Special Relativity, part 2:
The Minkowski Interval and the Lorentz Transformations.

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The Minkowski Interval ( 1908 ): deskrib'd.

'The Interval' of Minkowski is an extension of Euclid's jeometry for making mesurments in thry dimensions of spas. An exampl, of Euclid, is mesuring the hIt, lenth and bredth of a box-rwm from uon of its korners. Minkowski tryted tIm as a forth dimension of spas and integrated it into a for-dimensional Euclidean spas-tIm jeometry.

The advantaj of this nw mathematikal formalism was that al observers' diferent spas and tIm mesurments kud be re-stated in a formula, nown as 'the Interval', a konstant spatio-temporal mesurment, wich is the saim for al observers. This reflekts the fakt that, tho the observers wer yusing ther own ko-ordinat systems, they wer al sying the saim event.

( The qualifikation nyds to be mayd that the Interval only aplIs to observers moving in 'uniform relativ motion'. To remov this arbitrary restriktion, Einstein had to kom up with his 'prinsipl of equivalens' of akseleration to gravity, wich was the basis of a jeneral theory of relativity. )

The only mathematikal diferens betwyn the thry spatial dimensions and the uon temporal dimension is that tIm units ar multiply'd by the mysterius squar rwt of minus uon ( given the symbol i ).

We may sy solid shaips chanjing in tIm. We may imajin our thry dimensional world in a historikal perspektiv. How-ever, the thry spatial dimensions ar esentialy the saim in karakter and we nyd only uon of them to show how the tIm dimension relats to spatial mesurment. Uon spas ko-ordinat, x, and uon tIm ko-ordinat, t, kan be represented, diagramatikly on a tw-dimensional paj.

So, we ar relating the spatio-temporal mesurments of tw observers, working in tw diferent plases and tIms, and therfor with tw diferent ko-ordinat systems: the x and t system is distinguish'd from an x' and t' system, by the later's indises. Of kors, both systems hav tw other dimensions of spas to konsider. But for simplisity's saik, we'v asum'd ther mesurments dont difer in that respekt. In other words, ther spas mesurs on the y and z dimensions ar equal, or y = y' and z = z', komparing the tw observers' ko-ordinat systems.

The Interval derivs the Lorentz transformation for tIm, and for spas, in much the saim sort of way, partly with the yus of trigonometry. The Lorentz transformation kan be expres'd as an equation of uon observer's tIm, t', or the other observer's tIm, t. LIkwIs, for ther respektiv distans mesurments, x and x'.

The Interval points to a spas-tIm ko-ordinat position, that is both x + ict and x' + ict' Thys 4 valus ar esentialy the Lorentz transformations, we'v just byn spyking of. Hyr, they form tw diferent sets of ko-ordinat axes, that ar the tw observers' fraims of referens.

Yusing trigonometry, the Interval alows observers' varying spas and tIm mesurs to be drawn as the varying oposit and ajacent sIds, of a rIt-angl'd triangl, with respekt to uon of the other tw angls. For instans, a mor akyut angl maks the oposit sId, of the triangl, relativly short. Thys oposit and ajasent sIds wil vary akording to the rotation of the hypotenus, as the konstant radius of a sirkl.

Any position on the sirkl boundary, that the radius points to, is a given spas-tIm position, perhaps marking som mutualy observ'd fysikal event. Any number of observers kan chws diferent nInty degry ko-ordinat systems of spas and tIm, on the sirkl, given that ther axes al sher the saim orijin at the senter of the sirkl.

With thys diferent anglings of ther rektangular ko-ordinats, they kan al mesur any uon given spas-tIm position on the sirkumferens. Ther respektiv axes wil al show diferent spas-lenths and diferent tIm-lenths in relation to that komon sIting. In efekt, ych ko-ordinat system is lIk a triangl with varying oposit and ajacent sIds for varying distanses and tIms.

But al thys diferent triangls hav the saim hypotenus. By Pythagoras' theorem, ther varying lenths, on diferent spas and tIm axes, ad up to the saim unitary spas-tIm mesurment, given by the lenth and direktion of ther sher'd hypotenus, as a konstant radius pointing out any given spas-tIm event on the sirkl sirkumferens.

This spas-tIm radius vektor is 'the Interval'. ( A vektor myns a lenth or magnitude in a pointed direktion. ) The Interval aplIs not only in uon dimension of spas and uon dimension of tIm. For exampl, making mesurments in tw dimensions of spas, as wel as uon dimension of tIm, the radius kud vektor-in on a given thry-dimensional spas-tIm position, represented as a point on a sfyr, insted of a sirkl.

( Pythagoras' theorem, for finding the Interval as hypotenus, extends to aplI to thry or mor dimensions. )

For the ful for-dimensional spas-tIm trytment, the Interval wud, presumably, be a radius 4-vektor pointing out a position on a for-dimensional hyper-sfyr, wat-ever that is -- som-thing beyond the imajination of our thry-dimensional experiens, perhaps.

The Interval: diagram explain'd.

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Having explain'd the Interval in words, simpl equations wil giv a mor presis idea, with the help of the folowing diagram.

Diagram of the Interval

The Interval is given by the radius vektor OP, wich is the hypotenus of triangulations bais'd on both ko-ordinat systems. Ther-by, the point, P, is lokated by both systems, the tw sets of ko-ordinat positions, x + ict and x' + ict', being equal or diferently representing the saim position.

The diagram shows, skymatikly, the esential diferens betwyn tw observers mesuring, in spas and tIm units, the saim event at position, P. Special relativity rekognises lokal ruler or 'rod' lenths and lokal klok raits. Thys korespond to the diferent ko-ordinat systems the observers ar working with.

On the diagram, the tw observers ych hav a pair of nInty-degry axes. Uon observer mesurs distans and tIm, in units x and t ( or rather, ict, wer t is just multiply'd by c, or konstant spyd of lIt, and by i, som-tIms deskrib'd, in mathematiks, as an operator. Just as the plus sIn, +, myns the operation of ading, so the i sIn, as it hapens, kan myn the operation: turn thru nInty degrys ).

The other observer mesurs distans and tIm, in his units, x' and t', on his x' and ict' ko-ordinats. But the diagram shows the only ryl diferens betwyn this ko-ordinat system, and the other, mathematikly, is that it is a turn, thru angl, a, away from the x and ict axes system. ( Of the 4 angls, a, on the diagram, I refer to the angl a, at the orijin, O, and mark'd by the up-wards kurling arow, to show angl, a, is the angl of tilt betwyn the ( x, ict ) rIt-angl'd ko-ordinat system and the index'd ko-ordinat system. The triangl for this partikular angl, a, is kolor-koded yelow in the diagram. )

Position, P, wer the event is, the observers ar both mesuring, is equaly wel lokated by uon set of axes as an other: x' + ict' marks the spot as wel as x + ict. Ther is no privilej'd or prior fraim of referens. The fraims ar 'only relativ' to ych other. This wud be tru for any number of observers, with ther own ko-ordinat fraims, at varying angls, about orijin, O, from ych other.

But they wud al hav in komon wat thys tw observers hav. They wud sher the saim hypotenus, OP. This is 'the Interval', a konstant spas-tIm mesurment, that al observers ( in konstant relativ motion ) kan agry is the saim mesurment of the event at P. Yet, the observers ariv at the Interval from ther diferent mesurments of spas and tIm, tryted by ych observer as separat mesurments on a separat spas axis and a separat tIm axis.

It is in the kontext of the Interval, that special relativity is about an integrated konsept of 'spas-tIm'.

The valu of the Interval kan be found from any ko-od. system. Taking the ( x,t ) system, and yusing triangle OPJ: As shown on the diagram, the lenth of OJ is x = ut. The lenth of JP is ict. By Pythagoras' theorem, the lenth of the hypotenus, OP, wich is the Interval, is found from: ( OP ) = x + ( ict ).

So, the Interval equals ( t( u - c )) = t( u - c )

Taking the other observer's system, ( x',t' ), the Interval kan be found by the saim prosedur, this tIm yusing triangl OPM. The diagram shows that, as this other rIt-angl'd triangl has the saim hypotenus, OP, the Interval, as such, is obviusly the saim valu.

In the index'd ko-ods., the Interval equals t'( u' - c ).

Therfor, t( u - c ) = t'( u' - c ).

( Usualy, text bwks lyv the Interval in terms of the hypotenus squar'd, without taking the squar rwt. )

The Interval: deriving the Lorentz transformations.

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The Interval may deriv the Lorentz transformations of uon observer's distans, or tIm, mesurments, into thos of an other.

The Lorentz transformation of distans, x, into distans x', is klosly related to that for tIm ( wich was given in the preseding web paj ):

x' = ( x - vt )/( 1 - v/c )

This kan be render'd:

x' = x.(c/( c - v )) + ict.( iv/( ( c - v )).

The diagram of the Interval also provids an equation for x':

x' = x.cos(a) + ict.sin(a).

The lenth, x', is the lIn OM, wich the diagram marks as on the x' axis. OM kan be split into lenths OL and LM. By trigonometry, OL is the lenth, x.kos(a) ( deriv'd with angl, a, in triangl OLJ ), and LM is the lenth, ict.sIn(a), ( deriv'd with angl, a, in triangl PJK ).

( The former angl, a, is kolor-koded yelow, and the later angl, a, is kolor-koded gryn, on the diagram. )

The Lorentz transformation kan be deriv'd from the diagram of the Interval, by equating the abov tw equations for x'. This implIs that the faktors of ther x terms must be equal: naimly, c/( c - v ) = kos(a). LIkwIs, the faktors of ther ict terms must be equal, or: iv/( c - v ) = sIn(a).

To rylIs thys equalitys, sertain simpl konditions ar asum'd. An event, say a flash of lIt, hapens wer uon of the observers aktualy is. So, he mesurs zero distans from the event. Or, x' = 0 ( asuming this is the observer with the index'd ko-od. system ).

The other observer has byn moving away from him, at a relativ velosity, v, so that, by tIm, t, wen the event hapens, she is a distans, away, of x = vt. ( Normaly, the velositys ar u' = x'/t' and u = x/t. But in this instans, u' is zero and v, ther relativ velosity, kan be set equal to u. Tho, v is jeneraly not a simpl diferens betwyn observers' velositys, wen an observer's velosity aproches signifikantly klos to lIt spyd -- this being wen relativistik fysiks revises the komon sens asumptions of klasikal fysiks. )

Substituting x' = 0 and x = vt, in the Interval diagram's equation for x':

x' = 0 = vt.kos(a) + ict.sIn(a).

Or, iv.kos(a) = c.sIn(a).

Then, sIn(a) = ( iv/c )kos(a).

But, by Pythagoras' theorem, ( sIn(a)) + ( kos(a)) = 1.


(( iv/c )kos(a)) + ( kos(a)) = 1.

So, ( kos(a))( (iv/c) + 1 ) = 1.

Then, kos(a) = 1/( 1 - v/c ) = c/( c - v ).

And substituting this equivalent for kos(a), to get an equivalent for sIn(a):

c/( c - v ) + ( sIn(a)) = 1.

Then, sIn(a) = iv/( c - v )

Thys ar the tw equivalent valus we wer lwking for, to mak the Interval diagram equation for x' equal to the Lorentz transformation for x'.

The identitys for kos(a) and sIn(a) kan be substituted in the trigonometrik formula, sIn(b+a) = sIn(b).kos(a) + kos(b).sIn(a), wer angl, b, is shown in the Interval diagram. Then:

ict/t( u - c ) = ( ict'/ t'( u' - c )).( c/( c - v ) + ( u't'/t'( u' - c )).( iv/( c - v )).

This equation transforms into:

( u' - c )/( u - c ) = ( 1 + u'.v/c )/F = t/t', wer F = ( c - v )/c, the kontraktion faktor, in the Lorentz transformation of observers' tIms, given in the midl term's equation to the rIt sId term, t/t'.

Wer-as, t/t', equated to the left sId term, givs the Interval:

t'( u' - c ) = t( u - c ).

The Lorentz transformation of tIm kud hav byn dedus'd in a lIk way to the Lorentz transformation of distans. From the Interval diagram, we kud dedyus that

ict = x'.sIn(a) + ict'.kos(a).

This result derivs from the top-rIt angl, a ( kolor-koded oranj in the diagram ) as the angl in the x'.sIn(a) term.

The angl, a, kolor-koded blu, derivs the ict'.kos(a) term. I havn't mark'd the exakt triangls for the kalkulations, that thys angls belong to ( not wanting to kluter the diagram ).

The Lorentz transformation of tIm wud also be split into tw terms, that ar faktors of x' and t'. Then, the tw faktors of ych equation, for t, kud be respektivly equated, by the sort of simplifying asumptions mayd, abov, for the kais of observers' spatial mesurments.

At the end of the preseding web paj, the kalkulation for the Michelson-Morley experiment was shown as a restrikted form of the Interval. How wud this lwk on the Interval diagram?

On our diagram, x = ut is the lenth OJ. But in our Interval version of the Michelson-Morley kalkulation, u = 0. So, x = 0, and the lenth OJ shrinks to nothing. Insted of mesuring a rektangular ko-ordinat position, to mark the event at point, P, this fraim of referens konsists only of uon lIn, the tIm axis, ict, that points direktly to P, and is, therfor, also that fraim's Interval mesurment.

Michelson-Morley's ( x', t' ) system ( wich mesur'd the putativ greitest ether drag on lIt spyd ) has both spas and tIm komponents greiter than zero. It resembls the tilted rektangular system in the diagram. But its hypotenus, by wich that system mesurs the Interval, wil ko-insid with the vertikal temporal axis, in t ( of wich the putativ lyst-ether-drag system soly konsists ).

The Interval: Feiz-diferent fraims of referens.

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The Interval is a rotatabl radius ( tho, the diagram dosnt inklud a sirkl, showing the orijin, O, at its senter, and the Interval's partikular position, given at P, as on its sirkumferens ). No mathematician, I stil kudnt help notis that such a radius vektor draws out wavs, in either tIm or spas, or both. Such wav equations hav independent variabls of up to thry dimensions in spas and uon independent variabl for tIm, on the other sId of the equation. Both sIds akt on a dependent variabl of wav displasment. The Interval syms rather lIk this klasikal wav displasment, akted upon by 4 separat komponents of spas and tIm, but sumarising them mathematikly as a unify'd spas-tIm mesurment.

The Interval is a konstant for any given spas-tIm event but bekoms a variabl uons uon konsiders it moving thru a sirkl or sfyr of events ( or 4-D spas-tIm hyper-sfyr of events ).

The Interval was mark'd at P, on the diagram, by tw observers' ko-ordinat systems. But, in the proses, it was also mark'd by tw diferent kinds of ko-ordinat system: rektangular or rektilinear ko-ordinats and polar ko-ordinats.

The Interval was deskrib'd as a spas-tIm vektor, giving both lenth ( inkluding lenth of tIm ) and direktion. In our tw-dimensional exampl, any point on a playn, or shyt of paper, kud be indikated by this myns.

The saim kan be don with komplex numbers. Thys ar an extension and komplytion of the number system, with ruls of kounting som-wat diferent from traditional numbers. Komplex numbers ar taut to yung children as part of the 'nw maths' in skwls.

The tw observers ych deskrib'd the position, P, by a complex number: x + ict or x' + ict' ( wich may be set equal to z and z', respektivly. ) This ment that, in this instans, the tw komplex numbers wer equal. Thys komplex numbers ar in the system of rektilinear ko-ordinats. This works lIk a net. Depending on how fIn the net is, yu can mark any position, on a stretch of ground it kovers, by wich noud or not in the net lIs over it.

But the Interval diagram also yus'd another kind of ko-ordinat system, bekaus the tw rektangular ko-ordinat systems wer related to ych other, by shering the saim orijin, and having ther axes turn'd at a given angl to ych other. They also sher'd the saim hypotenus, a radius vektor, wich was the Interval itself.

A ko-ordinat system wich mesurs, in this kais a playn, by swyping out an angl, in a sirkl, with a radius, is kal'd a polar ko-ordinat system. The orijin is the 'pol'.

We kan also expres z and z' in polar ko-ordinats. From the Interval diagram, let the radius, from the orijin, O, pointing to position, P, be kal'd r.

Then, kos(b+a) = x/r.

i.sIn(b+a) = ict/r.

z = x + ict = r( kos(b+a) + i.sIn(b+a))

Thus, z is expres'd both in rektangular and polar ko-ordinats.

Similarly for z':

kos(b) = x'/r.

i.sIn(b) = ict'/r.

And z' = x' + ict' = r( kos(b) + i.sIn(b)).

A wel-nown result says the trigonometrik terms kan be replas'd by an equivalent term, using the exponent ( usualy riten 'e' or 'exp' ).

( So, komplex number, z', equals radius, r, tIms the exponent, e, to the power of an index, wich is the operator, i, tIms angl, b.
For, z, the only diferens is that the angl in the index is not angl, b, but angl, b+a. )

Thys varius expresions ar standard text bwk solutions for z or z' as a wav form, in a so-kal'd wav equation. Z kan be imajin'd as the point, P, in the Interval diagram, moving around the sirkl, hws sirkumferens it is on. This sirkular motion kan be charted, horizontaly on a graf, as a wav motion. The horizontal axis kan represent the laps of tIm or distans.

The ( positiv part of the ) graf's vertikal axis represents the hIt of the wav. Its ful hIt or krest is kal'd the amplitud. It is the saim lenth as the radius. Wen the radius ryches the northern vertikal position, on the Interval diagram, this koresponds to a krest of the wav. From ther, for every ful sirkl, the radius swyps out, the graf of the wav ryches an other krest.

Our Interval diagram only shows the positiv quadrant of the sirkl ko-ordinats. But wen the radius ryches the southern vertikal position, wer the ict-axis has chanj'd to -ict on the sirkl, this koresponds to a trof of the wav being drawn out on the graf.

This is wI a solution of the wav equation is typikly given as a funktion of: x ct. ( Wether the wav equation has a komplex solution, kontaining i, depends on sertain konditions, wich qualify the shaip of the wav. Thys ar taken into akount in the solution of a wav equation as a form of diferential equation. )

Both z and z' mark an event, P. Presumably, the tw observers kud both trak P, if it wer to deskrib a sirkl in spas-tIm, on the Interval diagram. In doing so, the tw observers wud trais out ther respektiv z and z' wav forms on the graf. But thys wav forms wud not be synkronis'd, bekaus ther is a diferens of angl, a, betwyn ther ko-ordinat systems.

This diferens of angl is kal'd the feiz. A diferens of feiz, betwyn fysikal wavs, is responsibl for interferens efekts, such as the paterns made by koliding water wavs.

On the Interval diagram, an angl, a ( kolor-koded yelow ) shows the tw observers' spatial axes, x and x', out of feiz to that degry. Konsequently, ther temporal axes, ict and ict', ar also out of feiz by angl, a ( kolor-koded gryn ).

In jeneral, this situation aplIs to the Interval as a for-dimensional spas-tIm. So, yu kud karakteris diferent observers, of relativistik efekts, as having ko-ordinat systems, with spas-tIm feiz diferenses from ych other.

It is only a feiz they ar going thru.


For the Michelson-Morley experiment, I relI'd on:

Sir James Jeans, The New Background Of Science. Scientific Book Club, 1945.

Isaac Asimov, Asimov's Guide To Science, Vol. 1, The Physical Sciences. Penguin 1972. The apendix also kontains a simpl derivation of E = mc.

Arthur Beiser, Concepts Of Modern Physics. McGraw-Hill 1973. Chapter 1.

For the Lorentz transformations, I kan not remember the first bwk that interested me in them, but I hav red apreciativly, on the way:

Milton Rothman, The Laws Of Physics. Penguin 1963.

Allan M Munn, From Nought To Relativity. Allen and Unwin 1974.

Paul Davies, Space And Time In The Modern Universe. Cambridge University Press 1977. This kerfuly explains away the twin 'paradox'. Davies has riten a gud many popular fysiks bwks sins then, most of wich I'v red.

For the Minkowski Interval, I relI'd on:

George E Owen, Fundamentals of Scientific Mathematics. The John Hopkins Press 1961. How-ever, I didnt folow the text's folowing of Lorentz's yus of matrises to ariv, formaly, at his transformations, but trI'd to mak do with trigonometry.

Richard Lung.

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Special relativity ( part 3 ): The Lorentz transformations.
An arithmetik exampl of the los of simultanus tIm betwyn observers in relativ motion komparabl to lIt speed.

For koments, kritisisms or korektions, regarding this paj, plys replI to the e-mail adres on my hom paj.

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