Septem b er 2018

Lecture Notes for 8.225 / STS.042, “Ph ysics in the 20th Cen tury”: Electro dynamics for Maxw ell and Loren tz

Da vid Kaiser

Center for The or etic al Physics, MIT

Lecture Outline:

1. W a v es for Maxw el l and Loren tz

2. The Mic helson-Morley Exp eri men t

3. Loren tz Con traction

In tro duction

T o d a y w e are going to con tin ue exa mining elec tro dynami cs the w a y that go o d mathematical ph ysicists in th e late 19th cen tury did, to consider ho w they approac hed certa in problems. This will ma k e the con trast with y oung Alb ert Einstein’s approac h, to whic h w e w ill turn in the next lectu re, more clear.

T o b egin, w e need to go bac k to a cen tral conclusion that James Clerk Maxw ell reac hed in the 1860s: all of spa ce is filled with a ph ysical, material substance (the “luminiferous ethe r”), and ligh t w as simply the “transv erse undulation” of electric and magnetic field s within the ether. Ho w did Maxw ell arriv e at this conclusion? Reviewing Maxw ell’s approac h will mak e more clear ho w la ter ph y s ic ists, suc h as the Dutc h mathematical ph ysicist Hendrik

A. Loren tz, sough t to w ork within Maxw ell’s program and extend it further. W e will see that Loren tz had at least t w o distinct c hallenges in mind when he b egan w orking on the electro dynamics of mo ving b o dies in the 1880s and 1890s.

1 W a v es for Maxw ell and Loren tz

1.1 Maxw ell’s “ T ransv erse Undulations” in the Ether

W e b egin with a quic k review of wave physics , and ho w Maxw ell’s equation s for electricit y and magnet ism led him to think ab out ligh t as w a v es of electric and magnetic fields. F or the sak e of brevit y , w e will write Maxw ell’s equations in their mo dern form, using the v ector

notation that a later Maxw ellian — Oliv er Hea viside — in v en ted in the mid-1880s sp ecifically to mak e Maxw ell’s equations easier to manipulate: 1

∂ B

r × E = − ∂ t ,

∂ D

r × H = ∂ t + J , (1)

r · D = 4 π ρ,

r · B = 0 .

Here E and B are the electric and m agnetic fields, and D and H are the electric displac emen t and magnetic field strength, resp ecti v ely . The fields D and H quan tify ho w a giv en material is affecte d b y ex ternal electric a nd magnetic fiel ds. In general, the fields D and H are related to E and B b y the prop erti es of the me dium or material:

D = ϵ E , B = µ H , (2)

where ϵ is the “dielectric constan t” and µ is the “magnetic p ermeab ilit y .” T o us, to da y , ϵ and µ quan tify ho w readily the in ternal constituen ts within a material — including its electrons, ions, and/or clusters of suc h ob jects, suc h as electric and magnetic dip oles — can reorien t themselv es when an external electr ic or magnetic field is applied. 2 T o Maxw ell and his British colleagues, on the other hand, the parameters ( ϵ, µ ) w ere related to the elastic prop erties of the underlying ether or other materials; that is, they w ere analogous to spring c onstants , indicating ho w quic kly a gi v en material could dissip ate stresses or tensions that arose from the presence of e lectromagnetic fields. 3

Maxw ell calculated ho w quic kly electric and magnetic disturbances w ould propagate within the ether — that is, in regions in whic h there w ere no external sources of c harge or cu rren t (so that ρ = J = 0). Being a highly t rained Cam bridge W ran gler, he had prac - ticed ho w to use v a rious ma thematical iden tities among the differen tial op erators, suc h as the follo wing handy iden tit y: for a n y v ector F , one ma y write 4

r × ( r × F ) = r ( r · F ) − r 2 F . (3)

He then found that within an otherwise empt y region of space that con tained only ether,

1 See Bruce Hun t, The Maxwel lians (Ithaca: Cornell Univ ersit y Press, 1991), 245-247.

2 See, e.g., Da vid Griffiths, Intr o duction to Ele ctr o dynamics , 4th ed. (New Y ork: Cam bridge Univ ersit y Press, 2017), c haps. 4 and 6.

3 See esp. Je d Buc h w ald, F r om Maxwel l to Micr ophysics (Chicago: Univ ersit y of Chicago Press, 1985); and Olivier Darr igol, Ele ctr o dynamics fr om A mp ` er e to E i n s t ei n (New Y ork: Oxford Univ ersit y Press , 2000). 4 On the training of Cam bridge W ranglers, see esp. Andrew W arwic k, Masters of The ory: Cambridge and

the R ise of Mathematic al Physics (Chicago: Univ ersit y of Chicago Press, 2003).

ƒ ƒ

x / λ t / T

Figure 1: The function f ( t, x ) of Eq. (6) is a solution to Eq . (5) in one spatial dimension. S ho wn here is f ( t, x ) with A = 0, B = 1. A t a giv en momen t i n time, the function f ( t, x ) os cillat e s in space with a c haracteristic w a v elength λ ( left ); at a giv en lo cation in space, the fu nction f ( t, x ) oscillates o v er time with a c haracteristic p erio d T ( right ).

the electric an d magnetic fields w ould eac h ob ey an equation of the same form :

2 ∂ 2 2 ∂ 2

where ϵ 0 and µ 0 (to Maxw ell) w ere the corresp onding “spring constan ts” of the ether itself.

Wh y did Maxw ell in terpret this result in terms of the pr opagation of light ? Becau s e (as he knew from his Cam bridge T rip os training) this is the general form of a wave e quation , whic h generically tak es the form:

1 ∂ 2

r − v 2 ∂ t 2

f ( t, x ) = 0 , (5)

where v is the sp eed of the tra v eling w a v e. Wh y is Eq. (5) kno wn as a “w a v e equation”? F or simplicit y , let us consider motio n in a single directi on of spac e, so that x → x . Then w e see that solutions of Eq. (5) ma y b e writt en in the form

f ( t, x ) = A sin( k x + ω t ) + B cos( k x + ω t ) , (6) where A and B are c onstan ts, and w e ha v e in tro duced the wavenumb er ( k ) and the angu lar

fr e quency ( ω ) as:

2 π 2 π

k λ , ω ≡

, (7)

T

in terms of the wavelength ( λ ) an d the p erio d ( T ). F rom Eq. (6), w e see that solutions to the w a v e equation of Eq. (5) simply oscil late in space and tim e. Moreo v er, the quan tities λ and T are related to the sp eed of the w a v e, v , whic h app ea rs in Eq. (5):

When Maxw ell applied the quan titativ e v alues for ϵ 0 and µ 0 that had b een inferred fr om v ari ous electromagnetic exp erimen ts, he found that

1

ϵ 0 µ 0 ' c 2 , (9)

where c ' 3 × 10 5 km / s w as the sp eed o f ligh t t hat researc hers had indep ende n tly inferred from v arious astronomical phenomena. That is what con vinced Maxw ell that ligh t consisted of “tra nsv erse undulations” of electric and magnetic fields in th e luminiferous e ther: the

fields E and B ob ey ed the simp le w a v e eq uation of Eq. (4), with the sp eed of the w a v es giv en b y v = 1 / √ ϵ 0 µ 0 = c . 5

1.2 H. A. Loren tz: Generalize for Mo ving Sources or Receiv ers

Maxw ell’s deep insigh t, whic h united optics with electricit y and m agnetism, had b een ac - complished b y assuming that b oth th e source of ligh t an d the receiv er of the ligh t w ere at rest with resp ect to the ether. Beginning in the 1880s, the great Dutc h ma thematical ph ysicist Hendrik A. Loren tz tried to generalize these re s u lts to the case in whic h either the emitter or receiv er of ligh t w as in motion relativ e to the ether. Hence Loren tz and his con temp oraries w ere concerned with the general problem of “the electro dynamics of mo ving b o dies.” Loren tz had b oth mat hematical and exp erimen tal motiv ations fo r his w ork. W e will first consider the mathematical side.

Loren tz knew ho w to relate the co ordinates of differen t frame s of refe rence whic h w ere mo ving with a constan t sp eed v with resp ect to eac h other: these w ere simply the co ordinate transformations that Galileo had first form ulated in the early 1600 s , and that Newton had co dified in the late 1600s:

x 0 = x + v t , t 0 = t. (10)

(Reference frames that mo v e with a constan t sp eed are called “inertial frames of reference.”) Note, in particular, that there w as no c hange for the time co ordinate: for Galileo and Newton (and their follo w ers righ t through the end of the nineteen th cen tury), time w as simply time, and all observ ers should agree on the rate at whic h time passes. 6 (Newton had famously written in the op ening pa s sag es of his great Principia Mathematic a in 1687 that “Absolute, true, and mathematical tim e, of itself, and from its o wn n ature, flo ws equ ably without relation to an ything external, and b y anot her nam e is called duration.” 7 In short: time is absolute.)

5 See, e.g., Darrigol, Ele ctr o dynamics fr om A mp ` er e to Einstein , 151-53, 162.

6 See, e.g., Alb erto A. Mar t ´ ınez, Kinematics: The L ost Origins of Einstei n ’s R elativity (Baltimore: Johns Hopkins Univ ersit y Press, 2009).

7 Isaac Newton, The Principia: Mathematic al Principles of Natur al Philosophy , trans. I. Bernard Cohen and Anne Wh itman (Berk eley: U n iv ersit y of Calif ornia Press, 1999), 408.

When Loren tz used this Galilean co ordinate transformation and lo ok ed at what the resulting equation for the E and B fields w o uld lo ok lik e in the mo ving frame, he found (restricting to a single spatial dimension for simplicit y):

" ∂ 2 1 ∂ ∂

2

0 0 0

∂ x 0 2 − c 2 ∂ t 0 − v ∂ x 0

E ( t , x ) = 0 , (11)

and a similar eq uation for B 0 ( t 0 , x 0 ). But Eq. (11) do es not lo ok lik e the sim ple form of a w a v e equation; in particular, solut ions to Eq. (11) w ould not b e simple sines and cosines, akin to the w a v e b eha v ior w e found in Eq. (6). 8

So what? Loren tz knew that w e (on the moving Earth) often do observ e ligh t b eha v ing as sines and cosines. So if the ethe r is real, a nd the Earth is really mo ving through it, ho w could it b e that w e observ e optical phenomena b eha ving as if the E and B fields ob ey ed Eq. (4) rather than Eq. (1 1)?

This presen ted Loren tz with a n imp ortan t mathematical c hallenge within the Maxw ellian tradition: ho w to bring the new equati ons for E 0 ( t 0 , x 0 ) and B 0 ( t 0 , x 0 ), whic h w ould hold in a mo ving reference frame, in to the familiar form that had w ork ed so w e ll when describing optical exp erimen ts.

Loren tz w as a v ery talen ted math ematical ph ysicist, and he realized th at if he in tro duce d a new set of h yp othetical rules for co ordinate tr ansformations — distinct from the familiar Galilean transformation — he could put the w a v e equation for E ( t 0 , x 0 ) and B ( t 0 , x 0 ) in the same form as for the stationary case in Eq. (4). In particular, if the new time co ordin ate, t 0 , b ecame a function of x , t , and v , just as x 0 transformed as a function of x , t , and v , then Loren tz could preserv e the fo rm of the w a v e equation. He called the new co ordin ate t 0 = t 0 ( x, t, v ) the “lo cal time,” and considered it little more than a mathematical trick : in the end , Loren tz concluded, t 0 w ould alw a ys b e referred bac k to the gen uine, absolute time t of the ether rest frame.

This w as Lore n tz’s mathematic al resp onse to the quandary o f the electro dynamics of mo ving b o dies: to alter the Galilean co ordinate-transformation rules in order to sa v e the form of a particularly imp ortan t set of equations. W e’ll see in the next s e ction that Loren tz also had an exp erimental reason for adop ting the new transformation rule as w ell.

8 See, e.g., Darrigol, Ele ctr o dynamics fr om A mp ` er e to Einstein , 327-28; and Arth ur I. Miller, A lb ert Einstei n ’s Sp e cial The ory of R elativity: Emer genc e (1905) and Early Interpr etation (1905-1911) , 2nd ed. (New Y ork: Springer, 1998), 18-37.

2 The Mic helson-Morley Exp erimen t

Lik e all go o d mathematical ph ysicists of the era, Loren tz knew that ligh t propagated in the ether: the ether w as the medium in whic h ligh t w a v es tra v eled. Hen ce there w as a natural question whic h o ccupied the efforts of m an y ph ysicists in the late 19th cen tury: could one detect the Earth’s motion thr ough this all-p erv asiv e ether? Clea rly the Earth mo v ed around the Sun, and it seemed extremely unlik ely that the ether w o uld b e mo ving exactly along with the Earth — so there m ust b e some relativ e motion b et w een the stationary ether and the mo ving Ea rth. Add on top of this the fact that the ethe r supp orted ligh t in its tra v els, and some researc hers sough t to use the b eh a vior of ligh t to measure the Earth’s motion through the ether. This w as the exp erimen ta l con text in whic h Loren tz sough t to extend Maxw ell’s study of ligh t and ether.

2.1 The In terferometer

The most sensitiv e exp erimen t p erformed to test for the Earth’s motio n through the ether w as devised and conducted b y the American exp erimen tal ph ysicist Alb ert Mic helson in the 1880s. Mic helson w as w orking at what is no w kno w n as Case W estern Reserv e Univ ersit y in Clev eland, Ohio, and w as already a reno wned exp ert in optics; in fact, he w as the first US-based ph ysicist to win the Nob el Prize in Ph ysics, whic h he recei v ed in 1907. A t the time, his resea rc h w as among the v ery few efforts in the US that Europ ean-based ph ysicists p aid an y atten tion to; the scien ti fic comm unit y in the US otherwise still seemed lik e a bac kw ater compared to t he leading cen ters in W estern Europ e. 9

Mic helson’s most significa n t con tribution w as to u s e the interfer enc e of light to test for the Earth’ s motion through the ether. T o understand his new instrumen t, the interfer ometer , it will help us to first consider t w o swimmers in a riv er. Imagine that eac h swimmer can only swim at a constan t sp eed c with resp ect to the w ater; mean while, the riv er flo ws with resp ect to the shore with a constan t sp eed v . One swimmer swims directly against the cu rren t for a length L , and then for he r return trip she swims directly with the curren t. The other swimmer swims across the riv er a length L and th en bac k. The question: who wil l win the r ac e?

F or eac h leg of eac h swimm er’s journey , w e kno w that

distance

time = . (12)

sp eed

F or the first leg of her journey , the first swimmer has a sp eed relativ e to the shore of ( c − v ),

9 See Daniel J. Kevles, The Physicists: The History of a Scientific Community in Mo dern A meric a , 3r d ed. (Cam bridge: Harv ard Univ ersit y Press, 1995), 27-29.

Figure 2: An analogy for the Mic helson-Morley in terferomete r: t w o swimmers race in a riv er w i th curren t v . They eac h swim at sp eed c with resp ect to the w ater. The first wimm er starts at p oin t A and swims a distance L to p oin t B , then swims bac k to A . The second swimmer starts at p oin t A and swims a distance L across the riv er to p oin t C , then swims bac k.

b ecause s h e is figh ting the riv er ’s curren t. So her time to swim the length L from lo cation

A to lo c ation B is

L

t AB = ( c − v ) . (13)

F or her return trip fr om B to A , her sp eed as measured from the shore is ( c + v ) — she pic ks up the sp eed of the riv er. So her time to swim from B bac k to A is

L

t B A = ( c + v ) . (14)

The total tim e for her l ap is then

L L 2 L 1

t AB A = ( c − v ) + ( c + v ) =

c 1 − � v 2 . (15)

The second swimmer m ust set off at a diagonal to offset the effects of the riv er’s curren t. Since the riv er is a distance L across, she will swim along the diagonal R in time t AC . During that time, the riv er’s curren t will push her straig h t do wn a le ngth v t AC . W e ma y t hen analyze her path in terms of a righ t triangle, and use the Pythagorean theorem:

R 2 = ( v t AC ) 2 + L 2 . (16)

But remem b er that she swims at sp eed c with resp ect to th e w ater, so R = ct AC . Substituting

γ

0.0 0.2 0.4 0.6 0.8 1.0 v / c

Figure 3: The function γ of Eq. (19). F or v c , γ ' 1, but γ 1 as v → c .

in to Eq. (16), w e then ha v e

( ct AC ) 2 = ( v t AC ) 2 + L 2

L 1 (17)

− → t AC =

.

c v 2

c 2

Her return trip is symmetrical, so w e find t C A = t AC , and hence

2 L 1

Let us define

t AC A = t AC + t C A =

1

c q 1 − �

. (18)

v 2 c 2

γ ≡ q

� 2

(19)

v

c 2

W e note that γ ≥ 1 for all 0 ≤ v ≤ c .

So who wins the race? W ritten in terms of γ , w e ha v e found

W e just sa w that for v / = 0, γ > 1, and hence if there is a cu rren t in t he riv er, then swimmer 1 tak es a longer time to return to the starting p oin t than do es swimmer 2. So swimmer 2, who set off across the riv er alo ng the diagonal, wins the race. Moreo v er, w e see that the difference in t heir times is

Δ t = t AB A

— t AC A

= 2 L γ ( γ − 1)

L v 2 v 4 (21)

The race is close: the difference in the t w o swimmers’ lap times is se c ond or der in the ratio ( v /c ); hence this race is sensitiv e to “second-order” effects in ( v /c ).

Alb ert Mic helson realized that this though t exp erimen t in v olving swimmers in a riv er with a curren t flo wing should hold true for ligh t as measured on a mo ving Earth as w ell. Lik e the swimmers, ligh t will tra v el a t a constan t sp eed through the stationary ether, at sp eed c . But if the Earth is mo ving through the ether, there should b e an “ether wind” that w e w ould feel on our mo ving platform — lik e the wind y ou feel on y our face when y ou ride a bicycle through calm air. When y ou’re at rest with resp ect to the air, y ou don’t feel an y wind, but as so on as y ou b egin to mo v e with resp ect to the air , y ou feel the wi nd on y our face. Mic helson built his interfer ometer to m easure this ether wind on Earth. One of the arms of the i n terferometer w ould b e aligned with the Earth’s motion, and he nce ligh t b eams tra v eling along that route w ould b e lik e the first swimmer, swimming directly a gainst the riv er’s curren t, and then directly with it for the return trip. The second arm of the in terferometer w ould b e p erp endicular to this ether wind; ligh t ra ys tra v eling along that arm w ould b e akin to th e swimmer who set off across the riv e r.

Just lik e the swimmers, ligh t that set off along the second arm (p erp endicular to the ether wind) should ther efore return to the starting p oin t b efor e ligh t from the first arm did. If the li gh t w a v es that follo w ed these distinct paths to ok differen t amoun ts of time to return to the starting p oin t, the n they w ould arriv e out of phase with eac h other: crests of one ligh t w a v e w ould not line up with crests of the second ligh t w a v e. Hence there should b e interfer enc e of the t w o w a v es — a pattern that w ould b e readily observ able as a series of brigh t and dark patc hes on a screen. The sp ecific in te rference pattern w oul d dep end on the magnitude of the tim e dela y b et w een the t w o ligh t w a v es, and hence on the Earth’s motion through the ether, v . Th us, reasoned Mic helson, he should b e able to measure precisel y the Earth’s sp eed through the stationary eth er.

He built a mo dest-sized instrumen t (with eac h arm ab out 1 meter long) and fou nd no in terference pat tern. But w e sa w in Eq. (21) that the time difference (and hence the d etails of the exp ected in terference pattern) scaled with L , th e length of the arms. So in 1887, Mic helson built a m uc h larger device, together with his assistan t, Edw ard Morley . Their new instrumen t had arms 11 meters long. They set the en tire apparatus floating on a h uge v at of mercury to damp en vib rations from the street , and they to ok careful measuremen ts o v er sev era l mon ths: sw apping the direction in whic h the arms p oin ted, lo oking for seasonal effects, and so on. 10

10 Gerald Holton, “Einstein, Mi c helson, and the ‘crucial’ exp erimen t,” in Holton, Thematic Origins of Scientific Thought: Kepler to Einstein , rev. ed. (Cam bridge: Harv ard Univ ersit y Press, 1988), 279-370; see also Darrigol, Ele ctr o dynamics fr om A mp ` er e to Einstein , 316-19.

Figure 4: Sc hematic of the Mic h e l s on in terferometer. Ligh t from a mono c hromatic s ou rc e ( S ) encoun ters a half-silv ered mirr or. Half of the ligh t passes through and tra v els a length L along path 1 b efore b eing reflected b y a mirror; half of that returning w a v e is reflected b y the half- silv ered mirror and reac hes the screen. In the m ean time, half of the original b eam is reflected b y the half-silv ered mirror and tra v els do wn the other arm of the in te r ferome ter for a length L along path 2, b efor e b eing reflected b y a mirror; half of that returning w a v e is transmitted b y t he half- silv ered mirror and reac hes the screen. If the ligh t w a v es fr om eac h path tak e differen t amoun ts of time b efore reac hing the screen, they will b e out of phase with eac h oth e r, rev ealing a c haracteristic in terference p atte r n on the scree n .

T o Mic helson’s sho c k and lifelong disapp oin tme n t, he nev er measured an y in terference pattern. His results w e re consisten t with the ra ce alw a ys b eing a tie, with Δ t = 0, time and time again. This w as a h uge puzzle: ligh t w as clearly a w a v e in the ether; the Earth w as clearly in motion through the ether; and Mic helson had built the most sensitiv e instrumen t ev er with whic h t o try to measure the effec ts of the Earth’s motion thr ough the ether. He considered himself a failure, ri gh t up to his death in 1927. (Ma y w e all win the Nob el Prize but remain un s a tisfied...)

3 Loren tz Con tra ction

The “n ull result” whic h Mic helson and Morley found time and time again b othered man y of the leading Europ ean theorists, m uc h as it b othered Mic helson. Hendrik Loren tz, in particular, lab ored to mak e sense of the exp erimen tal result. He concluded that t here m ust b e a physic al c ontr action of the en tire apparatus along the direction of motio n — and that this con traction should exactly offset what should ha v e b een a longer tra v el time for one of the ligh t w a v es. I f the arm of the in terferometer that w as mo ving directly in to the ether wind b ecame shortene d , then the ligh t w a v es that tra v ele d along that path w ould ha v e a shorter distance to tra v el, and hence they w ou ld arriv e bac k at the starting p o in t at the same time as the ligh t w a v es in the o ther arm; the race w ould b ecome a tie, and no in terference fringes should b e observ e d.

The main idea, whic h Loren tz first published in 1892 and to whic h he returned throughout the 1890s and early 1900s, w as th at the ph ysical material making up the apparatus actually shrunk (along the direction of motion) du e to the resistance of th e ether. Loren tz cautiously noted in his pap ers and in corresp ondence from the time th at it w as “not inconceiv able,” and “not far-f etc hed to supp ose” suc h a ph ysical deformation due to the ether. 11

F or one t hing, the effect w ould b e small — remem b er, Loren tz w as trying to accoun t fo r a se c ond-or der effect, prop ortional to ( v /c ) 2 . T o get an estimate for the exp ected size of suc h an effect, Loren tz appro ximated v (for the Earth’s motion through the ether) based on the Earth’s sp eed through the Solar System as it orbits the Sun: v ' 2 π r /T ∼ 10 11 m / (10 7 sec) =

10 4 m / sec. Giv en the sp eed of ligh t c = 3 × 10 8 m / sec, this yielded

v 2 ∼ 10 − 8 , (22)

meaning that the effect w ould b e ab out one part in a h undred m illion!

11 See, e.g., Darrigol, Ele ctr o dynamics fr o m A mp e ` er e to Einstein , 327- 28. The Irish mathematical ph ysicist George F. FitzGerald h ad made a similar suggestion in 1889, though Loren tz ap p ears not to ha v e kno wn ab out FitzGerald’s short pap er at the time.

Moreo v er, Loren tz reasoned b y analogy that one could see all kinds of ph ysical defor - mations when b o dies tra v eled through viscous, resistiv e media , suc h as a b eac h ball b eing dragged at high sp eed un der w ater: the b eac h ball w ould b ecome squeezed a long the direc - tion of motion. If the ether w as a ph ysical, elastic medium, then h yd ro dynami cal analogies lik e this should hold — leading to an exp ectation that the arm of the in terferom eter really should b ecome shorter along the direc tion of motion.

Loren tz therefore suggested his h yp othesis of length c ontr action : p erhaps the length of ob jects shrunk b y a small amoun t along their directions of motion through the ether, to b e

L 0 = L , (23)

γ

where γ w as the factor defin ed in Eq. (19). In that case, then the time for the first swimmer’s lap w ould b e adjusted. Instead of t AB A of Eq. (15), her return-trip time w ou ld b e

0

AB A

2 L 0

γ 2

c

= 2 L γ 2

c γ (24)

2 L

= γ

c

= t AC A ,

and hence Δ t = 0 — the race w ould b e a tie, afte r all, an d no in terference fringes w ould b e detected within Mic helson’s in terferometer.

Moreo v er, as w e sa w earlier, Loren tz had already figured out fr om his mathematical analysis that in order to retain the prop er mathematic al form for the w a v e equation for l igh t, he w ould need to c hange b oth his spatial co ordinates and his time co ordi nate, to in tro duce “lo cal time.” In p lace of the usual Galilean transformation of Eq. (10), Loren tz found what w e no w call th e “Loren tz transformations”:

x 0 = γ ( x + v t ) ,

t 0 = γ t + v x ,