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PHYSI CAL REVIEW VOI UME 124, NUMBER 3NOVEM HER 1, 1961
Mach's Principle and aRelativistic Theory of Gravitation*
C. BRANS) AND R. H. Dzcxz
Palmer Physical Laboratory, Princeton University, Princeton, ¹mJersey
(Received June 23, 1961)
The role of Mach's principle in physics is discussed in relation to the equivalence principle. The difhculties
encountered in attempting to incorporate Mach's principle into general relativity are discussed. Amodified
relativistic theory of gravitation, apparently compatible with Mach sprinciple, is developed.
INTRODUCTION small mass, its eGect on the metric is minor and can be
considered in the weak-field approximation. The ob-
server would, according to general relativity, observe
normal behavior of his apparatus in accordance with the
usual laws of physics. However, also according to general
relativity, the experimenter could set his laboratory ro-
tating by leaning out awindow and firing his 22-caliber
riQe tangentially. Thereafter the delicate gyroscope in
the laboratory would continue to point in adirection
nearly fixed relative to the direction of motion of the
rapidly receding bullet. The gyroscope would rotate
relative to the walls of the laboratory. Thus, from the
point of view of Mach, the tiny, almost massless, very
distant bullet seems to be more important that the
massive, nearby walls of the laboratory in determining
inertial coordinate frames and the orientation of the
gyroscope. 'It is clear that what is being described here is
more nearly an absolute space in the sense of Newton
rather than aphysical space in the sense of Berkeley
and Mach.
The above example poses aproblem for us. Ap-
parently, we may assume one of at least three things:
''T is interesting that only two ideas concerning the
~-nature of space have dominated our thinking since
the time of Descartes. According to one of these pic-
tures, space is an absolute physical structure with
properties of its own. This picture can be traced from
Descartes vortices' through the absolute space of
Newton, 'to the ether theories of the 19th century.
The contrary view that the geometrical and inertial
properties of space are meaningless for an empty space,
that the physical properties of space have their origin
in the matter contained therein, and that the only
meaningful motion of aparticle is motion relative to
other matter in the universe has never found its com-
plete expression in aphysical theory. This picture is
also old and can be traced from the writings of Bishop
Berkeley' to those of Krnst Mach.4These ideas have
found alimited expression in general relativity, but it
must be admitted that, although in general relativity
spatial geometries are aGected by mass distributions,
the geometry is not uniquely specified by the distribu-
tion. It has not yet been possible to specify boundary
conditions on the field equations of general relativity
which would bring the theory into accord with Mach's
principle. Such boundary conditions would, among other
things, eliminate all solutions without mass present.
It is necessary to remark that, according to the ideas
of Mach, the inertial forces observed locally in an ac-
celerated laboratory may be interpreted as gravitational
effects having their origin in distant matter accelerated
relative to the laboratory. The imperfect expression
of this idea in general relativity can be seen by consider-
ing the case of aspace empty except for alone experi-
menter in his laboratory. Using the traditional, asymp-
totically Minkowskian coordinate system 6xed relative
to the laboratory, and assuming anormal laboratory of
1. that physical space has intrinsic geometrical and
inertial properties beyond those derived from the matter
contained therein;
2. that the above example may be excluded as non-
physical by some presently unknown boundary condi-
tion on the equations of general relativity.
3. that the above physical situation is not correctly
described by the equations of general relativity.
These various alternatives have been discussed pre-
viously. Objections to the first possibility are mainly
philosophical and, as stated previously, go back to the
time of Bishop Berkeley. Acommon inheritance of all
present-day physicists from Einstein is an appreciation
for the concept of relativity of motion.
As the universe is observed to be nonuniform, it
would appear to be dificult to specify boundary condi-
tions which would have the eGect of prohibiting un-
suitable mass distributions relative to the laboratory
arbitrarily p/aced; for could not alaboratory be built
near amassive starP Should not the presence of this
massive star contribute to the inertial reaction)
The difhculty is brought into sharper focus by con-
*Supported in part by research contracts with the U. S.Atomic
Energy Commission and the Ofhce of Naval Research.
fNational Science Foundation Fellow; now at Loyola Uni-
versity, New Orleans, Louisiana.
'E. T. Whittaker, History of the Theories of A.ether and L~lec-
tricity (Thomas Nelson and Sons, New York, 1951).
I. Newton, Principia Mathematica Philosophiae Eatlralis
(1686) (reprinted by University of California Press, Berkeley,
California, 1934).
'G. Berkeley, The Prenoeptes of unman Enorotedge, paragraphs
111—
117, 1/10-De Motn (1726).
'E. Mach, Conseroatson of Energy, note No. 1, 1872 (reprinted
by Open Court Publishing Company, LaSalle, Illinois, 1911),an
The Science of Mechanics, 1883 {reprinted by Open Court Publish
ing Company, LaSalle, Illinois, 1902), Chap. II, Sec. VI.
d'Because of the Thirring-Lense eiiect, PH. Thirring and J.
Lense, Phys. Zeits. 19, 156 (1918)),the rotating laboratory would
have aweak effect on the axis of the gyroscope.
925
R. H. DlCKE
sidering the laws of physics, including their quantitative
aspects, inside astatic massive spherical shell. It is
well known that the interior Schwarzschild solution is
Hat and can be expressed in acoordinate system
Minkowskian in the interior. Also, according to general
relativity all Minkowskian coordinate systems are
equivalent and the mass and radius of the spherical
shell have no discernible effects upon the laws of physics
as they are observed in the interior. Apparently the
spherical shell does not contribute in any discernible
way to inertial effects in the interior. %hat would
happen if the mass of the shell were decreased, or its
radius increased without limits It might be remarked
also that Komar' has attempted, without success, to
find suitable boundary- and initial-value conditions for
general relativity which would bring into evidence
Mach's principle.
The third alternative is the subject of this paper.
Actually the objectives of this paper are more limited
than the formulation of atheory in complete accord
with Mach's principle. Such aprogram would consist of
two parts, the formulation of asuitable field theory
and the formulation of suitable boundary- and initial-
value conditions for the theory which would make the
space geometry depend uniquely upon the matter
distribution. This latter part of the problem is treated
only partially.
At the end of the last section we shall briefly return
again to the problem of the rotating laboratory.
Aprinciple as sweeping as that of Mach, having its
origins in matters of philosophy, can be described in
the absence of atheory in aqualitative way only. A
model of atheory incorporating elements of Mach's
principle has been given by Sciama. 7From simple
dimensional arguments'' as well as the discussion of
Sciama, it has appeared that, with the assumption of
validity of Mach's principle, the gravitational constant
Gis related to the mass distribution in auniform
expanding universe in the following way:
GM/Rc' 1.
Here Mstands for the finite mass of the visible (i.e.,
causally related) universe, and Rstands for the radius
of the boundary of the visible universe.
The physical ideas behind Eq. (1) have been given
in references 7—
9and can be summarized easily. As
stated before, according to Mach's principle the only
meaningful motion is that relative to the rest of the
matter in the universe, and the inertial reaction experi-
enced in alaboratory accelerated relative to the distant
matter of the universe may be interpreted equivalently
as agravitational force acting on afixed laboratory
'A. Komar, Ph.D. thesis, Princeton University, 1956
(unpublished).
'D. %V. Sciama, Monthly Notices Roy. Astron. Soc, 113, 34
I'1953); The Unity of the Universe (Doubleday 8z Company, Inc.,
New York, 1959), Chaps. 7-9.
'R. H. Dicke, Am. Scientist 47, 25 (1959).
'R. H. Dicke, Science 129, 621. (1959),
due to the presence of distant accelerated matter. '
This interpretation of the inertial reaction carries with
it an interesting implication. Consider atest body falling
toward the sun. In acoordinate system so chosen that
the object is not accelerating, the gravitational pull of
the sun may be considered as balanced by another
gravitational pull, the inertial reaction. 'Note that the
balance is not disturbed by adoubling of all gravita-
tional forces. Thus the acceleration is determined by the
mass distribution in the universe, but is independent
of the strength of gravitational interactions. Designating
the mass of the sun by ns, and its distance by renables
the acceleration to be expressed according to Newton
as a=Gm, /r' or, from dimensional arguments, in terms
of the mass distribution as amRc'/3A'. Combining
the two expressions gives Eq. (1).
This relation has significance in arough order-of-
magnitude manner only, but it suggests that either the
ratio of 3f to Rshould be Axed by the theory, or alter-
natively that the gravitational constant observed locally
should be variable and determined by the mass distribu-
tion about the point in question. The first of these two
alternatives is of course, in part, simply the limitation
of mass distribution which it might be hoped would
result from some boundary condition on the field equa-
tions of general relativity. The second alternative is
not compatible with the "strong principle of equiva-
lence"" and general relativity. The reasons for this will
be discussed below.
If the inertial reaction may be interpreted as agravi-
tational force due to distant accelerated matter, it
might be expected that the locally observed values of
the inertial masses of particles would depend upon the
distribution of matter about the point in question. It
should be noted, however, that there is afundamental
ambiguity in astatement of this type, for there is no
direct way in which the mass of aparticle such as an
electron can be compared with that of another at a
different space-time point. Mass ratios can be compared
at different points, but not masses. On the other hand,
gravitation provides another characteristic mass
(Ac/G)'=2. 16X10 "" g,
and the mass ratio, the dimensionless number
m(G/Ac)' —
5X10 "
(2)
provides an unambiguous measure of the mass of an
electron which can be compared at different space-
time points.
It should also be remarked that statements such as
"A and care the same at all space-time points" are in
the same way meaningless within the same context
until amethod of measurement is prescribed. In fact,
it should be noted that Aand cmay be defined to be
constant. Aset of physical "constants" may be defined
as constant if they cannot be combined to form one or
10 RHDicke~ Am yPhys 29 344 (1960)
MACH'S 0RI iN CI PLE 927
more dimensionless numbers. The necessity for this
limit. ation is obvious, for adimensionless number is
invariant under atransformation of units and the ques-
tion of the constancy of such dimensionless numbers is
to be settled, not by definition, but by measurements.
Aset of such independent physical constants which are
constant by definition is "complete" if it is impossible
to include another without generating dimensionless
numbers.
It should be noted that if the number, Eq. (3),
should vary with position and Aand care defined as
constant, then either mor G, or both, couM vary with
position. There is no fundamental diGerence between
the alternatives of constant mass or constant G. How-
ever, one or the other may be more convenient, for the
formal structure of the theory wouM, in asuperficial
way, be quite different for the two cases.
To return to Eq. (3), the odd size of this dimension-
less number has often been noticed as well as its ap-
parent relation to the large dimensionless numbers of
astrophysics. The apparent relation of the square of the
reciprocal of this number LEq. (3)) to the age of the
universe expressed as adimensionless number in atomic
time units and the square root of the mass of the visible
portion of the universe expressed in proton mass units
suggested to Dirac" acausal connection that would lead
to the value of Eq. (3) changing with time. The signifi-
cance of Dirac's hypothesis from the standpoint of
Mach's principle has been discussed. '
Dirac postulated adetailed cosmological model based
on these numerical coincidences. This has been criti-
cized on the grounds that it goes well beyond the empiri-
cal data upon which it is based. Also in another publi-
cation by one of us (R. H. D.), it will be shown that: it.
gives results not in accord. with astrophysical observa-
tions examined. in the light of modern stellar evolution-
ary theory.
On the other hand, it should be noted that alarge
dimensionless physical constant such as the reciprocal
of Eq. (3) must be regarded as either determined by
nature in acompletely capricious fashion or else as re-
lated to some other large number derived from nature.
In any case, it seems unreasonable to attempt to derive
anumber like 1023 from theory as apurely mathematical
number involving factors such as 4s/3.
It is concluded therefore, that although the detailed
structure of Dirac's cosmology cannot be justified by
the weak empirical evidence on which it is based, the
more general conclusion ths, tthe number LEq. (3)]
varies with time has amore solid basis.
If, in line with the interpretation of Mach's principle
being developed, the dimensionless mass ratio given by
Eq. (3) should depend upon the matter distribution in
the universe, with Aand cconstant by definition, either
the mass mor the gravitational constant, or both, must
vary. Although these are alternative descriptions of the
"P.A. M. Dirac, Proc. Roy. Soc. (London) A165, 199 (1938).
same physical situation, the formal structure of the
theory would be very diferent for the two cases. Thus,
for example, it can be easily shown that uncharged
spinless particles whose masses are position dependent
no longer move on geodesics of the metric. (See Ap-
pendix I.)Thus, the definition of the metric tensor is
different for the two cases. The two metric tensors are
connected by aconformal transformation.
The arbitrariness in the metric tensor which results
from the indefiniteness in the choice of units of measure
raises questions about the physical significance of Rie-
mannian geometry in relativity. "In particular the 14
invariants which characterize the space are generally
not invariant under aconformal transformation inter-
preted as aredefinition of the metric tensor in the same
space."Matters are even worse, for amore general
redefinition of the units of measure can be used to re-
duce all 14 invariants to zero. It should be said that
these remarks should not be interpreted as casting
doubt on the correctness or usefulness of Riemannian
geometry in relativity, but rather that each such
geometry is but aparticular representation of the theory.
It would be expected that the physical content of the
theory should be contained in the invariants of the group
of position-dependent transformations of units and co-
ordinate transformations. The usual invariants of
Riemannian geometry are not invariants under this
wider group.
In general relativity the representation is one in
which units are chosen so that atoms are described as
having physical properties independent of location. It
is assumed that this choice is possible.
In accordance with the above, aparticular choice of
units is made with the realization that the choice is
arbitrary and without an invariant significance. The
theoretical structure appears to be simpler if one de-
fines the inertial masses of elementary particles to be
constant and permits the gravitational constant to vary.
It should be noted that this is possible only if the mass
ratios of elementary particles are constant. There may
be reasonable doubt about this. "On the other hand,
it would be expected that such quantities as particle
mass ratios or the fine-structure constant, if they
depend upon mass distributions in the universe, would
be much less sensitive in their dependence' rather than
the number given by Eq. (3) and their variation could
be neglected in afirst crude theory. Also it should be
remarked that the requirements of the approximate
constancy of the ratio of inertial to passive gravitational
mass, "and the extremely stringent requirement of
spatial isotropy, "impose conditions so severe that it
has been found to be difFicult, if not impossible, to
'2 E.P. Wigner has questioned the physical significance of Rie-
mannian geometry on other grounds /Relativity Seminar, Stevens
Institute, May 9, 1961 (unpublished)g.
"B.Hoffman, Phys. Rev. 89, 49 (1953).
'4 R. Eotvos, Ann. Physik 68, 11 (1922).
~' V. W. Hughes, H. G. Robinson, and V. Beltran-Lopez,
Phys. Rev. Letters 4, 342 (1960).
928 C. BRANS AN DR. H. DICKE
construct asatisfactory theory with avariable fine-
structure constant,
It should be emphasized that the above argument in-
volving the large dimensionless numbers, Eq. (3), does
not concern Mach's principle directly, but that Mach's
principle and the assumption of agravitational "con-
stant" dependent upon mass distributions gives a
reasonable explanation for varying "constants. "
It would be expected that both nearby and distant
matter should contribute to the inertial reaction experi-
enced locally. If the theory were linear, which one does
not expect, Eq. (1)would suggest that it is the reciprocal
of the gravitational constant which is determined locally
as alinear superposition of contributions from the mat-
ter in the universe which is causally connected to the
point in question. This can be expressed in asomewhat
symbolic equation:
G—
'-P;(m;/r;c'),
where the sum is over all the matter which can con-
tribute to the inertial reaction. This equation can be
given an exact meaning only after atheory has been
constructed. Equation (4) is also arelation from
Sciama's theory.
It is necessary to say afew words about the equiva-
lence principle as it is used in general relativity and as
it relates to Mach's principle. As it enters general rela-
tivity, the equivalence principle is more than the as-
sumption of the local equivalence of agravitational
force and an acceleration. Actually, in general relativity
it is assumed that the laws of physics, including numeri-
cal content (i.e.,dimensionless physical constants), as
observed locally in afreely falling laboratory, are inde-
pendent of the location in time or space of the labora-
tory. This is astatement of the "strong equivalence
principle. '"' The interpretation of Mach's principle
being developed here is obviously incompatible with
strong equivalence. The local equality of all gravitational
accelerations (to the accuracy of present experiments)
is the "weak equivalence principle. "It should be noted
that it is the "weak equivalence principle" that re-
ceives strong experimental support from the Eotvos
experiment.
Before attempting to formulate atheory of gravita-
tion which is more satisfactory from the standpoint of
Mach's principle than general relativity, the physical
ideas outlined above, and the assumptions being made,
will be summarized:
1.An approach to Mach's principle which attempts,
with boundary conditions, to allow only those mass
distributions which produce the "correct" inertial
reaction seems foredoomed, for there do exist large
localized masses in the universe (e.g.,white dwarf
stars) and alaboratory could, in principle, be con-
structed near such amass. Also it appears to be possible
to modify the mass distribution, For example, amassive
concrete spherical shell could be constructed with the
laboratory in its interior.
2. The contrary view is that locally observed inertial
reactions depend upon the mass distribution of the uni-
verse about the point of observation and consequently
the quantitative aspects of locally observed physical
laws (as expressed in the physical "constants") are
position dependent.
3. It is possible to reduce the variation of physical
"constants" required by this interpretation of Mach's
principle to that of asingle parameter, the gravitational
"constant. "
4. The separate but related problem posed by the
existence of very large dimensionless numbers repre-
senting quantitative aspects of physical laws is clarified
by noting that these large numbers involve 6and that
they are of the same order of magnitude as the large
numbers characterizing the size and mass distribution
of the universe.
5. The "strong principle of equivalence" upon which
general relativity rests is incompatible with these ideas.
However, it is only the "weak principle" which is
directly supported by the very precise experiments of
Kotvos.
ATHEORY OF GRAVITATION BASED ON ASCALAR
FIELD IN ARIEMANNIAN GEOMETRY
The theory to be developed represents ageneraliza-
tion of general relativity. It is not acompletely geometri-
cal theory of gravitation, as gravitational effects are
described by ascalar field in aRiemannian manifold.
Thus, the gravitational eGects are in part geometrical
and in part due to ascalar interaction. There is aformal
connection between this theory and that of Jordan, "
but there are diGerences and the physical interpretation
is quite diGerent. For example, the aspect of mass crea-
tion'r in Jordan's theory is absent from this theory.
In developing this theory we start with the "weak
principle of equivalence. "The great accuracy of the
Eotvos experiment suggests that the motion of un-
charged test particles in this theory should be, as in
general relativity, ageodesic in thefour-dimensional
manifold.
With the assumption that only the gravitational
"constant" (or active gravitational masses) vary with
position, the laws of physics (exclusive of gravitation)
observed in afreely falling laboratory should be unaf-
fected by the rest of the universe as long as self-gravi-
tational fields are negligible. The theory should be con-
structed in such away as to exhibit this eGect.
If the gravitational "constant" is to vary, it should be
"P.Jordon, Schmerkruft and 8'eltall (Friedrich Vieweg and
Sohn, Brannschweig, 1955); Z. Physiir 157, 112 (1959).In this sec-
ond reference, Jordan has taken cognizance of the objections of
Fierz (see reference 19) and has written his variational principle
in aform which differs in only two respects from that expressed
in Eq. ($6}.See also reference 20.
For adiscussion of this, see H. Bondi, Cosmology, 2nd edition,
1960,
MACH'8 PRI NCI PLE 929
afunction of some scalar field variable. The contracted
metric tensor is aconstant and devoid of interest. The
scalar curvature and the other scalars formed from the
curvature tensor are also devoid of interest as they con-
tain gradients of the metric tensor components, and
fall o8 more rapidly than r'from amass source. Thus
such scalars are determined primarily by nearby mass
distributions rather than by distant matter.
As the scalars of general relativity are not suitable,
anew scalar field is introduced. The primary function of
this 6eld is the determination of the local value of the
gravitational constant.
In order to generalize general relativity, we start
with the usual variational principle of general relativity
from which the equations of motion of matter and non-
gravitational fields are obtained as well as the Einstein
field equation, namely "
f'
0=8)[R+(167rG/c') L](g)*'d'x—
.
Here, Ris the scalar curvature and Lis the Iagran-
gian density of matter including all nongravitational
fields.
In order to generalize Eq. (5) it is first divided by
G, and aLagrangian density of ascalar field pis added
inside the bracket. Gis assumed to be afunction of g.
Remembering the discussion in connection with Eq.
(4), it would be reasonable to assume that G'varies
as @, for then asimple wave equation for pwith ascalar
matter density as source would give an equation roughly
the same as (4).
The required generalization of Eq. (6) is clearly
[4R+(16 /'V- —(0 A"/0))(—
g)'~' (6)
Here Pplays arole analogous to G'and will have the
dimensions 3fL'T'. The third term is the usual Lagran-
gian density of ascalar 6eld, and the scalar in the de-
nominator has been introduced to permit the constant
cv to be dimensionless. In any sensible theory co must be
of the general order of magnitude of unity.
It should be noted that the term involving the
Lagrangian density of matter in Eq. (6) is identical
with that in Eq. (5). Thus the equations of motion of
matter in agiven externally determined metric 6eld
are the same as in general relativity. The difference
between the two theories lies in the gravitational field
equations which determine g;;, rather than in the equa-
tions of motion in agiven metric field.
It is evident, therefore, that, as in general relativity,
the energy-momentum tensor of matter must have a
vanishing covariant divergence,
"L. Landau and E. Liftschitz, Classical Theory of Fields
(Addison-Wesley Publishing Company, Reading, Massachusetts,
1951).
ay=a ', ,=(—
g)-'[(—
g)14 '];. (10)
From the form of Eq. (9), it is evident that pR and the
Lagrangian density of pserves as the source term for
the generation of pwaves. Remarkably enough, as
will be shown below, this equation can be transformed
so as to make the source term appear as the contracted
energy-momentum tensor of matter alone. Thus, in
accordance with the requirements of Mach's principle,
phas as its sources the matter distribution in space.
By varying the components of the metric tensor and
their first derivatives in Eq. (6), the field equations for
the metric field are obtained. This is the analog of the
Einstein field equation and is
R@ ztg;;R= (Szg '/c')T;;——
+( /~')(v, 'e,;-lg,,e, e')
+e-'(e,', ;—
g', oe). (»)
The left side of Eq. (11)is completely familiar and needs
no comment. Note that the first term on the right is the
usual source term of general relativity, but with the
variable gravitational coupling parameter P'. Note
also that the second term is the energy-momentum
tensor of the scalar 6eld, also coupled with the gravita-
tional coupling p'.The third term is foreign and results
from the presence of second derivatives of the metric
"M.Fierz, Helv. Phys. Acta. 29, 128 (1956).
where 7'"=[2/( —
g)'j(~/~C') r(—
g)'L3.
It is assumed that Ldoes not depend explicitly upon
derivatives of g;,.
Jordan's theory has been criticized by Fierz" on the
grounds that the introduction of matter into the theory
required further assumptions concerning the standards
of length and time. Further, the mass creation aspects
of this theory and the nonconservation of the energy-
momentum tensor raise serious questions about the
significance of the energy-momentum tensor. To make
it clear that this objection cannot be raised against
this version of the theory, we hasten to point out that
Lis assumed to be the normal Lagrangian density of
matter, afunction of matter variables and of g;; only,
tarot afunction of p. It is awell-known result that for
aly reasonable metric field distribution g;; (a distribu-
tion which need not be asolution of the field equations
of g;,), the matter equations of motion, obtained by
varying matter variables in Eq. (6), are such that
Eq. (7) is satisfied with T" defined by Eq. (g). Thus
Eq. (7) is satisfied and this theory does not contain a
mass creation principle.
The wave equation for Pis obtained in the usual way
by varying @and p,;in Eq. (6). This gives
2~4 '&0 —
(te/0')0'0 +R=o (9)
Here the generally covariant d'Alembertian is defined
to be the covariant divergence of g':
930 C. BRANS AND R. H. DICKE
tensor in Rin Eq. (6). These second derivatives are
eliminated by integration by parts to give adivergence
and the extra terms. It should be noted that when the
first term dominates the right side of Eq. (11),the equa-
tion divers from Einstein's field equation by the pres-
ence of avariable gravitational constant only.
While the "extra" terms in Eq. (12) may at first
seem strange, their role is essential. They are needed
if Eq. (7) is to be consistent with Eqs. (9) and (11).
This can be seen by multiplying Eq. (11)by Pand then
taking the covariant divergence of the resulting equa-
tion. The divergence of these two terms cancels the term
p,;R,'=it 'R;;. To show this, use is made of the well-
known property of the full curvature tensor that it
serves as acommutator for two successive gradient
operations applied to an arbitrary vector.
If Eq. (11)is contracted there results
—
~=(~~4 '/c')T (~/@')4 p4 —
"—
3e '&0 (12)
As in general relativity the metric tensor is written as
g~i'rl'i+Jr~i
where g;; is the Minkowskian metric tensor
Qpp 1p ger~ 1) (X 1) 2) 3e
(16)
(17)
—,
L(—
g) 'a"5,'l,
(—
g)" c)'$ 87rT
vp r)t' (3+2oi)c4 (18)
It is evident that a. reta, rded-time solution to Eq. (18)
can be written as
h;; is computed to the linear first approximation only.
In similar fashion let g=pp+$, where @p is aconstant:
and is to be computed to first order in mass densities.
The weak-field solution to Eq. (13)is computed first.
In this equation g;; may be replaced by p;;.
Equation (12) can be combined with Eq. (9) to give a,
new wave equation for qP: &=—
L2/(3+2oi)], 'I'd'x/rc4, (19)
for aQuid
so that
ds =g&&ds'ds~ and gpp(0,
T'i =(p+e)~ ~i+—
pg'i,
T= e+3pi
y= $8~/(3+2~)c'jT.
With the sign convention
(13)
(14)
(15)
where Tis to be evaluated at the retarded time.
The weak-field solution to Eq. (11)is obtained in a
manner similar to that of general relativity by introduc-
ing acoordinate condition that simplifies the equation.
As apreliminary step let
(20)
where eis the energy density of the matter in comovirig
coordinates and pis the pressure in the fluid. With
this sign convention and co positive, the contribution to
@from alocal mass is positive. Note, however, that
there is no direct. electromagnetic contribution to T,
as the contracted energy-momentum tensor of an elec-
troma, gnetic field is identically zero. However, bound
electromagnetic energy does contribute indirectly
through the stress terms in other fields, the stresses
being necessary to confine the electromagnetic field. "
In conclusion, co must be positive if the contribution
to the inertial reaction from nearby matter is to be
positive.
Equation (11) can be written to first order in h,,and
(as
p{Vij &i,j&j,~+gij&k, iri )'."
Sm
=LE,', ri': bjA '+
—4p '&'. —
c' (21)
and the notation ir'=&, Ap ',
n'i=&'i n'iH o.—
(22)
(23)
Equation (21) then becomes
Equation (21) can now be simplified by introducing the
four coordinate conditions
THE %'EAK FIELD APPROXIMATION
An approximate solution to Eqs. (11)and (13) which
is of first order in rnatter mass densities is now obtained.
This weak-field solution plays the same important role
that the corresponding solution fills in general relativity.
Pn;, =—
(16~/c4)yp 'T;,,-
with the retarded-time solution
n;,=(&p—
'/c') ~(T,,/r)d'a.
(24)
(25)
"There are but two formal differences between the field equa-
tions of this theory and those of the particular form of Jordan' s
theory given in Z. Physik 157, 112 (1959).First, Jordan has de-
fined his scalar field variable reciprocal to p. Thus, the simple wave
character of the scalar field equation LEq. (13)jis not so clear
and the physical arguments based on Mach's principle and leading
to Eq. (4) have not been satis6ed. Second, as aresult of its out-
growth from his five-dimensional theory, Jordan has limited his
matter variables to those of the electromagnetic 6eld.
"C, Misner and P. Putnam, Phys. Rev. 116, 1045 (1959).
1rom Eqs. (20) and (23),
i=n'i krl'in .rt'i&4
—
o—
—
1
Thus (26)
4P —
'(2' 4P —
'(1+pi )"(T
h;, =IP* ~~&;; '—
d'x. (27)
c4 ~rc' (3+2~) '~ r
MACH'S PRI NCI PLF.
For asta,tionary mass point of mass Mthese equa-
tions become
(28)
goo=i)'oo+hoo= —
1+(2M4o '/«')I 1+1/(3+2co)j
g=1+(2M& '/rcs)[1 —
1/(3+2o&)$, n= 1, 2, 3, (29)
g;,=0, i~2
The above weak-field solution is sufficiently accurate
to discuss the gravitational red shift and the deQection
of light. However, to discuss the rotation of the peri-
helion of Mercury's orbit requires asolution good to the
second approximation for gpp.
The gravitational red shift is determined by gpp
which also determines the gravitational weight of a
body. Thus, there is no anomaly in the red shift. The
strange factor (4+2'&)/(3+2&o) in goo is simply ab-
sorbed into the definition of the gravitational constant
Go=Co i(4+2o&) (3+2&o) '. (29a)
On the other hand, there is an anomaly in the deflection
of light. This is determined, not by gpp alone, but by
the ratio g/goo. It is easily shown that the light deflec-
tion computed from general relativity divers from the
value in this theory by the above factor. Thus, the
light deflection computed from this theory is
M= (4GoM/Rc') [(3+2&v)/(4+2'~) l, (30)
where Ris the closest approach distance of the light
ray to the sun of mass M. It differs from the general
relativity value by the factor in brackets. The accuracy
of the light deQection observa, tions is too poor to set
any useful limit to the size of ~.
On the contrary, there is fair accuracy in the observa-
tion of the perihelion rotation of the orbit of Mercury
and this does serve to set alimit to the size of co. In
order to discuss the perihelion rotation, an exact solu-
tion for astatic mass point will be written.
STATIC SPHERICALLY SYMMETRIC FIELD
ABOUT APOINT MASS"
Expressing the line element in isotropic form gives
be seen by substitution of Eqs. (31) and (32) into Eqs.
(13)and (11)that this is the static solution for spherical
symmetry when T;,=0.
To discuss the perihelion rotation of aplanet, about
the sun requires aspecification of the arbitrary constants
in Eq. (32) in such away that this solution agrees in
the weak-fleld limit, [first order in M/(c'rgo)j with
the previously obtained solution, Eqs. (28) and (29).
It may be easily verified that the appropria, te choice of
constants is
yo given by Eq. (29a);
~o=Po=o,
C—
—
1/(2+os),
8—
(M/2c'yo)[(2&v+4)/(2oi+3)]'*, (34)
with Xgiven by Eq. (33).
Remembering the previous discussion of Mach's
principle, it is clear that the asymptotic Minkowskian
character of this solution makes sense only if there is
matter at great distance. Second, the matching of the
solution to the weak-field solution is permissible only
if the sun is asuitable mass distribution for the weak-
field approximation. Namely, the field generated by the
sun must be everywhere small, including the interior
of the sun. With this assumption, the solution, Eqs.
(31), (32), (33), and (34), is valid for the sun. It does
not, however, justify its use for apoint mass.
The question might be raised as to whether amatch-
ing of solutions, accurate to first order only in M/ (foe'r),
has avalidity to the second order. It should be noted,
however, tha, tthis matching condition is sufhcient to
assign suAiciently accura, te values to all the adjustable
parameters in Eqs. (32) except XB, and that we do not
demand that XB be determined in terms of an integra-
tion over the matter distribution of the sun; it is deter-
mined from the observed periods of the planetary
motion.
With the a,bove solution, it is asimple matter to
calculate the perihelion rotation. The labor is reduced
if e' is carreid only to second order in M/(c'-rpo), and
e'& to first order. The result of this calculation is that
the relativistic perihelion rotation rate of a, planetary
orbit is
ds'= e' dt'+e' [dr +—
r'(do'+sin'e~') j(31) [(4+3oi)/(6+3o~) jX(value of general relativity). (35)
where nand pare functions of ronly. For oo) os the
general vacuum solution can be written in the form
where X=[(C+1)'—
C(1—
—,'ooC) j:, (33)
and no, Po, po, 8, and Care arbitrary constants. It may
~2 This form of solution was suggested to one of us (C. B.}by
C. Misner.
This is auseful result as it sets alimit on permissible
values of the constant co. If it be assumed that the ob-
served relativistic perihelion rotation agrees with an
accuracy of 8% or less with the computed result of
general relativity, it is necessary for oo in Eq. (35) to
satisfy the inequality (o&6. (36)
The observed relativistic perihelion rotation of Mercury
(after subtracting off planetary perturbations and other
effects presumed known) is 42.6"&0.9"/century. "For
s' G. M. Clemence, Revs. Modern Phys. 19, 361 (1947).
932 C. BRANS AND R. H. DICKE
co=6, the computed relativistic perihelion rotation rate
is 39.4". The difference of 3.2" of arc per century is
3.3times the formal probable error. It should also be
remarked that Clemence'4 has shown that if some re-
cent data on the general precession constant and the
masses of Venus and the Earth-Moon system are
adopted, the result is an increase in the discrepancy
to 3.7" while decreasing the formal probable error by
afactor of 2.
The formal probable error is thus substantially less
than 3.2" arc, but it may be reasonable to allow this
much to take account of systematic errors in observa-
tions and future modification of observations, adopted
masses, and orbit parameters. Apparently there are
many examples in celestial mechanics of quantities
changing by substantially more than the formal prob-
able errors. Thus, for example, the following is alist
of values which have been assigned to the reciprocal of
Saturn's mass (in units of the sun's reciprocal mass) by
authors at various times:
3f—
'=3501.6~0.8, Bessel (1833) from the motion of
Saturn's moon Titan;
Jeffrey (1954} and G. Struve
(1924-37) (Titan);
Hill (1895)Saturn's perturbations
of Jupiter;
Hertz (1953) Saturn's perturba-
tions of Jupiter;
Clemence (1960) Saturn's pertur-
bations of Jupiter.
=3494.8~0.3,
=3502.2+0.53,
=3497.64~0.27,
=3499.7&0.4,
While this example may be atypical, it does suggest that
considerable caution be used in judging errors in celes-
tial mechanics.
MACH'S PRINCIPLE
(37)
M/Ee'. (38)
It may be noted that in aRat space, with the bound-
GM/Ec'-1.
Equivalently
24 G. M. Clemence (private communication). One of ua (R.H.
D.)is grateful for helpful correspondence and conversations with
Dr. Clemence on this and other aspects of celestial mechanics.
Acomplete analysis of Mach's principle in relation to
the present scalar theory will not be attempted here.
However, because of the motivation of this theory by
Mach sprinciple, it is desirable to give abrief discussion.
Having formulated the desired field equations, it re-
mains to establish initial-value and boundary conditions
to bring the theory in accord with Mach's principle.
This will not be attempted in ageneral way, but in con-
nection with special problems only.
The qualitative discussion in the Introduction sug-
gested that for astatic mass shell of radius Rand mass
M, the gravitational constant in its interior should
satisfy the relation
e'I =e'~'P(B/r 1)/(B/r+—
1)1'~"
e"'=e'e'$1+B/rj'I(B/r 1)/(B/r—
+1)]"&~c'&'"', (41)
~=~oL(B/r 1)/(B/r+1) -j"".
It may be noted that this solution, interesting for r&R
and X)0only, results in space closure at the radius
r=B provided (42)
(X—
C—
1)/X)0.
In similar fashion at the closure radius, g—
+0, provided
C&0.
Equations (36) and (33) require that
C)2/Gl. (43)
That this boundary condition is appropriate to
Mach's principle can be seen by an application of
ary condition that &=0 at infinity, Eq. (13) has as a
solution for the interior r(E.
P=2M/(3+ 2')Rc' (39)
This is ahopeful sign and bodes well for Mach's
principle within the framework of this theory. One
should not be misled by this simple result, however.
There are several factors which invalidate Eq. (39) as
aquantitative result. First, space is not Qat, but is
warped by the presence of the mass shell. Second, the
asymptotic zero boundary condition may be impossible
for the exact static solution to the field equation. Third,
it may be impossible to construct such astatic massive
shell in auniverse empty except for the shell, without
giving matter nonphysical properties. This third point
is not meant to imply apractical limitation of real
materials, but rather afundamental limitation on the
stress-energy tensor of matter. In this connection it
should be noted that if Eq. (37) is to be satisfied, inde-
pendent of the size and mass of the shell, the gravita-
tionally induced stresses in the shell are enormous, of
the order of magnitude of the energy density of the
spherical shell. It is not possible to reduce the stress by
decreasing Mor increasing E, as the resulting change in
the gravitational constant compensates for the change.
Ke ignore here the above third point and assume for
the moment that such ashell can be constructed in
principle.
To turn now to the massive static shell, consider
first the solution to the field equations in the exterior
region, r&E..This solution is encompassed in the general
solution Eqs. (32) and (33). Note that the boundary
condition
@—
+0 as r—
+oo
is not possible.
On the other hand, it is possible to change the sign
in the brackets in Eq. (32) and absorb the complex
factor into the constant before the bracket. These
equations may then be assumed to hold for r(8 rather
than for r)Bas in Eq. (32). The equations now have
the form
MACH'S PRINCIPLE
y(xo) =)8m./(3+2(u) c4] gT (g)'d'x, —
(47)
Green's theorem. Introduce aCreen's function
satisfying
~= (—
g) '((—
g)'g"n, j,;=(—
g)-'&'(x —
xo),
also 4=L8~/(3+2~)c4)T. (45)
Combining Eqs. (44) and (45) after the appropriate
multiplications gives
L( g)'g—
"(n4.'4m —
')1
=(—
g)'*LSD/(3+2m&)c']Tg —
@8'(x—
xo). (46)
It is assumed that qis an "advanced-wave" solution
to Eq. (44), i.e.,g=0 for all time future to to The.
condition given by Eq. (42) implies a6nite coordinate
time for light to propagate from the radius 8to R, the
radius of the shell, hence to any interior point xp.
Integrate Eq. (46) over the interior of the closed
space (r(B) between the time $2)to and the space
like surface S~ so chosen that the gwave starts out at
the radius r=8 at times lying on this surface and that
the normal to the surface at r=8has no component in
the rdirection. The integral of the left side of Eq. (46),
after conversion to asurface integral, vanishes, for
gand go both vanish on t2, and both pand p; vanish
on Sg at r=8, with iQ1.
The integral over the right side of Eq. (46) yields
simplifies the expression for line element somewhat:
ds'= —
d8+a'Pdy'+ sin'x (d8+ sin'edqP) $
(closed space). (50)
The most interesting case physically seems to be the
closed universe.
Using Eq. (50) for interval and writing the (0,0)
component of Eq. (11),
Ro'—
-'R= —
(3/u') (d'+1)
(+,space closed; —,
space open)
8n.P'~d@
To' —qP+3—
—. (51)
c2qP a/
Assuming negligible pressure in the universe we have
—
T=—
Too=+pc', where the mass density is p(obser-
vationally pseems to satisfy, p) 10 "g/cm').
Again assuming negligible pressure, the energy den-
sity times ameasure of the volume of the universe is
constant. Hence pa'= poap' —
—
const.
Substituting these results in Eq. (51) yields
)a 1jy' Xpj's' 8~ )aq'
I-+--i+—
=-:(1+3M)]—I+—
po I—I.
(a 2$) a' 4Q) 3P Ea)
Here pp and Gp refer to values at some arbitrary fixed
time to. In similar fashion Eq. (13) becomes
or
(55)
The constant of integration, t, in Eq. (55), can be evalu-
ated by considerations of Mach's principle.
As before, we introduce Mach's principle into this
problem by expressing Q(t) as an advanced-wave inte-
gral over all matter. Equations (46) and (47) require
some assumption about the history of matter in the
universe. We assume that the universe expands from
ahighly condensed state. It is possible that in the intense
gravitational field of this condensed state, matter is
created. For a closed universe, matter from aprevious
cycle may be regenerated in this high-temperature state.
In view of our present state of ignorance, there seems
to be little point in speculating about the processes
involved. In any case the creation process lies outside
the present theory.
We assume, therefore, an initial state at the beginning
of the expansion (1=0) with a=0 and matter already
present. Although pressure would certainly be important
in such ahighly condensed state, with expansion the
pressure would rapidly fall and no great harm is done
to the model if pressure e6ects are neglected. In fact,
an integration of the initial high-pressure phase for a
COSMOLOGY
Aphysically more interesting problem to discuss
from the standpoint of Mach's principle is the cosmo-
logical model derived from this theory. It will be recalled
that the assumption of auniform and isotropic space
is supported to some extent by the observations of
galaxy distribution. The kinematics of the comoving
coordinate system is completely free of dynamical con-
sideration. In spherical coordinates, aform of the line
element is
ds'= —
dt2+a'(t)/dr /(1 —
Xr')+r (d8 +sin e~')], (48)
with X=+1 for aclosed space, X=—
1for open, and
)=0for aQat space, and where r&1for the closed space.
The Hubble age associated with the rate of expansion of
the univese and the galactic red shift is a/d= a/(da/dt)
The substitution
r=sinx (closed space, X=+1)
r=sinhx (open space, X= —
1)
01 (49)
y(xo) M/Rc'.
Note that this equation states that pat the point xo (d/d)(&~) L8~/(3+ ~)jp'~''
is determined by an integral over the mass distribution, Afte~ integration Eq (54)
with each mass element contributing awavelet which
propagates to the point xp. This is just the interpretation ja'= (87r/(3+2'&)]poco'(t —
t,).
of Mach's principle desired.
C. BRANS AND R. H. DICKE
9
40x 10
35
30
LLJ
LLI
&25
20
NE GATIV ECURVATURE that the other surface integral, over the surface 3=t»,
vanishes since qand all its gradients are zero on this
surface (advanced-wave solution).
Letting t,=O in Eq. (55) and combining with Eq.
(53) gives
L(a/a)+2(b/0) j'+~ a'
=l(1+l )(i/S)'+(1+l )8/S)(1/t), (57)
O
(A 15
CL
)C
UJ 10
20 25 I
30 x10
ja'= $8~/(3+2'))]ppao't. (58)
It can be seen that for sufficiently small time the
term 1/a in Eq. (57) is negligible and the solution dif-
fers only infinitesimally from the Oat-space case. The
resulting equations can be integrated exactly with the
initial conditions &=a=0; t=0 (59)
FIG. 1.The expansion parameter uas afunction of tfor
the three cases, closed, open, and flat space with cU=9.
1.6—
14—
Scalar
QJ=9
1.2
I.O
I—
.8
Ct
I-
CQ
0CURVATURE
.2—
I
10
TIME
I
20 I
25 I
30 X10 TEARs
FIG. 2. The scalar @, in arbitrary units, as afunction of tfor the
three cases, closed, open, and Oat space, with co=9.
particular cosmological model shows explicit. ly that it
may be neglected to good approximation.
It is assumed that the inertial reaction, and hence
p, at time tp is determined uniquely by the matter dis-
tribution from t=0 to t=tp. Hence, if Eq. (46) is inte-
grated over all 3-space from t=0 to t») tp, the surface
integral obtained from the left-hand side should vanish.
Initial conditions for Eqs. (53) and (55) in the form of
values of aand Pat t=0 and avalue of the constant
t, must be so chosen that the surface integral from Eq.
(46) vanishes. In order for this surface integral to be
meaningful at t=0, the amust be at least infinitesimally
positive on the surface, otherwise the metric tensor is
singular. If t,=O and /=0 on this surface, the surface
integral vanishes. This follows because the vanishing
factors pand a'P, oLsee Eq. (55)$ occur in the integral.
It is concluded, therefore, that the appropriate initial
conditions are a=P=0with t,=0. It should be noted
As both Eqs. (57) and (58) are now (in this approxima-
tion) homogeneous in (a,ap), the solution is determined
within ascale factor in aonly.
This solution, good for the early expansion phases
(i.e.,a))t), is
with r=2/(4+3(o),
(60)
(61)
aiid q= (2+2m))/(4+3m)),
yp 8n-L(4+
—
—
c3o)/ 2(3+2&v)jpptoo
(62)
(63)
For the Aat-space case, the solution is exact for all t&0.
It should be noted that Eq. (63) is compatible with
Eq. (1), for in Eq. (1) Mis of the order of magnitude
of ppc fp' and Ris approximately ctp. Thus, the initial
conditions are compatible with Mach's principle as it
has been formula, ted here.
For anonfat space, the only feasible method of
integrating Eqs. (57) and (58) beyond the range of
validity of the above solution is numerical integration.
An example of an integration is plotted in Figs. 1and
2, where aand pare plotted as afunction of time for
the three cases of positive, zero, and negative curvature
with co=9.
It should be noted that for ~&6, and the Rat-space
solution, the time dependence of adiffers only slightly
from the corresponding case in general relativity (Ein-
stein-deSitter) where at*. Consequently, it would be
difficult to distinguish between the two theories on the
basis of space geometry only. In similar fashion the mass
density required for aparticular Hubble age a/a (fiat
space) is the same as for general relativity if pI))1. For
op=6 there is only a2% difference between the two
theories.
On the other hand, stellar evolutionary rates are a
sensitive function of the gravitational constant, and
this makes an observational test of the theory possible.
MACH'S PRI NCI PLE 935
This matter is discussed in acompanion article by one
of us (R. H. D.)."
At the beginning of this article aproblem was posed,
to understand within the framework of Mach's principle
the laws of physics seen within alaboratory set rotating
within auniverse otherwise almost empty. We are now
in aposition to begin to understand this problem. Con-
sider alaboratory, idealized to aspherical mass shell
with amass mand radius r, and stationary in the co-
moving coordinate system given by Eqs. (50) with
Eqs. (60), (61), (62), and (63) satisfied. Imagine now
that the laboratory is set rotating about an axis with an
angular velocity np. This rotation aGects the metric
tensor inside the spherical shell in such away as to
cause the gyroscope to precess with an angular velocity'
Lalso see Eq. (27)]
In general relativity the equation of motion of a
point particle, without spin, moving in agravitational
field only, may be obtained from the variational principle
ol
0=8 m(gou*'u&)wads,
al (66)
(d/ds) (mu;) ,
'm—
g,
—~,
u'u, "=0.(67)
and one of us (C. B.)is indebted for advice on this and
other matters in his thesis. The authors wish also to
thank P.Roll and D, Curott for the machine integration
of the cosmological solutions LEqs. (49) and (50)), a
small part of which is plotted in Figs. 1and 2.
APPENDIX
n= (Sm/3rc'Pp)Qp, (64) If the mass in Eq. (66) is assumed to be afunction of
position,
where gp is given by Eq. (63). Equation (64) is valid
in the weak-field approximation only for which
m/(rc'go)((1. Substituting Eq. (63) in Eq. (64) gives
n=L2 (3+2~)/3~(4+3op) ](m/rc'poto') &o (65)
It may be noted that if the matter density pp of the
universe is decreased, with tp const, nincreases. Thus,
as the universe is emptied, the Thirring-Lense precession
of the gyroscope approaches more closely the rotation
velocity o,pof the laboratory. Unfortunately, the weak-
field approximation does not permit astudy of the
limiting process pp ~0.
In another publication by one of us (C. B.)other
aspects of the theory, including conservation laws, will
be discussed.
ACKNOWLEDGMENTS
The authors wish to acknowledge helpful conversa-
tions with C. Misner on various aspects of this problem,
"R.H. Dicke, Revs. Modern Phys. (to be published).
m= mof(~),
an added force term appears and
(d/ds) (mu, )',mg, g„u—
'u—' m; =0.—
(68)
(69)
Both equations are consistent with the constraint
condition u'I;=1. It should be noted that because of
the added force term in Eq. (69), the particle does not
move on ageodesic of the geometry.
If now the geometry is redefined in such away that
the new metric tensor is (conformal transformation)
and gag'= fgag',
o2 f2ds2 N4 flui— ,
(70)
Equation (69) may be written as
(d/do) (m p)N,'mpg;—
o;-I'ct'= 0. (71)
The particle moves on ageodesic of the new geometry.
With the new units of length, tinle, and mass appropriate
for this new geometry, the mass of the particle is mp, a
constant.
