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The principle of equivalence
ANNALS
OF
PHYSICS:
!%b,
383-390 (1964)
The Principle
of Equivalence*
A. L. HARVEY Department
of Physics, Queens College, Flushing,
New York
The principle of equivalence is examined in its role as a teat which a prospective theory of gravitation must satisfy. This point of view suggests that the commonly stipulated property of “locality” is inherent in the gravitational field and not in the principle. The original formulation of Einstein (1) is taken to be correct and complete. This is a “strong” principle in the sense of Dicke (8). The “weak” principle, i.e., the postulation of no more than equivalence of inertial and gravitational mass, can be satisfied by theories which predict a smaller gravitational acceleration for a rotating than for a nonrotating object. Consequently, only the strong form can be considered to be acceptable. The situation is illustrated by application (due originally to Einstein) to the first theory of Nordstrom (fO), a simple, special relativistic generalization of Newtonian theory, and the general theory of relativity. LIST
c g y y0 i 8 k H n1’ t 7 iii v IT Xi 2 w w.
OF SYMBOLS
speed of light acceleration of gravity Lorentz factor initial Lorentz factor positive square root of minus one angular coordinate constant constant ratio of circumference to diameter radial coordinate coordinate time proper time parameter components of four-velocity magnitude of three-speed gravitational potential function components of four position axial coordinate angular speed initial angular speed * Supported
in part by the Office of Naval Research. 383
384
HARVEY
I. INTRODUCTION
From the time that Einstein (1) first stated the principle of equivalence there has been a lack of unanimity among subsequent commentators as to its scope, significance, and physical content. For a fair sampling, one might consult Bergmann (2>, Bertotti, Brill, and Krotkov (S), Fock (+$), Moller (5), Pauli (6), and Rohrlich (7). Among other differences in statement and interpretation there appears to be a well defined distinction between some statement concerning only the equivalence of inertial and gravitational mass and a much broader statement such as the original one of Einstein. The distinction is sufficiently clear for a descriptive terminology to have come into use; Dicke (8) refers to the former as “weak” and the latter as “strong.” For the purposes of this article, it is sufficient to take the formulation of Pauli as being a representative strong statement of the principle. l‘For every infinitely small world region (i.e., a world region which is so small that the space- and time-variation of gravity can be neglected in it), there always exists a coordinate system &,(X1 , 5, , XS , X,) in which gravitation has no influence either on the motion of particles or any other physical processes.” A typical weak formulation would be one stating that the ratio of inertial to gravitational mass is the same for all objects and that no mass factor appears in the equations of motion of an object under the influence solely of gravitational forces. The roles of these alternative formulations, and indeed the role of any other formulation, is clarified by noting that the relationship of the principle to a theory of gravitation must be analogous to the relationship of the principle of special relativity to any dynamical theory not involving gravitation. That is, it is no more or less than a statement delimiting the structure of an admissible theory. This is just the sense in which the principle was first introduced by Einstein. Because of the almost universal acceptance of the Einstein theory, this purpose has been lost. Were the Einstein theory of gravitation the final word, it would seem that the principle has no other than pedagogic importance. But, there is the notoriously persistent failure to connect the Einstein theory with the remainder of physics. There are many questions which are yet to be satisfactorily answered. For this reason, if for no other, the principle of equivalence must be treated with circumspection. In its strong form, it makes a positive statement concerning the relationship of the laws of nature to gravitational phenomena. Therein lies one deficiency of the weak formulation. It makes no such affirmation. A question of crucial importance is that of “locality.” It has been said that the weak principle is applicable only to structureless point particles. That is, such objects as mass multipoles or particles possessing spin are outside the scope of the principle. This attitude has been extended to the strong principle so that it is
PRINCIPLE
OF
EQUIVALENCE
3%
stated to be valid only in an infinitesimal domain. This point of view is unnecessarily restrictive; it represents an unconscious attempt to build into the principle of equivalence the restrictions inherent in a physical situation. It is true that in any real gravitational field, only in an infinitesimal domain can the field be transformed away. But, why should this condition be built into the principle of equivalence? If, throughout some finite region, there is no gravitational field, there is no reason for supposing the inertial behavior of a free mass multipole to be any different from that of a structureless particle. So to suppose would be to insist on the lack of validity of the principle of special relativity. The same argument must be valid for the strong principle of equivalence. In short, “locality” must be considered a qualitative property of the field in question and should not be restrictive on the principle of equivalence. It must be emphasized that the parenthetic portion of Pauli’s formulation is of essential importance; the restriction to an “infinitely small world region” is typically the restriction due to a real field, nowhere homogeneous and uniform. The attendant parenthetic qualification that the principle is applicable to a “world-region which is so small that the space- and time-variation of the field can be neglected in it” is a two way condition and is the pertinent portion of the statement. It would be more accurate to use the phrase “sufficiently small” rather than infinitesimally small. The point concerning “locality” is nontrivial. The situations subsequently to be analyzed concern behavior in one-dimensional, uniform, static, homogeneous fields of precisely the type first considered by Einstein in 1911. In such fields, the region in which the field may be transformed away completely is arbitrarily large. It is contended that, if the principle of equivalence and the principle of special relativity are not empty concepts, the latter must be valid through the field free domain provided by the former. With this situation established it is then quite simple to show that the weak principle of equivalence is just too weak to be considered a principle at all. It is possible to construct theories of gravitation which satisfy the weak principle but not the strong. II.
THE
NORDSTROM
THEORY
The first theory of Nordstrom is one in which the weak principle of equivalence is satisfied in as elegant a manner as has ever been accomplished. Near the end of the paper in which this theory is presented, Nordstrom mentions that he received a letter from Einstein which indicated that Einstein had investigated a similar approach and had found that the theory predicted that a rotating object would fall with less acceleration than a nonrotating object. This is a clear violation of the principle of equivalence. For, if such rotating and nonrotating objects were
386
HARVEY
dropped in the same gravitational field then clearly there could be no single frame of reference in which both objects would be in free fall. That is to say, the field could be transformed away for the one but not the other object. The argument whereby Einstein arrived at the conclusion that the rotating object would fall more slowly is not given by Nordstrom nor has it been published elsewhere. The following is a demonstration of this feature of the theory. Consider a static homogeneous gravitational field given by (0, 0, g) in polar cylindrical coordinates. The equations of motion (in the Nordstrom theory) are then (as given by Behacker (9) ) :
(la)
(lc)
where the Ui are the components
of the four-velocity,
and X4 = ict. For a particle with initial conditions Xi = (0, 0, 0, 0) and Ui = (0, 0, 0, ic) the solutions of the equations of motion are
r=O e=o
(2a) (2b) 2
x = - 2. In cos -97 9
c
@cl
t =C-lntan c’ 1 9 (,4T+@ Instead, now, take the case of a dumbbell consisting of two equal mass points connected by a rod of length 2L and negligible mass. The rod is set spinning with angular speed w. in the plane z = 0 with the center of the rod as the center of rotation and coincident with the z-axis. The rod is then released to the action of the gravitational field. The subsequent motion of the dumbbell is such that the displacement of any element of the rod is normal to its length, and hence there is no Lorentz contraction.
PRINCIPLE
OF
387
EQUIVALENCE
The equations of motion of one of the mass points are the same as for the free particle with the exception of the initial conditions which are now Xi = (L, 0, 0, 0) and zip = (0, I&y0 , 0, icyO) where
The solutions are readily found to be 1’ = I;
e= !l!!fC ln tan % + $ ( >
t = E? In tan ;+c. 9 ( 1 Equations (4a) and (4b) when compared with (2~) and (Zd) show clearly that for the freely falling particle and the rotating dumbbell the proper time is the same for the same distance of fall but the elapsed coordinate time is greater for the dumbbell. Hence, the conflict with the principle of equivalence. III.
THE
GENERALIZED
NEWTONIAN
CASE
A similar analysis may be applied to the case where a trivial generalization is made from Newtonian dynamics to relativistic dynamics for the same type of field. This leads to the well-known hyperbolic motion (see e.g. Mpiller, p. 75) for the one dimensional motion of a free particle. The equations of motion are
These readily reduce to
388
HARVEY
& (my) = 0 $ c&r> = g. For a particle falling from rest under the same initial conditions motion is quite readily found to be
as before the
(7) For the motion of one of the masses of the rotating dumbbell, the initial conditions again being the same, Eqs. (6) integrate once immediately to vr
= 0
(gal
yve = lc
(8b)
yv, = qt.
(843)
The equation in r is trivial and integrates once again to the final result 1’ = L. It may be observed that k is merely the 0 component of the initial four-velocity. Further integration of the angular equation is not necessary. A simple algebraic manipulation of Eq. (8b) leads to the expression
l102L2 k2L2 [l - (v,2/c2)] c2 = c2 [i + (LW/C2)] which is useful for the further to set
integration
of the equation
(9) in x. It is convenient
X2 = -k2L2
c2 (1 + ~‘k’/cz)
in which case Eq. (8~) may be written as
((1 - H2)[l” (v,2/c2)]}l’2= qt. Equation (ll), with the exception of the factor (1 - K”), which is readily shown to be less than 1, is precisely that for the free falling particle and has the solution 2
1 )-
1 + (1 - JL2)g2t2li2 _ 1
z = g(l “_ X2)i[
C2
(12)
If the right hand side is expanded then zg(1
-&$gt2+
Clearly, the rotating object falls more slowly.
. .. .
(13)
PRINCIPLE
IT.
It is convenient
THE
to follow
OF
EINSTEIN
the work
cb2 = -dl”
389
EQUIVALENCE
THEORY
of Rohrlich.
- r2de2 - (V’,‘g)“dz*
The general line element is + V”dt2
(14)
where V is a real function of z only, has a nonrelativistic limit of 1 + 2gz, and primes represent derivatives with respect to z. The nonvanishing Christoffel symbols are
The general equations for the geodesics are ( 16a)
Equations
(16d) integrates
inmediately
to
dt l< - = -~. 1” dr
(17)
The initial conditions will determine k. Indeed the two different sets of conditions used previously lead to different fc’s. However, when Eq. (17) is used to eliminate 7 from i 16~) the result is 18) i.e., indepeudent of h-. Hence, motion in the x-direction takes place in the ~-0 plane even though the converse principle of equivalence is thus satisfied. V.
THE
PRINCIPLE
OF
is independent of what clearly is not’ true. The
EQUI\~ALENCE
It is clear, then, that unless one is prepared to insist that in a uniform houlogeneous field a rotating object may fall with less acceleration than a mass point, ouly the strong form is acceptable as a proper statement of the principle of equivalence. RECEIVED:
February
7, 1964
390
HARVEY REFERENCES
1. A. EINSTEIN, Ann. Physik. 36, 898 (1911). 2. P. G. BERGMANN, In “Handbuch der Physik,” Vol. 4, esp. pp. 204ff. Springer, Berlin, 1962. 3. B. BERTOTTI et al. In “Gravitation, An Introduction to Current Research,” pp. 8-9. Wiley, New York, 1962. 4. V. FOCK, “The Theory of Space Time and Gravitation,” esp. pp. 206 ff. Pergamon Press, London, 1959. 6. C. M$LLER, “The Theory of Relativity,” esp. pp. 75,220 ff. Oxford Univ. Press, London, 1952. 6. W. PAULI, “Theory of Relativity,” esp. p. 145. Pergamon Press, London, 1950. 7. F. ROHRLICH, Ann. Phys. (N. Y.) 22, 169 (1963). 8. R. H. DICKE, Science 129,621 (1959). 9. M. BEHACKER, Physik. Z. 14,989 (1913). 10. G. NORDSTROM, Physik. Z. 13, 1126 (1912).
OF
PHYSICS:
!%b,
383-390 (1964)
The Principle
of Equivalence*
A. L. HARVEY Department
of Physics, Queens College, Flushing,
New York
The principle of equivalence is examined in its role as a teat which a prospective theory of gravitation must satisfy. This point of view suggests that the commonly stipulated property of “locality” is inherent in the gravitational field and not in the principle. The original formulation of Einstein (1) is taken to be correct and complete. This is a “strong” principle in the sense of Dicke (8). The “weak” principle, i.e., the postulation of no more than equivalence of inertial and gravitational mass, can be satisfied by theories which predict a smaller gravitational acceleration for a rotating than for a nonrotating object. Consequently, only the strong form can be considered to be acceptable. The situation is illustrated by application (due originally to Einstein) to the first theory of Nordstrom (fO), a simple, special relativistic generalization of Newtonian theory, and the general theory of relativity. LIST
c g y y0 i 8 k H n1’ t 7 iii v IT Xi 2 w w.
OF SYMBOLS
speed of light acceleration of gravity Lorentz factor initial Lorentz factor positive square root of minus one angular coordinate constant constant ratio of circumference to diameter radial coordinate coordinate time proper time parameter components of four-velocity magnitude of three-speed gravitational potential function components of four position axial coordinate angular speed initial angular speed * Supported
in part by the Office of Naval Research. 383
384
HARVEY
I. INTRODUCTION
From the time that Einstein (1) first stated the principle of equivalence there has been a lack of unanimity among subsequent commentators as to its scope, significance, and physical content. For a fair sampling, one might consult Bergmann (2>, Bertotti, Brill, and Krotkov (S), Fock (+$), Moller (5), Pauli (6), and Rohrlich (7). Among other differences in statement and interpretation there appears to be a well defined distinction between some statement concerning only the equivalence of inertial and gravitational mass and a much broader statement such as the original one of Einstein. The distinction is sufficiently clear for a descriptive terminology to have come into use; Dicke (8) refers to the former as “weak” and the latter as “strong.” For the purposes of this article, it is sufficient to take the formulation of Pauli as being a representative strong statement of the principle. l‘For every infinitely small world region (i.e., a world region which is so small that the space- and time-variation of gravity can be neglected in it), there always exists a coordinate system &,(X1 , 5, , XS , X,) in which gravitation has no influence either on the motion of particles or any other physical processes.” A typical weak formulation would be one stating that the ratio of inertial to gravitational mass is the same for all objects and that no mass factor appears in the equations of motion of an object under the influence solely of gravitational forces. The roles of these alternative formulations, and indeed the role of any other formulation, is clarified by noting that the relationship of the principle to a theory of gravitation must be analogous to the relationship of the principle of special relativity to any dynamical theory not involving gravitation. That is, it is no more or less than a statement delimiting the structure of an admissible theory. This is just the sense in which the principle was first introduced by Einstein. Because of the almost universal acceptance of the Einstein theory, this purpose has been lost. Were the Einstein theory of gravitation the final word, it would seem that the principle has no other than pedagogic importance. But, there is the notoriously persistent failure to connect the Einstein theory with the remainder of physics. There are many questions which are yet to be satisfactorily answered. For this reason, if for no other, the principle of equivalence must be treated with circumspection. In its strong form, it makes a positive statement concerning the relationship of the laws of nature to gravitational phenomena. Therein lies one deficiency of the weak formulation. It makes no such affirmation. A question of crucial importance is that of “locality.” It has been said that the weak principle is applicable only to structureless point particles. That is, such objects as mass multipoles or particles possessing spin are outside the scope of the principle. This attitude has been extended to the strong principle so that it is
PRINCIPLE
OF
EQUIVALENCE
3%
stated to be valid only in an infinitesimal domain. This point of view is unnecessarily restrictive; it represents an unconscious attempt to build into the principle of equivalence the restrictions inherent in a physical situation. It is true that in any real gravitational field, only in an infinitesimal domain can the field be transformed away. But, why should this condition be built into the principle of equivalence? If, throughout some finite region, there is no gravitational field, there is no reason for supposing the inertial behavior of a free mass multipole to be any different from that of a structureless particle. So to suppose would be to insist on the lack of validity of the principle of special relativity. The same argument must be valid for the strong principle of equivalence. In short, “locality” must be considered a qualitative property of the field in question and should not be restrictive on the principle of equivalence. It must be emphasized that the parenthetic portion of Pauli’s formulation is of essential importance; the restriction to an “infinitely small world region” is typically the restriction due to a real field, nowhere homogeneous and uniform. The attendant parenthetic qualification that the principle is applicable to a “world-region which is so small that the space- and time-variation of the field can be neglected in it” is a two way condition and is the pertinent portion of the statement. It would be more accurate to use the phrase “sufficiently small” rather than infinitesimally small. The point concerning “locality” is nontrivial. The situations subsequently to be analyzed concern behavior in one-dimensional, uniform, static, homogeneous fields of precisely the type first considered by Einstein in 1911. In such fields, the region in which the field may be transformed away completely is arbitrarily large. It is contended that, if the principle of equivalence and the principle of special relativity are not empty concepts, the latter must be valid through the field free domain provided by the former. With this situation established it is then quite simple to show that the weak principle of equivalence is just too weak to be considered a principle at all. It is possible to construct theories of gravitation which satisfy the weak principle but not the strong. II.
THE
NORDSTROM
THEORY
The first theory of Nordstrom is one in which the weak principle of equivalence is satisfied in as elegant a manner as has ever been accomplished. Near the end of the paper in which this theory is presented, Nordstrom mentions that he received a letter from Einstein which indicated that Einstein had investigated a similar approach and had found that the theory predicted that a rotating object would fall with less acceleration than a nonrotating object. This is a clear violation of the principle of equivalence. For, if such rotating and nonrotating objects were
386
HARVEY
dropped in the same gravitational field then clearly there could be no single frame of reference in which both objects would be in free fall. That is to say, the field could be transformed away for the one but not the other object. The argument whereby Einstein arrived at the conclusion that the rotating object would fall more slowly is not given by Nordstrom nor has it been published elsewhere. The following is a demonstration of this feature of the theory. Consider a static homogeneous gravitational field given by (0, 0, g) in polar cylindrical coordinates. The equations of motion (in the Nordstrom theory) are then (as given by Behacker (9) ) :
(la)
(lc)
where the Ui are the components
of the four-velocity,
and X4 = ict. For a particle with initial conditions Xi = (0, 0, 0, 0) and Ui = (0, 0, 0, ic) the solutions of the equations of motion are
r=O e=o
(2a) (2b) 2
x = - 2. In cos -97 9
c
@cl
t =C-lntan c’ 1 9 (,4T+@ Instead, now, take the case of a dumbbell consisting of two equal mass points connected by a rod of length 2L and negligible mass. The rod is set spinning with angular speed w. in the plane z = 0 with the center of the rod as the center of rotation and coincident with the z-axis. The rod is then released to the action of the gravitational field. The subsequent motion of the dumbbell is such that the displacement of any element of the rod is normal to its length, and hence there is no Lorentz contraction.
PRINCIPLE
OF
387
EQUIVALENCE
The equations of motion of one of the mass points are the same as for the free particle with the exception of the initial conditions which are now Xi = (L, 0, 0, 0) and zip = (0, I&y0 , 0, icyO) where
The solutions are readily found to be 1’ = I;
e= !l!!fC ln tan % + $ ( >
t = E? In tan ;+c. 9 ( 1 Equations (4a) and (4b) when compared with (2~) and (Zd) show clearly that for the freely falling particle and the rotating dumbbell the proper time is the same for the same distance of fall but the elapsed coordinate time is greater for the dumbbell. Hence, the conflict with the principle of equivalence. III.
THE
GENERALIZED
NEWTONIAN
CASE
A similar analysis may be applied to the case where a trivial generalization is made from Newtonian dynamics to relativistic dynamics for the same type of field. This leads to the well-known hyperbolic motion (see e.g. Mpiller, p. 75) for the one dimensional motion of a free particle. The equations of motion are
These readily reduce to
388
HARVEY
& (my) = 0 $ c&r> = g. For a particle falling from rest under the same initial conditions motion is quite readily found to be
as before the
(7) For the motion of one of the masses of the rotating dumbbell, the initial conditions again being the same, Eqs. (6) integrate once immediately to vr
= 0
(gal
yve = lc
(8b)
yv, = qt.
(843)
The equation in r is trivial and integrates once again to the final result 1’ = L. It may be observed that k is merely the 0 component of the initial four-velocity. Further integration of the angular equation is not necessary. A simple algebraic manipulation of Eq. (8b) leads to the expression
l102L2 k2L2 [l - (v,2/c2)] c2 = c2 [i + (LW/C2)] which is useful for the further to set
integration
of the equation
(9) in x. It is convenient
X2 = -k2L2
c2 (1 + ~‘k’/cz)
in which case Eq. (8~) may be written as
((1 - H2)[l” (v,2/c2)]}l’2= qt. Equation (ll), with the exception of the factor (1 - K”), which is readily shown to be less than 1, is precisely that for the free falling particle and has the solution 2
1 )-
1 + (1 - JL2)g2t2li2 _ 1
z = g(l “_ X2)i[
C2
(12)
If the right hand side is expanded then zg(1
-&$gt2+
Clearly, the rotating object falls more slowly.
. .. .
(13)
PRINCIPLE
IT.
It is convenient
THE
to follow
OF
EINSTEIN
the work
cb2 = -dl”
389
EQUIVALENCE
THEORY
of Rohrlich.
- r2de2 - (V’,‘g)“dz*
The general line element is + V”dt2
(14)
where V is a real function of z only, has a nonrelativistic limit of 1 + 2gz, and primes represent derivatives with respect to z. The nonvanishing Christoffel symbols are
The general equations for the geodesics are ( 16a)
Equations
(16d) integrates
inmediately
to
dt l< - = -~. 1” dr
(17)
The initial conditions will determine k. Indeed the two different sets of conditions used previously lead to different fc’s. However, when Eq. (17) is used to eliminate 7 from i 16~) the result is 18) i.e., indepeudent of h-. Hence, motion in the x-direction takes place in the ~-0 plane even though the converse principle of equivalence is thus satisfied. V.
THE
PRINCIPLE
OF
is independent of what clearly is not’ true. The
EQUI\~ALENCE
It is clear, then, that unless one is prepared to insist that in a uniform houlogeneous field a rotating object may fall with less acceleration than a mass point, ouly the strong form is acceptable as a proper statement of the principle of equivalence. RECEIVED:
February
7, 1964
390
HARVEY REFERENCES
1. A. EINSTEIN, Ann. Physik. 36, 898 (1911). 2. P. G. BERGMANN, In “Handbuch der Physik,” Vol. 4, esp. pp. 204ff. Springer, Berlin, 1962. 3. B. BERTOTTI et al. In “Gravitation, An Introduction to Current Research,” pp. 8-9. Wiley, New York, 1962. 4. V. FOCK, “The Theory of Space Time and Gravitation,” esp. pp. 206 ff. Pergamon Press, London, 1959. 6. C. M$LLER, “The Theory of Relativity,” esp. pp. 75,220 ff. Oxford Univ. Press, London, 1952. 6. W. PAULI, “Theory of Relativity,” esp. p. 145. Pergamon Press, London, 1950. 7. F. ROHRLICH, Ann. Phys. (N. Y.) 22, 169 (1963). 8. R. H. DICKE, Science 129,621 (1959). 9. M. BEHACKER, Physik. Z. 14,989 (1913). 10. G. NORDSTROM, Physik. Z. 13, 1126 (1912).