# The so-called breakdown of spacetime

An assertion which is often made is that quantum gravity effects will become important at the Planck length — about meters — and the notion of spacetime will break down at that length. People like to wax lyrical about spacetime turning into some kind of “quantum foam” at that level.

This bugs me. If it really is the case, then why doesn’t the notion of mass doesn’t break down at the Planck mass, which has the rather hefty value of about kilograms? What’s the critical difference between mass and length?

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One reason for this is that mass is not expected to be a dynamic variable for future physical theories (what is the variable conjugate to mass?) whereas the metric is the dynamical variable of general relativity (or the connection or whatever generalization your local theorist prefers.)

Also the interpreation of the Planck mass is something like: general relativity allows black holes of any mass, but quantum theory says that if they are smaller than the Planck mass then some of the mass must lie outside of the event horizon of the black hole (remember the Planck mass is the mass at which the Compton length=the Schwarzchild radius.) So it seems that the notion of a black hole event horizon should break down for black holes less than this mass and this is probably equivalent to some quantum foam picture or whatnot.

In quantum gravity the metric fluctuates, so the question is what you mean exactly by various lengths- these are quantum observables, which have mean and some spread. However you can still do some kind of semiclassical WKB approximation around some fixed solution of Einstein equation, this defines for you approximately what you mean by the metric, and therefore what you mean by distances. Once the energy density becomes of order the Planck mass (to the fourth power), or the length become of order the planck length, or the time becomes of order the planck time, etc. then WKB approximation breaks down, and the geometrical quantities are no longer sharply peaked around their classical values. What really happens beyond that depends on precisely what you think quantum gravity is.

Note that the Planck mass itself is not that important, it is the Planck mass density that matters- you need to concentrate lots of mass in a small volume to make gravity strong.

One more comment…The criterion of breakdown of semiclassical approximation, as usual, is that the actions relevant in the problem become of order planck constant (h-bar), that would normally translate in Einstein gravity to some typical energy (length, time…) being of order planck scale. This is basically only dimensional analysis.