ian . As a esul , one migh wonde i special
ela i i y could ha e been de eloped ea lie wi h no no ion o pho on speed o Maxwell’s
elec omagne ic equa ions. We also gi e up he e he no ion o in a ian ames, e en hough
his idea is pu ely physical and could ha e been conside ed a New on’s ime.
The Impo ance o Desc ibing a F ee Pa icle by Bo h Kine ic Ene gy and Momen um
New on’s laws seem o ocus on momen um, which is a ec o . E en he second law:
dp/d = F ((1))
is based on momen um. Momen um is physically linked o impulse and so one migh hink ha i
is a comple e desc ip ion o a ee pa icle, al hough one migh wan o also include mo as
m1 1=m2 2. I is ue ha one may ob ain a second quan i y:
Mo d = Fd = Fdx/ —> In eg al F dx = .5mo (i ini ial =0) ((2))
This second quan i y, howe e , is used by New on in scena ios in which he e exis s a po en ial.
Fo a ee pa icle, momen um seems o almos always su ice and may be used o desc ibe
collisions wi h a a ge . The e is, howe e , a case in New onian mechanics, whe e bo h
momen um and kine ic ene gy a e used oge he and ha is in he case o elas ic sca e ing.
Al hough no all wo body sca e ing cases a e elas ic, he Maxwell-Bol zmann ideal gas is
based on hese elas ic collisions and so elas ic collisions a e impo an . The key idea we wish o
s ess is ha e en hough kine ic ene gy usually appea s oge he wi h po en ial ene gy V(x), i
is s ill pa o he desc ip ion o a ee pa icle. In o he wo ds, we sugges ha New on’s wo k
implies ha a ee pa icle canno be desc ibed by momen um alone, bu should be desc ibed
by:
Mo, .5mo and p = mo (non ela i is ic) ((3))
A pa icle a es has mo, .5mo =0 and p=0 ((4)). I is e y iny, o a i s app oxima ion, one
has mo. Thus, we sugges ha o small , mo and .5mo as well as as p ec o should
desc ibe a ee pa icle. This implies ha one should ha e:
mo bb + .5mo and p ec o ((4))
He e b is a cons an wi h uni s o speed which has he same alue as iewed om any ame.
P esumably i should ep esen a physical objec , bu o he p esen , i s exis ence su ices. We
gene alize ((4)) and pos ula e ha any ee pa icle s a e is ep esen ed by
Mo ( /b) bb and p ec o = mo h( /b) whe e and h a e unknown unc ions ((5a))
Gi en ha one only has mo in he es ame, i seems ha o one -dimensional mo ion, one
may c ea e he ec o :
(p=0, mobb) ((5b))
and a gue ha he e exis s a 2x2 ma ix which ans o ms i o c ea e p’b and mo ( /b) bb. We
do no assume any in a iance be ween a es ame and one mo ing a a cons an speed. In he
New onian pic u e, a pa icle a es di e s om a mo ing one (cons an speed) by he condi ion
ha wo k has been applied. Ne e heless, he ac ha ((5)) is linea in mo and ha one has
only mo in he es ame, one may conside :
| g( /b) g( /b) | ((6))
| g( /b) g( /b) |
A his poin , howe e , g( /b) is comple ely unknown, and one has also aken he libe y o
in oducing some a p io i symme y in o ((6)) which may o may no exis .
A key idea, howe e , is ha his ans o ma ion, like mobb, equi es he exis ence o cons an
speed b which is he same as iewed in all ames. This seems o imply an objec wi h no es
mass, bu o he han ha , i is unknown. The e y ac ha b exis s and has his p ope y,
howe e , implies ha clocks mus be di e en in a es and mo ing ame.
Clocks in a Res and Mo ing F ame
Gi en he exis ence o a cons an speed b ( o an objec wi h ze o es mass), he usual
a gumen s o special ela i i y show ha a clock canno ell he same ime in bo h ames. To
see his, conside an objec wi h speed b mo ing along he y axis and e lec ing be ween wo
pla es sepa a ed by dis ance L. In he mo ing ame, he o e all speed o he objec mus s ill be
b and i will a el a dis ance L in he y di ec ion, bu also a dis ance ’ along he x di ec ion.
Thus:
bb ’ ’ = b ( ’ ’ + b b ) → ’ = / sq (1- /bb) ((7))
As a esul , special ela i i y ollows om he exis ence o b which has he same alue as seen
in all cons an speed ames, bu he exis ence o b ollows om he idea ha one may
cha ac e ize a ee pa icle s a e by mo ( /b) and p=mo h( /b). Thus, i is ul ima ely New on’s
idea ha he e mus be bo h .5mo and p=mo , ex ended o he case o mobb +.5mo o
e y small which ul ima ely leads o special ela i i y, we a gue.
Gi en ((7)), one may e u n o ((6)) and show ha :
g( /b) = 1/sq (1- /bb) ((8))
A no poin ha e we a gued ha he e is in a iance be ween wo ames mo ing a cons an
speed wi h espec o each o he . We ha e simply a gued ha a ee pa icle mus be
cha ac e ized by a numbe p opo ional o mo which mus become mobb o =0 and a ec o
p opo ional o mo (and mul iplied by some unc ion h( /b)).
One may, howe e , show ma hema ically ha ((6)) ul ima ely leads o di e en ames mo ing
a a cons an speed ela i e o each o he being in a ian . In pa icula , gi en ((7)) and ((6)) one
may show ha :
X’x’ - b ’b ’ = xx - b b ((9))
As a esul , he wo ames mus be in a ian , which ma ches he physical obse a ion ha one
canno ell i one is in a mo ing ame o a es ame i one sees an objec mo e a cons an
speed. This idea was al eady p esen ed by Eins ein.
Using ((6)) one inds ha :
mobb/sq (1- /bb)) and p = mo /sq (1- /bb) ((10))
yield he wo alues (numbe and ec o ) which desc ibes a ee pa icle.
Conclusion
In conclusion, we a gue ha New on’s idea ha a ee pa icle may be desc ibed by bo h p
do p /2m and p ec o is he key idea which leads o he de elopmen o special ela i i y. In a
ee pa icle case, p may be used o desc ibe he impulse hi agains a a ge , bu one should
eally also ha e he numbe mo because m1 1=m2 2 yield he same impulse. P do p/2m is
usually used when he e is a po en ial V(x), i.e. one does no ha e a ee pa icle, bu in he
case o an elas ic collision, bo h p do p/2m and p a e key o desc ibe a pa icle, We sugges
ha one gene alize his idea and a gue ha any ee pa icle be cha ac e ized by mo ( /b) and
mo h( /b), whe e and h a e unknown unc ions. This equi es in oducing a cons an speed b
which has he same alue as seen in all ames mo ing a cons an speed. B p esumably
ep esen s an objec wi h ze o mo.
Gi en ha one only has mobb in he es ame, one may c ea e a 2x2 ma ix o ans o m i
in o mo ( /b) and mo h( /b) wi hou any no ion o he ames being in a ian . To ind he ac ual
alues o he ma ix elemen s, one may no e ha he exis ence o b equi es ha clocks in a es
and mo ing ame o di e by: ’ = /sq (1- /bb) wi h ’ini ial = ini ial = 0. This allows one o
cons uc he Lo en z 2x2 ma ix and ind ha mobb becomes: mobb/sq (1- /bb) and p=
mo /sq (1- /bb). Fo <<b, one has mobb +.5m and p=mo , he gene alized New onian
esul s. Thus, we a gue ha New on’s idea ha one mus ha e p do p /2m and p=mo ec o
bo h desc ibe a ee pa icle is he seed which leads o special ela i i y.
