Septem b er 2020

Lecture Notes for 8.225 / STS.042, “Ph ysics in the 20th Cen tury”:

E = mc 2

Da vid Kaiser

Center for The or etic al Physics, MIT

In tro duction

Alb ert Einstein submitted his pap er, “On the electro dynamics of mo ving b o die s ,” to the A nnalen der Physik in June 1905; thi s pap er laid the groundw ork for what w ould b ecome kno wn as the sp ecial theory of relativit y . 1 So on after completing the pap er, Einstein wrote to his friend Conrad Habic h t, a fello w mem b er of his informal “Olympia Academ y”: “a consequence of the w ork on electro dynamics has suddenly o ccurred to me, namely , that the principle of relativit y in conju nction with Maxw ell’s fundam en tal equations requires that the mass of a b o dy is a direct measure of its ener gy con ten t — that ligh t transfers mass. [...] This though t is b oth am using and attract iv e; but whether or not the go o d Lord laughs at me concerning this notion and has led me around b y the nose — that I cannot kno w.” 2 Einstein submitted a short pap er to the A nnalen der Physik in Se ptem b er 1905 with the t itle, “Do es the inertia of a b o dy dep end up on its ener gy con ten t?” 3 It w as in this short pa p er — a mere 3 pages! — that Einstein deriv ed the no w-famous equation, E = mc 2 .

Einstein returned man y times o v er the y ears to this equation, offering a v ariet y o f simple r

w a y s to re-deriv e the result. In these short notes w e will consider one of his later deriv ations. W e will also briefly consider the concept of “relativi s t ic momen tum,” in terms of whic h w e find the more general expression of Einstein’s equation, E = γ mc 2 , where γ is the usual factor w e ha v e encoun tered sev eral times: γ ≡ 1 / 1 − ( v /c ) 2 .

1 A. Einstein, “Zur Elektro dynamik b ew egter K ¨ orp er,” A nnalen der Physik 17 (1905): 891-921. An English translation is a v ailable in John Stac hel, ed., Einstei n ’s Mir aculous Y e ar : Five Pap ers that Change d the F ac e of Physics (Princeton: Princeton Un iv ersit y Press, 2005 [1998]), 123-160.

2 Alb ert Einstein t o Conrad Habic h t, undated (ca. summer 1905), as translated and quoted in Arth ur I. Miller, A lb ert Einstei n ’s Sp e cial The ory of R elativity (Reading, MA: Ad dison-W esley , 1981), p. 353.

3 A. Einstein “Ist die T r ¨ agheit eines K ¨ orp e rs v on se i ne m Energieinhalt abh ¨ angig?,” A nnalen der Physik 18 (1905): 639-41. An English translation is a v ailab le in Stac h e l, Einstei n ’s Mir aculous Y e ar , 161-164.

Figure 1: A b o x of mass M and length L in whi c h a burst of radiation, carrying energy E , tra v e ls from left to righ t.

The Energy of a Mo ving Bo x

Consider a b o x of mass M and length L floating in space and at rest w ith resp ect to us. Suddenly a burst of radiation, with energy E , is emitted from the left end of the b o x and tra v els to w ard the righ t , as in Fig. 1. F rom Maxw ell’s equations, w e kno w that this radia tion carries momen tum of mag nitude | p radiation | = E /c , where, as usual, c is the sp eed of ligh t. Prior to the release of t he radiation the system had zero total momen tum , and there are no external forces acting on t he b o x, so ev en after the rele ase of the radiation the system’s t otal momen tum m ust remain zero. T his means that the b o x m u s t r e c oil in the opp osite direction from the direction of motio n of the radiatio n, to preserv e p total = p b o x + p radiation = 0. The v elo cit y of the b o x’s recoil v is fixed b y the conserv a tion o f mom en tum. If w e co nsider a v ery massiv e b o x w e can exp ect the recoil v elo cit y to remain small, | v | c . Requiring p total = 0 then yields

M v + E x ˆ = 0 = ⇒ v = − E x ˆ . (1)

c M c

(Here x ˆ is a unit v ector p oin ting along the x axis.) If the radiation tra v els in the + x ˆ direction, then the b o x will recoil in the − x ˆ direction, with sp eed v = | v | = E / ( M c ). After the ligh t tra v els the length of the b o x it will hit the oth er end of the b o x. Since the radiat ion carries momen tum | p radiation | = E /c , it will deliv er an impulse to the righ t side of th e b o x that will halt the b o x’s recoil, bringing the b o x to rest again.

W e ha v e assumed v c , so w e can neglect the motion of t he b o x when calculating the duration Δ t , the time-of-fligh t for the radiation to cross the length of the b o x. In tha t limit, Δ t = L/c . Bet w een the emission and absorption of the radiation with in the b o x, t he b o x will ha v e mo v ed a distance Δ x to the left, as in Fig. 2:

Δ x = v Δ t = − E L = − E L

M c c M c 2

, (2)

where the min us sign reminds us that th e b o x has mo v ed in the − x ˆ direction during its recoil motion.

Throughout this en tire pro cess, the b o x has b een sub ject to no external forces. Hence the

Figure 2: After the radiation tra v e r s es the length of the b o x it will strik e the ri gh t side of the b o x, bringing the b o x’s recoil motion to a halt. During the time that the radiat ion tra v els within the b o x, the b o x mo v es a distance Δ x to the left.

c enter of mass of the system has n ot mo v ed. In gener al, the cen ter of mass for a collection of N ob jects is giv en b y

N

m i ( r i − R ) = 0 , (3)

i =1

where R is the p osition of the cen ter of mass, and the N ob jects ha v e masses m i and p ositions r i . W e ma y adopt co ordinates suc h that initially R = 0, that is, the cen ter of mass is initially lo cated at the origin. In the absence of external forces, d R /dt = 0. In our case, the massiv e b o x has clearly c hanged its p osition o v er time, recoiling a distance Δ x to the left. I n order for the system’s cen ter of mass to remain unc hanged, therefore, some mass e quivalent m ust ha v e tra v eled with the radiation the distance L from the left side of the b o x to the righ t. Let us call this mass equiv alen t m . Then Eq. (3) b ecomes

M

mL + M Δ x = 0 = ⇒ m = − Δ x . (4)

L

In other w ords, the mass equiv alen t m asso ciated with the radiation that mo v ed a length L

to the righ t has comp ensat ed for the motion of th e massiv e b o x M a distance Δ x to the left.

Com bining Eqs. (2) and (4 ), w e find

m = − M − E L = E , (5)

L M c 2 c 2

or, more famo usly ,

E = mc 2 . (6)

W e arriv ed at Eq. (6) b y neglectin g quan tities of order ( v /c ) 2 . If w e had calculated the effects to arbitrary o rder in ( v /c ), w e w ould ha v e found the result Einstein originally deriv ed:

E = γ mc 2 , (7)

with

1

γ ≡ q

. (8)

2

Relativistic Momen tum

In our class discussion of Hermann Mink o wski’s geometrical in terpretation of Einstein’s 1 905 w ork on the electro dynamics of mo ving b o dies, w e will see that Mink o wski found an invariant quan tit y on whic h observ ers in any inertial reference frame will agree, ev en though (as Einstein ha d demonstrated) t he observ ers will disagree ab out measuremen ts of length and of time. F or finite durations Δ t and displacemen ts Δ x , Mink o wski sho w ed that the quan tit y s 2 remains in v arian t across inertial reference frames:

s 2 ≡ c 2 (Δ t ) 2 − (Δ x ) 2

= c 2 (Δ t 0 ) 2 − (Δ x 0 ) 2 .

(9)

The quan tit y s is often called the “spacetime in terv al.” W e ma y define the pr op er time Δ τ for an ob ject as the time measured b y a clo c k that is mo ving with that ob ject:

c Δ τ ≡ s . (10)

F or a clo c k sitting at rest, Δ x = 0 and the prop er time simply equals the co ordinate time measured in that ob ject’s reference frame, Δ τ = Δ t . Ho w ev er, sinc e any inertial observ er will agree on the spacetime in ter v al s 2 — and since the sp eed of ligh t c is a univ ersal constan t, on whic h all inertial observ ers will also agree — an y inertial observ er will also agree on the prop er time Δ τ , ev en when they disagree ab out co ordinate-dep enden t durations Δ t .

Just as w e ma y consider an infinitesimal displace men t in time or space, dt or d x , w e ma y also consider an infinitesimal spacetime in terv al:

ds 2 = c 2 dt 2 − d x 2

= c 2 dt 2 " 1 −

1 d x 2 #

= c 2 dt c 2

2 v 2

2

(11)

= γ 2 dt ,

where I ha v e used v = d x /dt , v = | v | , and the definit ion of γ in Eq. (8). Using Eqs. (10) and (11), w e ma y fin d an expression relating the infinitesimal pro p er tim e dτ = ds/c to the co ordinate time dt in a giv en inertial reference frame:

dt

= γ , (12)

dτ

whic h is just another w a y of expressing the time dilation of one observ er’s measurem en t of the rate at whic h time passes on a mo ving clo c k.

In Newtonian mec hanics (that is, when w e ign ore sp ecial relativit y), w e de fine the mo - men tum of an ob ject of mass m to b e

d x

p nonrel ≡ m dt . (13)

Since all inertial observ ers will agree on the pr op er time asso ciated with a mo ving ob ject, ev en if they disagree on the co ordinate times asso ciated with v arious reference frames, w e ma y generalize Eq. (13) and write t he r elativistic momentum for an ob ject of mass m as

d x p rel ≡ m dτ

= m d x dt (14)

dt dτ

= γ m v ,

up on using v = d x /dt and Eq. (12).

No w w e can see that the g eneral expression in Eq. (7), E = γ mc 2 , is equiv alen t to taking in to acc oun t b oth the “rest energy” of an ob ject — a nonzero qua n tit y for an y massiv e ob ject, ev en if the ob ject is sitting at rest in a giv en referenc e frame — as w ell as its kinetic energy , due to its motion. Let us p osit that

E 2 = m 2 c 4 + | p rel | 2 c 2 . (15)

(As a quic k c hec k, note that ligh t has v anishin g rest mass, m = 0, and hence Eq. (15) implies that E = | p radiation | c , whic h is consisten t with Maxw ell’s equations and w as cen tral to the argumen t that Einstein used in his discussion of the momen tum carried b y the radiation within the b o x, whic h w e considered ab o v e.) Up on using Eq. (14), w e find

E 2 = m 2 c 4 + γ 2 m 2 v 2 c 2

= m 2 c 4 1 + γ

2 v 2

c 2

= m 2 c 4

v 2 /c 2

1 + [1 − ( v 2 /c 2 )]

(16)

m 2 c 4

v 2 v 2

= [1 − ( v 2 /c 2 )]

= γ 2 m 2 c 4 ,

or

1 − c 2 + c 2

E = γ mc 2 . (17)

An ob ject with mass m at rest ( with v = 0) will ha v e a “rest energy” E = mc 2 . An ob ject with mass m that is mo ving with some v elo cit y v will ha v e a relativistic momen tum p = γ m v and energy E = γ mc 2 .

MIT OpenCourseWare https://ocw.mit.edu

STS.042J / 8.225J Einstein, Oppenheimer, Feynman: Physics in the 20th Century

Fall 2020

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