## Tuesday, January 11, 2005

### New Scientist and minimal length

New Scientist, a UK science magazine, covered our work on minimal length and quantum gravity. (Done while I was on sabbatical at Caltech.) You can compare the following summary to what I posted on this blog last November.

The smallest measurable length announced

27 November 2004. From New Scientist Print Edition.

PHYSICISTS have set a limit on the smallest length that can ever be measured - and any device that tries to beat the limit will be crushed into a black hole of its own making.

The finding is based on an analysis of interferometry, a technique that uses interference of waves to measure small lengths. Quantum theory says that the more accurate the measurement you want, the more massive the interferometer you need.

But Xavier Calmet, Michael Graesser and Stephen Hsu of the California Institute of Technology in Pasadena point out that any very massive interferometer would have to be spread over an extremely large region of space. Otherwise, the large mass concentrated in a small area would produce strong enough gravity to form a black hole, sucking in the interferometer.

But as the mass and the size are made ever larger to measure ever smaller lengths, the interferometer eventually becomes so big that its various components would not be able to interact fast enough for it to work, even using signals travelling at the speed of light.

Mathematically, these constraints lead to an instrument that can accurately measure only down to about 10-33 centimetres, a distance known as the Planck length (Physical Review Letters, vol 93, p 211101).

Anonymous said...

Nice. I rather enjoyed this.

Anne

Carson Chow said...

Hi Steve,

What needs to be done to turn the hoop conjecture into a theorem?

cc

steve said...

Carson,

The hoop conjecture basically says that if you pack enough energy into a small enough region it will turn into a black hole:
i.e, if R < G E (we neglect coefficients), where R is the size of the region into which energy is confined, E is the energy and G is Newton's constant, then a horizon will form.

To give a rigorous proof for all possible initial data satisfying the conditions of the conjecture would be very hard.

But, in the last few years there has been some very interesting progress in showing that even in the *least favorable* circumstance, the conjecture holds. What would be the least favorable? Suppose the energy E is divided between two nearly massless particles, and is comprised completely of their kinetic energy. Unlike the favorable case, where E might be comprised entirely of rest mass (as in a collapsing star or dust ball), here the energy is concentrated on particles moving at nearly c, which want to escape to infinity. Yet, it has been shown rigorously that if the two particles pass within R = GE of each other (again, I neglect coefficients), a horizon will be formed. You can find references to this work, done over the last few years, in our paper. (Incidentally, this result also answers the question of whether and when black holes can be produced in collider experiments - don't hold your breath, though, because we are far from building Planck energy accelerators!)

So, to answer your question, no proof which would be satisfactory to a mathematician is forthcoming, but most physicists familiar with the subect would be very surprised if the hoop conjecture were wrong.