PHYSICAL REVIEW D VOLUME 46, NUMBER 10 15 NOVEMBER 1992

ARTICLES

Testing local Lorentz invariance of gravity with binary-pulsar data

Thibault Damour
Institut des Hautes Etudes Scientifiques, 91440 Bures sur Yvette, France
and Dipartement d'Astrophysique Relativiste et de Cosmologie, Observatoire de Paris-
Centre National de la Recherche Scientifique, 92195 Meudon C E D E X , France

Gilles Esposito-FarBse
Institut des Hautes Etudes Scientifiques, 91440 Bures sur Yvette, France
and Centre de Physique The'orique, Centre National de la Recherche Scientifique, Luminy,
13288 Marseille CEDEX 9, France.
(Received 30 April 1992)
As gravity is a long-range force, one might a priori expect the Universe's global matter distribution
to select a preferred rest frame for local gravitational physics. Two parameters a1 and a2 suffice
to describe the phenomenology of preferred-frame effects in post-Newtonian gravity. One of them
has already been very tightly constrained (la21 < 2.4 x We show here that binary-pulsar data
provide a bound on the other one (la11 < 5.0 x 90% C.L.) which is quantitatively comparable
to previous solar-system limits, but qualitatively more powerful because it is derived for systems
comprising strong-gravitational-field regions. Our results correct a previous claim that a1 could be
very tightly constrained via a purported semisecular effect in the orbital period of binary pulsars.
PACS number(s): 04.80.+z, 03.30.+p, 97.60.Gb

Local Lorentz invariance, i.e., the absence of preferred select a preferred rest frame for the gravitational inter-
frames in local experiments, is an essential ingredient of action. In the post-Newtonian limit all the gravitational
our present understanding of the constitution and inter- effects associated with the possible existence of such a
actions of matter and is verified every day in high-energy preferred cosmic frame are phenomenologically describ-
experiments. If gravity is mediated only by a second- able by two parameters crl and crz [I]. These parame-
rank symmetric tensor field (as assumed in general rela- ters contribute additional, non-boost-invariant, velocity-
tivity), or, more generally, by one symmetric tensor field dependent terms in the gravitational many-body post-
and an arbitrary number of scalar fields, the gravitational Newtonian Lagrangian, beyond the usual boost-invariant
physics of localized systems will also be boost-invariant terms obtained in general relativity and its minimal ex-
(at least within a good approximation). On the other tensions described by the Eddington parameters y and P
hand, it has been pointed out some time ago by Will (which correspond t o adding one or several scalar fields
and Nordtvedt [I]that if gravity is mediated in part by [2]). More precisely, the N-body post-Newtonian La-
a long-range vector field (or by a second tensor field) grangian reads
one expects the Universe's global matter distribution to

@ 1992 The American Physical Society

 TESTING LOCAL LORENTZ INVARIANCE OF GRAVITY WITH ... 4129

I
in which v i denotes the velocity of the mass mA with of the pulsar. From Eqs. (2) and (3) one sees that the
respect to the Universe's preferred rest frame. matter within the pulsar experiences an effective gravi-
Bounds on the magnitudes of a1 and a 2 have been tational constant [16,3],
obtained by several authors [3-71, based upon various
effects associated with Eqs. (3) and (4) in the weak-
gravitational-field context of the solar system. When
deriving such bounds, it is necessary to make a definite
assumption about the preferred rest frame entering the which is modulated, because of v l , at the orbital fre-
Lagrangians (3) and (4). The standard assumption [4-7) quency. This modulation causes a corresponding mod-
that we shall take up in the present paper is to take the ulation in the spin angular velocity of the pulsar,
frame defined by the cosmic microwave background. (If Awl/wl = &AGeff/Geff, which, after a time integration,
local Lorentz violation is due to an extra vector or ten- contributes additional terms in the timing formula, giv-
sor interaction, this assumption means essentially that ing the arrival times of the pulses. However, the integral
its range is infinite or, at least, of cosmological magni- of the term -alw.vl/c2 can be completely reabsorbed in
tude.) The final results are that the close alignment of the main ("Romer") term of the timing model [17],while
the Sun's spin axis with the solar system's planetary an- the integral of the term -alvf/2c2 can be reabsorbed in
gular momentum yields an extremely tight bound on a 2 the "Einstein" time delay. In terms of observables, these
reabsorptions lead to fractional modifications of the tim-
PI 9
ing parameters xtiming= a1 sinilc and ytimingof order
IN timing 12timing , QlnW/C and Aytiminglytimk3 , al&.
In view of the current and foreseeable precision of the
while combined orbital data on the planetary system [6] tests obtainable by combining the measurements of sev-
yield a much weaker bound on a l : eral timing parameters [13],these "internal" effects of a1
cannot compete with the existing solar-system limit (6)
and we shall not consider them any further.
Let us now consider the effects of a1 on the orbital
The limit (6) is only a factor five better than the motion of the pulsar. Decomposing the "absolute" ve-
present limits on the (more conservative) Eddington + +
locities according to vy = w v l , v! = w vg, where
post-Newtonian parameters P and y [8,9]. w and v l have the same meaning as above, and where
In view of this situation, it seems important to in- v2 denotes the velocity of the pulsar's companion with
vestigate whether or not binary-pulsar data, which have respect to the center of mass of the binary system, one
proven to be marvelous gravitational probes [lo-141, can- gets
not be used to set more stringent bounds on al. In
fact, it has been claimed [15] that a very stringent bound
(IalI 5 could be deduced from the agreement be-
tween the observed orbital period change of the binary
pulsar PSR 1913+16 and the general relativistic predic-
tion. Actually, we found that this claim was incorrect
(see below), and this motivated us to look in detail at We have added a caret to a1 in Eq. (8) to denote a
other ways of using binary-pulsar data to constrain the possible modification of the weak-field value of a1 by
Lorentz-invariance-violation parameter a1. strong-field-gravity effects in the pulsar and its compan-
The al-dependent terms, Eq. (3), have several different ion. Similarly G = Gl2 denotes the effective gravitational
types of observable consequences in the dynamics of a bi- coupling constant between the pulsar and its companion,
nary pulsar. First, let us consider a pair of mass elements including self-gravity effects (see, e.g., [2]).
( m ~m, ~ within
) the pulsar, and let us decompose (with, One can verify that Eq. (8) predicts no overall secular
sufficient, Newtonian accuracy) each "absolute" velocity acceleration of a binary system. One then defines w as
+ +
v i according to v i = w v l UA. Here, w denotes the the absolute velocity of the frames with respect to which
velocity of the center of mass of the binary system with the binary system is, on the average, at rest. In such
respect to the preferred rest frame, v l denotes the veloc- a frame the (usually defined) instantaneous relativistic
ity of the pulsar with respect to the center of mass of the center of mass of the binary oscillates around its fixed
binary system, and UA denotes the velocity of the con- average position by an amount A&,,.(t), obtained by
sidered mass element with respect to the center of mass integrating
 +
In Eq. (9) M = m l m2, X I r m l / M , X2 -- m2/M,
a is the semimajor axis of the relative orbit, vl2 v l -
v2 the relative velocity, and the angular brackets denote
secular advance of the periastron.] In the case of the bi-
nary pulsar PSR 1913+16 the secular variation of e is ob-
servationally constrained at the level Iel < 1.9 x 10-l4 s-I
the time average (e.g. (r;;) = a*'). Given the solar- [lo], while Eq. (12) yields e = 1.3981(w.a)/cx lo-' s-'.
system limit (6), and ~ 1 2 51 W/C ~ N the wobbling Even if a is favorably oriented (so that Iw. al/c
Ax,,,.(t) is easily seen t o give a negligible contribution
to the timing of a pulsar such as PSR 1913+16: At
AX,,,,/C 5 s.
- this limits &1 only a t the N level, which pales in
comparison with Eq. (6). [The observational constraints
on the variation of the timing parameter x = a x 2 sin i/c
It remains t o study the effects of a1 on the relative [lo] puts a limit on the change of the orbital inclination
motion of the pulsar around its companion. It is easy which, after using Eq. ( l l ) , yields an even weaker bound
to see in advance that, given the solar-system limit (6), on GI.]
all the periodic effects (that do not build up beyond one Fortunately the class of low-companion-mass, small-
orbital period) give negligible contributions to the timing eccentricity, long-orbital-period binary pulsars turns out
of the pulsar (similar to that associated with A+,,.). t o provide a better testing ground for a possible violation
Finally, our only hope of getting new, tight constraints of the local Lorentz invariance of the gravitational inter-
on d l is to study the secular effects in the relative motion. action. Taking into account the very small eccentricity
Adding the contributions from and y [Eq. (2)] t o
the a1 contributions from Eq. (8), and averaging over
one orbital period the time derivatives of the energy, the
angular momentum, and the Lagrange-Laplace (-Runge-
Lenz) vector, one finds the following equations for the
-
of these systems, we can simplify very much the secular
evolution system ( l l ) , (12). Equation (11) shows that
the orbital plane is fixed, (dcldt) = O(elk/) 0: while
Eq. (12) becomes

secular evolution of the Keplerian elements of the relative

orbit:
- +
where k L ( k . a)a (k . b)b is the projection of k onto
the orbital plane. Equation (15) shows that the main
new effect of an al-type Lorentz-invariance violation is
e to add a constant forcing term in the time evolution of
b x k ,
the eccentricity vector which tries to LLpolarize" the orbit
in the direction of the projection of w onto the orbital
plane, w l . The familiar relativistic periastron precession
term WR in Eq. (15) cuts off the build up of the constantly
polarizing term &kl and deflects its action by 90' (in a
gyroscopelike way). More precisely the general solution
of the linear evolution equation (15) can be written as
the vectorial superposition

In Eqs. (10)-(12) 1 5 (1 - e2)lI2c,e = ea,

C X W
-
- - X2)
4(2y-B+2+X1X281) na ' (17)

In Eq. (16) eR(t) is a vector of constant magnitude which

rotates in the orbital plane with angular velocity w~
A
(usual relativistic periastron advance). By contrast, e~
where e is the eccentricity, and n r 2?r/Pb = ( G ~ / a ~ ) l / is~ a fixed vector which represents a constant, &-induced,
is the orbital frequency, (a,b , c ) being an orthonormal polarization of the orbit (or "forced eccentricity"). Note
triad with a in the direction of the periastron of the that, because of the small denominator WR in Eq. (17),
pulsar orbit, and c = a x b along the orbital angular the velocity of light has dropped from the final expres-
momentum. Moreover, 5 and 3 in Eq. (14) denote the sion of e~ whose magnitude depends essentially on 21
effective values of the Eddington parameters for the rel- and the ratio w l l n a , where w l is the projection on the
ative orbital dynamics of two compact objects, including orbital plane of the absolute velocity of the center of mass
possible strong-field effects. (See [2] for the exact defini- of the binary system, and n a is the relative orbital ve-
tion of these quantities.) locity. Note also that, although the instantaneous form
The result (10) shows that a1 has no secular effect on of the al-type perturbing forces [derived from Eq. (8)] is
the orbital period, contrarily t o the claim of Ref. [15]. very different from that corresponding to a possible dif-
On the other hand, Eqs. (11) and (12) show that both ferential free-fall acceleration in the gravitational field of
the eccentricity and the spatial orientation of the Keple- the Galaxy (equivalence principle violation), their secu-
rian binary ellipse undergo secular changes when 81 # 0. lar effects in small-eccentricity binary systems have ex-
[Eqs. (12) and (14) exhibit also the influence of 81 on the actly the same structure [compare Eqs. (15)-(17) above
46
- TESTING LOCAL LORENTZ INVARIANCE OF GRAVITY WITH . .. 4131

to Eqs. (3) and (4) of Ref. [12]]. We can therefore take The experimental results of Ryba and Taylor [18] on
up the method of Ref. [12] for deriving an upper bound PSR 1855+09 yield e = 2.167 x Pb= 1.0650676 x
on IGII from the observations of binary pulsars having a lo6 S, i = 88.2g0, X1 - X2 = 0.690, M = 1.50Ma.
very small eccentricity and a long orbital period. Let us On the other hand, the experimental results of the Cos-
first remark that a recent work [14] has derived experi- mic Background Explorer (COBE) on the velocity of
mental constraints on the magnitude of the parameters the solar system with respect to the cosmic microwave
-
(p' and P") that drive (in a class of theories) the possi-
A

ble strong-field effects in G, P, and T. These constraints

background [19] give w = 365 kms-l, in the direction
(a,S) = (11.2 h, -7'), i.e., making an angle X = 117'
are sufficiently tight t o ensure that, in a system such as with the line of sight towards PSR 1855+09. (We are us-
PSR 1855+09 that we shall consider in the following, one ing the fact that PSR 1855+09 is a nearby system with
will have = G ( l f0.06) and 2~ - P+2 = 3(1 f0.04) at small observed apparent transverse motion as seen from
the solar system to estimate that W p s ~N wa.) Insert-
the 90% confidence level. Because of these results (which
will be probably tightened in the future), we shall sim- ing all numbers in Eqs. (18) we get Ii,x = 1.43, and the
plify Eq. (17) to l e ~ l= 611x1- x ~ I w L / ~ ~ ( G M ~ ) result
'/~.
Applying to this expression the probabilistic reasoning 1811 < 5.0 x (90% C.L., PSR 1855+09 data) .
of Ref. [12], we can conclude that the observation of an
(old) binary-pulsar system having a (small) observed ec-
centricity e allows one to set an upper bound on IGII
Our final (90% C.L.) upper bound (19) on a possible
given by
gravitational violation of local Lorentz invariance is in-
IS11 < (10/r) I;,xel?? (90% C.L.) , (184 teresting in two respects. First, it is quite comparable
with the best existing solar-system limit (6) which al-
where lows, at the 90% confidence level (1.64a), an a1 as big
2k as +5.2 x Second, it is the first limit obtained for
I,,* = (2r)-l d o [I - ( cos i cos r a gravitationally bound system which comprises strong-
field regions (namely the 1.27Ma neutron star seen as a
+
s i n i s i n ~ s i n f l ) ~ ],- ~ / ~ pulsar in PSR 1855+09). Recently, it has been shown
(18b) [2] that there existed classes of boost-invariant theories
which had the same Eddington parameters as general rel-

-
ativity in the weak-field limit but for which the effective
values of and T for a system containing compact objects
in which the full magnitude of the absolute velocity ap- would generically be given by power series in the com-
pears, w Iwl. The complete elliptic integral (18b) arises pactness parameters CA = -2dlnmA/dlnG of the type
from making a probabilistic argument about the a pri-
o r i unknown values of two angles in the problem: the
P^ + +
= 1 - iP1(X1c2 X2c1) . *,7 = 1 - P'(c? c;) + +. .. .
If a similar result holds for non-boost-invariant theories,
time-dependent angle 8 between e~ and e R ( t ) , and the we see that our limit (19), taken in conjunction with the
longitude of the node Cl of the binary orbit with respect solar-system limit (6), provides already tight limits on
to the line of sight (see [12]). The factor ( 1 0 / ~ ) Irep- ~ , ~ the coefficients of any conceivable strong-field modifica-
resents a quantitatively precise way of allowing for u n f s + + +
tion of cul: 51 = a1 a;(c1 c2) . . . (cl x 0.27 for a
vorable configurations of the angles 8 and 0 when trying 1.27Mo neutron star).
to estimate 61 from the observed e. All the quantities Let us also note that if one could extract from the
appearing in the final results (18) are (in favorable cir- PSR 1855+09 data a bound on the secular variation of
cumstances) measurable from Earth: i is as above the the eccentricity vector at a level Ideldtl < 2 x 10-l6 s-'
inclination of the orbital plane, while X is the angle be- (i.e., about a factor 10 below the present limit on deldt)
tween w and the line of sight. one could both render more secure (by freeing it from
Fkom Eqs. (18), one sees that ~ ~ ~defines / ~ a /figuree of the need to use probabilistic considerations) and tighten
merit for selecting the binary-pulsar systems giving the the limit (19). Indeed, we have from Eq. (16) deldt =
best limits on 21. A survey of existing small-eccentricity WR c x e~ SO that Ideldtl gives access to the magnitude
long-orbital-period binary pulsars show that the two sys- of e ~ The . level 2 x 10-l6 s-l quoted above corresponds
tems PSR 0655+64 and PSR 1855+09 have, by far, the to e ~ / < e ( 1 0 / ~ ) I=~ 4.55,
, ~ i.e., a level for e~ where the
highest figures of merit. We shall consider solely the lat- probability argument behind the derivation of the safety
ter system which is the only one for which all the needed factor ( 1 0 / ~ ) Iin~ Eq.
, ~ (18a) is becoming too pessimistic.
quantities have been measured (and which is known to
be old enough for our probabilistic argument to be ap- It is a pleasure to thank Ken Nordtvedt for stimulating
plicable). discussions.

[I] C.M. Will and K. Nordtvedt, Astrophys. J. 177, 757 [3] K. Nordtvedt and C.M. Will, Astrophys. J. 177, 775
(1972). (1972).
[2] T. Damour and G. Esposito-FwBse, Class. Quantum [4] R.J. Warburton and J.M. Goodkind, Astrophys. J. 208,
Grav. 9, 2093 (1992). 881 (1976).
 [5] C.M. Will, Theory and Experiment in Gravitational (1989).
Physics (Cambridge University Press, Cambridge, Eng- [ll] C.M. Will and H.W. Zaglauer, Astrophys. J . 346, 366
land, 1981). (1989).
161 R.W. Hellings, in General Relativity and Gravitation, 1121 T. Damour and G. Schafer, Phys. Rev. Lett. 66, 2549
Proceedings of the Tenth International Conference, (1991).
Padua, Italy, 1983, edited by B. Bertotti, F . de Felice, [13] T. Damour and J.H. Taylor, Phys. Rev. D 4 5 , 1840
and A. Pascolini (Reidel, Dordrecht, 1984), pp. 365-385. (1992).
[7] K. Nordtvedt, Astrophys. J . 320, 871 (1987); beware [14] J . H . Taylor, A. Wolszczan, T . Damour, and J.M. Weis-
that the latter paper uses a nonstandard normalization berg, Nature (London) 355, 132 (1992).
for ( Y Z : CYF
= [15] K . Nordtvedt, Astrophys. J . 322, 288 (1987).
[8] 1.1. Shapiro, in General Relativity and Gravitation, 1989, [16] C.M. Will, Astrophys. 3. 169, 141 (1971).
edited b y N. Ashby, D.F. Bartlett, and W . Wyss (Cam- [17] T . Damour and N. Deruelle, Ann. Inst. Henri PoincarC
bridge University Press, Cambridge, England, 1990), 4 4 , 263 (1986).
pp. 313-330. 1181 M.F. Ryba and J.H. Taylor, Astrophys. J . 371, 739
[9] C.M. Will, Int. J . Mod. Phys. A (to be published). (1991).
[lo] J.H. Taylor and J.M. Weisberg, Astrophys. J . 345, 434 [19] G . F . Smoot et al., Astrophys. J. Lett. 371, L1 (1991).