The uncertainty principle and black holes

Question

What are the consequences of applying the uncertainty principle to black holes?

Does the uncertainty principle need to be modified in the context of a black hole and if so what are the implications of these modifications?

Answer 1

The GUP (Generalised Uncertainty Principle): In view of the discussion generated on this question, and the answer by Dilaton, I have decided to add an edition to my answer in the hope that it will generate further discussion.

EDITION: UNCERTAINTY PRINCIPLE FOR A BLACK HOLE

The most famous effect where the uncertainty principle plays a very important part around a black hole is the Hawking radiation. In this, the usual quantum fluctuations of the vacuum just outside the event horizon of a black hole, generate particle–antiparticle pairs which are “separated” by the immensely strong gravitational field of the black hole. The phenomenon then evolves by having the negative energy particle (antiparticle) fall into the black hole, hence reducing the energy of the black hole. The positive energy particle moves away from the black hole to reach an observer at some distance from the event horizon. As long as the observer is concerned, the black hole appears to radiate energy in the from of particles – black hole vaporisation. There is yet another level of uncertainty, however, and in this gravity plays a very important part. This is a string theory result, the GUP (generalised uncertainty principle) and goes as follows

$\Delta x\ge \frac{\hbar}{\Delta p}+ \frac{G\Delta p}{c^3}$.

One can see the effect of gravity in the above GUP. We can observe that, in the usual “low” energy uncertainty principle, $\Delta x\Delta p\sim\hbar/2$, large uncertainty in measuring the momentum of an electron, large $\Delta p$, implies small uncertainty in the measurement of its position, $\Delta x$. However, from the above equation, at the Planck scale, near the singularity of a black hole, this no longer is the case! We see that as $\Delta p$ increases so does $\Delta x$ due to the second term in the GUP. Hence gravity introduces an extra level of uncertainty so that $\Delta x$ and $\Delta p$ do not mutually exclude each other. This can be interpreted that at the Planck scale, both wave and particle behaviour are manifest simultaneously.

By completing the squares in the above quadratic form for $\Delta p$ and taking the "equal" sign one gets

$(\Delta p-\frac{c^3\Delta x}{2G})^2=c^3\frac {c^3\Delta x^2-4G\hbar}{4G^2}$

Due to the square on the LHS of the above equation we can see that

$\Delta x^2 \ge 4\frac{G\hbar}{c^3}$

This result has also been written by Dilaton. This equation shows that gravity sets an ultimate accuracy in the measurement of the position of the electron, and this is the Planck length. This is what we should expect thinking in terms of string theory. $\Delta x$ can be interpreted as the wavelentgh of the electron field, which has to be $2L_p$.

FIRST EDITION

The strong gravitational field of the black hole has a "dual" effect. Outside the event horizon normal quantum fluctuations of the vacuum can give rise to particle-antiparticle pairs, which then can be separated by the strong gravitational field of the black hole to lead to the famous Hawking radiation. However closer to the black hole there is an extra source of uncertainty due to gravity. The GUP (generalised uncertainty princple) is a result of string theory, and the Planck length begins to make crucial contribution to the minimal action. An interesting analysis and discussion of the effects can be found in this link:

http://arxiv.org/abs/gr-qc/0106080

I hope it will make an interesting reading.

Answer 2

To put what JKL said in a slightly different way, in situations where quantum gravity or Planck scale physics can not be ignored, such as in the context of black holes (or the very early universe too), the second stringy part of the generalized uncertainty principle

$$ \Delta x = \frac{\hbar}{\Delta p} + \alpha' \frac{\Delta p}{\hbar} $$

where

$$ \alpha' = \frac{1}{2\pi T} $$

is the slope of the Regge trajectories (and T is the string tension), becomes important.

The second term can be explained by the fact that string theory introduces a very small (at most 1000 times the Planck scale as I have heard) minimum (string) length scale

$$ x_{min} \sim 2\sqrt{\alpha'} \sim \frac{l_{Planck}}{g^{\beta}_{closed}} $$

($l_{Planck}$ is the Planck length, $g_{closed} << 1$ is the coupling constant of closed strings, and $\beta > 1$) which can be neglected at low energy (or large length) everyday scales.

When trying to probe shorter and shorter distances down to the Planck lenght one has to put the energy of $10^{19}$ GeV into the colliding particles. Since the Schwarzschild radius of a particle with the corresponding Planck mass is the Planck length too, this means that one produces the smallest possible black holes by such Planck energy collisions. Increasing the energy further to try to probe distances even smaller leads to the production of larger black holes instead, and the length scale one attains by increasing the energy beyond the Planck energy starts to grow again.

My (if not correct please complain!) interpretation of the generalized uncertainty principle is that the second stringy term, which is proportional to the uncertainty in momentum (or energy) and which starts to dominate the short distance behavior already at the string scale which is assumed to be larger than the Planck length, correctly describes this at a first glance counter intuitive behavior.