IL NUOVO CIMENTO VoL. X X I , IN. 5 1 o Settembre 1961

The Definition of Electromagnetic Radiation ( ').

F. ROHRLICH

Department o] Physics erred Astronomy, State U~iversity o / Iowa - Iowa City, Ia.

(ricevuto il 6 Luglio 1961)

Summary. - - Momentum and energy of electromagnetic radiat ion emit ted from a charge are Lorentz-covariantly defined. The non-locM measu- rement of a Lorentz invariant , the rate of radiation energy emission is proposed as a necessary and sufficient criterion for the emission of radiat ion at a given time. This measurement is a field measurement which can be performed at any distance from the charge and need not be made in the wave zone. I t can be regarded as an operational definition of radiation in terms of rate of energy emission. These results permit a simple derivation of the equation of motion of a charged particle which avoids the self-energy difficulties. A question concerning the principle of equivalence and radiation in uniformly accelerated motion is answered.

1. - S t a t e m e n t of the prob lem.

D e s p i t e t h e generM consensus of op in ion t h a t c lass ical e l e c t r o m a g n e t i c t h e o r y

is well u n d e r s t o o d , t h e r e seem to be a n u m b e r of p r o b l e m s which a re r e p e a t e d l y

e n c o u n t e r e d , a n d wh ich ra ise f u n d a m e n t M ques t ions . One such p r o b l e m m i g h t

be b r o a d l y l abe l l ed ~ the de f in i t ion of r a d i a t i o n ~>. I t is c h a r a c t e r i z e d b y the

fo l lowing ques t ions :

1) is r a d i a t i o n a L o r e n t z - i n v a r i a n t p h e n o m e n o n ? I f one i ne r t i a l ob-

se rve r sees r a d i a t i o n , do a l l i n e r t i a l obse rve r s see i t , such t h a t t h e r e is no

L o r e n t z t r a n s f o r m a t i o n b y which r a d i a t i o n can be ~ t r a n s f o r m e d a w a y ~?

2) I n o r d e r to t e l l w h e t h e r a s y s t e m r ad i a t e s , is i t neces sa ry to go to

v e r y l a rge d i s t a n c e ( c o m p a r e d to t h e w~ve leng th )? A n d if not , w h a t m e a s u r e -

m e n t s a t sma l l d i s t a n c e can d e t e r m i n e t h e p resence of r a d i a t i o n ? Is th is meas -

u r e m e n t u n a m b i g u o u s ? I s i t L o r e n t z - i n v a r i a n t ?

(*) Supported in par t by the National Science Foundation.

812 F. ROIIRLICII

The first question was raised recently by SYNGE (1). I t was ~nswered to

a large extent by SCmLD (~). His invariance proof is basic to our later argu-

ments.

The second question has recently played a crucial role in certain arguments

concerned with the principle of equivalence (3). In these arguments it was

taken for granted that the presence or absence of radiation cannot be deter-

mined at arbitrarily small distance from an accelerated point <.har~'e. Since

it will be shown in Section 3 that this is not the case, some remarks will have

to be made concerning the principle of equivalence.

Finally, the eovariant formulation of radiation permits a simple deri-r

of the equation of motion.

2. - Th e m o m e n t u m f o u r - v e c t o r of radiat ion .

Given a point charge e of known trajectory z~'(T), where ~ is the proper

time, with velocity v ~-- d z ' / d v and acceleration a~-- - dv~'/dT. The electromag-

netic fields produced by the charge's motion, F "', are measured at a, point P

with co-ordinates x ~. Assuming for definiteness retarded fields (advanced fields

would do just as well), the point P is uniquely correlated with a point (2 on

the world-line .z"(z), such that R ~ ~ x U - - z ~' is a null-vector, R,R"----0. The

choice of retarded fields specifies furthermore R ~ 0 (we use a Minkowski

metric ~"" with signature +2 , and we measure time in light-seconds).

The inw~riant distance

(2.1) o~ - - - v~R ~ 0

can now be defined. I t is equal to the spatial distance between P and Q in the

special inertial system Sq in which Q is instantaneously at rest. v ~ will always refer to the retarded point Q.

We can define a, space-like vector n '

1 (2.2) )~u :-- R~ - - ,~,l,

9

which has the properties

(2.3) nzn ~ --~ 1 , n;v ~" = 0 ,

(1) j. l,. SY~('E: Relativity: The Special Theory (Amsterdam and New York, 1956). (2) A. SCmLD: Journ. Math. Analysis and .lpl)l., 9, 127 (1960), (a) T. I'~H~TOY and F. ROHRLICIt: Ann. Phys. , 9, 499 (1960).

T I l E I ) E F I N 1 T I ( ) N OF E L E C T R O M A G N E T I C R A D I A T I O N 813

as (,an eas i ly be ver i f ied. R" can lhe re fo re be w r i t t e n

(2.4) R f ' : o ( # ' + d') .

In th is n o t a t i o n the l A 6 n a r d - W i e c h e r t l )o ten t ia l s are (4) (Gauss ian un i t s ) ,

(2.5) W'(x) = ~ 'v" / , 2 .

So far on ly the nu l l -vec to r ( 'hara( ' te r (it' R ~' was spe(.ifie(l.

van( ' ed p o t e n t i a l s a re to be used, one wri tes

If retarded or ~d-

( 2 .~ ) 1~,' - ( ~-/ , ' , R ) , /,' = I R [ ,

where the u p p e r (lower) s ign refers to the r e t a r d e d ( advanced ) case. As iilell-

t i o n e d above , we shal l use r e t a r d e d p o t e n t i a l s in t he folh)wing.

F r o m (2.5) one o b t a i n s for the r e t a r d e d fields t he a n t i s y m m e t r i ( , t en so r (4)

2 r 2 t ' (2.7) FI'"(x) - - t~l l ' i '''l - [ r l / ' a "1 i - # l ' ( ' r ' l a . a"~)] .

(12 O

where a,, ~ a~n" a,nd the a n t i s y m m e t r i z ~ t i o n symbo l is as usua l def ined as

a[t~b"J �89 (a"b"-- a"M) .

The first t e r m in (2.7) is tht, (tz'eneralized) [ ' o u l o m b field, t he se(.ond t e r m ,

whi(.h va,nishes if and only if the a( ' ( 'e lerat ion ve( ' tor van ishes , is the r a d i a t i o n

field, b y def ini t ion.

The ele( . tronm,~'neti( . (qwr~,'y t enso r is def ined b y

(2.8) 4 ~ T t"' F~'aFa '' } 14 ~II"F'~F f~ .

Inser t in tz (2.7) in to th is express ion one f inds the e n e r g y t enso r a t a l)oinl P

due to the field emiss ion al the r e t a r d e d p o i n t Q on the wor ld l ine z~'(T) of a

p o i n t ( .barge e,

(,2 ~(J2 (2.!)) 4.~Tm'(a') (~tui/" r~r" �89 i - - [a,,ut'u" ~(:'(c")a,, + a")]

O 1 O 3

02 Uffi~,,(([2n (I).(~2") .

(b F. ROllltLI(:lI: 7'he r Elech'm~. published in W. E. I~mT'rlx and B. W. DowNs: Leeture,~' i*~ Theoretical Physics, vol. 2 (New York, 196(t). This refe- rem,e contains various details and derivatiolls of lhe basic eqm~tions used here.

814 F. ROHILLtCH

The velocity and acceleration vectors always refer to the point Q, and

(2.10) u l' ~- n" + v # .

Consider now a line element d r of the world line z~(r). This is the infinite-

simal invariant distance of two points Q and Q' on this line. If one draws

the forward light cones from Q and Q' and an arbi-

z~t~) t ra ry time-like cylinder surrounding the world line,

the light cones will cut a surface 2: from this cylinder

�9 (see Fig. 1). Z is a three dimensional surface with

space-like surface normal. I t forms a band around

the world line, which lies between the two light

cones.

One defines the ~our quantities

Fig. 1. (2.11) dP" = -- f T,~ dacr~.

2:

In a recent paper (2) SCHILD has shown that the conservation law

~l,l't'~ = 0

which holds in charge-free regions implies tha t

(2.12) dP*' ~ lim dP ~

is independent of the surface and is consequently a four-vector. The notat ion

Z--> oo is symbolic to indicate tha t the point of closest approach of 27 to the

world line, ~=i., is taken to the limit. In going to the limit, Z is moved out-

wards always remaining between the two light cones; the limit therefore cor-

responds to a simultaneous limit in space-like and time-like directions.

Schild's proof is simply an application of Gauss' theorem to the above

conservation law. But it is essential that the four-volume involved is a doubly

connected (annular) region between two surfaces of the type Z, so that one

never crosses or even reaches the world line z ' ( r ) on which the fields "~re singular.

Let us take for 27 the special surface Zo which is a sphere in tile inertial

system in which Q is instantaneously at rest,

(2.]3) dan" = n" d r ~o 2 d ~ .

THE D E F I N I T I O N OF ELECTROMAGNI~TIC R A D I A T I O N 8 1 5

F r o m (2.12) a n d (2.9) we f ind

d P " e 2 V - - ~ - l i ra I ul '(aaa ~ - ct~,~)d~ : ~e2a~a~v I' . (2.14) d r 4re e_~ j

Zo

This f o u r - v e c t o r is t he t o t a l r a t e of ene rgy a n d m o m e n t u m emiss ion in fo rm

of r a d i a t i o n . W e no t e t h a t i t is a t i m e - l i k e vec tor . The i n v a r i a n t r a t e of e n e r g y

emiss ion is

dP / ' 2 (2.15) ' ~ - - rt' d r -=:3 e2a~a~'"

This e s t ab l i shes t he L o r e n t z i n v a r i a n c e of e l e c t r o m a g n e t i c r a d i a t i o n . ~ is non-

n e g a t i v e a n d van i shes if a n d on ly if t he f o u r - v e c t o r of a c c e l e r a t i on van ishes .

3. - T h e c r i t e r i o n for r a d i a t i o n .

W i t h the spec ia l sur face Xo one can c o m p u t e t h e r a t e of ene rgy a n d mo-

m e n t u m which crosses th is sur face in t he d i r ec t i on n f' p e r u n i t sol id angle ,

(3 . ] ) (lr dr2 - - 8~ ~2 ~- 4no (a~' (~,~tt') + ~ ~"(axa x - a~) .

This q u a n t i t y is o b v i o u s l y a fou r -vec to r . The first two t e r m s a re b o t h space-

l ike vec to r s a n d m a y be i n t e r p r e t e d as t he Cou lomb f o u r - m o m e n t u m a n d the

cross t e r m b e t w e e n Cou lomb a n d r a d i a t i o n fields. The l a s t t e r m (cf. eq. (2.10))

is a ~ u l l v e c t o r a n d desc r ibes pu re r a d i a t i o n , i t i s i n d e p e n d e n t o/ ~, b u t does

d e p e n d on the d i r ec t i on n z. The nu l l c h a r a c t e r of t he l a s t t e r m co r r e sponds

to t he m o m e n t u m f o u r - v e c t o r of a p a r t i c l e of zero mass (photon) .

Eq . (3.1) d e p e n d s on ly on the c o m p o n e n t of t he a c c e l e r a t i o n which is o r tho-

gona l to ~ ' , s ince

(a ~" - - ann;)(a~ - - a,~n~) = a; a ~ - - a~ .

I n the i n s t a n t a n e o u s res t s y s t e m of Q, Sr t he m o m e n t u m r a t e pe r un i t

sol id an~'le is

d P ez ~ e2 a~ e 2 (3.2) - - - ,

dtd':2 S~R 2 + ~1~- + ~ ~a~

816 F. ROtlRLICH

where the separation

a = a i l @ a , a l i - -a '~3, a~=laX~, ]

is made relative to ~. At small distances the Coulomb field dominates. I t

gives rise to a spherically symmetr ic rate of momentum change directed to-

wards the source. At intermediate distances the cross term contributes a

vector in the direction a • which lies in a plane orthogonal to ~. At large dis-

tances the radiation field dominates and yields the Poynt ing vector in the

direction ~, i.e. away from the source.

Tile energy rate per nnit solid angle in SQ is

dW e~ - - a 2 ( i n ,'~) (3.3) dtd.Q 4z~ -~' "

Thus, only the pm'e radiation term contributes to the energy rate.

One now observes tha t the two space-like terms in (3.1) are both orthogonal

to tile velocity vector v'. Thus,

dP" e 2

(3.4) - - r " (tTdD = 4~ (a~aX-- a~),

whi(.h is the invariant generalization of the energy rate (3.3). I t is the in-

var iant formulation of Poynt ing 's fornmla for the radiation energy rate per

unit solid angle.

While (3.1) is a four-vector, its integral over the angles is no longer a four-

vector unless the limit 2J-+ oo is taken, in accordance with (2.12). However~

(3.4) is independent of ~o, so that

d P ~ ' d P .

' ~ = - - % itT = - '% dr '

or, with (2.11) and ( 2 . 1 3 ) ,

(3.5) ~ ( T ) = - II~,.T" ' , ~o~ d~9. .J

I t follows that the invariant radiation energy rate Call be expressed as an

integral over the surfaee •o (sphere in the system SQ) which is independent of 9.

The integral in (3.5) gives the inw~riant result without need to go to the limit ~V~0 ---> C ~ "

This result permits one to establish a criterion for testing whether a charge

is emitt ing radiation at a given instmlt, by measuring the fields only, and without

T I l E I ) E F I N I T I O N O F E 1 , E C ' I ' R O M A ( I N I , ; T I ( ~ R A I ) I A T I ( ) N N ] 7

h a v i n g to do so a t a d i s t a n c e large comi )a red to the e m i t t e d wave length . This

( , r i ter ion for r a d i a t i o n can be e x p r e s s e d as follows.

Criterion for radiation: G i v e n the wor ld line of a ( ,harge a n d an a r b i t r a r y

i n s t a n t T0 o n i t . (~onsi(ler a sphere of arbitrary radius r in the i ne r t i a l s y s t e m S e

whieh is t he i n s t a n t a n e o u s res t sys ten l of the charge a t the p r o p e r t i m e T,,

( t ime to in SQ) a n d a t whose ( 'enter the i .harge will be a t t h a t i n s t a n t ; me a su re

t he e l ec t romagne t i ( , fields P ' " on ~,, a l the t i m e to + r ; e v a l u a t e the i n t e g r a l

, ~ ( T , ) : / T % ,r 2 (1.(2 : = �9 ~ d 2 ~ , in SQ

L' o

where S is the Poynt ing" ve(, tor. The va lue of this inteta'ral is t i le i n v a r i a n t r a t e

of r a d i a t i o n ene rgy emiss ion .# a / t i m e t0 a n d van i shes if a n d on ly if the (qmrge

d id n o t r a d i a t e a t t h a t i n s t an t .

Obv ious ly , th is c r i t e r ion is also an o p e r a t i o n a l def in i t ion ()f the i n v a r i a n t

r a d i a t i o n enero 'y ra te .

Severa l r e m a r k s are in o rde r a t th is t)oint. F i r s t , the m e a s u r e m e n t is ca, r r i ed

ou t in an ine r t i a l sys t ( 'm (No) a n d the invar ian( .e of .# refers to Lorenlz t r ans -

f o r m a t i o n only. An a(~celerated o b s e r w ' r m a y ve ry well measu re a di]ferent r a t e (or no r a t e a t all). Second ly , the poss ib i l i t y (if ascer ta in ing ' w h e t h e r a

charta'e r a d i a t e s w i t h o u t having ' to ~'~ l:o the wave z(ine is a c t u a l l y a logical

necess i ty , sin(,e the r a d i a t i o n e m i t t e d will in o'eneral c on t a in Four i e r ~'ompo-

nen t s of all wave leno'ths, so t h a t one wouhl never be in the wave zone of all

waves e m i t t e d . T h i r d l y , the n w a s u r e m e n t is t<) be m a d e over the surface of

a sphere (~0). Thus , a l t h o u g h the ra(tius of the sphere can be elmsen a r b i t r a r i l y ,

i t can never be zero. The n w a s u r e m e n t is the re fo re in t r ins i ( .a l ly ~mn-local.

4. - T h e e q u a t i o n of m o t i o n .

The a b o v e der iw~t ion was en t i r e ly based on t i le M a x w e l l - L o r e n t z eqm~tions.

But i t p e r m i t s one to go far b e y o n d these e l e c t r o m a g n e t i c field equa t ions .

One ('an, in fact , ~, a l m o s t ,~ de r ive t he equa t i ons of m o t i o n of a c h a r g e d I ) a r tM e

f rom the above resul ts , m a k i n g use only of the field e q u a t i o n s a n d the con-

se rwt t ion laws.

( 'onsi( ter a p a r t M e of mass m a n d charge e. An e x t e r n a l force F ~ is a c t i ng

on it. L ike all forces in spec ia l r e l a t i v i t y , th is is a space- l ike v e c t o r o r t h o g o n a l

to the ve loc i ty of the p a r t M e ,

(4 .1 ) P'~:# 0 .

I f the law of c o n s e r v a t i o n of m o m e n t u m a n d ene rgy is to ho ld du r ing e v e r y

i n t e r v a l of p r o p e r t ime d r , the e x t e r n a l force m u s t a c c o u n t for the ( revers ible)

8 1 8 F . R O H R L I C I [

increase in some m o m e n t u m vector pS as well as for the four-vector of energy and m o m e n t u m lost i rreversibly in the form of radiation,

(4.2) F~ = dp,__ § ~ v " , dT

according to (2.14) and (2.15). 5Tote t ha t p~' cannot be identified wi th the

kinetic m o m e n t u m my ~, because mult ipl icat ion of (4.2) with v ~ would yield

~ = 0. This has to do with the fact t ha t the radiation-loss vector is time-like.

The ~ ansatz ~ (4.2) so far contains only one well-defined term, viz. the

last one. The other two te rms are wri t ten down first of all because one

~, guesses ,> a certain s t ructure for an equat ion of motion. This s t ructure is

the covar iant general izat ion of the s t a t ement of energy conservat ion (corres- ponding to # = 0) per unit t ime. Since (4.1) implies tha t F" is of the f o r m

F ~ : (Tv ' F , ~'F) ,

the s t a t ement is t ha t the work per uni t proper t ime done by the impressed

force consists of two parts , one which is a to ta l t ime derivative, and the other one which is not a to ta l t ime derivat ive. This is a separat ion into reversible

and irreversible energy change. The rever~ible energy change and its eovar ian t

generalization will be called ~ inertial t e rms ~>. These te rms are uniquely de-

fined in (4.2) as the difference between F" and the radiat ion term. I t is assumed

tha t this difference is a tota] derivat ive, i.e. t ha t radiation is the only dissi- pative process.

This establishes the formal s t ructure of (4.2). Bu t a physical specification of F ~ and the inertial t e rm is still lacking. We now specify tha t F ~ should

not contain te rms which depend only on the mot ion (v ~, a ~, etc.) and the characterist ic propert ies of the particle (m, e, etc.). Such te rms would have to be ~ on the r igh t -hand side ~> of this equation. Thus, a t e rm of the form

2a" cannot be tolerated in F ~, but (if /~ is a constant) i t belongs into dp' /d~. These considerations imply t ha t one mus t regard the equat ion of mot ion {4.2)

as ~ ~lready renormalized ~ in the language of field theory.

In fact , the force F ~ is assumed to be entirely due to the external fields.

(See (4.5) and (4.9) below.)

Combining (4.1) and (4.2) one finds for the ra te of fou r -momen tum changes

- - = ~ e a ~ a = - d r d r '

o r

(4.3) d p , _ _~_ e 2 da~ dr ~ -~ v~ ,

T H E I ) E F I N I T I O N O1" E I , E C T R O M A ( ~ N E T I C R A D I A T I O N 819

u is a vec to r o r thogomd to ~,', which mus t be expressible in te rms where v•

of v ~ and its der ivat ives and m u s t itself be a to ta l der iva t ive wi th respect to v.

The equa t ion of mot ion (4.2) can now be wri t ten

(4.4)

Since F ~ is in general no t a pure ly e lect romagnet i ( , force,

(4.5) = F e l m ~ - Fneu t , ' a I '

this equat ion m u s t reduce to the equa t ion of mot ion of a neu t ra l particle,

(4.6) m a f ' = I~ 't' neutral ,

in the l imit e - + 0. (We exclude here for s implic i ty higher e lec t romagnet ic

mul t ipole moments . ) This requ i rement , t h a t (4.4) should become (4.6) in the

l imit e - * 0 is ve ry basic ~md was elsewhere referred to (s) as (~ the principle of unde tec tab i l i t y of sufficiently small charge ~.

F r o m this r equ i remen t one deduces.

(4.7) ~'~ = mcd ' + v .

r u

The vec to r v• canno t be de termined , unless one requires t h a t (4.4) should

be a differential equa t ion of order not higher t han th i rd order. I n t h a t case

v~ ~ = 0 and (4.4) becomes

d (4.8) U' ~ (m~"- ~ e~a,') § ~ , ; .

This is the desired equa t ion of mot ion . F r o m the discussion of (4.2) it is clear

t h a t m represents the observed rest mass of the par t ic le and F " is the (~ re-

normal ized ~> in teract ion, i.e., the inter~wtion due to ex terna l effects only. I t

is o r thogona l to v s, re tarded, and in general of the fo rm (4.5). �9 /~

F r o m these remarks follows t h a t Fel., nlust be g iven b y the r e t a rded external

fields only, for example,

(4.9) F~m ) '~ --~ ( ~ F e x t V v .

(5) F. ROIIRLICH: Ann. l 'hys. , 10, 93 (1961).

t~20 F. ROHRLICH

The Lorentz force (4.9) can be combined with (4.8) to yield

(4 .1o)

which is the Lorentz-Dirac (G) equation. The above derivation of the equations of motion (4.8) from the field equa-

tions involved only the conservation law, the principle tha t the limit e--> 0 #l

should result in (4.6), and the simplicity assumption v• = 0. The self-energy difficulties have been avoided, bu t the form of the interaction with the external

fields (e.g. (4.9)) is not specified by this derivation. Finally, it must be remarked tha t the equations of motion (4.8) or (4.10)

are third order equations and are therefore to be supplemented by a suitable asymptot ic condition which effectively reduces them to second order equations. At the same t ime it eliminates the run-away solutions. Together with this condition (4.8) becomes (5)

; e x p [-- "r'/'co][F"(r ') -- Y2('r')~,,'~(r')] d'r' , d2z,(r) IT/r0] e x p

(~ .11) h~-~ = 30 .

T

where To = w Only this equat ion can be regarded as a true equation of motion, since it is of second order and determines the world-line from the initial position and velocity. Precise conditions for the existence and uni- queness of solutions of (4.]1) were given by HALE and S~OKES (7).

5 . - C o n c l u s i o n s .

1) The rate of change (per unit proper time) of field momentum and energy dP"/d~d~2 present at a given distance from the charge, and due to fields produced by the charge at a given instant and into a given solid angle, is a four-vector, given by (3.1) and containing ~-2, Q-l, and Qo terms. The

first two a.re space-like vectors not t ransmit t ing energy like particles of real mass, m 2 ~ - - p s p ~ , while the last t e rm is a null vector representing radiation.

2) The total (i.e. integrated over all directions) rate of change of field momentum and energy, dP"/d~, on a given surface 27 surrounding the charge and emit ted at a given instant, is not a four-vector, The limit dP"/d~ obtained

(") •). k. -~I. DIRAC: l'roc.. Roy. Soc., 167, 148 (1938). (7) J. K. HALE and A. P. 81'OKES: O• some physical solutions o] Dirac type equations,

preprint, RIA8, Baltimore, Md. (April 1961).

T I l E D E F I N I T I O N O F E L E C T R O M A G N E T I C R A D I A T I O N 8 2 1

for 2~--> c~, however, is a four-vector (Schild's theorem). This four-vector is

t ime-l ike and parallel to the (retarded) velocity four-vector.

3) The invar ian t to ta l ra te of enertzy emission (an irreversible process) ~ , defined in (2.15), is a non-ne~'ative invar ian t and vanishes if and only if the accelerat ion four-vector wmishes. I t is due to the radiat ion fields only. This establishes radiat ion as a Lorentz-inw~ri~mt phenomenon and makes the meas- m ' emen t of ~ a= necessary und sufficient criterion of the presence of radiation.

4) This measuremen t of -~ is a ~wn-tocat one and can be carried out

as a field measurement over a suitable sphere of arbitrary radius (ef. Section 3). This establishes t lmt radiat ion fan be detected ~rnd measured unambiguously

a t a rb i t ra ry distanee f rom the enlittintt charge.

5) The relativistic equat ion of mot ion of a charge can be derived wi thout

encounter ing self-ener~'y difficulties by assuming

a) the Maxwell-Lorcntz equations;

b) the Lorentz inwu' iant conservat ion laws of energy and m o m e n t u m ;

c) the principle of umh, ctabil i ty of sufficiently small charges;

d) the equat ion of mot ion is at most of third order.

All these assmnptions were also used by DmAC (r

6) The invariance of .J2 ment ioned in (3) refers to Lorentz invarianee.

Transformat ion to a non-inertial system does not leave ~ invariant . In par- t i tu lar , t ransformat ion to a sui tably uniformly accelerated sys tem S' will make

vanish. The proof of this sl~,tcment follows f rom the fact tha t this t rans- format ion can be brou~'ht aboul by a eollfornl~l t ransformat ion (3). Such a transfomm~tion leaves Maxwell 's equations invar iant , so t ha t the radiat ion criterion hohls also in the nnifornfly ae~.elemted, non-inertial, sys tem S'. On the other tlund, the acceleration in tha t sys tem wmishes identically. This

establishes tha t a mfiformly accelerated (relative to an inertiM system) charge, while radiating at a r rate ;)2 as seen f rom an inertial sys tem S, will

not be seen to emi t radiation by the co-moving (non-inertial) observer S'

at a~y di,s'ta~wc f rom the ehar~'e (a).

Consequently, there is no contradict ion between electromagnet ic theory and

the principle of cqt~i~,ab,~tcc, even when the la t ter is referred to constant homo-

geneous t travitational fields which extend over finite space-t ime regions. The

recourse to the local character of the principle of equivalence used by BONDI

and GOLD and adopt~'d in reference (~) is therefore not necessary. This is

essential, because in homotz'encous ~TavitationM fields (which do exits over ]inite space-t ime re~'ions) the principle is no longer ~ local principle.

~r 5 3 - i I ht~ovo ('i~lu'Mo.

822 F. ROHRLIC~

R I A S S U N T 0 (*)

Si definisce in te rmini eovar ian t i secondo Loren tz l ' impulso e l 'energia della radia- z ione e le t t romagne t ica emessa da una carica. Come cri terio necessario e sufliciente per la misura del l ' emiss ione di una radiazione a un dato t empo si p ropone la misura non locale di un invar ian te di Lorentz . La misura consiste di u n a misura di campo che pub essere esegui ta a q u a l s i a s i dis tanza dal la carica e non occorre sia f a t t a nella zona delle onde. Si pub considerare come una definizione operazionale della radiazione in t e rmin i del tasso di emissione del l 'energia . Quest i r i su l ta t i consentono una sempl ice der ivazione de l l ' equazione del moto di una par t ice l la carica che p e r m e t t e di ev i t a re le diffieolts dovu te a l l ' au toenergia . Si r i solve un quesito concernente il pr incipio d ' equ i - va l enza e la radiazione nel moto u n i f o r m e m e n t e accelerato.

(*) Traduzione a cura della Redazione.