One can choose other coordinates in which the metric is regular there.
26
r=0 singularity
I +
4
+
r=constant
3
1
I 0
r = 2M
2
_
_
r=0 singularity
I
future event horizon
past event horizon
The Carter-Penrose diagram has the form of a diamond with flattened top and bottom.
It is divided into four regions by the two null surfaces on which r = 2 M . The region on the right, marked
1 on the diagram is the asymptotically flat space in which we are
supposed to live. It has past and future null infinities I− and I+ like flat spacetime. There is another asymptotically flat region
3 on the left that seems to correspond to another
universe that is connected to ours only through a wormhole. However, as we shall see, it
is connected to our region through imaginary time. The null surface from bottom left to
top right is the boundary of the region from which one can escape to the infinity on the
right. Thus it is the future event horizon. The epithet future being added to distinguish
it from the past event horizon which goes from bottom right to top left.
Let us now return to the Schwarzschild metric in the original r and t coordinates. If one puts t = iτ one gets a positive definite metric. I shall refer to such positive definite metrics as Euclidean even though they may be curved. In the Euclidean-Schwarzschild
metric there is again an apparent singularity at r = 2 M . However, one can define a new 1
radial coordinate x to be 4 M (1 − 2 Mr− 1) 2 .
Euclidean-Schwarzschild Metric
2
2
dτ
r 2
ds 2 = x 2
+
dx 2 + r 2( dθ 2 + sin2 θdφ 2)
4 M
4 M 2
The metric in the x − τ plane then becomes like the origin of polar coordinates if one identifies the coordinate τ with period 8 πM . Similarly other Euclidean black hole metrics will have apparent singularities on their horizons which can be removed by identifying the
27
τ = τ2
period
τ = 8πΜ
τ = τ1
r=2M
r = constant
imaginary time coordinate with period 2 π .
κ
So what is the significance of having imaginary time identified with some period β.
To see this consider the amplitude to go from some field configuration φ 1 on the surface t 1 to a configuration φ 2 on the surface t 2. This will be given by the matrix element of eiH( t 2 −t 1). However, one can also represent this amplitude as a path integral over all fields φ between t 1 and t 2 which agree with the given fields φ 1 and φ 2 on the two surfaces.
φ = φ2; t = t2
φ = φ1; t = t1
< φ 2 , t 2 | φ 1 , t 1 > = < φ 2 | exp( −iH( t 2 − t 1)) | φ 1 > Z
=
D[ φ] exp( iI[ φ])
One now chooses the time separation ( t 2 − t 1) to be pure imaginary and equal to β.
One also puts the initial field φ 1 equal to the final field φ 2 and sums over a complete basis of states φn. On the left one has the expectation value of e−βH summed over all states.
This is just the thermodynamic partition function Z at the temperature T = β− 1.
On the right hand of the equation one has a path integral. One puts φ 1 = φ 2 and 28
period
β
t 2 − t 1 = −iβ,
φ 2 = φ 1
X
Z =
< φn | exp( −βH) | φn >
Z
=
D[ φ] exp( −i ˆ
I[ φ])
sums over all field configurations φn. This means that effectively one is doing the path integral over all fields φ on a spacetime that is identified periodically in the imaginary time direction with period β. Thus the partition function for the field φ at temperature T is given by a path integral over all fields on a Euclidean spacetime. This spacetime is periodic in the imaginary time direction with period β = T − 1.