( 1 - ( v² / c² ) ).( Kapital-i, in 'I, myself', now spels Il as in isle or aisle.
Leter y spels ryd for reed or read and partys for parties.
Leter w spels swn for soon. )
Links to sektions:
Supos som advans'd rais kreated Orbitsville, an idea taken up by siens fiktion rIters such as Bob Shaw. A fysisist sujested that al but a tiny proportion of a sun's enerjy is wasted in nurturing intelijent lIf on the od planet orbiting it. A vastly mor eficient way wud be to enjinyr an eg-shaip'd suround at a distans of the erth's eliptik orbit, about eit minuts' lIt-spyd distans from the sun. That way, lIf kud exist on the entIr in-sId surfas of the eg ( or rather an elipsoid and not tu diferent from a sfyr ).
( Orbitsville wud hav to be manuvrabl from kolisions with stelar mater. It wud be a mother ship to other spas-ships. )
Imajin Orbitsville and a spas-ship pasing by it ( or being pas'd by it, depending on yor point of viw ) with a uniform velosity, at a signifikant fraktion of lIt spyd. We kan yus this senario to ilustrat Einstein's demonstration that tw observers in relativ motion kan not agry on the tIm of an event: in the special theory of relativity, simultaneity is lost.
The sun is in the senter of Orbitsville ( we supos, rather than at uon of the tw fokuses of Orbitsville's eliptic hul ). The klimat on its narower part ( the minor axis ) wil be warmer than on the lenth-wIs direktion ( the major axis ). From thos living on the in-sId surfas, observers ar station'd at ych end of the major axis. Supos the enjinyrs of Orbitsville hav kreated a nIt-simulating mekanism, of eklipsing satelits or wat-ever, such that dawn breks at exaktly the saim tIm for the observers at the far ends of Orbitsville. For instans, internal 'astronomers' on the minor axis of Orbitsville mIt sy a simultanus swath of lIt upon either end of the major axis.
Say, both ends of Orbitsville ar about eity-nIn and uon-third milion
mIls from the sun. The spyd of lIt is about 670 milion mIls per hour.
Therfor, the internal astronomers wil mesur the sun-lIt taking ( ( 89
1/3 ) / 670 ) = 2 / 15 of an hour or eit minuts, to rych either end of
Orbitsville.
( LIt taks about eit minuts to rych the erth from the sun, and the
distans betwyn them, typikly quoted, is about nInty-thry milion mIls.
)
Orbitsville mIt hav a transparent band along the major axis of its
shel. The spas-ship, out-sId, is moving in lIn with this at a fair
fraktion of lIt spyd ( say, uon-haf the spyd of lIt ). Relativ to
itself, it sys Orbitsville heding off at that 1/2 of lIt spyd. But the
spas-ship dosnt agry with the internal astronomers that the shafts of
sun-lIt, traveling from the midl of Orbitsville, ariv the saim tIm at
either end.
Wat is, from the spas-ship, the forward-moving end of Orbitsville apyrs to
be distansing itself from the sun's rays. So, the spas-ship mesurs a tIm
longer than eit minuts for the sun-lIt to rych the end moving forward,
relativ to the spas-ship.
Wer-as, the aft end apyrs to be moving to myt the sun-lIt, so the
spas-ship mesurs it taking les than eit minuts. Orbitsville's kloks
show it twk the saim eit minuts for lIt to rych either end of
Orbitsville. But the spacs-ship's klok wil show an inequality, dependent on
how klos its relativ motion to lIt spyd.
Yusing the figurs given, this inequality kan be work'd out by the arithmetik of the Lorentz transformations. Thys transformations also show that the spas-ship wil sy Orbitsville's sun-lIt has a konstant spyd, wether a lIt ray is moving from, or towards, the spas-ship. This is an axiom of the special theory of relativity. We show this first.
Supos the spas-ship pases the exakt midl of Orbitsville, just as a nIt shyld, kovering its sun, opens befor it. LIt is mayd visibl to either end of Orbitsville, at the saim tIm. But, from the spas-ship's point of viw, that dos not myn both ends resyv the lIt at the saim tIm.
The komon sens ( of the Galilean transformations ) wud hav us belyv that if the relativ motion of Orbitsville and spas-ship is 1/2 lIt spyd, then we hav to ad or subtrakt that spyd, with respekt to the lIt spyd, depending on wether a lIt ray is moving with or against the direktion of Orbitsville, with respekt to the spas-ship.
How-ever, the Lorentz transformations, for the adition and subtraktion of velositys, ensur that the spas-ship sys lIt spyd stay the saim in both kases.
Tak the kais of the lIt ray moving with Orbitsville's away-ward moving direktion from the spas-ship ( paralel to it ). This requirs the Lorentz transformation for the adition of velosities. This is a bit mor komplikated than the Galilean transformation, that simply givs spyd of lIt plus Orbitsville's relativ motion of uon-haf the spyd of lIt:
Say u' is the spyd of away-moving lIt, syn from the spas-ship. And u
is that lIt ray, wich is no diferent, to thos station'd in Orbitsville,
from the ray moving to the other end of its shel. Thos standing in
Orbitsville hav no relativ motion to ( ad to or subtrakt from ) the lIt
rays going to both ends. Wich end is forward or bakward has no myning to
them. In this kais ( as wel as the other kais, we shal dyl with next ), the
velosity, u, of the lIt ray, within Orbitsville's stationary fraim-work, kan
be set at a 'c' for ( konventionaly spelt ) 'constant': u = c.
Let v equal the relativ motion of one-haf light speed or c/2.
Then, the Lorentz transformation for adition of velositys is:
u' = ( u + v ) / ( 1 + ( uv / c² )).
To simplify the working, lIt spyd is often tryted as unitary, or c = 1. This maks v = 1/2, and u = c, in this exampl. Therfor:
u' = ( 1 + 1/2 ) / ( 1 + ( 1 x 1/2 ) / 1 ) = 1.
Or, u' = c.
Therfor, the spas-ship mesurs u', the spyd of the lIt ray, moving with the away-moving Orbitsville, as just c, the konstant spyd of lIt, irespektiv of an adition of relativ motion.
The saim aplIs for the lIt ray traveling to the oposit end of
Orbitsville. This ray is traveling against the relativ motion of Orbitsville
from the spas-ship. This myns that the aft end of Orbitsville is koming, at
uon-haf lIt spyd to myt the unit lIt spyd of the bakward-moving ray,
from the spas-ship's point of viw.
The komon sens or Galilean transformation is that the kombin'd velosity wud be uon minus uon-haf equals a velosity of haf unitary lIt spyd for the bakward ray,
relativ to the spas-ship.
But the Lorentz transformation, this tIm, for the subtraktion of
velositys again ensurs the spas-ship's relativ motion dos not alow it to
sy lIt mov at other than its konstant spyd.
In our exampl, the terms hav the saim myning as befor:
u' = ( u - v ) / ( 1 - ( uv / c² ) )
u' = ( c - v ) / ( 1 - ( cv / c² ) )
u' = ( 1 - 1/2 ) / ( 1 - (( 1 x 1/2 ) / 1) )
u' = 1 = c.
Therfor, the Orbitsville sun-lIt moving with the spas-ship's direktion is stil mesur'd at lIt's konstant spyd. This is the saim velosity the spas-ship mesur'd for the Orbitsville sun-lIt's away-moving bym.
The Lorentz transformations of tIm show wI observers in relativ motion,
that is a fair fraktion of lIt spyd, kan not agry on the tIm they observ'd
an event.
Al standing in Orbitsville ar agry'd that both lIt rays, from senter to
either end, twk ( tIm, t, of ) 8 minuts, or 2/15 of an hour, to kros (
distans, X, of ) 89 1/3 milion mIls, at lIt's konstant spyd, c, wich is
about 670 milion mIls per hour.
But the spas-ship's klok wil show diferent tIms, t', that it twk the tw lIt rays to rych the ends of Orbitsville. The away-moving lIt-ray wil be tIm'd as taking longer than eit minuts, as the Lorentz adition of tIms shows:
t' = ( t + ( Xv / c² ) ) / ( 1 - ( v² / c² ) ).
Distans, X, equals velosity, u, tIms tIm, t. Also u = c. If c = 1, and v = 1/2, the squar of v equals 1/4. So:
t' = 8 ( 1 + 1/2 ) / ( 1 - 1/4 ).
t' = 8 3.
That is, the spas-ship's klok wil rekord the away-moving Orbitsville
lIt ray as taking about 13.9 minutes ( to uon desimal plas ) to katch up
with the 'forward-moving' end of Orbitsville, insted of the 8 minuts that
Orbitsville kloks rejister.
Now we find out the tIm the spas-ship's klok tels us it taks the lIt
ray, that is moving the saim way as itself, to myt the aft end of
Orbitsville. From the spas-ship's viw, the aft is moving, at haf the spyd
of lIt, toward this ray, and so redusing the tIm they tak to myt.
Yusing the Lorentz subtraktion of tIms:
t' = ( t - ( Xv / c² ) ) / ( 1 - ( v² / c² ) ).
Folowing a similar prosedur to the previus exampl for Lorentz adition of tIms:
t' = 8 ( 1 - 1/2 ) / ( 1 - 1/4 ).
t' = 8 / 3.
This tIm, t', of about 4.6 minuts ( to uon desimal plas ) is the
spas-ship's kloking of the lIt ray, in the saim direktion as the
spas-ship, hws relativ motion maks the aparent aft of Orbitsville apyr
to kom to myt the ray.
This is not quIt haf the 8 minuts tIm, t, that Orbitsville's klok shows
for the saim jurny. With this other Orbitsville lIt ray, direkted in the
oposit direktion, simultanus tIm has byn lost, again. Bekaus of ther
hI relativ motion, the spas-ship observers kan not agry with the
Orbitsville observers wen the lIt ray ryches from senter to end of
Orbitsville.
BesIds the dis-agryment over tIms betwyn the spas-ship and Orbitsville observers, they also dis-agry over distanses. Distans equals velosity tIms tIm. Nolej of any tw of thys variabls wil tel us the third. The previus tw sektions hav alredy found the respektiv velositys and tIms mesur'd by the tw lots of observers. So, the respektiv distanses, mesur'd by ther respektiv 'rods', folows from that information:
In Orbitsville, the distans, X, from senter to either end is eity nIn and uon-third milion mIls. The spas-ship sys uon lIt ray apyr to hav to chais the end of Orbitsville moving away from the spas-ship, thus elongating the distans, X', that the spas-ship's rod mesurs.
Or, X' = u'.t' = c.t'. This equals the lIt spyd, of 670 milion mIls
per hour, tIms ( 2/15 ) 3 hours. ( Eit minuts is 2/15 of an hour.
) Therfor, X' works out at about 154.7 milion mIls ( to uon desimal plas
).
For the saik of komplytnes, we kan repyt this result, by the Lorentz adition of distanses:
X' = ( X + vt ) / ( 1 - ( v² / c² ) ).
X' = ( ( 89 1/3 ) + ( 335 x 2/15 ) ) / ( ( 3 ) / 2 ).
X' = ( 134 x 2 ) / 3.
This again equals about 154.7 milion mIls for the spas-ship's 'expanded' rod ryding.
Now we kom to the kais wer the spas-ship's rod or spatial mesur apyrs
to 'shrink' to les than the Orbitsville rod's rejister'd eity-nIn and
uon-third milion mIls. Again the simplest solution of, X', now as the
shrunken distans, is u'.t', wer u' stil equals c, and wer t' equals 8 / 3 ( wich is about 4.6 ) minuts, or 2 / ( 15 x 3 ) of an hour.
This puts X', as at about 51.6 milion mIls.
Had we not had the benefit of both velosity and tIm information, we mIt hav had to find this result by starting with the Lorentz subtraktion of distanses:
X' = ( X - vt ) / ( 1 - ( v² / c² ) ). Or,
X' = t ( u - v ) / ( 1 - ( v² / c² ) ). Wer u = c = 670 milion
mIls per hour.
X' = ( 2/15 ) ( 670 / 2 ) 2 / 3.
X' = 268 / ( 3 x 3 ).
X' ~ 51.6.
This repyts the result we got direktly bekaus we hapen'd to hav the information for X' = u'.t'.
In the abov exampls, we ilustrated the special relativity prinsipl of the konstant spyd of lIt, despIt an observer ading or subtrakting the other observer's relativ motion to it. In jeneral, the observers' respektivly mesur'd velositys, u' and u, nyd not be of lIt byms, hws konstant spyd they ar both bound to equal, but kan mesur things at slower velositys, so that, lIk distans and tIm, velositys ar not jeneraly agry'd by observers in relativ motion, signifikantly aproching the velosity of lIt.
The valu of the Interval, for our exampl, also ilustrats its natur as a partikular kais. The previus web paj, part 2 on special relativity, related the Interval to the Lorentz transformations. The transformations, as exemplify'd abov, show that observers in hI spyd relativ motion usualy mesur diferent distanses and tIms for a given event.
The Interval rekovers a komon mesurment betwyn observers of an event. But it is a kombin'd spas-tIm mesurment, upon wich they depend for agryment as to the 'wer-wen' som-thing hapen'd.
The Interval mesurs a 4-dimensional spas-tIm komon to observers. Our simplify'd exampl only delt in uon dimension of spas, X, and the uon tIm dimension, t. From the abov sektions, the Orbitsville ko-ordinats ar X and t, and the spas-ship ko-ordinats ar X' and t'.
As deskrib'd on the previus web paj, the squar of the Interval equals:
X² - c².t² = X'² - c².t'².
Our exampl is of the Orbitsville observers and spas-ship observers of lIt byms. Special relativity asums both sets of observers must mesur the saim velosity for lIt. That is both u = c and u' = c.
In this partikular kais, sins X = ut and X' = u't', the Interval for both observers is zero. The general point is that the Interval is always the saim valu for both observers.
My explanations of special relativity wer in terms of 'kinematiks' or motion, not 'dynamiks', wich further involvs the konsept of mas or enerjy.
Fortunatly, the konversion of the Lorentz transformations, from ther kinematik form to ther dynamik form, is fairly strait-forward. The Lorentz transformations korespond observers' difering distans and tIm mesurments of an event. Kal the observers' difering distanses, x and x', and difering distanses , t and t', respektivly. Konsequently, the difering velositys of a body, they observ, wil be, say, u = x/t and u' = x'/t'.
If the mas of the
body is also to be konsider'd, then an observer, hw is at rest relativ to
the motion of the body, wil mesur its 'rest mas'.
In klasikal mekaniks, ther only is a rest mas: the mas is dym'd
konstant. But in relativistik mekaniks, the mas of a body is nown to
notisably inkrys as its motion notisably aproches the spyd of lIt. (
The relevant Lorentz transfomation shows that a body wud hav to be of
infinit mas to rych the spyd of lIt. And this is wI lIt spyd is
natur's spyd limit. )
Esentialy, the Lorentz transformations for distans and for tIm ar both multiply'd by the saim quantity, the mas divided by the tIm, to konvert them into Lorentz transformations for momentum and enerjy, respektivly.
Momentum, p, is mas tIms velosity. In the system of mekaniks, distans has thry dimensions, the x, y and z direktions, to refer to al spatial motions. Therfor, velosity, as distans over tIm, also has direktion with referens to the thry ko-ordinats. For simplisity, we hav konsider'd observers mesuring events, wich difer only in the x-direktion, akording to ther respektiv mesurments. That is to say wer distanses y = y' and z = z' betwyn respektiv observers. But ther respektiv mesurments do not mak x equal to x'. So, velositys in the x-direktion, u and u' also nyd not be equal.
Konsequently, the Lorentz transformation for momentum is in terms of the observers' difering mesurs of both mas and velosity, or: p = mu and p' = m'u'.
The difering tIms, observers mesur, ar proportionat to the difering mases they mesur: t'/t = m'/m. Nout, also, that E = mc² and E' = m'c², wer E and E' ar the respektiv enerjys mesur'd by the observers, akording to the famus equation, not hyr demonstrated.
Then, multiply the Lorentz transformation for distans, x, by m/t, and x' by m'/t', for the Lorentz transformation betwyn observers' difering mesurs of momentum ( in the x-direktion ):
mx/t = mv = p = m'(
x' + vt' )/ t'( 1 - v²/c² )
= ( p' + vE'/c²
)/( 1 - v²/c²
).
The saim prosedur, of multiplying by mas over tIm, is folow'd for konverting the Lorentz transformation of tIm into the Lorentz transformation of enerjy:
tE/c²t = ( t'E' +
vx'E'/c² )/ c²t'( 1 - v²/c² )
This simplifys to:
E = ( E' + vu'm' )/
( 1 - v²/c² )
= ( E' + vp' )/
( 1 - v²/c² ).
The Lorentz transformation of velosity folows from dividing the Lorentz transformation for distans by that for tIm:
u = ( u' + v )/ ( 1 + vu'/c² ).
In dynamiks, kompar the result of dividing momentum by enerjy. The denominators of both transformations ar the Lorentz kontraktion faktor, wich kansels:
p/E = ( p' + vE'/c² )/( E' + vp' ).
Or: u.m/mc² = m'( u' + v )/m'c²( 1 + u'v/c² ).
This reduses to the Lorentz transformation of velosity.
Further-mor, the Minkowski Interval, as wel as the Lorentz transformations, konvert from kinematiks to dynamiks, in a strait-forward maner. The Minkowski Interval is a for-dimensional spas-tIm jeometry. It is lIk Euclid's jeometry in thry dimensions. But tIm is tryted lIk a forth dimension of spas. The only formal mathematikal diferens betwyn the thry spas dimensions and the uon tIm dimension is that the tIm is multiplyd by the squar rwt of minus uon.
This faktor givs the tIm, t, the oposit sIn to the thry spas dimensions x, y and z, wen the Interval extends Euclid's jeometry, of Pythagoras theorem in thry dimensions, into a 4-D spas-time version of the theorem.
The Lorentz transformations ar the myns by wich an observer chanjes ther mesurments into thos of diferently situated observers of the saim event. The point of the Interval is that it kombIns ther respektiv mesurments into a 'spas-tIm' mesurment, of a given event, that is uon and the saim for al observers, in uniform relativ motion.
As for the Lorentz transformations, the Interval is simply given hyr for only uon dimension of spas and the uon dimension of tIm. As diskus'd on previus pajes, this is:
t²( c² - u² ) = t'²( c² - u'² ).
Or: t²c² - x² = t'²c² - x'².
The spas-tIm Interval chanjes to its momentum-enerjy version, as with the Lorentz transformations, thru replasing tIm by mas:
( mc )²t²( c² - u² )/t²c² = ( m'c )²t'² ( c² - u'² )/t'²c².
Or: ( E/c )² - p² = ( E'/c )² - p'².
Richard Lung.
To top.