Singularities in Space-Time Prove Hard to Kill

My understanding is that part of the issue with string theory is that the computations are extremely difficult. Like, yes, the landscape is very large, but it’s also very difficult to, even picking one particular option, it is very hard to get a prediction out of it. However, I think some correct but very loose postdictions have been made about some mass ratios?

There are many critics of string theory, and their voices deserve to be heard. However, I would also like to ask these critics: Can you come up with something better? String theory explains many phenomena with mathematical elegance, but at the moment, there is no way to test it.

The optimist in me believes that String Theory is the physics of universes, not just our own which is why it has so much fine-tuning needed.

First off, let's assume General Relativity (4d time-orientable manifold) is true. That means among other things that when I write "black hole" I mean an approximate solution which is a small perturbation of an exact solution like the McVittie metric, or even Kerr-Newman or Schwarzschild (which lack a cosmology part, but can be stitched into an expanding cosmology à la Darmois-Israel junction). Black hole solutions have inextensible curves or at least the distinction between curves vanishes at a "singularity". Curves are trajectories through spacetime that may in some places be accelerated and in some places may be geodesic. Physical objects bind to everywhere-non-spacelike curves and we further say these curves are everywhere future-pointing (no backwards time travel allowed). The "small" perturbations (e.g. adding outside matter) does not change any of these key features.

In that setting we could specify all the matter everywhere and solve the various partial differential equations and have an exact calculation for a particular spacetime. However anyone who has ever tried to do this by hand -- that includes many many grad students over the decades -- knows this is infeasible for complicated spacetimes. One can use automation with perturbation theory (e.g. https://bhptoolkit.org/ ) but then one discovers that perturbation theory breaks down deep inside black holes.

So the practical problem is that we want to do numerical relativity (NR) within black holes, and unfortunately singularities turn out to be numerically intractable with current methods.

How NR works, very roughly (and quite differently from your second-last paragraph) is that we slice up a model spacetime (with boundary and initial conditions) into 3 spatial + 1 time dimension, "foliating" on a time axis chosen (from the infinite possibilities) for pragmatic reasons. The axis is global for the whole spacetime. We have to note here that the relativity principle is that nobody's time axis is special, which means there is no right choice and no wrong choice here. However different choices come with different trade-offs, including in how straightforward it is to interconvert a system of coordinates adapted to our chosen time axis and those adapted to any other useful time axis.

We then canonicalize curves through this foliation into per-spatial-slice quantities reflecting position and momentum; we also make arbitrary choices about what represents "empty" spacetime so we can choose a shift vector (capturing how spatial coordinates differ from one slice to its neighbours) and a lapse function (capturing how coordinate time (e.g. proper-time of a massive particle) evolves from one slice to another). This gives us constraints (how quantities on a particular spatial slice relate to one another) and evolutions (how these quantities change from one slice to its infinitesimally future successor, and what was in its infinitesimally past predecessor).

There is no local "causality" clock ticking, there are only quantities on each whole-universe-spatial-slice and an ordering of slices. Suitable causal conditions -- notably globally hyperbolic solutions to a number of partial differential equations -- let us fill in the whole spacetime from infinite past to infinite future, and should be a formalism (the Initial-Value Formulation) of the full solution in general relativity up to numerical errors.

This is the most common approach to doing General Relativity on computers when perturbation theory breaks down (as it does deep inside black holes and in some cases where there are unusual gravitational-wave/gravitational-wave interactions and where matter waves strongly interact near one of these extreme events (this includes lensing close to a black hole)).

A concrete example of this is in SXS (Simulating eXtreme Spacetimes) numerical relativity kit, which you can begin reading about at https://www.black-holes.org/the-science/numerical-relativity...

It's not practical (and is likely highly error-prone) to use computers this way to calculate very close to a singularity, so various methods are used to "ignore" it, containing the inextensible curves inside a tiny region which we hope can be computationally smooth on the region's surface. This isn't totally new -- Gauss's gravity works that way too. But this raises the question: do we lose effects at the apparent black hole horizon (a "surface" we can obtain by doing local measurements, unlike the event horizon) when we blur the singularity inside the BH? And how do we calculate complete evaporation when relying on techniques like this? These and many related questions are active fields of study in NR.

A reminder: this is General Relativity, therefore the singularity is taken to be physically present. There is nothing that blocks the singularity from happening without adjusting the behaviour of stress-energy to introduce or substitute negative energy deep inside the black hole as matter moves inwards from one slice to the next (and there's no evidence that real matter does this), or substituting a global solution to the Einstein Field Equations which manifestly is not a black hole spacetime (even on the "initial" slice).

Physically, then, there is the question of whether matter somehow blocks the formation of singularities. Your "frozen star" idea is a very very very rough way of thinking about that question. There are many ideas in that space, and it's safe enough to generate those (although it's hard to keep them self-consistent) because there is no real hope for experimental verification of any of them in a human lifetime. However, there was in recent decades hope for exactly this sort of resolution when ideas in the particle physics space like (relatively) low-energy supersymmetry had not largely been killed off in contact with evidence from particle colliders like the LHC.

One can also find in the literature examples of "frozen star" ideas meaning that one doesn't use a black hole spacetime at all, for whatever reason. That raises lots of questions about why there are objects in our sky that radiate really really similarly to black hole spacetimes. "Frozen star" (of this kind) simulations tend to produce clearly wrong results far from the surface of the black hole.

> So what's the sense about talking about events that happen outside our universe?

Sure, this is the intuition between the puncture and excision approaches to the deep regions of black holes in numerical relativity. As long as whatever falls in also stays in, the approach is good. But what stops black holes from completely evaporating? In that case, what the hell is supposed to come out of the singularity / deep region / puncture region / excised region?

Sadly the lifetimes -- from our point of view -- of even low-mass primordial black holes or young stellar black holes is more than enough for anything crossing the apparent horizon (like primordial radiation, cosmic microwaves, distant starlight, and so on) to hit the singularity. This problem is even worse as we increase the black hole mass. There is no support in General Relativity for matter as we know it to "freeze" long enough in a black hole spacetime. The infalling matter doesn't care what we see happen from our perspective. We are allowed to experience optical illusions, or be misled by poor choices of systems of coordinates.

> our time reference

turns out to be a poor choice of time axis for many astrophysical events. One can choose more suitable systems of coordinates for events "over there" (or for a global foliation) and then do careful coordinate transforms from those coordinates to coordinates more in line with our day-to-day experiences.

For example one might attach Fermi normal coordinates to an infalling particle approaching an astrophysical black hole, and do a series of coordinate transformations to "cosmic time", from which we can do further transformations to TDB/TCB/TAI or whatever we want. The particle's collision with the singularity will be in our past. The flashes we detect from dust clouds and in our galaxy's central parsec or flashes from tidal disruption events in other galaxies are messages from matter which is already "in" their respective singularities.

it exists under superdeterminism, which is the most sane interpretation of wavefunction collapse

>Maybe the universe has some crazy repulsive force when atoms or subatomic particles get really really close (closer than in neutron stars, where atoms are femtometers apart).

The Celestial Ick

You can get this easy by reformulating gravity's effect on spacetime as slowing down the speed of light/causality and putting a natural bound that asymptotically approaches zero. It should agree with GR everywhere except at extremes like black holes.

Looking at gravity as a slowdown of c is appealing because it suggests a computational cost of massive particles. As stuff gets more dense, the clock of the universe must slow down.

> closer than in neutron stars, where atoms are femtometers apart

I suppose you meant neutrons instead of atoms.

Atoms do not exist in a neutron star. At least, non in any significant quantity.

That would mean is that what prevents them is some other mechanism doing it accidentally. My view is that they are prevented by the GR itself (by time dilatation).

We know something which looks exactly like a singularity though does exist - i.e. whatever black holes are, we can observe matching predictions very well.

So if singularities don't exist, then some other weird object must that naively looks like one.

There are no "atoms" in neutron stars. The density and temperature are so high that they are "crushed" and that the electrons and protons form neutrons. The result is tightly packed neutrons, hence the name.

I believe it is theorised that it might be possible to go even one step further to a "quark star" since neutrons are not elementary but made of quarks. No idea what a black hole might look like with no singularity...

  > a zero mass rock

I agree with you so much that I can't resist a nitpick. I'll also amplify your points in the last two paragraphs and justify "the popular scientists".

> the event horizon moves outwards to meet it

The event horizon is a global surface that might even be measurable locally in principle, and certainly might not be where one thinks. The event horizon is determined by what's inside and outside it, and arguably it's only "dynamical" if one takes a 3+1 view of the spacetime containing it.

Local measurements can determine which side of an apparent horizon the measurer is on. That's arguably more useful than the event horizon, since for all practical purposes it's the apparent horizon that is the point of no return. Anything inside that will quickly end up in a region where local measurements lose predictive power (roughly, "the singularity").

Visser 2014 <https://arxiv.org/abs/1407.7295> has more detail

The apparent horizon is dynamical in the sense that it will deform -- a small black hole orbiting a supermassive black hole raises a bump on the supermassive black hole's apparent horizon. (The event horizon has the same gross property, but you can't know the bump's details until you know the whole spacetime! Observed disturbances in the material surrounding such a binary represent the apparent horizon.)

Schwarzschild is a special case: an exact solution of a family of approximate solutions to astrophysical black holes which aren't isolated and which are immersed in a cosmology, and which are also probably rotating and may have some residual charge (not necessarily electromagnetic charge à la Kerr-Newman) from time to time. They may also radiate (but probably not isotropically like Vaiyda) and certainly intercept radiation (cosmic, from their local environment in their host galaxy, and otherwise) non-isotropically and with different inbound/outbound fluxes over time. And so on.

There is lots of literature about the stability of e.g. K-N to small perturbations (e.g. if you shine a very bright light on half of a K-N black hole, is K-N (with new parameter-values) still a good representation of the black hole once the light is off?), and it is pretty reasonable to think that an OS black hole even with lots of extra charges decorating the scene (and with other masses making a mess of the asymptotically flat region) settles down to a state very well modelled by K-N or even Schwarzschild.

i.e. Black holes go bald.

(but then lurking around the barber shop are the gravitational memory effect, BMS supertranslations, and so on ...).

With the current theories, I don't think it matters (pun intended) if a matter and anti-matter black holes merge.

E=mc2 (+change)

In normal space that would have liberated a huge amount of energy, but in sufficiently curved space, the energy stays inside the back hole, mass is conserved, nothing to see (literally).

> A "matter black hole" and an "antimatter black hole" will combine to form a larger black hole.

I think they'll annihilate forming photon black hole, which might be unstable.

> but it does have a notion of before and after.

Was forming of any singularity before present day on Earth?