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On classical scalar field theories and the relativistic kepler problem
ANNALS
OF PHYSICS:
62, 120-134 (1971)
On Classical Scalar Field Theories and the Relativistic Kepler Problem C. M. ANDERSEN AND HANS C. VON BAEYER Department
of Physics,
College
of William
and Mary,
Williamsburg,
Virginia
23185
Received June 25, 1970
Two versions of classical relativistic field theory corresponding to massless scalar exchange are compared. It is found that they lead to identical field equations but different equations of motion. The motion of one particle bound by the field of another, infinitely massive particle, i.e., the Kepler problem, is examined. In one case the path is an ellipse precessing through an angle equal to minus one-sixth the value predicted by general relativity. In the other case the path is an ellipse which does not precess. The relativistic Kepler problem in the latter case is identical to that formulated by Fronsdal, who started from an approximation to the Bethe-Salpeter equation.
1. INTRODUCTION Classical relativistic theories with scalar interactions are often discussed in conjunction with Lorentz-invariant theories of gravitation. The history of such theories of gravitation is long and often confusing, and the problems involved are not only mathematical but philosophical [l] as well. In this paper we do not concern ourselves with gravity or even with the tensor field theories. Instead we wish to point out some relations among different scalar field theories (without commenting on their physical significance). For convenience of identification we associate with two of these theories the names Bergmann and Thirring. Among the many papers on scalar field theories, one which stands out for clarity and simplicity was written by 0. Bergmann [2]. Although the theory described there certainly has roots in older formulations,l we shall call it simply Bergmann theory. A very detailed attempt at constructing a tensor field theory of gravity is due to Thirring [4].2 One of the theories mentioned by Thirring, but rejected as a candidate for gravity, is a scalar theory. A footnote suggeststhat this is just Bergmann theory, but part of the aim of the present paper t Some of these are referred to in a review article by Sex1 [3]. 2 See also the critique by Halpern [5].
120
RELATIVISTIC
KEPLER PROBLEM
121
is to show that it is not. In order to avoid confusion we call it Thirring theory, without any implication that Thirring was the first to state it or that he attaches any great significance to it. The fact that Bergmann’s and Thirring’s theories differ is not of great import in a field where theories abound. What makes Thirring theory interesting is that, with certain modifications which are quite in the spirit of Thirring’s approach, it predicts closed orbits for a particle bound to a stationary central source. Thus the Kepler problem permits a relativistic generalization in which the orbits remain ellipses. This fact contradicts the often-quoted claim that relativity breaks the degeneracy of the nonrelativistic Kepler problem [6, 71. The trouble with that claim is that it implicit1.y refers to the best known relativistic version of the Kepler problem, viz., the Coulomb problem in which the field is a four-vector with spatial components equal to zero. In Bergmann theory, where the field is scalar, this claim still holds; but in Thirring theory, where the field is again scalar, it doesnot. In Thirring theory a conserved Runge-Lenz vector exists and offers the possibility of couching the problem in group theoretical language analogous to that used for the nonrelativistic Kepler problem.3 An interesting observation is that the equations of motion derived from Thirring theory for the Kepler problem are identical to those found by Fronsdal [9] as a classical limit of a certain approximation to the Bethe-Salpeter equation. It must be emphasized, however, that Fronsdal’s theory also allows motion of the source and predicts ellipseseven in the two-body problem. A symmetry preserving extension of Thirring theory to the two-body case is described in Ref. [lS]. After presenting and comparing the theories of Bergmann and Thirring we summarize the equations in Table I. For convenience of comparison we include the analogous formulas for the nonrelativistic case and for the relativistic vector theory, which is fully discussedby, for example, Synge [lo]. In the appendix we sketch the reduction of Fronsdal’s equations to ours.
2. BERGMANN
THEORY
Bergmann’s theory [2] is a theory of particles interacting via a masslessscalar field in R4inkowski space.” Particles are labeled by a: and each has “charge” g, . The action principle is 6 s L(z) d4z = 0,
(2.1)
3 References to this approach appear in GyGrgyi [S]. We explore it further in Ref. [18]. 4 Notation: a, = Q&‘; 700 = -1111 = -722 = - r/33 = 1, otherwise T,,” = 0; fi = c = 1; X‘ = dxldx; a& = ala.+.
122
ANDERSEN
AND
VON
BAEYER
where 6 refers to variation with respect to the field and particle coordinates separately. The function L consists of three pieces corresponding to field, particle, and interaction contributions, L = LF + LP + 4,
(2.2)
where LF
=
++&>
(2.3)
audz),
Lp = - 1 m, f dAa[xr(au)ux:(OL)]1’2SO(x’“‘(h,)
LI = - 5 g,J dA,[x’(~‘)%X:(“)]~/~ v(z)
- z),
8(4)(x(~)(Aa) - z).
(2.5)
In these equations the path of each particle (Yis parametrized by a parameter A, which is monotonic increasing with time but otherwise arbitrary. Since L, and Lp are independent of these parameters A, we can, after the variation with respect to x(a), set $du.p u = 1. (2.6) This identifies A, as the proper time T, . We shall use dots only to refer to differentiation with respect to proper time. Variation of (2.1) with respect to x(“) and use of Eq. (2.6) yields the equation of motion (2.7)
These equations can also be derived from an action-at-a-distance analogous to that of Wheeler and Feynman [12]. Variation of (2.1) with respect to CJJyields the field equation dTa 8(4)[x(“)(~,)
- z].
theory [ll]
w3)
Since this equation is linear we can let 9, = c (PLIY a
(2.9)
where q’oIis the field produced by particle 01.By neglecting the self-force, i.e., terms proportional to gm2, the field which enters in Eqs. (2.7) is that produced by all particles except 0~. In action-at-a-distance theory this feature of the equations of motion is automatic.
RELATlVISTIC
The symmetric and is given by
KEPLER
conserved energy-momentum
TUY(Z) = $29 2,9, rj”” -
123
PROBLEM
tensor is derived by Bergmann [2]
i;u‘p 2”‘p - c g, i’ L/T, %(ww+?~ a
+ cp) s(,+(T,)
- z) (2.10)
with 8,FLD = 0. It is important to note that the source term in Eq. (2.8) contains only one piece of Twu, viz., that which refers to free particles. This means that only particles, not the field, act as sources of the field. For a theory of gravity this is undesirable and hence one might replace the right side of Eq. (2.5) by a term proportional to TMu, thus introducing nonlinearities in the field equations. In a tensor theory it is in fact necessary for consistency to make such a replacement [4, 51. (In that case it is Tp", not T,", which is inserted.) In Bergmann theory this replacement is not necessary and is omitted. The arbitrariness of A, permits a different interpretation of the theory. If we let pyxlo),
X’(a)
) = nlor[.Y”~)~X~ql~~ [l + g,m,‘&‘~‘)],
(2.11)
then
and the equation of motion for the a-th particle derived from (2.1) reads (2.13) Now let us choose, instead of Eq. (2.6),
(2.14) With this choice we can interpret metric
Bergmann theory in a Riemann space with the
Cal-= q,,,(l + g,n?,$)‘, sUl’ so that
The orbits are geodesics in this space.
(3.15)
124
ANDERSEN AND VON BAEYER
It is often convenient to use, instead of (2.16), the equivalent form (2.17) The proof [13] that this form is equivalent to (2.16) depends on the particular choice of h implied by (2.14). While it is clear that this is consistent with (2.16), since !iP) dA, is independent of A,, we must prove the consistency of (2.14) with (2.17). To this end we simply note that
is a consequence of the equation of motion. This means that g$x’(~)“x’(“)Y = constant.
(2.19)
We chose the constant to be unity to obtain Eq. (2.14). Note that the analogous expression
contains no information. The geometrical interpretation is useful if, as in gravitational theory, the charge g, is proportional to m, . Then m, drops out of the equations of motion (2.13) and one can define a metric tensor independent of the particle label CL.
3. THE KEPLER PROBLEM IN BERGMANN
THEORY
The special case of an infinitely heavy particle (p) located at the origin with a light particle (e) moving around it is solved by Bergmann [2] in a number of steps. i. Potential Only the potential qp is of interest. The coordinates of p are xfp) = (tP , 0, 0,O) so that Eq. (2.8) reduces to @P(Z) atp2-
V2FP(4
=
-gpw
(3.1)
with the solution FP(r)
=
-gpl(~rl.
(3.2)
RELATIVISTIC
KEPLER
PROBLEM
125
k inP
v
-y=
rnka -2,12
V
J,_k
E=mdl
-9
Ii714
E, = nr dl
rI:k
- kz/(n2-t k2)
fv2
E,, :
177
~‘1
-
~(k,‘77)’
r = k (1 + 9) rn$
J+l
-= E=dlm+v~
” Energy is denoted by E and is the total energy, except in the nonrelativistic case where rest mass has been neglected. The potential energy Vis given by -k/r throughout: 7 = (1 - I.~)-~/~ and p = I p 1.The conserved Runge-Lenz vector is denoted by R and the conserved angular momentum is J : r Y p in all four theories. The orbit equation is written in terms of II =: r-l and the azimuthal angle y.The perihelion advance per revolution is denoted by dv.
En=
J, z n
Circular Bohr orbits
y=-
J2
E=
Circular orbits
gz
f
u
5
RELATIVISTIC
KEPLER
127
PROBLEM
We shall drop the subscript p and let the potential energy be denoted by V, v(r)
= gepp(r)
= -gpg&4nr)
= -k/r.
(3.3)
ii. Equations of motion for e We drop the subscript e. The fourth equation of motion (2.7) can be integrated to give thLeconserved energy E = h + W,
(3.4)
where y ;= (1 - z9-l12. The other three equations then reduce to (3.5) with p = Ev.
(3.6)
Equations (3.4) and (3.6) can be solved to give the interesting relation E2 = p2 + (m + V)3,
(3.7)
which should be compared to the analogous expression for the vector potential in Table 1. The difference is simply that in this case V must be lumped with the scalar m while in vector theory it is lumped with the zero component of a fourvector, viz., E. The equ.ations of motion (3.4) to (3.6) can also be derived from the noncovariant Lagrangian L(x, v) = -(m
+ V(x)) 1/l - v2
(3.8)
or Hamiltlonian H(x, p) = z/p” + (112+ V(x)>“. The conserved angular momentum considered. in Table 1 is
G.9)
for this theory as well as the other theories
J=rxp.
(3.10)
iii. The bound state problem In spherical coordinates (r, 0, p) Eqs. (3.4) and (3.10) can be solved for dr/dt
128
AN~~EN
AND
VON
BAEYER
and dp/dt and hence their ratio. The result, after differentiation is the equation of the orbit
with respect to gs,
(3.11) where u = r-l. This equation is of the type (3.12) which has a general solution given by
u = a/(1 - c) + B cos(~‘l
- E y).
(3.13)
(One constant of integration has been fixed by the orientation of the coordinate axes. The other one, B, is fixed by the first-order differential equation from which (3.11) is obtained.) Equation (3.13) represents an ellipse whose perihelion advances by an angle 2~(1 - c) per revolution. Applied to Eq. (3.11) this means that the perihelion advance is (3.14) for almost circular orbits. This is minus one-sixth the prediction
of general relativity.
iv. Circular orbits For the case of circular
orbits Eq. (3.5) reads
yEv2 = k/r = -V,
(3.15)
E = 1112/l - 9.
(3.16)
which together with (3.4) gives
This suggestive equation, which reduces to the nonrelativistic form E - m = - $nv2, has been noted several times [14, 111 and was the subject of some speculation [15]. It occurs also in the vector theory. A similar calculation yields for the angular momentum
J = k d1 - v2jv
(3.17)
r = k/(mv”).
(3.18)
and for the radius
RELATIVISTIC
129
KEPLER PROBLEM
These expressions are useful for determining the minimum values of J and r which occur in the limit v = 1. The circular orbits can be quantized a la Bohr by setting J = n with the resulting energy spectrum E, = nz(l + P/~z~)-~/~. (3.19) This differs from the expression for E, in the vector theory (see Table I), which is identical to the spectrum obtained from the Dirac equation for orbitals with maximum angular momentum (most nearly circular orbits) [ll].
4. THIRRING
THEORY
Thirring theory [4] is cast in Riemann space but can be interpreted in Minkowski space. In order to contrast it with Bergmann theory we proceed just as in Section 2. The only difference is that instead of Eqs. (2.4) and (2.5) we now have Lp + LI = - 1 tn, 1‘ c/Au[X’(%zu‘(qw
[l +
2g
u
t+(p(ql” JI a
~(4)(X.(a)-
z),
(4.1)
or analogously to Eq. (2.1 I), Q(y) z t~la[.~‘(~)ug~(“)]l;2 [l + 2gp~‘&“)]“’
(4.2)
with $alw$n) = 1. Ll
(4.3)
When interpreted in Riemann space we have pa1 = 111 1l/2 o![gkdX’~alu~y’(“lv “!J
(4.4)
Ca;)= n,,,[l + 2g&1q$x(m))] g IL”
(4.5)
with
and .Y’(%:(a)(l
+ 2gmtt7,1ql)= 1.
(4.6)
Finally, if we use the equivalent form $!(a) = ~ttlRg~)X’(a)i’g’(OI)Y together with the metric (4.5), we have recovered Thirring’s 595/641-o
(4.7) starting point. (The
130
ANDERSEN AND VON BAEYER
fact that he attaches the opposite sign to g, is immaterial since the physics depends only on products of two charges.) Comparing Eqs. (2.11) and (4.2) it appears that Bergmann’s theory is a linearized version of Thirring’s. But comparing (2.15) with (4.5) it seems the other way around. In any case, the two theories lead to different predictions. The equations of motion derived from (4.2) are & ~“dx’a’) [(l + 2g,m;1q)1’2 @)I = (I + 2galn;lcp)‘~” a rather than (2.7), and the field equation is m, $-
(4.8)
au a’“rp(z) = - c g, J’ c/r, [I + 2g,ln,1cp(z)]-1’2 8(4)(.P(T) - z) (4.9) a rather than Eq. (2.8). This looks like a very nonlinear equation but the delta function in the numerator allows us to replace y(z) by y(x(“)) and to renormalize the charges by setting ,& = gJ1 + 2g,r?r,1+(‘))]-1’“. The value of 9 at the position of the charge is infinite for point charges, but if the 6 functions in (4.9) are replaced by extended densities p(z) = p(I z - xca) I), then the part of y(z) due to the ol-th term on the right side of (4.9) is still proportional to /z - x(m) j-l outside the c+th source but finite within the source. This implies a finite charge renormalization similar to that which is necessary in Thirring’s tensor theory. In principle the field at the particle 01, IYJI(x(“)),depends upon all particles, but the contribution due to particle LYitself is expected to be overwhelming. For this reason ja can be considered constant (i.e., independent of the positions and motions of the other particles). One can avoid this renormalization by replacing the source term in (4.9) by the trace of the energy-momentum tensor for free particles. This is, in a sense, the opposite of what Thirring did. It linearizes the intrinsically nonlinear theory. When this is done, Eq. (4.9) is replaced by Eq. (2.8) and the only surviving difference between Bergmann and Thirring theories lies in the equations of motion.
5. THE KEPLER PROBLEM IN THIRRING
THEORY
We repeat the steps of Section 3. i. Potential The potential energy is given by (5.1)
RELATIVISTIC
where k is either renormalized
KEPLER
131
PROBLEM
or not, as explained above.
ii. Equation of motion The fourth
equation of motion (4.8) yields E = ~1 \/I
+ 2tc1V
y,
(5.2)
while the other three read L/P -zy-&=z- 4 dr
c
.!2
,+A!-’ m
VV
1
(5.3)
with
p = Ev. The key observation
(5.4)
is that Eq. (5.3) can be written 4 dt=
-pv.
(5.5)
This has the structure of the nonrelativistic equations of motion for the Kepler problem and hence the orbits are closed. By straightforward differentiation one finds that the conserved Runge-Lenz vector is given by (5.6) which is exactly the same expression Again the angular momentum is
as used in the nonrelativistic
Kepler problem.
J=rxp. Equations
(5.2)-(5.4)
(5.7)
may also be derived from the noncovariant
L(x, v) = -tn
dl
t 2m-IV(x)
-\/I - v2
Lagrangian (5.8)
or Hamiltonian
H(x, p) = dp” + 1112+ 21?2V(X).
(5.9)
iii. The bound-state problem The remaining entries in the fourth column of Table I are obtained by trivial manipulations of Eqs. (5.1)-(5.7). The magic formula E = nl dm no longer holds for (circular orbits, but the Bohr spectrum is just that of the vector theory.
132
ANDERSEN
VON
AND
BAEYER
Actually the energy spectra for both the nonrelativistic theory and Thirring theory are the correct quantum-mechanical expressions. In the former case the energy equation (5.10)
E=&+V
becomes the time-independent Schrodinger equation if p is interpreted as an operator. The solution of this equation leads to the well-known spectrum -__tnk”
En=
2d
I? = 1, 2,...,
’
(5.11)
with I = 0, I).. .) I2 - 1.
J1 = I,
(5.12)
In the Thirring theory the energy equation can be written E2 - m” =&IV. 201
(5.13)
The similarity of (5.13) to (5.10) permits us to read off5 E,” - t# 2m
-
-
tnk” 2tP ’
Il. = 1, 2,...,
(5.14)
together with (5.12) and leads to the last entry in Table 1. It is interesting to note that Eqs. (5.4) and (5.5) were obtained by Fronsdal as the relativistic classical limit of an approximation to the Bethe-Salpeter equation. In the appendix we sketch the derivation from Fronsdal’s equations of motion. Note added in proof: P. Kustaanheimo (Commentationes Physica Mathematicae of the Sociatas Scientiarum Fennica, Vol. XXI No. 3 (1958)) has postulated the general scalar equation of motion miiL = (a&v - +ml
+ g,(w?z) + soJh>” + ....I
with arbitrary real g;, i = 1, 2 ,.... The perihelion advance for this equation of motion is A4= For Bergmann’s theory g = -1; Bergmann for this reference.
-(2+g,)$.
for Thirring’s
g, = -2. We are indebted to Professor 0.
6 Professor E. Remler has pointed out in an unpublished paper (Princeton, 1966) that whenever one has solved the two-body Schrodinger equation, then one has simultaneously solved the Lorentz-invariant two-body problem with Hamiltonian Hz = j P I2 + {[I k I2 + ml2 + 2i~z,V]“~ + [I k I2 + mp2 + 2r71,V]‘~~}*, where m, is the reduced mass, and P and k are the total and relative momenta, respectively. In the limit m, - ao this reduces our Eq. (5.9). He has related this Hamiltonian to both the Bakamjian-Thomas theories (cf. Ref. [16] and references contained therein) and to the Blankenbecler-Sugar equation [17].
RELATIVISTIC
KEPLER
133
PROBLEM
APPENDIX
Fronsdal [9] considers two particles, labeled 1 and 2, with positions xlU, xZu = xU and mornenta plu = q” and pZU = p” - q”. We think of particle 1 as the heavy one at the origin and find the equations of motion of particle 2. The theory contains a negative coupling constant y with dimensions of mass squared. In order to recover nonrelativistic theory we let
y= -2mlm,k. With this identification
(A.1)
Eq. (F.47) (where F stands for Fronsdal)
becomes
Ez2 = pz2 f mz2 - 2mnkr-‘.
64.2)
This is just the relation between energy and momentum identify I* as the distance between particles. Equation (F. 18) reads p2 = Ezvz ,
listed in Table I if we can
(A-3)
which is tour Eq. (5.4). To obtain the equations of motion for particle 2 we turn to Eq. (F.45) which reads in our notation
dp, yzz.7 &
+
YS4
2m12r3E2 =
-~__km2 sq E, mIr3 ’
where our (s4)i equals Fronsdal’s si (i = 1, 2, 3). In passing from (F.45) to (A.4) one must realize that Fronsdal operates in the center-of-mass frame and thus q E p1 =: -pZ . Now (F.55) identifies sq as s* = -m,y
= -ml(xl
- x2),
(A.5)
so that the equation of motion becomes
&2 -=dt,
km, 6, - ~2) E2
r3
.
w-9
If we let :x1 = 0 we have
4 -zzz dt, which is our Fronsdal’s differs from potential. It
km, x2 -Yg->
64.7)
Eq. (5.5) if we can properly identify Fronsdal’s r. r is a relativistic generalization of the interparticle distance which the usual one which appears, for example, in the Lienard-Wiechert is given by (F.59) as r = [(x1 - x2)2 - (tl - t2)2]1/2,
64.8)
134
ANDERSEN AND VON BAEYER
where (tI - tz) is given by (F.60) and (F.20) as y. = t, -
t.,- = in;2qvs
OV
= q2ho(q”~‘,)
+ yoWq,N
=
-~q”q,Wy,)
+ ~0 (A.91
(here we have used q”q” = p12 = ml”). Now since by (A.9) 0 = 4”Y” = PlOYO - Pl . Y
(A.lO)
and since in our limit plo - a with p1 and y finite, we have y. = 0. Thus r = 1x, - x, j and Fronsdal’s equations become identical with those of Section 5.
REFERENCES 1. P. HAVAS, Foundation problems in general relativity, in “Delaware Seminar in the Foundations of Physics” (M. Bunge, Ed.) Springer-Verlag, New York, 1967. 2. 0. BERGMANN, Amer. J. Phys. 24 (1956), 38. 3. R. H. SEXL, Fortschr. Phys. 15 (1967), 269. 4. W. E. THIRRING, Ann. Phys. (New York) 16 (1961), 96. 5. L. HALPERN, Ann. Phys. (New York) 25 (1963), 387. 6. D. F. GREENBERG, Amer. J. Phys. 34 (1966), 1101. 7. H. BACRY, H. RUEGG, AND J. M. SOURIAU, Con~mun. Math. Phys. 3 (1966), 323. 8. G. GYBRGYI, Nuovo Cimento A 53 (1968), 717. 9. C. FRONSDAL, “The Relativistic Kepler Problem,” UCLA preprint, March, 1970. 10. J. L. SYNGE, “Relativity: The Special Theory,” p. 396, Interscience, New York, 1956. 11. C. M. ANDERSEN AND H. C. VON BAEYER, Ann. Phys. (New York), 60(1970), 67. 12. J. A. WHEELER AND R. P. FEYNMAN, Rev. Mod. Phys. 17 (1945), 157; 21(1949), 425. 13. R. ADLER, M. BAZIN, AND M. SCHIFFER, “Introduction to General Relativity,” p. 53, McGrawHill, New York, 1965. 14. A. SCHILD, Phys. Rev. 131 (1963), 2762. 15. J. DORLING, Amer. J. Phys. 38 (1970), 510. 16. R. FONG AND J. SUCHER, J. Math. Phys. 5 (1964), 456. 17. R. BLANKENBECLER AND R. SUGAR, Phys. Rev. 142 (1966), 1051. 18. C. M. ANDERSEN AND H. C. VON BAEYER, “Closed Orbits and SO(4, 2) Symmetry in Relativistic Two-Body Theory,” in “Proceedings of the Symposium on De Sitter and Conformal Groups,” Boulder, Colorado, June, 1970, to be published.
OF PHYSICS:
62, 120-134 (1971)
On Classical Scalar Field Theories and the Relativistic Kepler Problem C. M. ANDERSEN AND HANS C. VON BAEYER Department
of Physics,
College
of William
and Mary,
Williamsburg,
Virginia
23185
Received June 25, 1970
Two versions of classical relativistic field theory corresponding to massless scalar exchange are compared. It is found that they lead to identical field equations but different equations of motion. The motion of one particle bound by the field of another, infinitely massive particle, i.e., the Kepler problem, is examined. In one case the path is an ellipse precessing through an angle equal to minus one-sixth the value predicted by general relativity. In the other case the path is an ellipse which does not precess. The relativistic Kepler problem in the latter case is identical to that formulated by Fronsdal, who started from an approximation to the Bethe-Salpeter equation.
1. INTRODUCTION Classical relativistic theories with scalar interactions are often discussed in conjunction with Lorentz-invariant theories of gravitation. The history of such theories of gravitation is long and often confusing, and the problems involved are not only mathematical but philosophical [l] as well. In this paper we do not concern ourselves with gravity or even with the tensor field theories. Instead we wish to point out some relations among different scalar field theories (without commenting on their physical significance). For convenience of identification we associate with two of these theories the names Bergmann and Thirring. Among the many papers on scalar field theories, one which stands out for clarity and simplicity was written by 0. Bergmann [2]. Although the theory described there certainly has roots in older formulations,l we shall call it simply Bergmann theory. A very detailed attempt at constructing a tensor field theory of gravity is due to Thirring [4].2 One of the theories mentioned by Thirring, but rejected as a candidate for gravity, is a scalar theory. A footnote suggeststhat this is just Bergmann theory, but part of the aim of the present paper t Some of these are referred to in a review article by Sex1 [3]. 2 See also the critique by Halpern [5].
120
RELATIVISTIC
KEPLER PROBLEM
121
is to show that it is not. In order to avoid confusion we call it Thirring theory, without any implication that Thirring was the first to state it or that he attaches any great significance to it. The fact that Bergmann’s and Thirring’s theories differ is not of great import in a field where theories abound. What makes Thirring theory interesting is that, with certain modifications which are quite in the spirit of Thirring’s approach, it predicts closed orbits for a particle bound to a stationary central source. Thus the Kepler problem permits a relativistic generalization in which the orbits remain ellipses. This fact contradicts the often-quoted claim that relativity breaks the degeneracy of the nonrelativistic Kepler problem [6, 71. The trouble with that claim is that it implicit1.y refers to the best known relativistic version of the Kepler problem, viz., the Coulomb problem in which the field is a four-vector with spatial components equal to zero. In Bergmann theory, where the field is scalar, this claim still holds; but in Thirring theory, where the field is again scalar, it doesnot. In Thirring theory a conserved Runge-Lenz vector exists and offers the possibility of couching the problem in group theoretical language analogous to that used for the nonrelativistic Kepler problem.3 An interesting observation is that the equations of motion derived from Thirring theory for the Kepler problem are identical to those found by Fronsdal [9] as a classical limit of a certain approximation to the Bethe-Salpeter equation. It must be emphasized, however, that Fronsdal’s theory also allows motion of the source and predicts ellipseseven in the two-body problem. A symmetry preserving extension of Thirring theory to the two-body case is described in Ref. [lS]. After presenting and comparing the theories of Bergmann and Thirring we summarize the equations in Table I. For convenience of comparison we include the analogous formulas for the nonrelativistic case and for the relativistic vector theory, which is fully discussedby, for example, Synge [lo]. In the appendix we sketch the reduction of Fronsdal’s equations to ours.
2. BERGMANN
THEORY
Bergmann’s theory [2] is a theory of particles interacting via a masslessscalar field in R4inkowski space.” Particles are labeled by a: and each has “charge” g, . The action principle is 6 s L(z) d4z = 0,
(2.1)
3 References to this approach appear in GyGrgyi [S]. We explore it further in Ref. [18]. 4 Notation: a, = Q&‘; 700 = -1111 = -722 = - r/33 = 1, otherwise T,,” = 0; fi = c = 1; X‘ = dxldx; a& = ala.+.
122
ANDERSEN
AND
VON
BAEYER
where 6 refers to variation with respect to the field and particle coordinates separately. The function L consists of three pieces corresponding to field, particle, and interaction contributions, L = LF + LP + 4,
(2.2)
where LF
=
++&>
(2.3)
audz),
Lp = - 1 m, f dAa[xr(au)ux:(OL)]1’2SO(x’“‘(h,)
LI = - 5 g,J dA,[x’(~‘)%X:(“)]~/~ v(z)
- z),
8(4)(x(~)(Aa) - z).
(2.5)
In these equations the path of each particle (Yis parametrized by a parameter A, which is monotonic increasing with time but otherwise arbitrary. Since L, and Lp are independent of these parameters A, we can, after the variation with respect to x(a), set $du.p u = 1. (2.6) This identifies A, as the proper time T, . We shall use dots only to refer to differentiation with respect to proper time. Variation of (2.1) with respect to x(“) and use of Eq. (2.6) yields the equation of motion (2.7)
These equations can also be derived from an action-at-a-distance analogous to that of Wheeler and Feynman [12]. Variation of (2.1) with respect to CJJyields the field equation dTa 8(4)[x(“)(~,)
- z].
theory [ll]
w3)
Since this equation is linear we can let 9, = c (PLIY a
(2.9)
where q’oIis the field produced by particle 01.By neglecting the self-force, i.e., terms proportional to gm2, the field which enters in Eqs. (2.7) is that produced by all particles except 0~. In action-at-a-distance theory this feature of the equations of motion is automatic.
RELATlVISTIC
The symmetric and is given by
KEPLER
conserved energy-momentum
TUY(Z) = $29 2,9, rj”” -
123
PROBLEM
tensor is derived by Bergmann [2]
i;u‘p 2”‘p - c g, i’ L/T, %(ww+?~ a
+ cp) s(,+(T,)
- z) (2.10)
with 8,FLD = 0. It is important to note that the source term in Eq. (2.8) contains only one piece of Twu, viz., that which refers to free particles. This means that only particles, not the field, act as sources of the field. For a theory of gravity this is undesirable and hence one might replace the right side of Eq. (2.5) by a term proportional to TMu, thus introducing nonlinearities in the field equations. In a tensor theory it is in fact necessary for consistency to make such a replacement [4, 51. (In that case it is Tp", not T,", which is inserted.) In Bergmann theory this replacement is not necessary and is omitted. The arbitrariness of A, permits a different interpretation of the theory. If we let pyxlo),
X’(a)
) = nlor[.Y”~)~X~ql~~ [l + g,m,‘&‘~‘)],
(2.11)
then
and the equation of motion for the a-th particle derived from (2.1) reads (2.13) Now let us choose, instead of Eq. (2.6),
(2.14) With this choice we can interpret metric
Bergmann theory in a Riemann space with the
Cal-= q,,,(l + g,n?,$)‘, sUl’ so that
The orbits are geodesics in this space.
(3.15)
124
ANDERSEN AND VON BAEYER
It is often convenient to use, instead of (2.16), the equivalent form (2.17) The proof [13] that this form is equivalent to (2.16) depends on the particular choice of h implied by (2.14). While it is clear that this is consistent with (2.16), since !iP) dA, is independent of A,, we must prove the consistency of (2.14) with (2.17). To this end we simply note that
is a consequence of the equation of motion. This means that g$x’(~)“x’(“)Y = constant.
(2.19)
We chose the constant to be unity to obtain Eq. (2.14). Note that the analogous expression
contains no information. The geometrical interpretation is useful if, as in gravitational theory, the charge g, is proportional to m, . Then m, drops out of the equations of motion (2.13) and one can define a metric tensor independent of the particle label CL.
3. THE KEPLER PROBLEM IN BERGMANN
THEORY
The special case of an infinitely heavy particle (p) located at the origin with a light particle (e) moving around it is solved by Bergmann [2] in a number of steps. i. Potential Only the potential qp is of interest. The coordinates of p are xfp) = (tP , 0, 0,O) so that Eq. (2.8) reduces to @P(Z) atp2-
V2FP(4
=
-gpw
(3.1)
with the solution FP(r)
=
-gpl(~rl.
(3.2)
RELATIVISTIC
KEPLER
PROBLEM
125
k inP
v
-y=
rnka -2,12
V
J,_k
E=mdl
-9
Ii714
E, = nr dl
rI:k
- kz/(n2-t k2)
fv2
E,, :
177
~‘1
-
~(k,‘77)’
r = k (1 + 9) rn$
J+l
-= E=dlm+v~
” Energy is denoted by E and is the total energy, except in the nonrelativistic case where rest mass has been neglected. The potential energy Vis given by -k/r throughout: 7 = (1 - I.~)-~/~ and p = I p 1.The conserved Runge-Lenz vector is denoted by R and the conserved angular momentum is J : r Y p in all four theories. The orbit equation is written in terms of II =: r-l and the azimuthal angle y.The perihelion advance per revolution is denoted by dv.
En=
J, z n
Circular Bohr orbits
y=-
J2
E=
Circular orbits
gz
f
u
5
RELATIVISTIC
KEPLER
127
PROBLEM
We shall drop the subscript p and let the potential energy be denoted by V, v(r)
= gepp(r)
= -gpg&4nr)
= -k/r.
(3.3)
ii. Equations of motion for e We drop the subscript e. The fourth equation of motion (2.7) can be integrated to give thLeconserved energy E = h + W,
(3.4)
where y ;= (1 - z9-l12. The other three equations then reduce to (3.5) with p = Ev.
(3.6)
Equations (3.4) and (3.6) can be solved to give the interesting relation E2 = p2 + (m + V)3,
(3.7)
which should be compared to the analogous expression for the vector potential in Table 1. The difference is simply that in this case V must be lumped with the scalar m while in vector theory it is lumped with the zero component of a fourvector, viz., E. The equ.ations of motion (3.4) to (3.6) can also be derived from the noncovariant Lagrangian L(x, v) = -(m
+ V(x)) 1/l - v2
(3.8)
or Hamiltlonian H(x, p) = z/p” + (112+ V(x)>“. The conserved angular momentum considered. in Table 1 is
G.9)
for this theory as well as the other theories
J=rxp.
(3.10)
iii. The bound state problem In spherical coordinates (r, 0, p) Eqs. (3.4) and (3.10) can be solved for dr/dt
128
AN~~EN
AND
VON
BAEYER
and dp/dt and hence their ratio. The result, after differentiation is the equation of the orbit
with respect to gs,
(3.11) where u = r-l. This equation is of the type (3.12) which has a general solution given by
u = a/(1 - c) + B cos(~‘l
- E y).
(3.13)
(One constant of integration has been fixed by the orientation of the coordinate axes. The other one, B, is fixed by the first-order differential equation from which (3.11) is obtained.) Equation (3.13) represents an ellipse whose perihelion advances by an angle 2~(1 - c) per revolution. Applied to Eq. (3.11) this means that the perihelion advance is (3.14) for almost circular orbits. This is minus one-sixth the prediction
of general relativity.
iv. Circular orbits For the case of circular
orbits Eq. (3.5) reads
yEv2 = k/r = -V,
(3.15)
E = 1112/l - 9.
(3.16)
which together with (3.4) gives
This suggestive equation, which reduces to the nonrelativistic form E - m = - $nv2, has been noted several times [14, 111 and was the subject of some speculation [15]. It occurs also in the vector theory. A similar calculation yields for the angular momentum
J = k d1 - v2jv
(3.17)
r = k/(mv”).
(3.18)
and for the radius
RELATIVISTIC
129
KEPLER PROBLEM
These expressions are useful for determining the minimum values of J and r which occur in the limit v = 1. The circular orbits can be quantized a la Bohr by setting J = n with the resulting energy spectrum E, = nz(l + P/~z~)-~/~. (3.19) This differs from the expression for E, in the vector theory (see Table I), which is identical to the spectrum obtained from the Dirac equation for orbitals with maximum angular momentum (most nearly circular orbits) [ll].
4. THIRRING
THEORY
Thirring theory [4] is cast in Riemann space but can be interpreted in Minkowski space. In order to contrast it with Bergmann theory we proceed just as in Section 2. The only difference is that instead of Eqs. (2.4) and (2.5) we now have Lp + LI = - 1 tn, 1‘ c/Au[X’(%zu‘(qw
[l +
2g
u
t+(p(ql” JI a
~(4)(X.(a)-
z),
(4.1)
or analogously to Eq. (2.1 I), Q(y) z t~la[.~‘(~)ug~(“)]l;2 [l + 2gp~‘&“)]“’
(4.2)
with $alw$n) = 1. Ll
(4.3)
When interpreted in Riemann space we have pa1 = 111 1l/2 o![gkdX’~alu~y’(“lv “!J
(4.4)
Ca;)= n,,,[l + 2g&1q$x(m))] g IL”
(4.5)
with
and .Y’(%:(a)(l
+ 2gmtt7,1ql)= 1.
(4.6)
Finally, if we use the equivalent form $!(a) = ~ttlRg~)X’(a)i’g’(OI)Y together with the metric (4.5), we have recovered Thirring’s 595/641-o
(4.7) starting point. (The
130
ANDERSEN AND VON BAEYER
fact that he attaches the opposite sign to g, is immaterial since the physics depends only on products of two charges.) Comparing Eqs. (2.11) and (4.2) it appears that Bergmann’s theory is a linearized version of Thirring’s. But comparing (2.15) with (4.5) it seems the other way around. In any case, the two theories lead to different predictions. The equations of motion derived from (4.2) are & ~“dx’a’) [(l + 2g,m;1q)1’2 @)I = (I + 2galn;lcp)‘~” a rather than (2.7), and the field equation is m, $-
(4.8)
au a’“rp(z) = - c g, J’ c/r, [I + 2g,ln,1cp(z)]-1’2 8(4)(.P(T) - z) (4.9) a rather than Eq. (2.8). This looks like a very nonlinear equation but the delta function in the numerator allows us to replace y(z) by y(x(“)) and to renormalize the charges by setting ,& = gJ1 + 2g,r?r,1+(‘))]-1’“. The value of 9 at the position of the charge is infinite for point charges, but if the 6 functions in (4.9) are replaced by extended densities p(z) = p(I z - xca) I), then the part of y(z) due to the ol-th term on the right side of (4.9) is still proportional to /z - x(m) j-l outside the c+th source but finite within the source. This implies a finite charge renormalization similar to that which is necessary in Thirring’s tensor theory. In principle the field at the particle 01, IYJI(x(“)),depends upon all particles, but the contribution due to particle LYitself is expected to be overwhelming. For this reason ja can be considered constant (i.e., independent of the positions and motions of the other particles). One can avoid this renormalization by replacing the source term in (4.9) by the trace of the energy-momentum tensor for free particles. This is, in a sense, the opposite of what Thirring did. It linearizes the intrinsically nonlinear theory. When this is done, Eq. (4.9) is replaced by Eq. (2.8) and the only surviving difference between Bergmann and Thirring theories lies in the equations of motion.
5. THE KEPLER PROBLEM IN THIRRING
THEORY
We repeat the steps of Section 3. i. Potential The potential energy is given by (5.1)
RELATIVISTIC
where k is either renormalized
KEPLER
131
PROBLEM
or not, as explained above.
ii. Equation of motion The fourth
equation of motion (4.8) yields E = ~1 \/I
+ 2tc1V
y,
(5.2)
while the other three read L/P -zy-&=z- 4 dr
c
.!2
,+A!-’ m
VV
1
(5.3)
with
p = Ev. The key observation
(5.4)
is that Eq. (5.3) can be written 4 dt=
-pv.
(5.5)
This has the structure of the nonrelativistic equations of motion for the Kepler problem and hence the orbits are closed. By straightforward differentiation one finds that the conserved Runge-Lenz vector is given by (5.6) which is exactly the same expression Again the angular momentum is
as used in the nonrelativistic
Kepler problem.
J=rxp. Equations
(5.2)-(5.4)
(5.7)
may also be derived from the noncovariant
L(x, v) = -tn
dl
t 2m-IV(x)
-\/I - v2
Lagrangian (5.8)
or Hamiltonian
H(x, p) = dp” + 1112+ 21?2V(X).
(5.9)
iii. The bound-state problem The remaining entries in the fourth column of Table I are obtained by trivial manipulations of Eqs. (5.1)-(5.7). The magic formula E = nl dm no longer holds for (circular orbits, but the Bohr spectrum is just that of the vector theory.
132
ANDERSEN
VON
AND
BAEYER
Actually the energy spectra for both the nonrelativistic theory and Thirring theory are the correct quantum-mechanical expressions. In the former case the energy equation (5.10)
E=&+V
becomes the time-independent Schrodinger equation if p is interpreted as an operator. The solution of this equation leads to the well-known spectrum -__tnk”
En=
2d
I? = 1, 2,...,
’
(5.11)
with I = 0, I).. .) I2 - 1.
J1 = I,
(5.12)
In the Thirring theory the energy equation can be written E2 - m” =&IV. 201
(5.13)
The similarity of (5.13) to (5.10) permits us to read off5 E,” - t# 2m
-
-
tnk” 2tP ’
Il. = 1, 2,...,
(5.14)
together with (5.12) and leads to the last entry in Table 1. It is interesting to note that Eqs. (5.4) and (5.5) were obtained by Fronsdal as the relativistic classical limit of an approximation to the Bethe-Salpeter equation. In the appendix we sketch the derivation from Fronsdal’s equations of motion. Note added in proof: P. Kustaanheimo (Commentationes Physica Mathematicae of the Sociatas Scientiarum Fennica, Vol. XXI No. 3 (1958)) has postulated the general scalar equation of motion miiL = (a&v - +ml
+ g,(w?z) + soJh>” + ....I
with arbitrary real g;, i = 1, 2 ,.... The perihelion advance for this equation of motion is A4= For Bergmann’s theory g = -1; Bergmann for this reference.
-(2+g,)$.
for Thirring’s
g, = -2. We are indebted to Professor 0.
6 Professor E. Remler has pointed out in an unpublished paper (Princeton, 1966) that whenever one has solved the two-body Schrodinger equation, then one has simultaneously solved the Lorentz-invariant two-body problem with Hamiltonian Hz = j P I2 + {[I k I2 + ml2 + 2i~z,V]“~ + [I k I2 + mp2 + 2r71,V]‘~~}*, where m, is the reduced mass, and P and k are the total and relative momenta, respectively. In the limit m, - ao this reduces our Eq. (5.9). He has related this Hamiltonian to both the Bakamjian-Thomas theories (cf. Ref. [16] and references contained therein) and to the Blankenbecler-Sugar equation [17].
RELATIVISTIC
KEPLER
133
PROBLEM
APPENDIX
Fronsdal [9] considers two particles, labeled 1 and 2, with positions xlU, xZu = xU and mornenta plu = q” and pZU = p” - q”. We think of particle 1 as the heavy one at the origin and find the equations of motion of particle 2. The theory contains a negative coupling constant y with dimensions of mass squared. In order to recover nonrelativistic theory we let
y= -2mlm,k. With this identification
(A.1)
Eq. (F.47) (where F stands for Fronsdal)
becomes
Ez2 = pz2 f mz2 - 2mnkr-‘.
64.2)
This is just the relation between energy and momentum identify I* as the distance between particles. Equation (F. 18) reads p2 = Ezvz ,
listed in Table I if we can
(A-3)
which is tour Eq. (5.4). To obtain the equations of motion for particle 2 we turn to Eq. (F.45) which reads in our notation
dp, yzz.7 &
+
YS4
2m12r3E2 =
-~__km2 sq E, mIr3 ’
where our (s4)i equals Fronsdal’s si (i = 1, 2, 3). In passing from (F.45) to (A.4) one must realize that Fronsdal operates in the center-of-mass frame and thus q E p1 =: -pZ . Now (F.55) identifies sq as s* = -m,y
= -ml(xl
- x2),
(A.5)
so that the equation of motion becomes
&2 -=dt,
km, 6, - ~2) E2
r3
.
w-9
If we let :x1 = 0 we have
4 -zzz dt, which is our Fronsdal’s differs from potential. It
km, x2 -Yg->
64.7)
Eq. (5.5) if we can properly identify Fronsdal’s r. r is a relativistic generalization of the interparticle distance which the usual one which appears, for example, in the Lienard-Wiechert is given by (F.59) as r = [(x1 - x2)2 - (tl - t2)2]1/2,
64.8)
134
ANDERSEN AND VON BAEYER
where (tI - tz) is given by (F.60) and (F.20) as y. = t, -
t.,- = in;2qvs
OV
= q2ho(q”~‘,)
+ yoWq,N
=
-~q”q,Wy,)
+ ~0 (A.91
(here we have used q”q” = p12 = ml”). Now since by (A.9) 0 = 4”Y” = PlOYO - Pl . Y
(A.lO)
and since in our limit plo - a with p1 and y finite, we have y. = 0. Thus r = 1x, - x, j and Fronsdal’s equations become identical with those of Section 5.
REFERENCES 1. P. HAVAS, Foundation problems in general relativity, in “Delaware Seminar in the Foundations of Physics” (M. Bunge, Ed.) Springer-Verlag, New York, 1967. 2. 0. BERGMANN, Amer. J. Phys. 24 (1956), 38. 3. R. H. SEXL, Fortschr. Phys. 15 (1967), 269. 4. W. E. THIRRING, Ann. Phys. (New York) 16 (1961), 96. 5. L. HALPERN, Ann. Phys. (New York) 25 (1963), 387. 6. D. F. GREENBERG, Amer. J. Phys. 34 (1966), 1101. 7. H. BACRY, H. RUEGG, AND J. M. SOURIAU, Con~mun. Math. Phys. 3 (1966), 323. 8. G. GYBRGYI, Nuovo Cimento A 53 (1968), 717. 9. C. FRONSDAL, “The Relativistic Kepler Problem,” UCLA preprint, March, 1970. 10. J. L. SYNGE, “Relativity: The Special Theory,” p. 396, Interscience, New York, 1956. 11. C. M. ANDERSEN AND H. C. VON BAEYER, Ann. Phys. (New York), 60(1970), 67. 12. J. A. WHEELER AND R. P. FEYNMAN, Rev. Mod. Phys. 17 (1945), 157; 21(1949), 425. 13. R. ADLER, M. BAZIN, AND M. SCHIFFER, “Introduction to General Relativity,” p. 53, McGrawHill, New York, 1965. 14. A. SCHILD, Phys. Rev. 131 (1963), 2762. 15. J. DORLING, Amer. J. Phys. 38 (1970), 510. 16. R. FONG AND J. SUCHER, J. Math. Phys. 5 (1964), 456. 17. R. BLANKENBECLER AND R. SUGAR, Phys. Rev. 142 (1966), 1051. 18. C. M. ANDERSEN AND H. C. VON BAEYER, “Closed Orbits and SO(4, 2) Symmetry in Relativistic Two-Body Theory,” in “Proceedings of the Symposium on De Sitter and Conformal Groups,” Boulder, Colorado, June, 1970, to be published.