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This chapter deals two closely

debates general relativity

1916-1918, one on gravitational waves, the other on the correct formulation

of conservation laws. Both issues involve-the definition of a quantity rep-

resenting the stress-energy of the gravitational field. Such definitions were

typically proposed in the context of deriving the gravitational field equa-

tions from a.variational principle. A proper understanding of the debates

on gravitational waves and conservation laws therefore requires some dis-

cussion of the rather complicated history of attempts to derive gravitational field equations from a/variational principle. 1

We will trace Einstein's work on gravitational waves and work

on conservation laws during the years 1916-1918 in this more complex

network. 'We

objections to Einstein's approach

Levi-

Civita, Schrodinger,

at alternative approaches suggested by

Lorentz Levi-Civita; and at Einstein's response'to of them. In

particular, we examine 1917 correspondence between Einstein

and Levi-Civita. We will see how Levi-Civita's criticism of Einstein's

formulation'of conservation laws strengthened Einstein his· conviction

physical considerations force one to adopt a noncovariant .II. '-'.II..II..II..II.'-'L.II.\l.4.\I,..a.'-J.Il...Il.

of conservation laws for matter plus gravitational field.

Einstein and Grossmann 1914 and Einstein 1914, Einstein used a variational method to derive field equations of limited covariance of his

64 Carlo Cattani and Michelangelo De Maria

so-called Entwurf theory (Einstein and Grossmann 1913). He used conservation of energy-momentum of matter plus gravitational field~the stressenergy of the latter'being represented by apseudotensor rather a tensor-to define the Lagrangian for the gravitational field to restrict the covariance of his theory. Einstein believed he had found a very general argument to fix the Lagrangian for the gravitational field. This Lagrangian leads to the field ~quationsof the Entwurf theory.
By substituting the gravitational tensor into the law of conservation of energy-momentum of matter (with stress-energy tensor ~ V), Einstein was able to derive certain constraints on H that he thought uniquely fixed its form. Imposing conservation of energy-momentum of matter and unaware of the contracted Bianchi identities, he obtained a set of equations to be satisfied by the gravitational field:

-8,8.x-VS'av

-
'

B0'=0,

(a, v, .' .. = 0, 1,2,3)

(1)

Einstein Cllhr"'Il'lrr.clril

So' v.. =

(3)

and used these conditions to define the form of Entwurf field equations in form3
aC:a(~_ggafJr~fJ) = -X('T</ +'tu V),

obtained (4)

where

stress-energy tensor for the 4 to' v

tnl"1I"'1Jl"il.TlIi"'Jltll.f""l,nIJlD

is as, riloll,n.clril

r Pl-t~-2,°1<tT'V grafrtp .rlp-ta)/

(5)

~~o' beingthe Christoffel symbols. Differentiatingequation (4) with respect

to x v, Einstein obtained the conservation law for matter plus gravitational

field in the form

a

a.xv(~V +tu V) O.

(6)

It~ust be stressed, however,

1914, noticed ........,J1.JI..IJlU"".....JUI.J1.

Conservation Laws and Gravitational Waves 65

to' v does not transform as a tensor under arbitrary justified transforma-

tions, but only underlinear transformations; nevertheless, we will call

to' v the [stress-]energy tensorS of the gravitational field. Something anal-

ogous holds for the components r~p of the gravitational field strength.

(Einstein 1914, p. 1077)

.

In the spring of 1915, private correspondence with Einstein, Levi-Civita

sharply attacked Einstein's proofs of the covariance of certain. fundamental

quantities of his Entwurf theory (Cattani

1989b); however,

he did not explicitly criticize the pseudotensor character of ta v.

5)
1915, Lorentz published a paper (Lorentz 1915) in which he criticized both the Entwurf theory and the variational formulation Einstein had given to it in 1914. In the second part of his paper, Lorentz proposed a more general variational derivation of gravitational field equations. Lorentz did not specify the form of the Lagrangian; he just assumed it to be a function of the metric tensor and its first-order derivatives. Requiring thatthe action integral be stationary not only for arbitrary infinitesimal variations of the coordinates, as Einstein required, but also for arbitrary infinitesimal variations of the components of metric. tensor~ Lorentz obtained the gravitational field equations in form
aR* (7)
agj1V
V\There R* and M are the Lagrangians for the gravitational field and mat-
ter, respectively. Furthermore, Lorentz showed that equations (7) tum into the Entwurf field. equations when the function chosen by Einstein
is. substituted for R*. As is well known, Einstein himself later realized
that his· choice of a Lagrangian was, in fact, quite arbitrary (Cattani and De Maria 1989b).Unlike Levi-Civita, Lorentz at this point was unaware of the mathematical mistakes Einstein made in his early variational approach, and praised for "his ingenious mode of reasoning" (Lorentz 1915, p. 1089).

a paper, entitled "The Founda- p]reS~~ntf~C1 A A..II.................j..... he discussed a variational princi-
both Einstein (1914, 1915a, 1915b,

66 Carlo Cattani and Michelangelo De Maria

1915c) and Mie (1912), the former for his gravitational field equations,

the latter for his work on nonlinear electrodynamics and his electromag-

netic theory of matter.

restricted his investigation to

situation of an electromagnetic in the presence of a gravitational field.

Hilbert was critical of Einstein's 1914 variational approach as the fol-

lowing quotation·from his paper illustrates:

Einstein gave the fundamental original idea of general invariance a simple expression; however, for Einsteinthe Hamilton principle only plays a subordinate role and his function H is not. at all generally invariant Moreover, the electrical potentials are not included [in his theory]. (Hilbert 1915, I, po 396, footnote)

............'L....,.............. proceeded as follows. He assumed

the "1IUIULJUlll-Jil'II-Jil"-'Ul

acterizing the fields are the' ten gravitational potentials gj1v and the

electromagnetic po~entialsqj1. defined a

world

tion according

following axioms:

Axiom 1 (of Mie about the world function). The law of physical events
is determined through a world function [Lagrangian] 1-[ = A H that
contains fonowing arguments:

and specifically variation of the action integral must vanish for

[changes everyone of the 14 potentials g/-LV, qa 0

Axiom 2 (of general invariance). The

1{ is invari-

ant with respect to arbitrary transformations of the world parameters

[coordinates]

x lX •

I, p. 396)

two ....""".,..,....... functions, one u,"-'Jl,JlJlJl'-"u,

JL1-I.I1.I1.04-.... JL

gravitational field

and one for matter.

used the Riemann .f">1l111l"''\{Tn1l"1I111l'''O

scalar R. For

a function As long as

gravitational

no derivatives of gj1V higher than of

second order,

1t must be

sum of these two

functions:

(8)

By ev~luating "Lagrangian derivatives"

I, p. 397) of /H

respect to various

obtained the evolution

tions for both gravitational electromagnetic potentials. next step

was to show that Axiom 2 allows one to give explicit proof of the cavan-

anceof these evolution equations. Splitting the Lagran.gian into

the scalar curvatureinvariant for the gravitational field and a Lagrangian

Conservation Laws and Gravitational Waves 67

the electromagnetic field, ...... ~L'-'_JELlI,. arrived at correct gravitational field

equations:

-x Gil-V =

r1::::;;~v, v-g

(9)

where (10)

Finally,

the evolution equations for electrodynamics in

a curved space-time by generalizing Mie's

for

space-time.

In conclusion, we want to stress the

of

method:

(1)

derived

equations in the context of

electro-

magnetic

of matter. As a consequence, his variational method

not

be generalized to other matter. To accomplish

have to specify how matter Lagrangian depends on the

potentials '-J'JLJLI~ .... lI,.lI.41l.-,....

....

gJ1,v'

(2)

generally covariant field equations, he made

use of Lagrangian derivatives were not generally covariant.

(3)

was unaware of contracted Bianchi identities, so he

arrived at the explicit form ofthe gravitational tensor in a rather clumsy

way.

In 1916, DutHISJl1ea .L.J'-J'JI.'''''JLllll.-1L.J

ity (Lorentz

field equations

gravitational

gravitational

As ~pposed' to the unspecified Lagrangian of his 1915 article, Lorentz

now chose

curvature scalar n as the Lagrangian for grav-

itational field.

come to realize the Lagrangian to be a

generally covariantscalar (Lorentz 1916, I, p. 248,p. 251; see also Janssen

1992). Lorentz

n the variation of the action into two parts. The first part,

which is no longer a scalar

leads to gravitational field equations;

the second vanishes identically on account of the boundary conditions.

Moreover, he showed that the form of his gravitational tensor coincided

with Einstein's "onlyfor one special choice of coordinates" (Lorentz 1916,

68 Carlo Cattani and Michelangelo De Maria

p. 281, italics in the original). Lorentz

the correct gravitational

field equations (Lorentz 1916, p. 285). We want to stress, however, that

Lorentz made some assumptions in deriving mGlth~~m~atH;alJlV ll1l'lnl'll'll:,rfJI1l'"1l"'fJI'lnI1t"arll

his results. He assumed that

variations of the components

of the metric tensor have tensor character. Moreover, he to make a

special choice of coordinates.

Lorentz also discussed the conservation of energy-momentum of matter

plus gravitational field, and arrived at the equations (6) obtained by Einstein

in 1914 (Lorentz 1916,

292). Lorentz too was aware of the fact

the complex'ta V is nota

(Lorentz 1916, p.294). Whereas this

was p-erfectly acceptable.to Einstein, Lorentz wrote that

[e]vidently. it would be more satisfactory if we could ascribe a stress-energy-tensor to the gravitation field. Now this can really be done. (Lorentz 1916" III,p. 295~ italics in the original)

A "natural" candidate for this tensor, according to Lorentz, was gravitational tensor GJlvof Einstei~'s generally covariant field equations. Therefore ,he suggested one interpret these equations as conservation laws. In Lorentz's opinion this interpretation of the field \,.1\..11 Qo.l1U.lI.-ll.VlI.JlO

and the conception to which they have led, may look some"what starAccording to it-we should have to imagine behind the directly
obseryallie world with its· stresses, energy etc. ,', the gravitation field is hidden with stress~s, energy etc. that are everywhere equal and opposite to the former; evidently this is in agreement with the interchange of momentum and energy which accompanies the action of gravitation. On the way of a lightbeam, e.g.,· there would be 'everywhere in the gravitation field an energy current equal and opposite to the one t:?xisting in the beam. If we remember that this hidden'energy-current·can be fully described mathematically by the quantities gab and that only the interchange just mentioned makes it perceptible to us, this mode of viewing the phenomena does not seem unacceptable. At all events we are forcibly led to it if we want to preserve the advantage of a stress-energy-tensor also for the gravitation field. (Lorentz 1916, III, p. 296, italics in the original)

In part IV' of his paper, Lorentz compared'his definition of the stress~nergy components of the gravitational field with the definition given by Einstein. While expression contained first and second order derivatives ofthemetric, "Einstein on the contrary has given valuesfor the stress-energy componefltswhichcontainthe derivatives only and which therefore are in many respects much more fit for application" (Lore,ntz 1916, IV, p. 297). Thus Lorentz defin,ed a stress-energy complex withcomponents to' v' are homogeneous·and'quadratic functions of the first-order derivatives of the me~ricanddo not contain any higher-order derivatives. The divergence of

Conservation Laws and Gravitational Waves 69

Lorentz's complex coincides the divergence of Einstein's ta- v. Lorentz

-H showed when

= 1 and gOlfJ = DOlfJ his complex is the same as

Einstein's. He added that "it seems very

agreement will

exist in general" (Lorentz 1916, IV, p. 299).

In conclusion, we want stress Lorentz showed, for the first time,

the quantity representing gravitational stress~energy was not uniquely

defined.

In 1916, Einstein- returned to a variational approach to derive his gravi-

field equations. 1I"01l"11r'hndJln

remarked that both Lorentz and Hilbert had

succeeded giving general relativity a clear form by deriving the field

equations from a single variational principle. His aim now was to present

the basic relations of the theory as clearly as possible and a more general

way. In fact, he considered his new approach more general and "in contrast

especially with Hilbert's treatment" (Einstein 1916b, p. 1111), since he

rejected some of

restrictive hypotheses' on the nature of matter.

H , starting point was the universal function 1t ~ H

assumed

to be a function of the metric tensor and its first-order derivatives and a

linear function of its second-order ~erivatives. Furthermore, he generalized

the variat~onal principle to any physical phenomenon by assuming 1-l to be

dependent on matter variables qp (not necessarily ofelectromagnetic origin)

and their first-order derivatives. Thus, he replaced his 1914 Lagrangian by

(11)

Integrating a Lagrangian of this form one arrives at variational principle

the usual boundary conditions,

D 1t*dr = 0,

(12)

where 1{* no longer depends on the second-order derivatives of the metric. Einstein had to start from a function of the form of (11) because, according to his principle of general relativity, the Lagrangian 1{ must be invariant under arbitrary coordinate transformations. However, the reduction of 1{ to 1t* (i.e., the reduction to a quadratic function of the metric's first-order derivatives) enabled Einstein to make use of the mathematical machinery developed in his 1914 paper. Meanwhile, the problems he had struggled

70 Carlo Cattani and Michelangelo De Maria

in 1914 been overcome: the theory was now generally covariant

and his choice ofa Lagrangian was no longer

(Norton 1984;

Cattani

1989b).

Einstein's next step was to

the Lagrangian into a

gravitational and a matter part (see equation (8) above). Einstein concluded

that in order to satisfy his principle of general relativity, gravitational

part of the Lagrangian "(up to a constant factor) must be the scalar of

the Riemann curvature tensor; since there is no other invariant

required properties" (Einstein 1916b, p. 1113). Clos~ly following

variational approach, Einstein showed, using an infinitesimal- coordinate

transformationx~/-= x~+!:ix~,

conditionBI-t = o(see equation (3)

above) still holds. fact, Einstein proved that this condition--could be
obtained by showing that li.J Rdr = 1.5. JR* dr where

Theref9re, the relation BJt='O now

every coordinate system,

to the invariance·of R and to the principle of general

Bit played a

fundamental role Einstein's new derivation of conservation laws. In

fact,; according to Einstein,

v...I\.U"~>"/"llULv...I\.\I,.JI.'U'.lI..lI.U' (7). ·These equations ,allowed

way, conservation laws.

,.,... a + axa

(aagR~f*L

g

V~)
.

=

v
X (.ler

v
ter ),

(13)

where

conditions (2)-(3) are JI..Il..Il..Il.llJ'U'U'~__q it follows

(R*8~ -

a~:g~a)'.
aga

(15)

"'Whenequation (13) is

with respect to xv, the left-hand,side

tumsinto Bf-l.Since B~ vanishes,

obtained in this way is

equation (6), expressing conservation of t(}talenergy-momentum.

Conservation Laws and Gravitational Waves 71

As in his previous theory, Einstein

~ v as· representing

stress-energy density for matter and t(j v as representing the stress-energy

density of the gravitational (Einstein 1916b, p. 1116). He concluded

that although· t(j v was not a tensor, the equations expressing the conserva-

tion of total energy-momentum are generally covariant, since they were

directly from the 'U'VQ.l\.ll..ll.ll..Jl.""'-'

of general relativity (Einstein 1916b,

p. 1116). As we see,this claim led Levi-Civita, in 1917, to dispute not

only the tensor character of t(j v also equations

used as his

conservation laws for matter gravitational field

De

1989a).

on

In

paper from 1916, Einstein tried to compute components of

t(j v for special case of a weak field,

doing so discovered the

existence of

waves. The metric for the weak is written,

as

in form

(16)

Minkowski metric YJLV (and its first-order derivatives)

are Inl1nlteS:imcal ~U".Il..II.\L..ll.\L.jl..""'0.

weak-field approximation the equa-

tions reduce to

(17)

where

Y

'. JLV

.=

YJLV -

21:y8JLv ,

JL
Y YJL·

(18)

The

Y~v are defined only up to a gauge transformation. Einstein

therefore imposed gauge condition

way, found solutions of the weak-field equations,vanishing are the analogs of retarded potentials in electrodynamics.. There-
fore, according to Einstein, "gravitational fields propagate as waves speed of light" (Einstein 1916a,p. 692). Multiplying equation (17)
by aY~v / 8x(j , Einstein obtained the conservation law for the total energy-
mome:ntu:m in the usual (6), where

aY~f3 a.Y~f3 _ 1.8 v·",·.(aY~f3.). 2

(19)

axJL 8x V 2JL LJ ax r.

a{3r

72 Carlo Cattani and Michelangelo De Maria

deriving the conservation law, however, Einstein made a trivial math-

ematical error used y/Ol/3 instead of yOl/3 in conservation law for

matter). As we shall see, two years elapsed before

discovered

this "regrettable error in computation" (Einstein 1918b, p. 154). The error

caused some "strange results" (Einstein 1916a, p. 696). Einstein obtained

three different types of gravitational waves compatible with

(17):

not just longitudinal and transversal 'ones but also a "new type" of wave

(Einstein 1916a, p. 693). Using equation (19) to compute the energy carried

by these waves, he found the paradoxical-result that no energy transport

was associated with either the longitudinal or the transversal waves.

tried to explain this absurdity by'treating these waves as fictitious:

The strange result that _there should exist gravitational waves without energy transport ... can easily be explained. They are not "real" waves, but "apparent" ones, because we have chosen as the coordinate system the one vibrating ~sthe waves. (Einstein 1916a, p. 696)

Einstein found only the

kind of waves transport energy. He

concluded, however, that the mean value of the energy radiated by this new

type of waves was very small, because of a damping factor Ijc4 and because

of the small value of the gravitational constant X 1.87 · 10-27)

entered into its expression.

the possibility of gr2lvlt:atlOtlcll l

JL\\-I1o-.J1.\L..Q.lL.J1.,",,'.II..B.

was bothersome. As Einstein.stated in his paper:

Nevertheless, due to .the motion of the electrons in the atom, the atoms should radiate not only electromagnetic energy, but also gravitational energy, though in a little quantity. Since, this does not happen in nature, it seems that the quantum theory should modify not only the electrodynamics of Maxwell, but also the new theory of gravitation. (Einstein 1916a,p.696)

80

Einstein's choice of a noncovariant stress~energy complex (Einstein 1916b)

and strange results on

waves (Einstein 1916a) motivated

Leyi-Civita to try

a satisfactory definition of a gravitational stress-

energy

theory (Levi-Civita 1917). In

opinion, it was Einstein's use of pseudotensor quantities

physically unacceptable results on gravitational waves. He wrote:

The idea of a gravitational [stress-energy] tensor belongs to the majestic construction of Einstein. But the definition proposed by the author is unsatisfactory. Firstof all, from the mathematical pointof,view, it lacks ~he invariant character it should have in the spirit of general relativity.

Conservation Laws and Gravitational Waves 73

More serious is the fact, noticed also by Einstein, that it leads to a clearly unacceptable physical result regarding gravitational waves. He thought that the way out of this last problem was through the quantum theory.... Indeed, the explanation is closer at hand: everything depends on the correct form of the gravitational [stress-energy] tensor. (Levi-Civita 1917,p.381)

In Levi-Civita's opinion, general relativity called for a generally covariant

gravitational stress-en1ergy tensor. Since no. differential invariants of the

first order exist, one cannot have a stress-energy tensor containing only first-

order derivatives of the metric; since the definition of ta v in (Einstein 1916b) only contains first-order derivatives, Levi-Civita concluded that

"Einstein's choice the gravitational tensor is not justified" (Levi-Civita

1917, p. 391). Levi-Civita, in fact, showed that Einstein's stress-energy

complex was covariant under linear transformations only. He proposed a

new

for the gravitational stress-energy tensor, and, consequently,

a new

for the conservation law.

Starting from the Ricci tensor RJ-lv, Levi-Civita, like Hilbert in 1915,
= GJ-lV ril.a.lI"1n.alril RJtv - ~ gJ-lV R and wrote the gravitational field equations

in

of (9). Using, for the first time, the contracted Bianchi iden-

tities, Levi-Civita showed that the covariant divergence of GJ-l v vanishes:

VvGJ-lv = O. Consequently, Vv~v = O. This conservation law for matter

will

Levi-Civita pointed out, since "~v includes the complete con-

tribution of all phenomena (but gravitation) which take place at the point

in

consideration" (Levi-Civita 1917, p. 389).

Levi-Civita now made·a move similar to the one we saw Lorentz make

earlier: proposed to interpret equation (9) both as field equations and as

conservation laws. Defining the stress-energy tensor for the gravitational

field as'--

= = + def 1
Ajtv -Yjtv

-~v

=}

AJ-lv ~v = 0,

X

(20)

he identified

A/lV as the components of a [stress-]energy tensor of the space-time domain, Le., depending only on the coefficients of ds2 • Such a tensor
can be called both gravitational and inertial, since gravity and inertia shnultaneously depend on ds 2• (Levi-Civita 1917, p. 389)

Acco~?ingtRLevi-Civita, A/Lv completely characterizes the contribution of gravityto the local mechanical behavior. With this interpretation, it follows from equfltion (20) that no net flux of energy can exist. This equilibrium is guaranteed by the "real" existence of both quantities which, being tensors, are independent of the choice of coordinates. Hence,

74 Carlo·Cattani and Michelangelo De Maria

[n]ot only the total force applied to every single element vanishes" but also (taking into account the inertia of the Aj.tv) the total stress, the flux, and the energy density. (Levi-Civita 1917, p. 389)

So, for Levi-Civita, gravitational stress-energy is characterized by the

only element independent of the coordinates, the Riemann tensor.

In Levi-Civita's approach, the problems

Einstein ran into are

avoided. Einstein to

the possibility that gravitational waves

transporting energy are generated the absence of sources. Einstein's
weak-field equations h~ve solutions for ~v = 0 representing such spon-

taneous gravitational waves. Moreover, the energy flux, computed on the

basis of equation (17), could be zero in one coordinate system and nonzero

in another. Einstein invoked the

of

theory to solve these

problems. Levi-Civita ,claimed that it was enough to define the gravitational stress-energy. tensor the way sugg~sted to reinterpret

field equations accordingly.. This precludes

situations

of the sort Einstein encountered, for, according to (20),

stress-energy tensor ,AJLv vanishes whenever the stress-energy tensor ~v

for

vanishes.

the summer of 1917, the Great

a vacation to

country,

gave him a copy ofLev~-Civita's paper (Levi-

published in Rendiconti dell'Accademia

o n August 2, JJ...4 .....·..........JLlLlL...... '1

Einstein wrote a long

was very close to war front), in

order to rebut

criticism of his theory, especially use of a

pseudotensor to represent gravitational stress-energy. Einstein gave

physical considerations to show

stress-energy of the

field cannot be represented by a generally covariant tensor.

Einstein began letter··expressing his

for

work":

I admire the elegance of your

of calculation. It must be nice

toride throughthese fields upon the horse of true mathematics, while

people like me have to make their way laboriously on foot. . .. I still

don't understand your objections to my view of the gravitationalfield.

I would like to tellyou again'what causes me to persist· in my view.

, (Einstein to Levi-Civita, August 2, 1917,p. 1)

Conservation Laws and Gravitational Waves 75

He proceeded to discuss the example of a counterweight pell0UUUlTI

clock to show that Levi-Civita's choice of a tensor to represent the stress-

energy of the gravitational field is problematic from a physical

of

view:

I start with a Galilean space, i.e~, one with constant g/-tv. Merely by changing the reference system [i.e., by introducing an accelerated reference system], I obtain a gravitational field. If in K' a pendulum clock driven by a weight is set up a state in which it is not working, gravitational energy is transformed into heat, while relative to the original system K, certainly no gravitational field and thereby no energy of this field is present.7 Since, in K, all components of the energy "tensor" in question vanish identically, all components would also have to vanish in K', if the energy of gravitation could actually be expressed by a tensor. (Einstein to Levi-Civita, August 2, 1917, p. 1)

stress-energy could be expressed by a tensor, no gravita- j:;".Il.f..lI.'If.ll.\\,U\\...Il."-JJl.Jlll.4.1l.

occur in , in which case, contrary to experience,

gravitational energy

be transformed into heat. In short, the pen-

example shows that it should be possible for the components of

gravitational stress-energy to be zero in one reference frame nonzero in

U.D.J1.'-,\\...D..ll\",1.1l.. Therefore, gravitational stress-energy cannot be represented by a

generally covariant tensor. Notice how Einstein's reasoning here is deeply

rooted in conception of equivalence principle.

To the physical argument of the pendulum clock, Einstein adds an ar-

gument against the tensor character of gravitational stress-energy of a more

mathematical

In general, it seems to me that the energy components of the gravitational
field should only depend upon the first-order derivatives ofg/-tv, because this is also valid for the forces exerted by the fields. 8 Tensors of the first order (depending only on Bg/-tv/8xa = g~V), however, do not exist.
(Einstein to Levi-Civita, August 2, 1917, pp. 1-2)

In his letter,

went on to criticize Levi-Civita's interpretation of

the gravitational field equations (20) as conservation laws. .Einstein gave

some examples. showing such conservation laws would have strange

and undesired consequences. He wrote to Levi-Civita,

You think that the field equations ... should be conceived of as energy
equations, so that [Q;:] would be the [stress-]energy components of the
gravitational field. However, with this conception it is quite incomprehensible how something like the energy law could hold in spaces where gravity can be disregarded. Why, for example, should it not be possible on your view for a body to cool off without giving off heat to the outside? (Einstein to Levi-Civita, August 2, 1917, p. 2)

76 Carlo Cattani and Michelangelo De Maria

On Levi-Civita's proposed

of the conservation laws, the

for matter to lose energy, it seems, is to transfer it

to

It does not seem to allow for possibility of energy ...m. .......~lJlU.m.'''''.m.

one place to another.

At the same time, Levi-Civita's proposal did seem to allow for processes

one would like to rule out. Einstein wrote:

The equation

gt + 7;.4 = 0

(21)

allows~4 to decrease everywhere, in which case this change is com-

pensated for by. a decrease of the, physically not perceived, absolute

91.... value of the quantity

I maintain, therefore, that what you [Levi-

Civita]. call the ep.ergy law has nothing to do with what is otherwise so

designated in physics. (Einstein to Levi-Civita, August 2, 1917, p. 2)

these grounds, Einstein rejected Levi-Civita's .Il.ll..ll.IL,.~""'.IlfIJ.a.VII,.'4\L..Il'-".ll..Il of

equations as conservation laws, and on to

tion of the conservation layvs (6).He argued

this was .Il"-".Il..ll..IlJl.Ul.ll.tl.4\L..lI."-".Il.lL

perfectly sensible from a physical point of view, even though it

a

pseudotensor representing gravitational stress-energy:

[My] conclusions are correct, whether or not one admits that the t~ are

"really" the components of the gravitational [stress-]energy. That is to

~ay, relation

d

dx~

=0

holds true with the vanishing of 4r v and ta v at [spatial] infinity, where
the integral is extended over the whole three-dimensional space. For
my conclusions, it is only necessary 144 be the· energy density of

matter, which neither one of us doubts. (Einstein to Levi-Civita, August

2, 1917, p. 2)

Finally, Einstein lIJ"-".l!..Jl..8.II,.,-'-, out that, in his definition, the· gravitational stress-energy exhibits desired behavior at spatial infinity:

... (in the static case) the field at infinity must be completely determined by the energy of matter and of the gravitational field (taken together). This is the case with my interpretation.. .. (Einstein to Levi-Civita, August2, 1917, p. 2)

Levi-Civita's
At the end of August 1917, Einstein received Levi-Civita's answer,9 flattery as well as criticism:

Conservation Laws and Gravitational Waves 77

I am very grateful that you kindly appreciate the mathematics of my last articles but the credit of having discovered these nevv fields of research goes to you. (Levi-Civita to Einstein, August 1917, draft, p. 1)

letter, Levi-Civita criticized Einstein's

the gravitational

energy, wondering why a

of first-order derivatives of

tensor

be taken as stress-energy (pseudo)tensor, and asking

for a more convincing motivation of choice.

the other

granted Einstein his interpretation

of field equations as conservation laws was not very fecund:

I recognize the importance of your objection that, in doing so, the energy principle would lose all its heuristic vC:\lue, because no physical process (or almost none) could be excluded a priori. In fact, [in order to get any physical process] one only has to associate it a suitable change of the ds2 • (Levi-Civita to Einstein,August 1917, draft, p. 1)

tensor

seems to be referring to Einstein's example of a stress-energy whose energy component decreases everywhere. Ein-

stein's conservation laws (4)

such a stress-energy tensor. It looks

as if Levi-Civita's conservation laws, I.e., the gravitational field equations,

do It looks as would be possible for almost any matter stress-energy

tensor to a metric field such the field equations are satisfied. The

conservation laws thus seem to lose "heuristic value" of restricting

the range of acceptable matter stress-energy tensors. Of course, through

the contracted Bianchi identities" the field equations do, in fact, restrict the

range of acceptable

stress-energy tensors.

In his letter, Levi-Civitastressed having no prejudice against a definition

gravitational stress-energy dependent on the choice of coordinates, or, as he it,

dependent on the expression of ds2, in analogy with what happens for

the notion of force of the field. . . . In the case of the equations of motion,

written in the forf!l

x v}. 2
d

v

_

{

dx'" dx v

& ' ds 2 - - (f {t ds

one can explicitly connect the right-hand side (which does not define either a covariant or a contravariant system) with the ordinary notion of force. According to you, the same should happe~ for your ta v (which do not constitute a tensor). I am not in principle opposed to your point of view. On the contrary, I am inclined to presume that it is right as are aU intuitions of geniuses. But I would like to see each conceptual step [canceled: logical element] to be clearly explained and described, as is done (or, at least, as is known can be done) in the case of the equation above, where we know how to recover the ordinary notion of force. (Levi-Civita to Einstein, August 1917, draft, pp. 1-2)

78 Carlo Cattani arid Michelangelo De Maria

At the same time, Levi-Civitainsisted that, at least from a logicalpoint of

view, there 'Has

wrong his own choice of a generally covariant

tensor to represent gravitational stress-energy:

[canceled: Let me add some opinions for a logical defense]. While I
maintain an attitude of prudent reserve and wait, I still want to defend the
logical flawlessness of my tensor 9JLV. (Levi-Civita to Einstein, August
1917, draft, p. 2)

Next, Levi-Civita attacked the· counterweight pendulum-clock example:

I want to'.stress that, contrary to. whatyou claim,'thereis no contradiction between the accounts of the pendulum-clock in the two systems K and K', the first one fixed (in the Newtonian sense),the second one moving with constant acceleration. You say that:

(a) K, the- energy·tensor zero because the gJLV are constant; (b) in K", thisis not the case; instead, there.is a physical phenomenon
with·an observable transformation of energy into heat; (c) due to the .invariant. character of a tensor, the simultaneous
validity of (a) and (b) implies that there is something wrong with the premises'.
contest (a), since we can assume .... gJLv. constant outside of the ponderable bodies, but [not] in the space taken by your pendulumclock. (Levi-Civita to Einstein, August 1917, draft, p. 2)

to Einstein's comment on 1!"'£:l!IC''lI'''Ilr\\1'l''lIrU£:l!IrfI

behavior

regard to the last consideration of your letter (point 4), if I am

not wrong, it [the behavior of the gravitational field at infinity] is not

a consequence of the special form of your ta v, is equally valid for my AJLv. It.seems to me that the behavior at infinity can be obtained

from [our equation (20)] by using the circumstance that the divergence

of the tensor A JLV is identically zero; therefore, the divergence of ~v

also. vanishes,

it red~ces asympto.tically to. ~a7xVirv =0, because

the gJLV tend to the values EJLv the constant Minkowski values of the

metric tensor]. (Levi-Civita to Einstein, August 1917~draft, p. 2)

So, Levi-Civita invoked the contracted Bianchi identities to show his conservation laws, like Einstein's, exhibit the desired 'behavior at infinity.

Conservation Laws and Gravitational Waves 79
In an addendum, Levi-Civita finally remarked:
An indication in favor [of our equation (20)] is the negative value of the
energy density of the gravitational field Aoo (assuming 100 > 0). This is
in agreement with the old att¥mpts to localize the potential ellergy of a Newtonian body, and explains the minus sign as due to the exceptional role of gravity compared to all other physical phenomena. (Levi-Civita to Einstein, August 1917, draft, p.2)

on

Waves .a. ..._ ....a.'"'JII..lL_A

18)

After Levi-Civita's August 1917 letter, the polemic between two scien-

tists stopped Einstein in 1918 published a new paper on gravitational

waves (Einstein 1918b). In introduction, he recognized

earlier

approach to gravitational waves (in Einstein 1916a)

was not transparent enough, and it was lIlarred by a regrettable error in computation. ,Therefore, I have to tum back to the same argument. (Einstein 1918b, 154)

Because of this error, he had obtained wrong expression for his stress-

energy complex. Correcting the error, Einstein could easily derive the

correct expression for the stress-energy complex. As a consequence, he

only two n.hllrlJlll1l''IIal"1l

of waves, thereby resolving

physical para-

doxes of his previous results. ~instein could now assert with confidence

[aJ mechanical system which always maintains its spherical symmetry cannot radiate, contrary to the result of my previous paper, which was obtained· on the basis of an erroneous calculation. (Einstein 191 ~b, p. 164)

the last section of (Einstein 1918b),

"Answer to an objection

advanced by Mr. Levi-Civita,"lO Einstein publicly gave his reply to

Levi-Civita's objections. Einstein gave improved versions of some of

arguments. already given in his August 1917 letter to Levi-Civita. He

(6) must be looked upon '-""1lUlU\I...A.1iIo...I.1l..II.

as

tVa cannot be considered components of

tensor.

In this section of his paper, Einstein gave ample credit to Levi-Civita

his contributions to general relativity:

In a recent series of highly interesting· studies, Levi-Civita has contributed significantly to. the clarification of some problems in general relativity. In one of these papers [Levi-Civita 1917], he defends a point

80 Carlo Cattani and Michelangelo De Maria

of view regarding the conservation laws different from mine, and disputes my conclusions about the radiation of energy through gra"itational waves. Although we have already settled the issue to the satisfaction of both of us in private correspondence, I think it is fitting, because of the importance of the problem, to add some further considerations concerning conservation laws.... There are different opinions on the question whether or not tVa should be considered as the components of the [stress-]energy of the gravitational field. I consider this disagreement to be irrelevant and merely a matter of words. But I have to stress that [our equation (6)], about which there are no doubts, implies a simplification of views that,is important for the signific'ance of the conservation laws. This has to be underscored for the fourth equation (a = 4), which I want to define as the energy equation. (Einstein 1918b, p.166)

Without entering into the· mathematical details of ta v, Einstein oelt'en<leCl his energy equation the following argument:

Let us consider a spatially bounded material system, whose matter den-
sity and electromagnetic field vanish outside some region. Let S be the boundary surface, at rest, which encloses the entire material system. Then, by integration of the fourth equation over the domain inside S, we get

'£(14 - ~4

4

-f-

4 t4

)dV=

+ + cos(nXt) t42 COS(nx2) t43 COS(nx3») dO'.

Oneis notentitled todefine t44 a~_the energy density of the gravitational field and (t4 1, t42 , t43) as the cOlnllonents of the flux of gravitational energy. But one can certainly maintain, in cases where the integral of t44
is small compared to the integral of the matter energy density 744 , that
the right-hand side represents the material energy loss of the system. It
was only this result that was used in this paper and in my first article on
gravitational waves. (Einstein 1918b, pp. 166-167)

Einstein then considered Levi-Civita's main objection against choice of conservation laws:

Levi-Eivita (and prior to him, although less sharply, H.A. Lorentz) proposed a different formulation ... of the conservation laws. He (as wen as other specialists) is against emphasizing [equations (6)] and against the above interpretation because ta V is not a tensor. (Einstein 1918b, p.166)

A1though Einstein obviously to cluded:

t(J'V is not a tensor, con-

I have to agree 'with this last criticism, but I do not see. why only those
quantities· with the ·transformation properties of the components of a
, tensor should have a physical meaning. (Einstein 1918b, p. 167)

Conservation Laws and Gravitational Waves 81

Finally, Einstein stressed that, even though there is no "logical objection" (Einstein 1918b, p.·167) against Levi-(~ivita's proposal, it has to be dismissed on physical grounds.

I find, on the basis of [equation (20)], that the components of the total energy vanish everywhere. [Equation (20)] , (contrary to [equation (6)]), does not exclude the possibility that a material system disappears completely, leaving no trace of its existence. In fact, the total energy in [equation (20)] (but not in [equation (6)]) is zero from the beginning; the conservation of this value of the energy does not guarantee the persistence of the system in any form. (Einstein 1918b, p. 167)

fact, this result is due to the algebraic form of Levi-Civita's "conser-

(according to

the

stress-energy is equal to zero

everywhere). In Levi-Civita's opinion, the local vanishing of the matter

stress-energy does not allow any energy flux.. From a mathematical point

of view, Levi-Civita's

with a generally covariant gravitational

stress-energy tensor, was ,certainly more general than Einstein's, and ap-

parently more in line the spirit of general relativity. Einstein's choice,

on the other

was more convincing on the basis of physical arguments,

as Levi-Civita himself admitted,. At the time, Einstein stood alone in his de-

fense of a noncovariant definition of gravitational energy. Modern,general

relativists, hov/ever, follow Einstein's rather than Levi-Civita's approach to

conservation laws.

Lorentz l..,evi-Civita were not the only two scientists to criticize Einstein's definition of gravitational stress~energy. In November 1917, Erwin Schrodingershowed, a straightforward calculation, that, given a symmetrical distribution of matter, Einstein's gravitational stress-energy complex ta v can be~ero in a suitable c.oordinate system. Schrodinger evaluated the stress-energy complex, starting from the Schwarzschild metric for the case of an incompressible sphere of matter, and noticed
to determine ta v, we must always specify the co()~dinate system, since their values do not have tensor. character and do not vanish in every system, but only in some of them. The result we get in this particular case, i.e. the possibility of reducing ta v' to be identically zero, is so surprising that I think it will need a deeper analysis..... Our calculation shows that there are some real gravitational fields whose [stress-]energy components vanish; in these fields not only the momentum and'the energy flow but also the energy density and the analogs ofthe Maxwell

82 Carlo Cattani and Michelangelo De Maria

stresses can vanish, in some finite region, asa consequence of a suitable choice of the coordinate system. (Schrodinger 1918, p. 4)

Thus, Schrodinger concluded,

This result seems to have, in this case, some consequences for our ideas about the physical nature of the gravitational field. Since we have to renounce the interpretation of tu V •.•.• as the [stress-]energy components. of the gravitationalfield, the conservation law is lost,and it will be our duty to. somehovyr~place this esselltialpart in. the foundation [of the theorY].,(Schrodinger 1918, pp. 6-7)

Abouttwo andahalfmonths later (on February 5, 1918), Einstein replied to

Schrodinger in the same journal (Einstein 1918a). Oddly enough, Einstein

started by raising further doubts about his choice of the

to

represent gravitationalstress-energy:

Sllrt~Ss-· leIfH:~n~ v C~Jm'DOIlents of

T;, represent a tensor,

for the "[stress-]energy.components" of the

tU v ;
tci (2) .the qUantities ht.==' X:;~.'l7rv gvi aresYII,J1lle c in the
r, while this not true for tUT:. = X:;vtuVgv-c.

For the same reason as mentioned in point (1), Lorentz and Levi-Civita alsoraised doubts about interpreting ta a as the [stress-]energy components of the gravitational field. Even though I can share their doubts, I
am still convinced that it is helpful to give a more convenient expression
for energy components ofthe gravitational field. (Einstein .1918a,

Conservation Laws and Gravitational Waves 83
[t]hese considerations hold mutatis mutandis in all those cases where the field transmits exchange effects between different bodies. But this is not the case for the field considered by Schrodinger. (Einstein 1918a, p. 116)
concluded peremptorily:
Hence, the formal doubts (1) and (2) cannot lead to a rejection of my proposal for the expression of the energy-momentum. It does not seem justifiedto put any further formal demands [on the properties ofa quantity representing gravitational stress-energy]. (Einstein 1918a, p. 116)

one

after Einstein's reply to Schrodinger, Hans Bauer at-

tacked Einstein's choice of to'v (Bauer 1918). discussed an example

complementary to Schrodinger's. ···Schrodinger had shown that·Einstein's

gravitational stress-energy sometimes vanishes despite the presence of a

Bauer now s.howed that it does not always vanish in

absence of a gravitational

He stressed

the partial nonvanishing of the [stress-]energy components has nothing to do with the presence of a gravitational field, but it is due only to the choice of a coordinate system.... This behavior is not surprising, since
is not a tensor. (Bauer 1918, 165)

thrown another stone

physical plausibility

we have to conclude that the "[stress~]energy components" ta v are not related· to presence of a gravitational field as they depend only on the choice of coordinates. They can vanish in presence of a field, as shown . by Schrodinger, and do not always vanish in absence ofa field, as shQwn below. Hence, their physical. significance seems to be very dubious. (Bauer 1918, p. 165)
Einstein replied to Bauer's criticism without delay. In May 1918, published a new reply to Schrodingerand Bauer (Einstein 1918c). once again justified his choice physical arguments. In his opinion,
the. theory of general relativity has been accepted by.most theoretical
physicists and mathematicians, even though almost an colleagues stand
against my formulation of the energy--momentum law. Since I am convinced that lam right, I will in the following present my point of view on these. matters in more detail. (Einstein 1918c, p. 448)

84 Carlo Cattani and Michelangelo De Maria

Einstein reminded his readers how special

combines

conservation laws of energy·and momentum

one (l1tterenl1al i

V\\.IlIl..U,.\l.Il,.JLVJUl

(i.e., the vanishing of the four-divergence of the stress-energy tensor)

is equivalent to the integral form of these conservation laws

in

experience. The generalization ofthis conservation law to general relativity,

he explained, was particularly delicate. Einstein showed how, with his

choice, "the classical concepts of energy and momentum are established as

concisely as we are accustomed to expect classical mechanics" (Einstein

1918c, p. 449). Then he demonstrated the energy and momentum of a

closed system are uniquely determined only when the motion of the system

(considered as a whole)· is expressed "with respect to a given coordinate

system" (Einstein 1918c, pp. 449-450). In particular, he. showed the

stress-energy

closed systems can only be expected to 1t1l"'4Jl1l'''lIC''1t'r,,~

as a tensor

coordinate transformations, viz. those coordinate

transformations that reduce to

at infinity. The

transformationsl.lsedin Schrodinger and Bauer's examples do not

requirement, so they do not'count as counterexamples.

After this article b'yEinstein, the debate on correct

of

conservation. laws. in general relativity· apparently came

U.V~J)V.Il...Il.lU',""u. ...ll.JlJlV_JJ,.a~JLH~';.Il..l~Il..IlJ··Ifl.V' between

Levi-

conservation

general relativity

during the years 1917-1918. Prompted by a mistake

made his

first paper

waves, ·Levi-Civita

the use of non-

covariant

a generally covariant theory.

stimu-

lated Einstein to give a new correct description of gravitational waves.

Meanwhile,

there is no unique definition of the

stress-energy of the gravitational field in general relativity. Following up

on this .insight, Lorentz proposed to interpret field equations as con-

servation· laws. .Levi-Civita independently made same

in a

mathematically more satisfactory way, using the contracted HlI-:111"1\1"'1hl1

tities.

on to

fonnulation of. the ·conservation laws

involving the pseudotensor ta v to represent gravitational stress-energy.

$chrodinger

showed

certain cases, -Einstein's choice of

t(1 v led/to paradoxical results.

This episode makes for interesting case study history of general

relativity for at least two reasons:

clarifies the connections between

variational methods and conservation

general relativity

cross-fertilization; (2) it shows

of Einstein's ""' """

"'" 1'IC""..".Urtl1I"11"..".1n\

Conservation Laws Gravitational Waves 85

in his efforts to complete edifice of general

1916-1918.

Some of most celebrated mathematical physicists, as Lorentz

Levi-Civita, attacked his choice of a pseudotensor to

gravitational

stress-energy on the basis of formal mathematical arguments very

in

spirit of general relativity. Moreover, two young theoretical physicists,

Schrodinger and

came up some

damning counterex~

amples against Einstein's choice. Yet

exploiting the

equivalence principle as a heuristic tool, stubbornly

choice

and justified it

strong physical arguments.

today's UI\,.\l.4Il..!l._II..l\.Jl_U'Il

was right.

ACKNOWLEDGMENTS.

reading

version

useful suggestions and his thorough editing of

b..JI "1I.l~1\"I.!l..!l.~"'.I!. for his critical M. Janssen for many
article.

NOTES

1 See also Cattani's chapter "Levi-Civita's Influence on Palatini's Contribution

to General Relativity" in this volume.

2

his 1914 choice of BJ.L explicitly is

_
BJ.L -

82 8x v (Jx ot

((1 -g/) 2gotfJ gaJ.L

U~8gxJ.fLJV

)

•

3 For a more extensive discussion of these calculations, see Norton (1984). 4 Einstein defined the pseudotensor t~ as (Einstein 1914, p. 1077)

.!. (_ t v ~
a- X

v, g

agar:

a _ v, aH<_g)1/2)

got

aT:

'

got

in order to show explicitly its dependence on H.

5 In this period physicists meant stress-energy tensor when they said energytensor.
6 Einstein to Levi-Civita, August 2, 1917, Einstein Archive, Boston (EA 16-253). English translation by J. Goldstein and E.G. Straus with some modifications.
7 Let us examine Einstein's pendulum clock example a moreclosely. In the reference frame in which there is no gravitational field, the clock is not working since the counterweight that should drive it is not subjected to a gravitational field.

Let us take a concrete example. Suppose our clock is in a spacecraft far from any masses with its engines turned off (frame In this case, the clock is in a situation

of "absence of weight," and consequently cannot work. When the engines are on,the spacecraft accelerates (frame K'). Consequently, objects inside
the spacecraft experience an apparent gravitational field. Our clock will want to start working under the influence of this field. If, in K', we want to prevent this, the clock's gravitational energy be transformed into heat.
8 Here Einstein presumably alludes to the fact that in general relativity gravitational forces are expressed in terms of the Christoffel symbols, which contain first-order derivatives of the metric only.

86 Carlo Cattani and Michelangelo De Maria
9 Levi-Civita to Einstein, August 1917. Only a draft of this letter survives (Levi-Civita Papers, Accademia dei Lincei, Rome). It seems reasonable, though, to assume that the actual letter was not that different from the draft.
10 "Antwort auf einen von Hm. Levi-Civita herrtihrenden Einwand," Einstein 1918b,pp.166-167.

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