**The Euclidean version of a black hole solution naturally encapsulates a lot of black hole wisdom**

The Euclidean spacetime with signature \(({+}{+}{+}{+})\) is a natural arena for Feynman's path integrals in which many mathematical objects become more well-behaved than they are in the Minkowski spacetime. See Wick rotation.

In the Euclidean spacetime, you don't need to worry about signs of \(i\epsilon\). Moreover, Feynman's path integral may actually be rigorously defined in the Euclidean setup while the Lebesgue-like functional measure may be "rigorously" shown not to exist in the physical, Minkowski signature.

Also, spheres are more compact (and lead to more convergent momentum integrals) than hyperboloids, their Minkowskian counterparts.

In string theory, Riemann surfaces with the Euclidean signature are more easily parameterized and described as genus \(h\) surfaces with \(h\) handles, \(b\) boundaries, and \(c\) crosscaps: discussing and computing with Minkowski-signature world sheets would be a mess.

Physical effects are linked to some properties of the objects in the Euclidean spacetime. We may view the Euclidean spacetime as the "intrinsically primary one" while the Minkowski-signature spacetime is a mere physical reflection of the "truly fundamental spirits".

I am sure Stephen Hawking would agree about this point.

That's one of the reasons why, in 1977, Stephen Hawking teamed up with Gary Gibbons (see the picture above that I took in what used to be my office, as demonstrated by my blue jacket as well haha), extended a paper by Perry and Gibbons (sorry, this more important paper for this blog entry won't be discussed in detail because they don't celebrate 70th birthday!), and studied the Hawking radiation of the black holes in the Euclideanized setup:

Cosmological event horizons, thermodynamics, and particle creation (PRD 1977)This Euclidean way of looking at black holes – and de Sitter space-like cosmologies – brings us a new perspective that has arguably not been fully exploited when it comes to our understanding of the information preservation and other things.

Google books version of the article (via Laurent)

Russian pirate text version (full text), Wikipedia

What is the Gibbons-Hawking method about? Imagine that you study a quantum field theory at (absolute) temperature \(T\). This can be done by considering a Euclidean spacetime with a periodic time coordinate \(\tau\sim it\) – now being spacelike – whose period is equal to

\[ \delta \tau = \beta = \frac{1}{kT} \] where \(k\) is Boltzmann's constant. Even in ordinary non-relativistic mechanics, you should know that the expectation values computed using this path integral will express the thermal expectation values. This fact boils to the equivalence between the evolution operator \(\exp(-iHt)\) and the thermal density matrix \(\exp(-\beta H)\) for \(t=-i\beta\); the tracing contained in the thermal expressions \({\rm Tr}\,(\rho L)\) is the source of the periodic identification of the imaginary time coordinate. If you don't already know this machinery, I doubt you will understand it from my ultra-concise explanation. So let me assume you know it.

On the other hand, Stephen Hawking has realized that black holes have a nonzero temperature \(T_H\). That can be translated to some periodic imaginary time, too. Does it play some role in the path integral?

You bet!

The original derivation of the Hawking temperature involved the operator formalism, especially the Bogoliubov transformation mapping creation and annihilation operators into their linear superposition. But the temperature itself may be seen more easily in the path integral formalism.

To see how it works, consider the Schwarzschild solution (in \(c=\hbar=1\) units)

\[ ds^2 = -\left( 1- \frac{2GM}{R} \right) dt^2 + \frac{dR^2}{1-\frac{2GM}{R}} + R^2 d\Omega^2 \] where

\[d\Omega^2 = d\theta^2 + \sin^2\theta\,\, d\phi^2. \] Let's change the sign of the \(dt^2\) from \(-dt^2\) to \({+d\tau^2}\) to get the Euclidean version of the solution. For \(R\gg 2GM\), i.e. well outside the black hole, we should have an ordinary flat spacetime in which we may define the temperature by the periodic identification

\[ \tau \approx \tau + \beta,\qquad \beta=\frac{1}{kT}. \] Everything looks ordinary far away from the event horizon. But what happens if we approach the event horizon? The factor of \(1-2M/R\) goes to zero and its inverse goes to infinity. How does it affect the geometry in the close vicinity of the horizon \(R=2GM\)?

First of all, the angular dimensions will simply span a sphere, \(S^2\), of radius \(a=2GM\). So the total four-dimensional geometry is very close to a Cartesian product of this \(S^2\) and a 2-dimensional manifold describing the \(R,\tau\) dimensions. We want to understand what is the shape in the \(R,\tau\) plane. Let us define \(\rho=R-2GM\). The metric becomes

\[ ds^2 = +\frac{\rho}{R} d\tau^2 + \frac{R}{\rho} d\rho^2 + R^2 d\Omega^2 \] I could have made the formula pretty concise by using both \(R\) and \(\rho\) although they're not independent. Near the horizon, \(R\to 2GM\) while \(\rho\to 0\). We see that the first two terms are singular: their coefficients are either going to zero or infinity. We need to redefine the coordinate \(\rho\) again. A clever choice is

\[ \rho = \Theta^2 \] which means \(d\rho = 2\Theta\,d\Theta\) and gives us, after a simple cancellation,

\[ ds^2 = +\frac{\Theta^2}{R} d\tau^2 + 4R\,\, d\Theta^2 + R^2 d\Omega^2. \] That's great because we see that \(d\Theta^2\) term has acquired a regular, finite coefficient near the horizon \(\Theta=0\). On the other hand, the coefficient of \(d\tau^2\) is \(\Theta^2/R\) and goes to zero quadratically with \(\Theta\) near the horizon \(\Theta=0\). Does this 2-dimensional geometry remind you of something?

It should.

The flat plane in polar coordinates has

\[ ds^2 = dr^2 + r^2 d\phi^2 \] and it is completely non-singular near \(r=0\), the origin, as long as \(\phi\) has the right periodicity, namely \(\Delta \phi = 2\pi\). For a wrong periodicity, the geometry would have a singularity at the origin linked to the deficit (or excess) angle, much like a cone that you may create out of a sheet of paper.

Can we determine what is the right periodicity of \(\tau\) for which the geometry above is equally non-singular at the event horizon, the \(S^2\)-shaped locus at \(\Theta=0\)? Yes, we can. Let us rewrite our metric once again

\[ ds^2 = +4R \frac{\Theta^2}{4R^2} d\tau^2 + 4R \,\,d\Theta^2 + R^2 d\Omega^2. \] Now, the first two terms share the \(4R\) prefactor which is a universal rescaling of the metric (the choice of units, if you wish) and doesn't affect the required periodicity of \(\tau\). The rest of the \(\tau\)-\(\Theta\) plane metric is

\[ ds^2 \sim \frac{\Theta^2}{4R^2} d\tau^2 + d\Theta^2 \] which is easily seen to be the flat plane in polar coordinates if we define

\[ \phi = \frac{\tau}{2R}. \] It means that the required periodicity is

\[ \phi\sim \phi+2\pi\quad \Leftrightarrow \quad \tau\sim \tau+ 4\pi R.\] But \(\tau\) is just the Euclidean time and the periodicity is the inverse temperature \(\beta\). We have found out that

\[ T = T_H\equiv \frac{1}{\beta k} = \frac{1}{k\,\Delta \tau} = \frac{1}{4\pi k R} = \frac{1}{8\pi k GM} \] where we finally substituted the regular value of \(R\) at the horizon, namely \(R=2GM\). We have determined that for for the temperature \(T=1/8\pi kGM\), the Euclideanized black hole solution is perfectly non-singular near the event horizon.

And indeed, this is the right Hawking temperature for the Schwarzschild black hole!

It had to work. The Euclidean spacetime with the right periodicity of \(\tau\) is allowed to have a different topology in a finite volume somewhere in the middle, so that implies that the thermal ensemble must also include the summation over the states including the black hole at the right temperature!

A cool thing is that the same procedure may be applied to arbitrary black holes (and black branes) in flat and AdS spaces (and maybe others), regardless of the spacetime dimension, charges, or angular momentum. In all cases, one may find a perfectly nice Euclidean solution whose regularity near the event horizon produces the desired black hole temperature.

A funny feature of the solution is that the interior of the black hole is completely absent in the Euclidean edition of the black hole. If there are some fields inside the black hole, you should understand them as analytic continuations to exotic values of \(\Theta\). Recall that

\[ \Theta=\sqrt{R-2GM} \] which means that \(\Theta\) has a pure imaginary value for \(R\) smaller than the black hole radius \(2GM\). If you want to know what's happening inside the black hole, it must be encoded in the continuation of the Euclideanized spacetime to imaginary value of \(\Theta\), the well-behaved radial coordinate near the non-singular Euclideanized event horizon!

I believe that there are many new insights one may obtain from the analytic continuations of this sort and from allowing various topologies of the spacetime and from the "gluing rules" governing such topologies. It's another thing that is mathematically pretty and that many people in the field avoid which slows down the progress.

Gee I don't know, Lubos, how one gets "gravity" in some spacetime of signature +4. It is anything but obvious to me.

ReplyDeleteDear Brian, the Euclidean spacetime is different than the Minkowski one. They have different physical phenomena.

ReplyDeleteBut the point is that whatever calculable structure exists on the Minkowski side has a counterpart on the Euclidean side.

For example, the Schwarzschild black hole is a Ricci-flat solution of the equations. This is true even for the Euclidean Schwarzschild solution: it's still a Ricci-flat geometry of purely positive signature!

There are still deviations from the flat metric that decrease with the distance from the source; this *is* an aspect of gravity. One would discuss the effect of this gravity - acceleration etc. - differently than in the Minkowski spacetime but there is *a* Euclideanized discussion that contains all the information about the dynamical laws.

I understand that Lubos, the issue has to do with transformations possible in Minkowski space that correspond to affine transformations in E^4 that cannot represent quantities related to covariant second-order transformations("potentials") from which gravitational "forces" (geometry) can be derived.

ReplyDeleteAnyway I think I'll stick to spaces for which hyperbolic conservation laws have meaning. Things get too crazy for me to contemplate without them.

[Example: It is certainly possible to enter a "closed" space in E^2 (the Euclidean plane) from E^3 without "breaking" or crossing a boundary. So it should be possible to enter a "closed" space in E^3 (provided with a material boundary) from E^4. It certainly is, or would be, were it not for conservation of matter. But if we don't have conservation laws, as provided only by hyperbolic forms, then all bets are off.]

Dear Brian, most likely, I don't understand this "entering" difference between the signature.

ReplyDeleteConservation laws have pretty much the same form in both signatures. It's still the same \(\partial_\mu J^\mu=0\) equation.

What isn't right to consider in the Euclidean space are timelike geodesics. After all, all lines in the Euclideanized spacetime are spacelike. So all the physical states become "poles away from the physical axes", but they're still encoded in the maths of the theory.

The transfer of the physical states away from the visible space is a good thing for the calculations - that's, for example, the reason why we don't have to solve \(i\epsilon\) prescriptions in the Euclidean spacetime.

Have the same form, BUT in the Euclidean case, there is an arbitrary constant or function that is not present in Minkowski form

ReplyDelete(Looking at an ordinary source free conservation law in the form "Divergence of (something) = 0" - this came from a hyperbolic conservation law that cannot have some arbitrary constant or function associated with it. Meaning, there is no good reason not to fill that something in with anything you please.)

We're getting far afield here, sorry. Sorry for rambling and wasting web page