ñòð. 90 |

;

Note that this potential has almost the same form as the Newtonian potential VN (r), the

main di erence is that the fth force decays exponentially with distance over a length

and that it is weaker than Newtonian gravity by a factor . This idea was prompted by

measurements of gravity in mines, in the ice-cap of Greenland, in a 600m high telecom-

munication tower and a number of other experiments that seemed to disagree with the

gravitational force that follows from the Newtonian potential VN (r).

Problem a: E ectively, the fth force leads to a change of change of the gravitational

constant G with distance. Compute the gravitational acceleration g(r) for the com-

^=r2

r

bined potential VN + V5 by taking the gradient and write the result as ;G(r)M

17.8. EPILOGUE, THE FIFTH FORCE 261

to show that the e ective gravitational constant is given by:

1 + r e;r=

G(r) = G 1 + : (17.71)

The fth force thus e ectively leads to a change of the gravitational constant over a

characteristic distance . This e ect is very small, in 1991 the value of was estimated

to be less than 10;3 for all estimates of longer than 1cm 23].

In doing geophysical measurements of gravity, one has to correct for perturbing ef-

fects such as the topography of the Earth's surface and density variations within the

Earth's crust. It has been shown later that the uncertainties in these corrections are much

larger than the observed discrepancy between the gravity measurements and Newtonian

gravity 44]. This means that the issue of the fth force seems be closed for the moment,

and that the physical world appears to be governed again by only four fundamental forces.

CHAPTER 17. POTENTIAL THEORY

262

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