ANNALS OF PHYSICS: 42, 334-3-12 (19Gij

Energy in an Expanding Universe

NATHAN ROSEN

Department of Physics, Technion-Israel Institute of Technology, Haifa, Israel

One can define as the energy of a particle, or of a system of particles, two quantities W and W * = Wf, where f is proportional to the radius of the uni- verse. The first is related to the active gravitational mass, the second to the inertial mass and the passive gravitational mass. W is the value of the Hamil- tonian and is very nearly constant for bound atomic energy levels. Hence it is reasonable to call it the energy, while IV* might bc called simply the mass, in the case of an isolated system. During a short time interval, the conserva- tion of energy can be described in terms of either W or W*, but in the general case neither one is conserved. However, in the case of radiation, W’” is con- stant, while W changes with time in a way that is related to the Hubble red shift.

In a recent paper (i ), the mot,ion of a particle in a homogeneous expanding universe was investigated. It was found that the behavior of t.he particle was such that an observer working in the framework of the special theory of relativity would assign to it an inertial mass which increases with time as the universe expands. The purpose of the present paper is to go more deeply into the mechanics of such a particle and, in particular, to consider the question of its energy as well as that of a system of particles.

I. PBRTICLE MECHANICS

We continue with the same model of the universe as in the previous paper ( 1)

and use the same notation and the same approach. We are Oherefore concerned here with the motion of a particle in a space having a background metric given by

ds’? = c1t2 - f( CL?? + c17J2 + &), (1)

wheref = f(t), withf’(t) > 0. Let us consider the case in which the particle is acted on by a force described

by the four-vector K”. According to the general theory of relativity, the equa- tions of moOion are given by

The space components of these equations (CL = k) can be put into the form

dPk/ds = f2K”, (3)

334

ENERGY IN AN EXPANDING UNIVERSE 335

where, as before,

P” = mU’, (4)

112 = mof, (5)

and

u” = f(dxk/ds), u4 = dt/ds. (‘3)

We see that from the standpoint of special relativity, the motion is that of a particle having a variable rest mass ~2 and acted upon by a force f2Kk.

Let us now go over to the Newtonian form of the equations of motion. Using the metric velocity

Vk = f( dcck/dt), (7)

and the relation

clt/cls = (1 - V’)-“2, (8)

with 1/” = Tixvk , we can write Eq. (3) in the form

dP”/dt = F”, (9)

where the momentum can be written, in the three-dimensional form,

Pk = ?ttvk( 1 - v2)--l12, (10)

and the Newtonian force is given by

Fk = f2Kk(l - V2)1’“e (11)

II. ENERGY CONSIDERATIONS

Let us now turn to the question of energy. From the special relativity stand- point it is natural (1) to identify P* with the energy of the particle, i.e., the sum of its kine-tic and rest energies. From Eq. (a) with p = 4, one finds with the help of (1) that

dP4/ds = dm/dt + fK*. (12)

Now, frl3mEqs. (I), (4), (5), and (6) one arrives at the relation (1)

~,yU’( dP’/ds) = clm/ds, (13)

where ?I,,” is t,he special-relativity metric tensor, and if one substitutes into it Eqs. (3) :and (12) one finds that

k’” = fVkKk. (14)

Eq. (12) can then be put into the form

dP4/clt = (1 - v”)*‘2 (dm/&) + V/3”. (15)

336 ROSEIi

If we assume the existence of a potential energy function C(.r, g, z, t) such that

F” = -f-y a U/d.c”) ) ( 16)

then, on wrking

W* = I’” + Ii, ( Ii)

we get

(1s)

Let us now introduce a canonical formalism. To get the equations of motion in the case of a force given by Eq. ( 16), we take a Lagrangian of t,he form

L = --mo(l - T~“)‘~2 - U/f. (19)

One sees that aL/a$ = rk, (20)

with kk = clxk/dt, so that the Lagrangian equations can be written as

clPk iau -=--- clt f'a?$' (‘21)

corresponding to Eq. (9). From the Lagrangian one obtains the Hamiltonian in the usual way, and it is

found to be given by

H = f’[(d + PkPk)1’2 + U]. (22)

If this is rewritten in terms of the ve1ocit.y components and denoted by W, one finds that

w = w*/.f. (‘3)

One readily calculates that

We see that we have arrived at two different expressions for the energy, one appropriate to the framework of the special relativity theory, t,he other a con- sequence of the canonical formalism. Either one can be used in suitable cir- cumstances to describe the conservation of energy, provided one is dealing with a time interval very much shorter than f/j’. In general, over a long time inOerva1 neither one is conserved.

To get a better understanding of the situation, let us consider the case of the

ENERGY IN AK EXPANDIKG UNIVERSE 337

free particle. Here we have

w” = p4 z m( 1 - v”)-“z. (25)

This can be interpreted as the inertial mass of the moving particle since it is the coefficient of the metric velocity V, in the expression for the momentum Pk. Then m is the inertial rest mass and also the passive gravitational mass (1). On the other hand, we have

2 -112 w = mo( 1 - v ) ) (26)

and one is tempted to call this the energy of the particle ieven if it is not strictly constant), because it is equal to the value of t,he Hamiltonian. Accordingly,7,1, is the rest energy of the particle, and it has been shown (a), (1) that this is also its active gravitational mass, i.e., it determines the gravitational field of the particle at, rest. If, for simplicit’y, we call TV* the mass and W the energy, then Eq. (23) t-ells us that the proportionality (or equality) of the tlvo, valid in the special theory of relativity, is no longer valid in the expanding-universe moclel provided we consider a time interval during which f changes appreciably.

Let us nom go over to the nonrelativist’ic approximation, corresponding to the case of small particle velocities. If we write

W” = m + E”, (27)

and, with E = E*/f,

w = rno + E, (2s)

we find that for V2 << 1, we have

and

dE -ES dt

(29)

(30)

with

E = smoV2 + U/f. (31)

Let us now consider the case of a charged particle moving periodically in an electrostatic Coulomb field. If the field is clue to a charge Q at the origin, and if the moving particle has a charge 4 then, accorcling to (1)) the force on the latter is

Fk = qQxk/fy3. (3”)

33s ROSEi%

Hence t)he potential energy T: is given by

so that aI?‘,‘& = 0. For the present purpose it will be sufficiently accurate to replace the expressions

in parentheses in Eqs. (29) and (30) by their time averages over a period of the motion, E, E”, and f being practically constant during this interval. Denoting such an average value by angular brackets, we know from the virial theorem that

(moV) = -(U/f). !34)

Hence we get

dE/& = 0, (35)

so that l? is constant, and

E* = Ef. (36)

These considerations can be extended to a system of particles. If we interpret E as the energy, we are led to expect that the spacing between energy levels should not change with time. This means that, if the present discussion is valid, one can rule out the possibility that the observed Hubble red shift might be partly due to a change in atomic energy levels with time (something which would n.ffect the estimate of the time scale of the universe).

Let us now go back to the relativistic relation, Eq. (24). If one looks for a general expression for U to describe the instantaneous interaction between two particles (insofar as t,his is possible), the obvious generalization of Eq. (33) is

u = j-u, u = u(fz, fy, fx). (37)

This expresses the viewpoint that the interaction should depend on the actual distance (or spatial interval) between the particles.

In t,his case one finds that

If one puts this into Eq. (24) and makes use of (7) and ( lo), one gets

dW f’ 1au h -=- dt [

-v-x - f faxk

p” ,y 1 , and, by Eq. (21),

(40)

ENERGY IN AN EXPANDIKG UNIVERSE 330

Let us assume that t’he particle is undergoing periodic motion. If we replace the expression in the brackets by its t’ime average over a period of the motion, we get zero. Hence we find that, to a good approximntjion,

clU’~clt = 0. (41)

Thus FV is constant, and

TV* = TVJ ( 4”)

These considerations can be extended to t)hc case of an isolated syst#em of interacting particles in periodic motion by taking for TV a sum of terms of the form of ( 26) for bhe various particles plus inter&ion energy t’erms. If each intcr- action energy is of the form

7:12/g = ,U(f(Xl - n),fCy1 - y2),f(x1 - 22)) (43)

then one obtains Eqs. (41) and (42) for the system. The above assumption concerning the form of the interaction appears reason-

able and, as we have seen, holds in the electrostatic case. Whether it holds in the case of nuclear interactions is somet.hing that’ may be known eventually if one succeeds :in observing the Hubble effect for nuclear gamma rays.

It might be remarked that the rate of change of the energy can slso be in- vestigated by t$hc use of a canonical transformation of the form

-1. x = fxk, p = p/f,

(44)

This is found to lead to the same results as above.

III. SCHR~DIXGER EQUATION

One ca.n raise the question as to whet,her a classical discussion of the behavior of atomic energy levels is valid. Hence it is desirable to consider the descript.ion of the particle behavior as given by the Schrijdinger equation.

To simplify the discussion let us Oake the nonrelativistic case and let us now

denote by H the Hamiltonian given by Eq. (22) minus N,, . Hence we have, in t,he nonrelativistic limit,

The corresponding Schrbdinger equation can then be writ,ten

7x( 8+/c%) = - ( fiZ/217L0f2)V 2$ + ( U/f)$, (46)

with V2# = d21C,/8xkdxk. If now we introduce new variables 5’ = fx”, it takes the

340 ROSEN

If U/f is a function only of c?, then on neglecting the second term in the pwren- theses of the left-hand side, we get an equation corresponding to a time-independ- ent Hamiltonian. Therefore the energy levels are constant with time, in this approximation, in agreement wit’h the results of the previous discussion based on classical mechanics.

The connection between the change of variables in the wave equabion and the canonical transformation of Eq. (44) is obvious.

IV. ENERGY-RIOMFXTUM DEXSITY TENSOR

One can also discuss the question of energy by making use of the energy- momentum density tensor 2”“. In the general relativity theory this satisfies the relation

1’;; = 0, (45)

which can also be written

[( -g1’2)TpYl,v = ?4’( -g1’2)TaPgq3,r. (49) In our case g,, is given by the line element of Eq. (1)) so that (-g)“’ = f”.

Let us consider the case in which T”’ differs from zero only in a finite region of space, and let us integrate Eq. (49) over a spatial region sufficiently large so that 7”” vanishes everywhere on the boundary. Then, on using Gauss’s theorem, one obtains for p = lc [in which case the right-hand member of Eq. (49) vanishes], d

<It s 1’; dr = 0,

or

(50)

where dr = f” dx dy dz, the metric value element. The conservation law of Eq. (51) is consistent with the previous result for the

momentum conservation of a free particle. For if we write

pk4 = f2Tk4, (52)

then in the case of dust, for which

db dx” ! I ! “ ” = PO ds ds , (53)

ENERGY IX AiT ESPAiYDING UKIVERSE 331

where po is the proper densit,y of the rnatt,er and tlz”/tls its vclocit(y, we have

Tk4 = p1;kL4, (54)

wi t,h P = ./PO . (55)

Eyuntion (55 ) for p corresponds to E(l. i 5) for ~1. We set that p is the proper inertial mass density, and hcnac t#hnt, in accordance with special relativity, Fk’ is the density of t’he momentum P’; as given by Rq. (4). This suggests that we should take, along with (54), the relations

p4 = j2,44, rI’k1 = f:i/,kl, (56)

in order that, for dust, KC should have i;PV 1 = p[T@jy”, (57)

corresponding to the special-relativity theory. If we now go hack to E(l. (49) nud integrate it over an arbitrary spatial region,

we find that itO can bc put into the form

(5s)

where rlS, is n metric element of area (i.e., f’ t,irnes the coordinate clement of area), and the right-hand integral is to be taken over the boundary of the spatial region. We can identify t,his equation wit,h E(l. (9) on writing

Kow let, us take Ey. C.49 j for p = 4. If we int’egrat’e it over a sufficiently large region we get the relation

which can also be written

The correspondence with the particle case is obtained by writing

w” = j !P44 clr, W = j T4” c/r,

so that Elq. (23) holds.

(62)

342 ROSEN

,4n inter&ing case is that of elect,ronl:1guet,ic radiation, for wliich

Q”p” = ST,” = 0. (, 63)

From ( 60) and ( 61)) we get

16 = ll-“;j’ ( 117(, = const), i 04)

,zlld

rlW”/tlt = 0. (65)

Eq. (65) corresponds to Eq. (1s) with V2 = 1, U = 0. It might be pointed out that the Hubble red shift for radiation coming to us

from a distant galaxy can bc described by means of the frequency relation

v = vg/f (VO = const). (66)

For the correct result for the red shift (8) can then be obtained by comparing the value of v at t,he time the radiation is received with that at the t,ime it is emitted :

JQf1 = v& . (67)

Equation (66) is consist’ent with (64) if we write

w = hv, ((33)

where Planck’s constant h is assumed to remain constant with time.

RECEIVED: February 9, 1966

1:EFEKEYCES /

1. N. ROSEN, Am. Phys. 36, -126 (1965). ,g. G. C. MCVITTIE, Monthly Xoles Roy. Jslr. Sot. 93, 325 (1933). 3. R. C. TOLMAN, “Relativity, Thermodynamics and Cosmology,” p. 391. Oxford Univer-

sity Press, Oxford, 1934.