The cosmological constant and classical tests of general relativity

Volume 97A, number

6

PHYSICS LETTERS

5 September 1983

THE COSMOLOGICAL CONSTANT AND CLASSICAL TESTS OF GENERAL RELATWITY J.N. ISLAM Department ofMathematics, The City University, London EC1 V OHB, UK Received 4 June 1983 Revised manuscript received 6 July 1983

2is placed on the absolute value of the cosmological constant by comparing with the preAn upper of 10~2cm diction of thelimit perihelion shift of Mercury. It is shown that the bending of starlight near the sun gives no limit on the cosmological constant since the equation for a null geodesic takes the same form with or without the cosmological constant.

1. Introduction. Recently there has been some interest in the cosmological constant, particularly with regard to its implications for quantum gravity. Hawking [1] has shown that if N = 8 supergravity is the correct theory of the universe, a phase transition occurs at a certain critical value of the coupling constant and below this critical value the ground state is an anti-de Sitter space with a negative cosmological constant while above the critical value the ground state has an “apparent” cosmological constant which is exactly zero. It is thus useful to have limits on the value of the cosmological constant. By putting rough limits on the Hubble constant and the deceleration parameter one can get a limit of i0—~cm2 on the absolute value of the cosmological constant A [1]. In this paper we derive a limit of 10—42 cm2 on the absolute value of A from the perihelion shift of Mercury, ignoring the quadrupole moment of the sun, which is not known. Although this limit is 12 orders of magnitude larger, it is smaller than one would expect and it is interesting to derive this limit from completely different phenomena where observations are more precise, if only to satisfy oneself that it is larger. Besides, it is a well-defined problem and the solution is an instructive exercise. We show further that the bending of light near the sun places no limits on A since the equation for a null geodesic takes the sameform withor without A. It is mteresting that one can find exact solutions for geodesics.

2. Form ofmetric. Einstein’s equations with the cosmological constant are as follows R R ÷A = 8~G/c4\ T 1 —

U~.W

~

~

/

It follows at once that where 0 we must have R = A 2 ~ Most of the simple exact solutions can easily be generalized to include the A term and several authors have considered such solutions (see e.g., ref. [2]). The Schwarzschild solution is modified as follows: ds2 = c2 (1 2M/r + ~ Ar2)dt2 —

—(1

2M/r

+

4~Ar2)1 dr2

r2(d02 + sin2O d~2), M = Gm/c2 —

(3)

3. Geodesics. With (x0, x1, x2, x3) = (t, r, 0, ~), the non-zero Christoffel symbols for the metric (3)are ~ 0 = M + ~ Ar3 r 1 = r 0 01 r(r 2M ÷~ Ar3) 11 01

0 031.9163/83/0000—0000/s 03.00 © 1983 North-Holland

—

[‘22

r

—

+ 2M

—

~Ar3,

[‘33

[‘22 smO,

—

—

=

(c2/r2)(M + Ar~)(l 2M/r

1

2

12

= =

~

1/r, i /r,

—

+~

Ar2),

2

[‘33 —sin0 cosO, ~‘2~ = cot 0

(4)

239

Volume 97A, number 6

PHYSICS LETTERS

5 September 1983

One can now write down the geodesic equations: (5) d2x~+ r ~ dxv dx° = 0. dS2 ~° cls -di~ In the following it is helpful to keep in mind the standard derivation of geodesics in the Schwarzschild geometry (see e.g. ref. [3]). It is easily seen that the motion is confined to the equatorial plane 0 = ir/2, so that (5) with ~z= 2 is identically satisfied. Further, we can ignore (5) with ji = 1 in favour of (3), which is a first integral of the geodesic equations. The resulting equations can be written as

where ~ is a constant and e is the eccentricity of the orbit. As a second will be the adequate to substitute in theapproximation right-hand sideitof(l3) value of u given by (14). Now for Mercury e is approximate. ly 0.2 so that powers of e higher than the first can be neglected. Expanding in powers of e and retaining only the first term, (13) reduces to the following equation: d2u/d~2+ u = Mc2/h2 + 3M3c4/h4 + Ah4/3M3c4

d2t + 2(M + ~ Ar3) dr dt ds2 rfr 2M + ~ Ar3) ~ ds

The terms without e have no observable consequence. The term proportional to e leads to the perthelion shift. Now for Mercury the shift is known to an accuracy of about percent (see e.g. ref. [4]). and Thusobto accord withhaIfa the agreement between theory

0,

(6)

=

—

d2~ 2 dr d~ 0 2 + ds ~ c2(l —

—

(1

2M/r

+

~Ar2)~(dr/ds)2

hr + (1

[—

dw/dr + (2/r)w

=

+

Ar2)/(r

—

—

r2 (d~/ds)2 1. (8)

2M + ~Ar3)] v =

0,

(10)

which can be integrated to yield v ar/(r 2M + ~ Ar3), w = b/r2 —

,

(11)

where a and b are arbitrary constants. Substituting in (8) we get the following equation after putting u = r1: (du/dØ)2

+ u2

=

—

Ah4/M3c4)e cos(Ø

—

~o)

.

(15)

must be less than half percent in absolute value of the

Setting v = dt/ds and w = d~/ds,(6) and (7) can be written as dy/dr +

(6M3c4/h4

servation the A term in the coefficient of the e term

~Ar2)(dt/ds)2

+

2M/r

—

(7)

+

(c2a2

—

1)/b2

+

(2M/b2)u + 2Mu3

sponding amount. For bending of light near the sun we have to find null geodesics in the metric (3). For this we replace

Differentiation with respect to 0 leads to the following: d2u/d02

the following equation

+u

M/b2

+

(12)

Now we have h2 = lMc2, where 1 is the semi-latus recturn of the orbit. For Mercury 1 5.79 X 1012 cm, while for the sunM = 1.475 km. Using these values in (16), we get about 10—42 cm2 as the upper limit for the absolute value of A. In this analysis we have ignored the quadrupole moment of the sun. Some portion of the perihelion shift could be due to the quadrupole moment of the sun, if it turns out to be significant. If this is the case the limit found above would be higher by a corre-

s by a parameter X in (6), (7) and (8) and put zero instead of 1 in the right-hand side of (8). Following similar steps as before we find that (13) is replaced by

—

~(1 /b2u2 ÷1)A.

term immediately preceding it, that is, we must have IAI<(6X0.005)M6c8/h8. (16)

3Mu2

+

A/3b2u3.

(13)

The first term on the right-hand side of (13) leads to the newtonian orbit. By comparison with the latter one can show that cb = h, where h is the angular momenturn per unit mass of the planet. As a first approxirnation we can take the solution of (13) to be the newtonian solution:

d2u/d02 + u = 3Mu2 (17) which does not involve A. Thus we get the same result as before for bending of light near the sun. One can probably get better limits by considering the outer planets but data for the outer planets are less precise. One can also get a limit on A by consider-

u

ing the galaxy as a bound system, but calculations there are not so precise. I am not aware of any detailed

=

240

(Mc2/h2)[1

+e

cos(Ø

— 0~)J~

(14)

Volume 97A, number

6

PHYSICS LETTERS

attempt it would be interesting to know if the limit on A is better or worse than the one we have derived here. —

My interest in this topic was aroused by listening to a talk by S.W. Hawking at the Royal Society, London. I am grateful to M.J. Rees and an anonymous referee for interesting comments.

5 September 1983

References [1J S.W. Hawking, The cosmological constant, talk at meeting on The constants of physics at the Royal Society, London (May 25, 26, 1983). 121 B. Carter, Commun. Math. Phys. 10 (1968) 280. [31 D.F. Lawden, An introduction to tensor calculus and relativity, 2nd Ed. (Methuen, London, 1967) sects. 53, 54. [4] C.M. Will, Theory and experiment in gravitational physics (Cambridge Univ. Press, London, 1981) p.181.

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