Page 30 - Nature Of Space And Time

3
2
I[g]= I[g 0]+ I 2 ( g) + I 3 ( g) + :::
The linear term vanishes because the background is a solution of the eld equations. The
quadratic term can be regarded as describing gravitons on the background while the cubic

and higher terms describe interactions between the gravitons. The path integral over
the quadratic terms are nite. There are non renormalizable divergences at two loops in
pure gravity but these cancel with the fermions in supergravity theories. It is not known
whether supergravity theories have divergences at three loops or higher because no one
has been brave or foolhardy enough to try the calculation. Some recent work indicates

that they may be nite to all orders. But even if there are higher loop divergences they
will make very little di erence except when the background is curved on the scale of the
Planck length, 10 −33 cm.

More interesting than the higher order terms is the zeroth order term, the action of
the background metric g 0 .

Z Z
1 1 4 1 1 3
2
2
I = − R(−g) d x + K(±h) d x
16 8
The usual Einstein-Hilbert action for general relativity is the volume integral of the scalar
curvature R. This is zero for vacuum solutions so one might think that the action of the
Euclidean-Schwarzschild solution was zero. However, there is also a surface term in the
action proportional to the integral of K, the trace of the second fundemental form of the

boundary surface. When one includes this and subtracts o the surface term for
at space
one nds the action of the Euclidean-Schwarzschild metric is 2 where is the period in
16
imaginary time at in nity. Thus the dominant contribution to the path integral for the
−β 2
partition function Z is e 16π .

2
X
Z = exp(− E n)= exp −
16
If one di erentiates log Z with respect to the period one gets the expectation value
of the energy, or in other words, the mass.

d
<E >= − (log Z)=
d 8

So this gives the mass M = . This con rms the relation between the mass and the
8
period, or inverse temperature, that we already knew. However, one can go further. By

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