raattgift

Not sure what you mean; you can't have any mass in a flat spacetime and obey the Einstein Field Equations for a Lorentzian spacetime (because T_{\mu\nu} doesn't vanish everywhere).

There are a variety of types of variable speed of light. If we foliate to 3+1 the usual picture is that c is constant on all spatial slices. Some VSL theories have the same c at all points on a given slice, but introduce a time variation of c. Other VSL theories introduce spatial variation as well (or instead). These families of theories all have significantly different equations of motion or actions from one another (cf. <https://en.wikipedia.org/wiki/Einstein%E2%80%93Hilbert_actio...>). There's no obvious reason why c couldn't relate in a more complicated way to the stress-energy tensor than the Einstein gravitational constant does, but there's also no obvious reason to think such an alternative theory should produce free-fall trajectories similar to those from GR.

In any case, I think you have to choose your function on c, obtain the field equations, decide which energy conditions and constraint equations you want to impose, set appropriate boundary conditions, choose a curve along which to foliate, and run with enough different initial-value surfaces (each of which must satisfy the constraints initially), that eventually an intuition develops. A Will-like parameterized post-Newtonian formalism approach would also be a good idea (<https://en.wikipedia.org/wiki/Parameterized_post-Newtonian_f...>).

Unfortunately I'm unable to guess your choice of "c(m) formula".

The problem is probably connecting an image of a tilted causal cone at p near but outside a regularized r_s in Schwarzschild spacetime with the idea that from p there is a limited number of null geodesics escaping to the asymptotically flat region, that the fraction decreases with decreasing regularized r coordinate, and that from p' at the trapping surface there are no such null geodesics, as all non-spacelike curves (accelerated or not) at any point on or interior to the trapping surface terminate at the singularity. The idea that the singularity inevitably lies in the future of any observer at p' is behind the "spacetime swap" notion.

Some of the problem is that Schwarzschild coordinates have surprises buried in them, and what \Delta r and \Delta t mean are not what most people tend to think.

Someone should do an ELI12 of Unruh's (ca. 2014) excellent (give or take varying the spelling of Martin Kruskal's surname) Schwarzschild BH global coordinates pedagogic review <http://theory.physics.ubc.ca/530-21/bh-coords2.pdf> and add in a bit on Fermi normal coordinates as a maybe-obvious not-a-chart follow-on to the commenary just above eqn (55). But on "maybe-obvious", Unruh has the choice line: "Since in a large number of cases, the single horizon coordinates were discovered long before Schild’s coordinates, this is an exercise in alternate reality – what could have so easily happened if only the generators of those coordinate systems had recognized what they had."

I think given time at a blackboard we could walk through Newton's cannon in the context of Poisson gravity, and for extra credit with the cannonball inducing a perturbation of the Poisson vector field. Even without the cannonball's backreaction, the Poisson picture offers a nice image of the gravitational potential energy at the top of the cannonball's inertial (ballistic) curve. We would then consider a cosmology like our own but with a recollapse: at maximum extent there is some (quasi-)Newtonian notion of gravitational potential energy for all the galaxies, since they are at the point where they begin free-falling back into a denser configuration. It's then the usual story of relating kinetic and potential energy, and recognizing that the standard cosmological frame is close to Newtonian by design. (We also have to stop this approach when the galaxies are merging enough that radiation pressure and gas ram pressure become relevant, because the errors become astronomical).

Since we don't have a blackboard in front of us to interact with, I can suggest Alan Guth's lecture notes on Newtonian cosmology. (Guth is credited with discovering cosmic inflation.) https://web.mit.edu/8.286/www/lecn18/ln03-euf18.pdf See around eqn (3.3). You could also borrow a copy of Baumann's textbook <https://www.cambridge.org/highereducation/books/cosmology/53...> which studies the Poisson equation for various spacetimes, however a static spacetime gets most of the focus.

A universe which expands forever, or which expands faster in the later universe, makes a mess of this sort of approach to calculating a gravitational potential energy. So does any apparent recession velocity that's a large fraction of c (inducing significant redshift, whatever the recession (pseudo-)"force" might be).

However, the general idea is that there is a relationship between the kinetic energy a receding galaxy (in a system of coordinates -- a "frame" -- in which these kinematics appear) and a gravitational potential energy still occurs in a non-recollapsing universe. It's just that the potential energy climbs forever, and you get an equivalent to gravitational time dilation between galaxies at different gravitational potentials (i.e., between early-universe galaxies and higher-potential modern-times galaxies).

Accelerometers in galaxies will not show a cosmic acceleration for any galaxy; they're all really close to freely-falling (local galaxy-galaxy interactions are real -- collisions and mergers and close-calls happen -- but wash out over cosmological distances; look up "peculiar velocity" for details). Therefore we can conclude that there's no real force imposing acceleration on the galaxies. However that's also true of a cannonball in a ballistic trajectory, including one on an escape trajectory or one that enters into a stable orbit. Consequently one can draw some practical comparisons between a ballistic launch from Earth into deep space and galaxies spreading out from an initially denser early part of an expanding cosmos.

> Dark energy as energy being extracted out of the universe

No, it's just a way of thinking about whatever is driving the expansion, and that doesn't dilute away with the expansion as ordinary matter and radiation does. It's not even a "real" energy in the sense that it is only an energy in the cosmological frame, and is a frame-dependent scalar quantity, whereas in the fuller theory it's just a multiplier of the metric tensor. So it's the full relativistic metric doing the work but we absorb some of that into cosmological coordinates in the cosmological frame of reference, carving up the metric tensor into a set of vectors including an expansion vector identical at every point in spacetime.

The expansion vector can also be thought of in terms of pressure: in a collapsing cosmological frame, a pressure drives galaxies together into a denser configuration. The inverse of pressure is tension, so in an expanding cosmological frame, it's a tension that pulls galaxies apart into a sparser configuration. (The reason one uses pressure or its inverse is that the matter fields are idealized as a set of perfect fluids at rest in the cosmological frame; each such fluid has an associated density and internal pressure which evolve with the expansion or contraction of the cosmos, generally becoming less positive in the time-direction of expansion (i.e., in the future direction in a universe like ours). Another way of thinking about pressure is as a measure of isotropic inflow of energy-momentum into a point; increasing pressure at a point therefore increases the curvature at that point. Tension is an isotropic outflow, and so positive tension is repulsive as opposed to the attraction from positive pressure.)

> that explanation felt unsatisfactory to me

Hopefully the above helps a bit. Unfortunately there's only so much teaching one might do in a series of HN comments, and ultimately one probably is better served in developing some grounding in the full Einstein Field Equations / Friedmann-Lemaître equations before thinking in quasi-Newtonian ways. Going the other direction tends to lead to misunderstandings and developing false intuitions when running into situations where the quasi-Newtonian picture needs post-Newtonian correction terms.

It's cool that you have all sorts of questions. You could consider signing up for part time / non-business-hours courses in relativity at a nearby community college or the equivalent, depending on where you are, or maybe just bringing a hot lunch to a lecturer there in exchange for a quick informal tutorial. Anything like that is bound to get you to better answers than raising comments on HN threads about astrophysics in the broadest sense, as answers here are often somewhere between non-standard and unreliable.

Physicists (and in particular the subset doing physical cosmology) don't usually explore parametrizations of c because they're not clearly physical (and sometimes even clearly unphysical), or alternatively don't help solve astrophysical or cosmological problems.

Relativists sometimes like to explore things that make using the Tangherlini transformations rather than the Lorentz transformations look positively benign. (To be clear, the Tangherlini synchronization system is clearly unphysical, requiring infinite speeds. His thesis also proposed using a distinguished global frame, which is not really philosophically different from how the standard cosmological frame is used, and seems OK because the distribution of stress-energy can pick out useful systems of coordinates in standard relativity. Unfortunately his method frustrates and probably outright breaks comparisons between inertial reference frames related by a boost, which the standard cosmology does not, and it's hard to see an alternative method that preserves his central ideas.)

But why even be stuck with 3+1d spacetime like Tangherlini? He was trying to do physics. But an unphysical metric signature with 47 plusses and no minuses is really cool!

In our observed universe, FAPP, c is the same everywhere after recombination, and we get that from spectral lines. You have to play really weird games to preserve the Lyman-alpha forest's apparent isotropy while introducing spacetime (or redshift-space, here) anisotropy. Things like BAOs make the problem even harder.

If we strip away all that pesky radiation and the information its structure encodes, analysing variations of c gets a lot easier. A relatively recent paper (Lewis & Barnes 2021) I enjoyed considered anisotropy in the one-way speed of light in an FLRW cosmology with zero energy density (well, really the convenient Milne model, which is also far from spatially flat). "So far, we have considered two cases, where either the speed of light is isotropic, or the extreme case where the anisotropic speed is 1/2 in one direction, and infinite in the other. The question remains whether this holds true in general case, for an arbitrary κ": https://www.cambridge.org/core/journals/publications-of-the-... (arxiv: <https://arxiv.org/abs/2012.12037>). "For more general cosmological models, where the presence of mass and energy results in curved space-time, the picture is more complicated as there is no simple mapping of the modified Lorentz transformations into the general relativistic picture. We leave this discussion for a future contribution."

Sadly there doesn't seem to be a future contribution yet, at least going by published citations (<https://scholar.google.com/scholar?cites=2012575105829699847...>). (Of those, I've put the Chamberlain paper on my to-read pile; you'd appreciate how it relates to Tangherlini, "credence is given to one-way infinite light-speed inward to each particle in direct comparison against Einstein’s isotropic (c=constant) light-speed").

Of course there's also the excellent Magueigo 2003 VSL overview <https://iopscience.iop.org/article/10.1088/0034-4885/66/11/R...> copy <https://cds.cern.ch/record/618057/files/0305457.pdf> preprint <https://arxiv.org/abs/astro-ph/0305457>.

And even if you can make sense of an f(c) cosmology in the early visible universe, you will get to epochs before recombination and try to make sense of the later universe's chemistry, which of course relates to big bang light nucleosynthesis, baryogenesis and electroweak ssb. How do you abolish Lorentz symmetry in those epochs? Good luck!

(I mean, I think if you are doing physical cosmology you ought not to ignore gauge theory...)

No, there's no change in dimensionality.

The swapping of timelike and radial dimensions are a "game" frequently played with families of coordinates, including Schwarzschild coordinates. One can apply any system of coordinates on a physical system without changing the behaviour of the physical system: coordinates are unphysical. Think of navigating around in a neighbourhood: you can talk about going forward a few blocks then turning left, after which you go forward two more blocks; or for the same journey, going "city north" a few blocks then going "city west" two blocks. Here assuming that (initially) "forward" is in the "city north" direction (and "city north" is not necessarily exactly magnetic north nor a section of a meridian of longitude). After the left turn, "forward" is "city west". There's an analogue to the discussion's (ab)use of Schwarzschild coordinates.

In Schwarzschild spacetime, without applying any system of coordinates, just floating in free-fall far from a black hole extremizes your travel in the timelike dimension. (You can do this at home: you stay put at some point on Earth (whether you use GPS latitude/longitude/altitude or some other system of coordinates) but your wristwatch keeps ticking). Inside the black hole horizon, just floating in free-fall extremizes your travel in the direction of the singularity. Far from the black hole, accelerating as strongly as you can in any direction takes travel from the timelike dimension and puts it into one or more spatial dimensions. In particular, you have the freedom to increase the spacetime interval between you and the singularity. Within the horizon, however strongly you accelerate the spacetime interval between you and the singularity shrinks. This behaviour seems to invite the use a different set of coordinates applied to a patch of space around an observer far from the black hole and a patch of space around an observer inside the horizon. It's some human cognition thing, and in the early 20th century it took decades to discover systems of coordinates that work for observers far from the black hole, at the horizon, and inside the horizon. And even today, most people don't seem to try to enhance physical intuitions by swapping among arbitrary systems of coordinates (including no coordinates) on a single physical system like a black hole and a pair of observers (one inside the horizon and one far outside the black hole).

The Schwarzschild black hole interior is still locally Lorentz-invariant everywhere (because the whole Schwarzschild spacetime is a Lorentzian manifold).

The various local interactions of the Standard Model will keep working inside a black hole. In a really tiny patch around every point, everything behaves as if its in Minkowski space (flat 4-d (3 spatial + 1 time) spacetime).

(That's one of the problems of quantum field theory on curved spacetime: the "focusing-pressure" [for experts: this is encoded in the Weyl curvature tensor; my "scare quotes" take a view of this in a Raychaudhuri equation way] gets so high that the unknown ultraviolet behaviour of the Standard Model (a quantum field theory) becomes relevant. The Weyl behaviour in Schwarzschild is that quasispherical objects are strongly prolated with the long axis aligned radially: a soccer ball or basketball starts looking like an American or Canadian or Rugby football ball. The radial stretching "spaghettifies" by ripping apart weaker bonds (like intermolecular ones, and molecular ones, and ionizing atoms), but the tangential squashing ("focusing") must eventually generate more nuclear interactions, probably up to quantum chromodynamics (QCD) energies possibly before the radial stretching starts generating hadronization.

How this works in the Standard Model is just unknown. However simpler "test" quantum field theories (fewer, or even no, interactions; and often no colour-confinement-like processes) raise really difficult questions.

Finally, back to the Standard Model as local theory: how does any allegedly quantum nonlocality behaviour work? Local here in the sence that states can be distinguished by local measurements alone. Related questions: can you entangle particles deep inside a black hole? If an entangled pair fall in together, how does the entanglement evolve? Or obsessing black hole information people, what if you throw in only one half of an entangled pair and locally measure the outside half? Nobody has great answers for these sorts of questions at present, and there's no near-term hope of testing any proposals in laboratories or via astrophysical observation.

The practical sense is the binary-merger waveforms detected at LIGO, Virgo, and Kagra, which were emitted in the detectors' (and humanity's) past lightcones. We don't even have to consider "the ... future history of human race"; the gravitational-wave ringdown is from after the two horizons touched (and in particular the intermediate stage of the ringdown encodes the crossing of light from the light or photon ring structure around the not-yet-merged black holes into the final merged object, and the recapture of quite a lot of gravitational radiation scattered in the strong-gravity zone around the not-quite-in-contact binary).

> Except in GR the math itself provides the reason why the part below the event horizon should be outside of our consideration

No it doesn't, and this has been known about since the 1920s (and reasonably understood since 1939 and well-understood since the early 1970s), and you can find out about this in any GR textbook, for example §12.2 of Wald, and problems 1 and 4 at the end of chapter 12. You should try solving those two exercises like any grad student (being sure to read problem 4 and to think about why it's asking for an upper limit) before making the sort of wrong claim about "the math itself" you made above.

A couple of things to understand: (1) in general black hole solutions do not superpose linearly [this is the thrust of problem 4]; (2) there are multiple lines of evidence supporting black hole mergers; (3) gravitational radiation is generic to any quasicircular orbit (not just black holes), with the emitted frequency and amplitude inversely proportional to the orbital diameter; (4) frequencies and amplitudes for some detections (e.g. GW150914) are too high for solid self-gravitating bodies anywhere in the sky.

You can't put a rock in a 100-millisecond orbit around the Earth; LIGO has ample data in the ~ 30 - 7000 Hz range. The gravitational wave ringdown of a neutron-star/neutron-star, neutron-star/black-hole, or black-hole/black-hole merger is totally unlike what one gets from a rock hitting the Earth.

The curves of rocks thrown in Earth's gravity don't become incomplete just because they transition from free-fall (geodesic motion) to accelerated (while resting on the surface). One can literally lift bits of meteorite out of craters. There's enough cosmic dust <https://en.wikipedia.org/wiki/Cosmic_dust> around launch sites that it's inevitable that some bits of meteorite have been taken back to space and are now in free-fall somewhere in or near the solar system (or as another possible example, some of the nickel or iridium in Voyager 1 components, if the ore was mined in Sudbury, Canada).

The curve of every part of a rock chucked through a black hole horizon ends inside the horizon. That curve-confinement is the characteristic feature of a black hole in General Relativity: no trapping surface, no black hole (Wald again, proposition 12.2.3). Where there is a trapping surface,no known high-energy behaviour of matter avoids collapsing gravitationally into a configuration in which every (non-spacelike) curve of the matter inside the horizon ends up becoming future-inextendible (that's what the singularity does: any curve touching it cannot be extended further within the horizon).

> coordinate system so that infinities don't spoil our fun

Curvature scalars don't diverge just below the horizon of even a stellar-mass black hole. Coordinate divergence is not the same as curvature divergence, and different systems of coordinates on a single arrangement of mass can diverge at arbitrary radial distances, including very very large ones.

> physics to philosophy

"Does the ringdown overtone of a merger of two black holes encode any of the accretion history of each black hole?" is a physical rather than philosophical question and is an avenue for testing solutions of the Einstein Field Equations (including approximate ones solved numerically) against astrophysical data. See for example https://www.ru.nl/en/research/research-news/the-observation-...

The National Film Board of Canada's documentary on black flies is an ear-worm: https://www.nfb.ca/film/blackfly/

> everywhere in the universe

No, that's not correct. It only has to happen at one place in the spacetime, namely at the horizon itself. General Relativity is a theory of point-coincidences, after all.

Additionally there are ordinary observers (cosmic microwave background radiation, in particular) on hyperbolic trajectories grazing the infall point; there are observers ultraboosted towards the black hole; and there are observers orbiting other black holes whose proper time is less tilted with respect to a free-falling infaller than your proper time is.

The key point is that one obtains the free-faller's geodesic by solving the EFEs in the block spacetime, and one notes that some of that geodesic is outside the black hole, some of it is inside the black hole, and that portion inside the black hole at no point in the future exits the black hole (barring complete evaporation that allows the infaller's worldline to be extended outside the black hole, in which case there isn't an event horizon but there's still an apparent horizon and probably other trapping surface structure).

Solutions of the Einstein Field Equation give rise to the geodesic equation for each such solution. In an exact, analytical solution like Schwarzschild or Kerr, every geodesic is solved for the whole spacetime. The spacetime is a block universe, and one can pick out individual geodesic worldlines as a sort of thread that runs from the infinite past to the infinite future. The most interesting worldlines are lightlike and timelike geodesics; the latter are those which a physical observer (with mass) would follow in eternal free-fall. Timelike geodesics have the property that the time dimension is longest dimension of their thread-like presence in the block spacetime.

So far there are no coordinates in place at all. There's just talk of a block fully described by a system of differential equations. We can then proceed to apply coordinates to the block, and can use any coordinates that we might want. No choice of coordinates can change the geodesics through the block, only the way in which they talk about it. For instance, let's look at using different 2-d coordinates on the a restaurant where you and a friend are looking at each other across a dinner table. One could say you are north of the friend looking south, or your friend is in front of you looking back at you, or you are in front of your friend and looking back at him. Someone else in the restaurant could say that you are to the left of your friend, or that you are slightly northeast and your friend slightly southeast; a different person elsewhere in the restuarant could describe it exactly opposite: you are to the right, your friend to the left. And so on. But you aren't actually moving around the table or restaurant: the configuration of you and your friend isn't changed as we apply different sets of coordinates. And we can move the origin of Cartesian or polar or whatever coordinates anywhere in the restaurant, so you're at x=0,y=0 and your friend is displaced in the y direction. Or you're bothed displaced in x and y from the x=0,y=0 preferred by an onlooking diner at a different table.

In a black hole spacetime, there are geodesics which are interior to a set of horizons: once they cross such a horizon they stay crossed. It does not matter at all what the coordinates are that are used to describe the horizons or the geodesics.

> relative to us

This is just you applying your coordinates to a block.

Doing so doesn't change the geodesics which cross a horizon in one direction only: their eternal past might be "free" but their eternal future is within the horizon.

> time dilation relative to us

Again, this is applying "our" coordinates. However even those might differ: you are probably thinking in terms of the standard Schwarzschild coordinates or something close to them, whereas I might reach for Kruskal coordinates instead, which absorb timing differences. Just like someone in the restaurant might prefer north-south/east-west coordinates instead of a personal left-right/foreground-background system.

> theoretical black holes

Well, yes, we assume General Relativity is sufficiently correct that astrophysical candidates are compared to black hole solutions in General Relativity rather than something different in a different theory.

We also make some simplifying assumptions which are known to be nonphysical but which represent very minor perturbations of black hole solutions to the Einstein Field Equations. Otherwise we would have no hope of calculating anything, and could not even approximate astrophysical candidates' behaviour.

> Never as measured by our clocks

There's nothing special about our clocks. That seems to be the problem you are wrestling with.

You could get a hyperbolic clock on your smartwatch that slows down as you age, and ultimately becomes so slow that a tick on your wristwatch is more than long enough for a free-falling body to cross a black hole horizon. The physics doesn't change; the free-falling body's worldline in the block universe has a portion outside the horizon and a larger portion inside the horizon. You're just using different algorithms on your wristwatch to apply timelike coordinates to a part of that free-falling body's worldline.

Because one can use any sort of coordinates on a block universe with a set of solved-by-the-equations worldlines (or at least timelike and lightlike geodesics reasonably near a subsystem of interest) one can make poor choices about what set of coordinates to apply to a particular subsystem of the block universe. "Our [linear] clocks" is a poor choice for describing systems with strong gravitation, speeds comparable to c, strong acceleration, or any combination of those, because there is always a nontrivial element of hyperbolicity in such systems' worldlines. The hyperbolicity comes from the geometry of the spacetime, and is always there in any Lorentzian spacetime (meaning it has 3 spacelike and 1 timelike dimension where the latter is related to the former in line elements by the constant c and a change of sign).

Ok, what I take from your comment is that you're identifying General Relativity with certain solutions of GR's Einstein Field Equations. That's like deciding that algebra is just a handful of popular equations.

(Aside, I have run out of time for an editing pass on this comment, so hopefully I didn't leave in ridiculous typos or whatever).

Firstly, let's restrict ourselves to General Relativity as a physical theory. That means we don't have arbitrarily many dimensions with wild metric signatures. We have three spatial and one timelike dimension, so take a metric signature of (+,-,-,-) or (+,+,+,-) which are equivalent but end up with things being written down in different form. That's as opposed to (+,+,+,+,+,+) or (+,+,-,-), all of which can be studied using Einstein's mathematics. Indeed, it's popular particularly among quantum gravity people to work with fewer dimensions (+,+,-) or (+,-) or to go from a Lorentzian (and thus semi-Riemannian) manifold (+,+,+,-) to a Euclidean (Riemannian) one (+,+,+,+). "Quantum gravity people" here include Hawking and 't Hooft. (duck duck go or wikipedia search "Metric signature" for more)

General Relativity as a physical theory of gravitation in our universe 3-spatial-dimension-and-one-time-dimension admits all sorts of really weird spacetimes which are wildly wildy unlike anything in our universe. (In fact relativists have over the decades invented energy conditions in an attempt to remove a few "wildly"s from consideration as possible physical systems: if your spacetime doesn't fulfil some energy conditions everywhere in it, your spacetime is probably not a good match to systems in our universe. For instance, one energy condition requires that energy-density is nowhere negative; another requires that energy is nowhere observed to flow faster than c.) (search term here is "energy condition").

There are a few books worth of exact, analytical solutions to the Einstein Field Equations (those equations define a whole spacetime), some of which resemble astrophysical systems. One such 700-page non-exhaustive book: <https://www.cambridge.org/core/books/exact-solutions-of-eins...>. However there are many many more approximate solutions which have no closed form solution: that's the realm of numerical relativity.

The Schwarzschild black hole spacetime is an example of an exact analytical solution. But there is no superposition of such spacetimes available in General Relativity, so two Schwarzschild black holes in the same universe is not just some linear combination -- instead, we have approximate solutions which can only be solved numerically. That's just with the external properties of black holes: their horizon structures, at least when studying the gravitational waves such systems emit in their last bunch of orbits before merger. It doesn't matter what's inside the horizons for that.

Likewise, where isolated astrophysical black holes give us data is outside the horizon -- what's inside is pretty much irrelevant.

So, the exterior part of an exact solution like Kerr-Newman, with some small perturbations, is at least soluble such that the perturbed KN is an excellent approximation of astrophysical observations of black holes. However we have no observations of the interior part of any black hole, so no way of knowing if Kerr-Newman's interior is wildly unphysical!

(In fact Roy Kerr has said from time to time that because of the presence of matter inside a Kerr BH, the interior part of the solution he arrived at for spinning black holes is probably wrong, even though the exterior part is a remarkably good basis for modelling astrophysical black holes. An example is in his excellent 2016 talk which you can find on youtube at <https://youtu.be/nypav68tq8Q?t=2880> immediately after the 48 minute mark of the video and again at 49:20, however he develops that theme and repeats that point across much of the talk.)

The repair of an unphysical black hole interior might be to stitch together (using a thin-shell method like Darmois-Israel) the physically useful external Kerr solution with something very different inside the horizon but which is more physical. That is not the same, at all, as declaring General Relativity wrong!

Why would one do this? As Kerr implies, there are several invariants ("symmetries") of black hole solutions which are broken by the presence of matter on either side of the horizon. Is the inside part of the Kerr solution fragile to perturbations by matter? That's a work in progress. The outside is pretty clearly stable to such perturbations, mathematically: if you throw in a blob of gas, or star, or shine a very bright light at it, or throw some gravitational waves in its direction, the outside departs from Kerr for a time but soon enough returns to being very well modelled by a Kerr solution with different mass and/or spin. The stability of the inside is not settled. So maybe matter's presence forces a departure from the interior Kerr solution to something else inside, with the Kerr solution remaining in place outside.

Stitching together metrics is something we do all the time. One puts a collapsing patch of spacetime representing a galaxy cluster (in which things tend to move towards the centre, which is where one finds ginormous elliptical galaxies), or a black hole, into a an expanding cosmology (where one finds galaxy clusters flying away from one another) using methods like this. It's all GR, it's just not one single metric line element everywhere.

Of course, there might not be a useful set of metrics that covers the whole of a black hole spacetime, because some region (e.g. near the singularity) demands a theory that differs from General Relativity. For example, the coupling between gravitation and matter might "de-universalize" near the singularity, with some matter moving differently (in particular not falling inwards) compared to matter that moves on GR's trajectories, perhaps because they couple to an auxiliary gravitational field that is so weak away from the centres of black holes that it's never noticed. (This is one approach taken by people who attempt to build relativistic MOND: when gravity is at its weakest, one gets a strengthening of an auxiliary field). [auxiliary here means in addition to the metric tensor, e.g. a vector field on the left hand (curvature) side of the Einstein Field Equations].

However there is no reason from astronomers to prefer alternative theories over General Relativity. The presence of singularities inside black holes might depend on our present toolkit of exact analytical black hole solutions, which are almost all matter-free (vacuum solutions).

Down that line of thinking is confrontation with the work of Penrose and others that shows that singularities are found pretty generically in 3+1d General Relativity. And that's a topic for another time.

The key parameter is mass.

The no-hair conjecture says that a Kerr-Newman black hole (KN BH: stationary, eternal) has only position, linear momentum, angular momentum, and electromagnetic charge.

3 components of linear momentum and three of position can be removed by keeping the KN BH at the origin of a system of coordinates. The KN BH doesn't evolve with time so we can remove two related time components as well.

This leaves us with 3 components of angular momentum ("spin"), electromagnetic charge (because by definition the KN BH is immersed in an electromagnetic field), and mass.

Schwarzschild BHs are a special case of KN BH where spin and electromagnetic charge vanish.

But we could add other fields with charges, and make those charges more complicated thanks to interactions among the matter fields. That's not a Kerr-Newman black hole any more, though. In practice the other standard model fields don't really make much difference: the charges will tend to neutralize before gravitation is relevant, and won't build up around the BH itself. KN BHs are in that sense electromagnetically quasi-neutral.

Things falling into a KN BH cause a perturbation that decays quickly away, changing the mass, and possibly spin, of the BH held at the coordinate origin. A change in electromagnetic charge will probably "reach out" and capture a charged particle in order to neutralize. Electromagnetic attraction is much stronger than gravitational attraction, while also being long range. However neutralization is not necessarily instant (the closest proton might be several light-minutes away, for example) or completely matched (oops, two protons are electromagnetically nudged into an infalling trajectory in response to the small negative charge, so when the second one arrives there'll be a small positive charge for a bit until a further electron is pulled in...). So quasi-neutral.

These parameters are only about the horizon. It says nothing about whether the mass was a bunch of individual protons and electrons or a bunch of neutral hydrogen or a bunch of heavier molecules. (The same "says nothing" also exists classically without reference to particles: start with a Schwarzschild BH and drop in a uniform spherical shell of mass M or two concentric uniform spherical shells of mass M/2 each, and after some "balding" time we cannot tell whether our increased-mass Schwarzschild BH had one or two shells dropped into it).

More prosaically, "no hair" is a statement about the stability of black holes in the face of small perturbations. If you throw something (a star, a big gravitational wave) into a black hole and wait a bit, does the resulting configuration still look like a black hole in the sense that the trajectories one grinds out of the Kerr-Newman metric (with adjusted mass, spin and charge parameters) accurately represent the trajectories around the new configuration?

There is a lot of literature on black hole stability justifying a "yes" answer.

The mean density of the interior is much much much lower than you think, because the Schwarzschild interior volume is enormously larger than that of a closed Euclidean 3-ball of the Schwarzschild radius.

Some details:

Christodoulou & Rovelli 2014 showed that the maximal interior volume of a Schwarzschild black hole increases with time <https://arxiv.org/abs/1411.2854>. There is plenty of follow-on work in the literature for other varieties of black hole, and it is pretty generic that a black hole gets REALLY BIG inside as it gets older.

I have previously aimed HNers at DiNunno & Matzner 2008 <https://arxiv.org/abs/0801.1734> (this is also good teaching material for thinking about different systems of coordinates on a Schwarzschild BH spacetime).

One might get some intuition by thinking about pouring material into a (non-static) black hole. When do you saturate the black hole? Does it ever fill completely up? How does its outer boundary (the horizon) evolve as you pour more and more material in?

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 ...).

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.

What do you do if some incident halts full trains (possibly depowering them but for things like emergency lighting) near the midpoints of longer sections of 40 degC deep tunnels?

Sure but using a physical pendulum as a frequency standard is unreliable; an unreliable frequency standard is a bad basis for any sort of time-of-travel definition of length.

Many difficulties of using pendulum clocks (and in transporting any sort of chronometer) in real circumstances were also known before the revolution, with French clockmakers competing for the prize money in Britain's Longitude Act 1714 (13 Ann. c. 14) and the ancien regime's various prize offers in the 1740-1770s.

Prior to Harrison's marine chronometers, minimum longitude errors introduced in multi-degree changes of latitudes were indeed on the order of 10% across an oceanic part of a great circle or other more favourable route under cloudy conditions, and sufficient that in the early 18th century it was common for ships to navigate by dead reckoning along a single line of latitude -- a boon to pirates and other enemies, and also often adding many days to the travel time, in an effort to avoid the common problem (eg. HMS Centurion, 1741) of not knowing whether one was west or east of a landmark at a known latitude.

Prominent pre-revolutionary figures also disliked the idea of relying on chronometry for position/length/angle measurements generally -- most notably the excellent geometer and astronomer Pierre Bouguer (after whom the relevant <https://en.wikipedia.org/wiki/Bouguer_anomaly> is named) -- so it's not as if messing up a seconds-pendulum-based definition of a metre (and its consequences for the neat pole-to-equator 1/4 great circle length or mass of a cm^3 of water at STP, both of which now are just approximately round numbers) would have been universally outrageous.

And anyway surely one could consider a solution in which the half-period of the metre pendulum might not be exactly one decimal second. After all, at the time in practice one had to measure across many swings to obtain the effective length with reasonable precision. And Earth's rotation was known to be unstable (Richer, Newton, Maupertuis).

Neither Jupiter nor Saturn is close to thermal equilibrium, whereas the sun is. Bounded self-gravitating gas spheres in thermodynamic equilibrium can show negative specific heats [The classic LBLB, Lynden-Bell & Lynden-Bell, 1977 <https://adsabs.harvard.edu/full/1977MNRAS.181..405L>]. A negative specific heat capacity reducews the gas's volume as its temperature increases. Temperature in stars and gas giants is mostly lowering due to outward radiation driven by internal processes. Unlike a star, Jupiter's specific heat capacity is positive. Very roughly the sun's excess power output will cause it to grow (this handwaves a complex balance of temperature, pressure, mass, and nuclear fusion as it rises in the main-sequence part of the H-R diagram <https://chandra.harvard.edu/graphics/edu/formal/variable_sta...> -- as it climbs in that region with similar temperature the sun gets brighter because it gets bigger), while Jupiter's power output has been higher (presently about 2.5x) than its solar radiation input yet the planet has probably been shrinking.

The energy input and internal heat budgets are under active study for Jupiter <https://www.nature.com/articles/s41467-018-06107-2> (open access), and will supply further evidence for various hypotheses about "primordial Jupiter", one of which is the topic here. One of the major points of comparison with a star here would be how the former is much more like an ideal blackbody than our local gas giants. And of course there is a dark side of Jupiter, while there is no dark side of the sun.

Latitude, mass concentrations, and climate also messed with the half-period/metre ("seconds pendulum") definition; with increasing frequency precision, one would need an almanac, an accelerometer, and probably other tools. Additionally, stabilizing the length of the pendulum under environmental conditions was already known to be tricky, with materials science unable to produce reasonably low thermal-expansion rods prior to the 20th century.

Consequently, the seconds-pendulum/metre relationship gets in the way as one might want to go to sub-millimetre length precision for parts made in different locations or at different times of the day or year. Precision copies of a prototype was more reliable in practice.

(In practice we mostly still generate precise and accurate physical artifacts and make copies from those, it's just that there one can in principle generate such an artifact just about anywhere and anywhen, calibrating with (for example) interferometry <https://iopscience.iop.org/book/edit/978-0-7503-1578-4/chapt...>)

Finally, the Trinity Clock <https://clock.trin.cam.ac.uk/main.php?menu_option=theory> is a neat examination of a well known pendulum clock that's surprisingly accurate (if not really precise; it's been reliably accurate to within two seconds over the course of a month for a very long time, but it's not going to give you a 10MHz sine-wave, and it's not a good for disciplining an oscillator which does so). Do check out the various plots.

Why quarks? There are untold bazillions of those inside each proton, and there's no quark conservation law (rather than conservation of (for example) isospin and strangeness, but only under electromagnetism not under weak interactions, so quark counts get furiously complex in bigger nuclei).

https://profmattstrassler.com/articles-and-posts/largehadron...

For a single proton, though, one always measures (with available measurement technology) a small excess of quarks: two excess up quarks and one excess down quark. That the valence quark model of hadrons works is weird. Who ordered that?

The excess quarks are not "the same" quarks every time you probe your carefully selected and isolated and cold sample proton. Indeed, today's valence quarks in your pet proton are not guaranteed to exist tomorrow, even if the proton stays trapped -- particle creation and annihilation are furious inside, and there are all sorts of other disturbances of quarks that go on in there.

Why atoms? While much calmer, there's still plenty of crazy stuff happening in atoms -- even a neutral hydrogen atom has a bunch of photons and positrons and excess electrons floating around "inside", with an energy fraction proportional to the fine structure constant and with no guarantees that they were there yesterday. Is it the "same" atom at that level? Also, for most of the hydrogen in an exhalation, it probably will be in and out of various electron-swapping configurations over the years. Water gets pretty crazy with its ions, for example.

Hawking 1974, "Black hole explosions?" (Nature paywall) <https://www.nature.com/articles/248030a0> (sci-hub antipaywall) <https://sci-hub.se/https://www.nature.com/articles/248030a0> ("... the time dependence of the metric during the collapse will cause a certain amount of mixing of positive and negative frequencies [...] Part of this wave will be scattered by the curvature of the static Schwarzschild solution outside the black hole ... Another part of the wave will propagate backwards into the star") was certainly not for science journalists and is not at all "far-fetched". The only missing piece here is possibly discussion of his particular approach to second-quantization of the matter fields and how one might interpret that; Hawking's peers fifty years ago did not need assistance there.

This was followed in detail by Hawking 1975 (open access via Project Euclid) <https://projecteuclid.org/journals/communications-in-mathema...> where the most relevant bit surrounds eqns (2.27-28) and there is nothing wrong with talking about signed "contributions to the probability flux into the collapsing body" (as opposed to identifying those contributions as particles).

Hawking himself wrote simpler pop-sci explanations in a number of places, including his book, where a reader would not be expected to understand how canonicalizations of wave functions in other than position space and directly-measured particles differ. However none of those works is the equivalent of the brief "explosions?" letter and its follow-on.

> accelerating observer

Note that this is proper acceleration which is measurable locally with e.g. a weighted spring, contact with a piezoelectric scale, or any other sort of accelerometer apparatus.

> a black hole reuires an accelerated observer to not be pulled in

There are an infinite number of free-fall trajectories outside a black hole, most of which never go near the black hole in the first place. There are also infinite numbers of hyperbolic orbits which "graze" a black hole and an infinity of various elliptical and (quasi-)circular orbits.

An accelerometer on one of these trajectories will report zero proper acceleration. Yet none of these trajectories cross the horizon from the outside.

There are additionally free-fall trajectories which cross the black hole horizon from the outside. Who knows what happens not long after that: flat-space physics like the Standard Model of Particle Physics curved spacetime physics like General Relativity give conflicting answers.

Finally there are also an infinity of trajectories which are somewhere properly accelerated. Most of these won't cross the horizon, but some can: one can turn on one's rocket engines and carefully steer a course that crosses the horizon.

(There are multiple types of horizon; the interesting one is the apparent horizon which can be measured locally with various types of apparatus. An event horizon -- if there is one -- can only be determined with reference to the entire global spacetime and all its contents from infinite past to infinite future. There's a variety of other horizons too. Visser catalogued some of them in <https://journals.aps.org/prd/abstract/10.1103/PhysRevD.90.12...>).

Note that trajectories in my paragraphs above correspond to everywhere non-spacelike curves of all sorts. Free-falling trajectories are geodesics (timelike or lightlike) and one can grind out the geodesic equation for the central mass in a Schwarzschild spacetime, for example.

Finally, proper acceleration is difficult to maintain for long, let along perpetually. (Although one can stand on the surface of a rocky planet and with an accelerometer measure over a very very very long term practically constant acceleration, and might for practical reasons want to assign that approximately constant acceleration to e.g. little "g"). So most curves where there is some proper acceleration also have some free-falling segments eventually.

One consequence is that a traveller who undergoes different accelerations (including none -- free-fall) along its path through spacetime will count different numbers of particles at different points along the traveller's sometimes-geodesic/sometimes-accelerated timelike curve.

A black hole doesn't really change this other than that the formation of some types of horizon induces an acceleration between freely-falling observers before the horizon forms and observers after. The later freely-falling observers see more particles than the earlier ones, and they bunch up not very far (but not preciesly on) the appropriate horizon.

The Unruh effect introduces an Unruh horizon attached to an observer undergoing proper acceleration. That observer sees more particles when it is accelerating than when it is/was freely-falling.

Some of the radiation is massless or has so little mass that it flies out to infinity. Around some types of horizon near a central mass, we'd call that Hawking radiation (as opposed to e.g. Unruh radiation, or radiation associated with e.g. a cosmic horizon in an expanding universe). And as Baez notes, over lonnnnnnnnnng times one would want to take into consideration how the central mass evolves as these particles fly away. Over much shorter durations (fuel is limited!) one would want to take into consideration how Unruh radiation and proper acceleration co-evolve.

The idea in the paper (which Baez dismantles in the linked blog entry at the top) is that no type of horizon is necessary for an apparently particle-free vacuum to look like a particle-rich patch of spacetime. That's a pretty wild claim. Why don't electrons evaporate on their own, in that case?

> See https://www.youtube.com/watch?v=XRr1kaXKBsU for the video where he does it.

I'd be grateful if you take me to the time stamp, because in a casual watch of the video there was nothing like your:

  Gravity pulls things in by causing space-time
  to accelerate in a particular direction. In
  other words we accelerate towards the Earth
  at 9.8 meters per second per second
  because that is what space-time itself does.

The closest thing I noticed was just after the 10 minute mark, where he points at Christoffel symbols and essentially says that you can choose a set of accelerated coordinates such that to remain at the spatial origin you have to undergo proper acceleration. "Your acceleration must be equal to this curvature term ... in curved spacetime you have to accelerate just to stand still". Which is totally fine, and even finer if he made it clear that you are "standing still" at some spatial coordinate after an arbitrary splitting of spacetime into space + time. But I don't know how or if those approx 90 seconds connect to what I quoted from you above.

(Even finer still if he removed the coordinates and completed the equation: see e.g. slide 20/50 at https://slideplayer.com/slide/12694784/ - his whiteboard is the term labelled in cyan, adapted. approx 10m45s "You don't have to worry about the details here. The point is...". The point is that I'm not his intended audience, and his presentation is fine enough, so there's no good reason for me to take you up on your suggestion to contact him.)

> At what point in the orbit does it not work as a description of what's going on locally where the orbiting body is?

I don't know that it doesn't work - it's just that to me it's such an odd way of putting it that the consequences of your "describing it that way" are unclear to me. An obvious probe is solving for an eccentric orbit.

Shoot a slower timelike observer on a ~secant line hyperbolic trajectory across the quasicircular orbit, comparing proper-time-series accelerometer and chronometer logs from their first kiss before the latter's periastron to their last kiss after. How does your "accelerated spacetime" vary by position and initial velocity? How does it work as we take v->c?

How do you propose to measure the ultralocal Riemann curvature? I agree with you that it can be there in an exact solution, but I don't really agree that it's necessarily physical (there are many many exact solutions which aren't, for starters). Even with a broader definition of local, a Synge 5-point like process is not going to find a failure of parallel transport in a sufficiently small region. Would you be satisfied with "effectively flat" (in an EFT sense) or even "FAPP flat"?

> You need to be near huge, dense masses, or on/in them to see LIV violations

A sufficient LLI violation in a compact object is likely to lead to a difference in pressure/contact line broadening, thanks to a modified dispersion relation. Ok, there's optical depth issues there, but looking at metal-rich WDs is a start (it gets you to your "or on...them", at least). Neutrino fluxes probably carry some Lorentz-symmetry-related information from multmessenger events too.

Additionally, binaries and multiples might show various equivalence breakdowns if there are LLI violations, with enhanced ellipticities or periastron precessions (by altering orbital polarization and spin precession parameters in the PPN).

But also there are plenty of theories which slightly violate local Lorentz-invariance in the Newtonian limit, bulding up over distances, and PTA data and GRB data are already constraining those.

> I don't know enough about physics

I'll try to keep this understandable, but can expand or ELI5 bits of it if that would help you.

Physically, local flatness is a statement about the local validity of Special Relativity. Practically, a failure of the local validity of Special Relativty -- a Local Lorentz Invariance violation (often abbreviated LLI violation or local LIV or local LV) -- would be apparent in stellar physics and the spectral lines of white dwarfs and neutron stars and close binaries of them. Certainly we haven't been able to generate local LIV in our highest-energy particle smashers, so the Lorentz group being built into the Standard Model is on pretty safe footing.

(For example, we need tests of Special Relativity -- and notably those of the Standard Model, which bakes in the group theory of Special Relativity -- to work for material bodies in free-fall, even if that free-fall is an elliptical path around and close to a massive object. That's everything from our atomic-clock navigation satellites to gas clouds and stars near our galaxy's central black hole or distant quasars.)

It wasn't a piece of math, which would involve writing out an Einstein-Cartan or Palatini action that let one break out the local Lorentz transformations and diffeomorphisms into a mathematical statement, as one can find in modern (particularly post-Ashtekar in the late 1980s) advanced graduate textbooks. Nobody wants that scribbled out in pseudo-LaTeX here on HN. :-)

Here is an interesting and very slightly contrarian (they do arrive at Theorem 1: it and most of the following text explaining it is beautifully stated orthodoxy -- and note Corollary 4) view by a pair of philosophers of mathematics (they both have also done physics, they are not cranks) at <https://philosophyofphysics.lse.ac.uk/articles/10.31389/pop....> (their rather orthodox part 2 is at <https://philosophyofphysics.lse.ac.uk/articles/10.31389/pop....>).

The choice quote from part 1: "[our] final interpretation says that every spacetime is locally approximately flat in the sense that near any point of any spacetime (or near sufficiently small segments of a curve), there exists a flat metric that coincides with the spacetime metric to first order at that point (or on that curve) and approximates it arbitrarily well," [emphasis mine].

You might prefer to emphasise "approximately" in that quote, but the approximation is much better than that of, say, a square millimetre of your floor.

Next, from a historical perspective: General Relativity was built with making gravitation Special-Relativistic, following Poincaré's 1905 argument about the finite-speed propagation of the gravitational interaction. Einstein (and others) had several false starts marrying gravitation and Special Relativity in various ways before ultimately arriving at spacetime curvature. (At that point, in the 1920s, one finally had the vocabularly to describe Special Relativity's Minkowski spacetime as flat; the Lorentz group theory came later). But making sure Special Relativity didn't break on around Earth -- where it had been tested aggressively for two decades -- was terribly important to Einstein. Additionally, he did not want to break what Newtonian gravitation got right. The mathematics follow somewhat from this compatibility approach where Newtonian gravitation and Special Relativity are correct in the limit where masses are moving very slowly compared to the speed of light and are not compact like white dwarfs or denser objects.

The regions in which there is no hope in many many human lifetimes for finding a deviation from Local Lorentz Invariance are huge (there are interplanetary tests with space probes in our solar system, and interstellar tests using pulsar timing arrays), even if General Relativity turns out to be slightly wrong. This is an area which invites frequent experimental investigation: <https://duckduckgo.com/?t=ffab&q=local%20lorentz%20invarianc...>.

Finally, it is precisely your intuition that big curvature must be built up from small curvature that is the point of investigating local LIV. So far, and to great precision, those intuitions are wrong. Nature builds up impressive spacetime curvature (e.g. in white dwarfs and neutron stars) without showing any signs of softening the local validity of Special Relativity (i.e., the interactions of matter within those compact stars). And that's part of why quantum gravitation is nowhere near decided.

> Gravity pulls things in by causing space-time to accelerate in a particular direction.

Ok, how does this sketch work for a low-ellipticity eccentric orbit?

> "The space-time that is in our frame of reference"

Isn't throwing out general covariance (and manifold insubstantivalism) rather a high price for a simplification of Einsteinian gravitation?

> the Big Bang wasn't a flat space-time

Sure, it's a set of events in a region of the whole spacetime. If we take "Big Bang" colloquially enough to include the inflationary epoch, always assuming GR is correct, then at every point in that "Big Bang" region of the whole spacetime there is a small patch -- a subregion -- of exactly flat spacetime. However, these small patches must be small because most choices of initially-close pairs of test objects can only couple to timelike curves that wildly spread in one direction (and focus in the other).

I don't know how to understand your two final sentences: how do you connect the period just before the end of inflation and the expansion history during the radiation and matter epochs?

tl;dr: drop the use of tensors in the wording and the slogan works better.

This sort of "classic" line only applies to a 3+1 slicing or 1+3 threading of the fully-solved Einstein Field Equations, and only in a certain limit.

A complete solution of the Einstein Field Equations relates the field-values (the curvature and matter tensors) at every point in the manifold, and therefore there's no moving or bending to speak of in the so-called block universe - there's just tensor-values at every point, and the tensor-values of each tensor have a strict relationship.

The choice of any curve along which to slice is totally arbitrary (it doesn't have to be a timelike geodesic, even!), although it certainly helps if for every slice there is no everywhere-non-spacelike curve which passes through the slice more than once, and (where the full solution allows) every point in the manifold is in exactly one slice.

Once you've sliced spacetime, you can consider the evolution of a field quantity from one slice to its neighbours - this is 3+1 general relativity (3-dimensional spacetime-filling fields exist on every spatial (or really "constant time coordinate") slice, the slices ordered by some choice of time axis & given coordinates: this is called a foliation), of which there are some formalisms. Taken along an infinite timelike geodesic, this approach gives you quantities that "look like" 3-d objects evolving in time, moving from point to point in spacetime, and interacting (feeling and generating) the gravitational interaction.

Threading is similar, and is often done in a 1+3 formalism, but decomposes the tensors in terms of the "radar distance" among sets of initially parallel and initially close-to-each-other curves. Threading is a lot less famous; Landau & Lifshitz's Classical theory of fields goes into some detail. For the purposes of this HN comment, threading also lets one see the evolution of 3-d objects' trajectories as they interact with each other, even though it is just applying "radar coordinates" among a set of curves traced through the fully solved block universe. (Arguably one can trace actual physical objects that are coupled to these curves, and the physical objects could beam light to one another, so it could be a little less arbitrary than selecting one curve (with an object on it or not) as the basis for of a foliation, but then one can get into ratholes involving real interacting matter fields where local interactions can delay "radar" signals... so one probably wants to add back in some arbitrariness by e.g. gauge fixing).

Perplexingly humans do not normally think of themselves or their parts (like those carbon atoms you ingested at lunch the other day but just exhaled) coupled to curves through a 4-d manifold, and struggle with the consequences of taking constant-time slices based on e.g. their wristwatches (c.f. practically any HN discussion dealing with recent detections of astronomical events, especially things like stellar deaths).

Of course also perplexingly humans do not remember their future as well as their past (although they're honestly not very good at remembering their past in detail, and tend to reconstruct it seemingly similarly to how they predict their future).

Finally I guess you could think of the decomposition of the tensors in a 3+1 or 1+3 as producing other tensors, just like you could think of scalar or vector quantities being tensor quantities, but then they are not the metric tensor and the stress-energy tensor of the Einstein Field Equations.

Also of course there are other tensors and scalars on the LHS of the EFEs, and they inevitably enter into equations of motion (dissipation from tides raised on partners in relativistic few-body relativistic systems explain certain orbital characteristics!), so the metric and its encoding of changes in angles and displacements isn't the full story in the slogan anyway.

That's the problem with slogans. And there are rather a lot of them that get thrown about wrt Einsteinian gravitation.

P.S.: I brought up threading (which few working scientists will ever really run into) principally because presently the top comment in this discussion essentially wonders -- somewhat off topically -- how the expansion of the universe could happen instead of everything collapsing into black holes. I was tempted to answer that in terms of the Raychaudhuri equations (two pre-Standard-Model particles of matter initially close together are taken away from each other by e-folds of inflation, and other factors (angular momentum and inelasticities from phase changes / symmetry breaking that gives rise to electrons, photons, protons, and so forth) also tend to keep their trajectories from ever "focusing"). However, I didn't have the time or energy to ELI5 it and scrolled along until deciding to think about your unusual wording of a usual slogan that I have repeated myself.

Sundials are really really cool, and the shadow sweeps in the clockwise direction at mid-latitudes in the northern hemisphere, but

https://en.wikipedia.org/wiki/Sundial#In_the_Southern_Hemisp...

So "clockwise" is because the Earth's axis of rotation about itself is not too far from perpendicular to its orbital plane of rotation around the sun, and because the Earth rotates counter-clockwise from the perspective of someone hovering above the planet's north pole. This is the right-hand rule.

Earth's axis of rotation (and orbital plane) is fairly well aligned with most of the round bodies in the solar system, except notably for oddball tilted Uranus and anti-aligned Venus (and the pretty different orbital planes of the minor planets like Pluto and Eris).

The sun's north pole (following the right-hand rule) and south pole do not point to anywhere particular in the galaxy, and its axis of rotation is highly tilted with respect to the axis through the central bulge of the Milky Way. A nice diagram that seems to have originated from the European Southern Observatory: <https://www.physicsforums.com/attachments/motion-of-earth-an...>

Most disc-like structures we've found tend to be randomly oriented compared to each respective host galaxy's polar axis.

For spiral galaxy rotations we have much better data from mostly edge-on views because we can measure the doppler shifts of molecular clouds at their margins; the advancing side will be less red-shifted than the trailing side. Spectral lines also broaden with the magnitude of rotation. AFAIK there is nothing at all unexpected about the distribution of trailing vs leading edges on our sky; the mystery is in the magnitude of the rotation of these outer gas clouds compared to things like their galaxy's apparent optical brightness or other markers of mass.

Also fun is that for elliptical galaxies, these gas clouds don't rotate around the equatorial bulge of those that have them. They instead tend to move mostly radially deeper and shallower within their host galaxy.

So if galaxies don't choose the rotation of their internal components like star systems or radio-loud objects like pulsars and relativistic-jet-equipped black holes, why (accepting for the sake of this argument that this garbage paper is correct about there being a bias in face-on spiral galaxies) would a distant galaxy affect us more than our own?

When two black holes collide, gravitational radiation shoots out into space. The origin of the radiation is in the dynamical spacetime outside each black hole's horizon, however. This is what the gravitational wave detectors operated by LIGO, Virgo, KAGRA, and others look for.

Similarly, the dynamical spacetime around a black hole not near any other black hole can couple with quantum fields -- even fields in a no-particle "vacuum" state as measured by an observer, for example one in orbit around the black hole -- with the result that Hawking radiation is produced.

Both gravitational radiation and Hawking radiation carry away energy (in the sense of ability to do work, per the "sticky bead" argument) from the environment immediately around a black hole. This in turn means that the horizon radius will be less than it could be.

So as a Hawking-radiating isolated black hole will tend to shrink (if it's not fed by hotter cosmic microwave background radiation, for example), the mass of a post-merger binary black hole will be less than the sum of the unmerged binary.

Just because things can't cross from the inside of a black hole horizon to the outside doesn't mean the horizon is always the same -- the horizon can grow and shrink dynamically when interacting with other self-gravitating bodies, with matter like dust or starlight, or with "the quantum vacuum".

In most mathematical circles, Bolyai-Lobachevsky geometry is strictly a (family of non-Euclidian) hyperbolic geometry.

János Bolyai, not Bolshai. And certainly in the 1820s-1830s he investigated the Euclidean parallel postulate, arriving at a hyperbolic geometry in which it does not hold true (i.e., initially close parallel lines diverge), and eventually studying geometries which take no position on the parallel postulate. Lobachevsky also independently arrived at a hyperbolic geometry, and continued develop a substitute postulate for the parallel postulate.

However, all of the above is several long steps before developing differential geometry with its inner products encoding angles and distances on the tangent spaces at each point on an arbitrarily curved smooth manifold of higher dimensions, and an even longer one from the pseudo-Riemannian manifold of 3+1d General Relativity. Although many hands contributed to the positive inner product -> non-degenerate inner product, the reason anyone was doing that was because of Einstein's work on gravitation (in turn provoked in part by Poincaré's 1905 argument about Lorentz-invariance of the wave equation for gravitation, in the spirit of Special Relativity).