**Quantum corrections to Newton's law of gravity**Newton said the force of gravity drops off as the inverse square of the distance. But Einstein's theory of gravity says that's not quite true. And

*quantum mechanics* gives

*further* corrections. That's why this formula has three terms in the brackets! And there are more, which we can't calculate yet...

We don't fully understand how quantum mechanics and gravity fit together. But with a lot of hard work, people have been able to figure out the first quantum correction to Newton's inverse square force law. It's a hard calculation - you can see that from the weird number 41/10π. People got it wrong a few times before getting it right.

The formula here is not for the

*force* of gravity. It's the formula for the

*potential energy* due to gravity when you have two masses, m1 and m2, at rest a distance r from each other. G is Newton's constant describing the strength of gravity.

Newton's theory says the potential energy is:

V = - G m1 m2 / r

This is the first term in the formula. This gives Newton's inverse square force law.

The second term involves the speed of light, c. This comes from Einstein's theory - general relativity. This term has an extra r in the denominator, so it gives an inverse

*cube* correction to Newton's force law.

The third term also involves Planck's constant, ħ. This comes from quantum mechanics! When we take quantum mechanics into account, the force of gravity is carried by particles called

**gravitons**. One mass can send a single graviton to another; this gives Newton's law. But fancier things can happen, too! These give a quantum correction to Newton's law - the third term here.

These fancier things are hard to explain, but let me try. A mass can send two gravitons to another, one after another, and this gives

*part* of the third term, with the number -47/3π in front. A mass can also send off two gravitons at the same time! If they reach the other mass at different times we get another part of the third term, with 28/π in front. If they reach the other mass at the same time we get -22/π. And a graviton can also split into two gravitons en route. This gives another part of the answer, namely 7/π. And finally, if a graviton splits in two and the two gravitons rejoin to form a single graviton we get yet another part of the third term, with the number -43/30π in front.

Adding these up we get

-47/3π + 28/π - 22/π + 7/π - 43/30π = 41/10π

You can see pictures of the different graviton processes here:

• N. E. J. Bjerrum-Bohr, John F. Donoghue and Barry R. Holstein, Quantum gravitational corrections to the nonrelativistic scattering potential of two masses,

http://arxiv.org/abs/hep-th/0211072.

If you look at earlier papers you'll see different answers. People made lots of mistakes. But in 2002 two separate teams redid the calculations and got the same answer! The other team is:

• I. B. Khriplovich and G. G. Kirilin, Quantum power correction to the Newton law,

http://arxiv.org/abs/gr-qc/0207118I believe the higher quantum corrections to Newton's law of gravity cannot be computed without choosing a theory of quantum gravity. These will be proportional to higher powers of Planck's constant, ħ. It's a minor miracle that the correction proportional to ħ can be computed using what we know today.

#spnetwork #recommend arXiv:hep-th/0211072