Can black holes explode?

Ahem .... I didn't mention the other aspect of observation that inflation solves, not the 'horizon problem' mentioned below, that is : the 'flatness problem'. Ignoring stuff below the scale of clumps of matter, like black holes or galaxies, then the universe is as per Euclid. By that I mean, in large scale indeed, the interior angles of triangles add to 180 degrees. That sort of thing. Why should this be so?

I'll go back to some basic calculus. The main idea of taking a derivative ( thinking of one dependent variable of one independent variable now ) is to progressively approximate a tangent to a curve by a secant line. A tangent line is one which just 'touches' a curve, intersecting at one point only, in a way which yields the concept of slope. So we say the gradient of the tangent line ( rise divided by run ) is the slope of the curve at that point of tangency. Now a secant line runs b/w two points on the curve, so it cuts the corner so to speak. So imagine two points on a curve A and B, keep A fixed but slide B back and forth along the curve. As B gets nearer to A then the slope of the secant line becomes nearer to the slope of the tangent line at A. If you can find a single number that the secant slope can be as close as you like, provided you let B get close enough to A ( but not equal ), then you have performed a limit process - with that single number being given the name limit and thus said to be the slope of the tangent and hence the curve at the point where you performed the limit process nearby. Whew!

The moral of the story is : provided that you have a curve for which you can perform the above limit process upon, then any such curve will look straighter and straighter as you zoom in toward it. In algebraic terms one can write :

delta_y ~ f'(x) * delta_x

meaning a slight change in y is well approximated by a slight change in x times the slope of the tangent. Or in Leibniz notation ( with some background I'll leave out ) :

dy = f'(x) dx

In physics we nearly exclusively model reality with smooth functions, these are curves which are as differentiable as we please. One can create a beast called a Taylor's Series that gives the value of the curve at points nearby to a chosen point, based upon values of derivatives at that chosen point. The above formula is the first part of the Taylor's.

Back to inflation. Take any 'shape' of the universe in the early stages, pick a 'point' and then expand the shape by 10^(50) in size. It'll look locally flat for sure regardless of any wiggles and bumps it had beforehand. Mind you, what we see of the universe today isn't the whole universe. It's just the bit we can see, and so 'local' in this discussion means a bubble of some fourteen odd billion light years in radius. We just don't know what's beyond that.

Inflation also solved another issue as regards 'monopoles', which I don't understand but mention for completeness.

Cheers, Mike.

Yeah the margin was too small to contain the proof. Cheeky lad has kept us busy for centuries. He hinted at a method of 'infinite descent', a sort of Principle of Mathematical Induction in reverse, plus Proof by Contradiction. Roughly : assume some Proposition is true for some ( unspecified ) huge number(s), deduce that it must therefore be true for a case with lesser magnitudes - rinse, lather, repeat - arriving at some exact case with low numbers, but such that the Proposition is known to not apply. Therefore, flipping all the dominoes in reverse, it can't have been true in the original ( huge ) case. Trouble is he never specified the detail, so the truth ( via this method ) is unknown ie. how many trucks can you drive through this logic?

The easier answer is to avoid the depth inherent in the question and say 'that we just do models'. The current dominant paradigm for non-gravitational forces is that of particle interaction. So when I push my hand against your chest this is essentially electromagnetic. The other forces are there but they just are no where near the same magnitude for this situation. Hence we would posit a blizzard of ( virtual ) photons surrounding charges labelled as being in/on my hand interacting with a similiar photon fuzz around charges labelled as being in/on your chest. By simply speed limiting the photons then you get Special Relativity et al. You can run this logic across the distance between you and me, or trot all the way to the Great Attractor.

For the weak and strong nuclear forces one replaces photon by intermediate vector boson and gluon respectively. But .... you have to thrash the mathematics a tad to wrestle non-infinite answers out of the full quantum mechanical horror - see Feynman diagrams and sum over histories, normalisation, re-normalisation, effective theories etc.

However : these tricks ( for photon/intermediate-vector-boson/gluon ) assume a spacetime background, a structure within which they operate. Call this 'background dependence'. Not so with gravity. In addition to being a classical ( non quantum mechanical ) theory, we are talking about the behaviour of spacetime - the background itself. So if you want to speak of gravitons ( gravitational quanta ) then they don't operate within spacetime as they are spacetime! Call this 'background independence', meaning gravitons have no background. Up to a point you could call this semantics, wordplay and what not.

But a special implication here is that one graviton must affect another graviton, and it does so by using - guess what? - gravitons. Arrghh ! To say this is non-linear is an understatement. Two photons will pass by each other, and apart from the possibility of some other particle creation ( like an electron/positron pair ), they won't upset each other at all. The cloud of virtual photons alluded to above ( hand pressing to chest ) are really interacting with the charges from both of us. Our electrons and protons, regardless of labels. So photons aren't surrounded by 'their own' photon fuzz, so if one photon is choofing along undisturbed between here and the Andromeda Nebula then it can well be alone. But gravitons are surrounded by other gravitons ..... this gives rise to, at present, some unconquerable infinities in the mathematics.

There's some ancient cosmology that says the Earth is sitting on a tortoise ( not speaking of Terry Pratchett here ), but that is sitting on another tortoise, which is sitting on ....... well you get tortoises all the way down, but without a bottom. An infinite stack. With non-gravitational forces you can approximate quite well by ignoring the tortoises that are well down. Sums converge, so if you stop considering them below some level then any error ( because you did that ) is bounded. The gravity turtles don't do that, you can stop accounting for more complicated mathematical terms, but your error for that choice will not subside and you will have no idea how near or far you are from the 'true' result that would be obtained by summing over the lot.

QCD lattice theory ( strong nuclear force ) was difficult but tractable and eventually, with supercomputer level computing power, the sums converged to within a quotable range. So you didn't do infinite tortoises but you knew how far off you would be by stopping at some level. So they wound up with a terrifically accurate number ( with respect to measurement ) for the mass of a proton : two up quarks plus one down quark sitting in a blob of gluons. A significant fraction of the total proton mass is the energy of the gluons being exchanged, which is a manifestation of the degree of 'slowness' of convergence when extra levels of tortoise terms are added. In electromagnetism the 'anomalous' magnetic moment of the electron was calculable with fewer tortoise levels than QCD and gave even more decimal places of impressive agreement with measurement.

But there's a sense of incompleteness for sure, like dividing the circumferences of lots of circles by their respective radii, and coming up with an amazing constancy in that ratio. Why should a universe do that?

Cheers, Mike.