State of "matter" of a singularity

[joigus]

13 hours ago, Mitcher said:

My concern there was about previous comment concerning a "density of some stuff". Properties of ST are mainly properties of mass. Most simplest models of BH, with their "incompressible sphere of matter" or 'point-mass' do not take matter itself into account, only the effects it creates on SP and fields. BH is about matter and energy content: no matter, no BH, hence the density towards their center should probably have something to do with Planck's density, at least in theory since my point here is that matter-energy in its known states cannot realistically exist past the horizon. And central singularities even less as it can be cancelled according to some theories.. I read and assimilated past comments though.

OK. Let me tell you you've been raising some interesting points here, and you're considerably better informed than the average occasional "speculative people" that come and go around here. I'm sorry that I was a bit grumpy some posts above.

GR is not at all about "mass." It's not even about "energy" really. If you stick to your guns in this way, you're going to get very confused. IMO --and I make room for other member's disagreement here--, it's about curvature, perhaps torsion, and the weirdest elements of all: Diffeomorphism invariance (smooth distorsions of the coordinates that preserve the essential information coded in the metric.) It's also about singularities and horizons.

Horizons are a peculiarity of geometries with non-definite metric. Trivial observation, but important to keep in mind. Horizons appear whenever there is a singularity in your theory.

They always hide a singularity behind.

Horizons always have an entropy associated with them.

Horizons always appear when we try global coordinates that are assymptotically or locally flat.

I'm a firm believer in Dirac's motto: The equation knows best. What did Dirac mean? I think he meant something like this: Once you have the right equation, it's never gonna lie to you. If it gives you a headache, don't despair, because it's a symptom and a clue that your initial intentions were corrupted in some way. So it's a priceless instrument in pointing to the limitations of your initial assumptions.

Mass arises in GR because we insist in obtaining a set of coordinates that is closest to solving ST globally, is locally or assymptotically flat, and on top of that, is a solution to Einstein's vacuum field equations. Now, isn't that perhaps asking too much? Not totally happy with that, we even demand that the solution be spherically symmetric.

So the equations behave as best they can. They reach a compromise. They give you back this parameter, M, which is totally meaningless in any real situation, but encapsulate all these properties in one parameter.

When you stick to the overly-demanding program that I've just described, you end up with more than you initially bargained for, because the theory gives you things that are completely extraneous to your initial dream (describing everything in terms of pure geometry.) You're stuck with this "mass," horizons, and entropy.

What does entropy tell you (always, always,...) in physics?: That you're missing something; that there are some variables that, quite simply, are not in your description. That you want to describe your phenomenology in terms of, say, (A,B,C,...), but there are other things, say, (p,q,r,...) that are not in your equations, but some fundamental principle operates in such a way as to impose constrictions on them.

I've left out vacuum energy, not because I think it's unimportant, but because it requires another discussion of itself.

 

1 hour ago, studiot said:

Lovely and simple that even my dog can understand it. +1

Agreed.

51 minutes ago, joigus said:

I'm a firm believer in Dirac's motto: The equation knows best.

Apparently it was Heisenberg who said this. Opps.

Quote

Whatever it is, it will be hard to believe without putting faith in theory and in Heisenberg's wistful dictum: ''The equation knows best.''

James Gleick, Genius; the Life and Science of Richard Feynman.

Edited September 5, 2022 by joigus
minor correction

[Mitcher]

18 hours ago, MigL said:

You don't seem to realize that you've just given an example of a co-ordinate singularity.

If you use latitude and longitude as your co-ordinate system, the North Pole and South Pole are singular. You cannot go any further North ( or South ) from those points, yet there is no 'edge', and nothing stops you from going further ( around the globe ).

That is the case with the Schwarzschild solution; a co-ordinate singularity ( at the horizon ), but nothing stops you from going further. It is not an 'edge' either.

I gave the pole exemple for the central singularity, not the one at the EH. Where did I say it is not possible to go around ?

11 hours ago, studiot said:

Lovely and simple that even my dog can understand it. +1

But please note that whilst all lines of longitude are geodesics, the equator is the only line of latitude that is one.
So lat andm long constitute a mixed coordinate system, not a pure geodesic one.

@Mitcher

This reminds me of another mathematical oddity.

The Hairy Ball Theorem and the Mermin-Ho Theorem and their relation to singularities and Index Theory.

https://en.wikipedia.org/wiki/Hairy_ball_theorem

https://www.parabola.unsw.edu.au/files/articles/2000-2009/volume-43-2007/issue-2/vol43_no2_3.pdf

http://www.maths.adelaide.edu.au/hang.wang/resources/What-is-Index-theory-1.pdf

https://physics.stackexchange.com/questions/656025/how-does-the-mermin-ho-theorem-handle-singularities-arising-from-the-hairy-ball

 

 

 

These are hairy subjects indeed, thanks, will take me awhile to go trough it. It reminds me of a question I asked myself long ago, I might need to open a new thread but to briefly summarize it was about whether a solid sphere was made to slowly rotate simultaneously around its 3 orthogonal axes. What would be the trajectories of points on its surface then, or would there be fixed points somewhere (at faster rates the moments of inertia would probably create uncontrolable vibrations i guess) but by using particular ratios between the 3 rotation rates maybe some combinations would result in all points of the surface to follow geodesics simultaneously (in relation to coordinates on an external fixed sphere of course).

[Mitcher]

10 hours ago, joigus said:

I'm sorry that I was a bit grumpy some posts above.

GR is not at all about "mass.

No problem, thanks for your time and detailed explanations here.  I was specifically speaking about matter and BH more than mass and GR though. Some scientists are studying wormhole structure trough the observation of nanotubes, which is fine due to the impossibility to observe real BH but they will have no real matter in their observations, contrarily to the teams at the Large Hadron collider working on real stuff at least. To me the Horizons are not only a virtual concept, there really exists a limit under which light will not escape, we cannot escape that but any theory which concludes with an inescapable central singularity is a doomed dead-end. I'am looking for theories that avoid it.

12 hours ago, studiot said:

the Mermin-Ho Theorem and their relation to singularities and Index Theory.

the Mermin Ho theorem posits singularities will vanish but i cannot comprehend it.

[MigL]

1 hour ago, Mitcher said:

but any theory which concludes with an inescapable central singularity is a doomed dead-end. I'am looking for theories that avoid it.

Not necessarily.
There are many instances in modern Physics where theories become intractable and produce infinities, or diverge, in certain 'out-of-bounds' applications.
Before the 'modification' of re-normalization, Quantum Electrodynamics was producing all sorts of 'close range' infinities.

See here      Renormalization - Wikipedia

Should QED, and its extremely accurate results/predictions, have been discarded, and the search begun for alternate theories/treatments ?

[studiot]

3 hours ago, Mitcher said:

These are hairy subjects indeed, thanks, will take me awhile to go trough it. It reminds me of a question I asked myself long ago, I might need to open a new thread but to briefly summarize it was about whether a solid sphere was made to slowly rotate simultaneously around its 3 orthogonal axes. What would be the trajectories of points on its surface then, or would there be fixed points somewhere (at faster rates the moments of inertia would probably create uncontrolable vibrations i guess) but by using particular ratios between the 3 rotation rates maybe some combinations would result in all points of the surface to follow geodesics simultaneously (in relation to coordinates on an external fixed sphere of course).

You need Euler's equations of motion for this.

https://en.wikipedia.org/wiki/Euler's_equations_(rigid_body_dynamics)

For any shape other than a perfect sphere rotation about one of the three axes will lead to a chaotic motion.

[studiot]

11 hours ago, studiot said:

You need Euler's equations of motion for this.

https://en.wikipedia.org/wiki/Euler's_equations_(rigid_body_dynamics)

For any shape other than a perfect sphere rotation about one of the three axes will lead to a chaotic motion.

Wiki has another more detailed article on rotations

 

https://en.wikipedia.org/wiki/Rotation_formalisms_in_three_dimensions

[joigus]

14 hours ago, Mitcher said:

To me the Horizons are not only a virtual concept, there really exists a limit under which light will not escape, we cannot escape that but any theory which concludes with an inescapable central singularity is a doomed dead-end. I'am looking for theories that avoid it.

I agree. I'm not implying that horizons are not real. I'm implying that choosing a coordinate chart in which the horizon is manifest will not necessarily tell you anything useful about the global aspects of ST, or some other aspects. Eg, coordinates that are locally Minkowkian along the trajectory of a falling observer, tell you nothing about the horizon. For all he knows, his signals are reaching the totality of space behind him. A falling observer through the horizon of a BH that's big enough that tidal forces are small, will notice nothing peculiar; will see no horizon.

This question of local charts is deeper than it looks.

It's even possible that the question is more general than it looks...

[Mitcher]

On 9/6/2022 at 12:29 AM, MigL said:

Not necessarily.
There are many instances in modern Physics where theories become intractable and produce infinities, or diverge, in certain 'out-of-bounds' applications.
Before the 'modification' of re-normalization, Quantum Electrodynamics was producing all sorts of 'close range' infinities.

See here      Renormalization - Wikipedia

Should QED, and its extremely accurate results/predictions, have been discarded, and the search begun for alternate theories/treatments ?

A singularity is created inside a BH if one insist to have space and time to trade place, but these do not necessarily have to. They could become undifferentiated, or even merge with matter itself into a new dimensional entity. Even this odd hypothesis would be less strange than a physical central singularity.

On 9/6/2022 at 1:08 AM, studiot said:

You need Euler's equations of motion for this.

https://en.wikipedia.org/wiki/Euler's_equations_(rigid_body_dynamics)

For any shape other than a perfect sphere rotation about one of the three axes will lead to a chaotic motion.

Euler's equations is only about a main axis of rotation isn't ? I never found anything about more than one axis. And yes, in regard to a perfect sphere of course, it would already be sufficiently complicated like that.

[MigL]

30 minutes ago, Mitcher said:

A singularity is created inside a BH if one insist to have space and time to trade place

Space and time d not trade places. This is a misconception due to the re-orientation of light cones in strongly curved space-time.

Rather, all geodesics, paths through space-time, within the event horizon, lead to the central singularity; and none extend past that point.

[Mitcher]

On 9/6/2022 at 3:13 PM, joigus said:

I agree. I'm not implying that horizons are not real. I'm implying that choosing a coordinate chart in which the horizon is manifest will not necessarily tell you anything useful about the global aspects of ST, or some other aspects. Eg, coordinates that are locally Minkowkian along the trajectory of a falling observer, tell you nothing about the horizon. For all he knows, his signals are reaching the totality of space behind him. A falling observer through the horizon of a BH that's big enough that tidal forces are small, will notice nothing peculiar; will see no horizon.

This question of local charts is deeper than it looks.

It's even possible that the question is more general than it looks...

In regard to the more general question I noted that the linear density (kg/m) or temporal density (kg/sec) of a BH is a constant, furthermore it seems to be very close for the Universe itself and itt could well be exactly the same, meaning that in terms of linear density the Universe would be indiscernible from any BH, the interface between them and the Universe would not exist. It would not have been the case in the past if the expansion theory holds true.

3 minutes ago, MigL said:

Space and time d not trade places. This is a misconception due to the re-orientation of light cones in strongly curved space-time.

Rather, all geodesics, paths through space-time, within the event horizon, lead to the central singularity; and none extend past that point.

Yes, I understand, but surely it cannot be right. Something is wrong, or else time itself and the speed of light have to stop.

On 9/6/2022 at 3:13 PM, joigus said:

For all he knows, his signals are reaching the totality of space behind him.

But if he sends a light signal to a mirror attached to a pole behind him (above him), surely at some point the signal will cease entirely to come back, even if he shortens the pole more and more. At that point everything will stop being functionnal inside the starship.

[MigL]

13 minutes ago, Mitcher said:

Yes, I understand, but surely it cannot be right. Something is wrong, or else time itself and the speed of light have to stop.

Agred. 
And that's what makes the central singularity unphysical.
We obviously need something better suited to describing hi-energy/lo-separation behaviour of collapsed matter than GR is capable of.

[joigus]

1 hour ago, Mitcher said:

In regard to the more general question I noted that the linear density (kg/m) or temporal density (kg/sec) of a BH is a constant, furthermore it seems to be very close for the Universe itself and itt could well be exactly the same, meaning that in terms of linear density the Universe would be indiscernible from any BH, the interface between them and the Universe would not exist. It would not have been the case in the past if the expansion theory holds true.

I'm not sure what you mean here. The linear density?

I'm not familiar with the concept of temporal density either. I'm having problems understanding "the interface between them (BH's?) and the universe." I'm not saying it's meaningless; I'm saying I don't know what it means.

1 hour ago, Mitcher said:

But if he sends a light signal to a mirror attached to a pole behind him (above him), surely at some point the signal will cease entirely to come back, even if he shortens the pole more and more. At that point everything will stop being functionnal inside the starship.

I think I understand what you mean here. This is similar to questions about how a man sees his own body as he crosses the horizon -if he enters feet first, he wouldn't be able to see his own feet.

My best guess is that he would see the mirror as a shape obstructing the light from behind, but not reflecting his own image. So it wouldn't be able to work as a mirror for him. But again, for all he knows, the space around him would "feel" quite normal if the BH is big enough that tidal forces are negligible.

[studiot]

4 hours ago, Mitcher said:

Euler's equations is only about a main axis of rotation isn't ? I never found anything about more than one axis. And yes, in regard to a perfect sphere of course, it would already be sufficiently complicated like that.

If you read the wiki article you should note that there are 3 separate angular velocities,   [math]{\omega _1};{\omega _2};{\omega _3}[/math]     about three mutually perpendicular axes.

You can choose any such set of 3 suitable axes, even not perpendicular ones, but the transformation equations normally work the other way because the 3 principal axes of inertia present the formulae in simplest form (diagonal matrices if you work in matrix format).

Of course all that is moot for a perfect sphere because any axis is a principal axis and the others follow by symmetry.

 

[Markus Hanke]

On 9/5/2022 at 8:49 AM, Mitcher said:

On the contrary I keep saying that geodesics are not bounded at the EH.

Ok, so we are in agreement on this point 👍

On 9/5/2022 at 8:49 AM, Mitcher said:

What's the point of those sophisticated geometries since most seem to possess one inside, which it is completely unphysical.

You mean they possess a singularity? Yes, I agree, it’s unphysical. 

When a singularity appears in a model that cannot be removed mathematically, then that is generally taken to imply that the model breaks down at that point. For GR, that means that the final stages of a gravitational collapse are outside its domain of applicability, so it must break down there. This isn’t a surprise, since GR is a purely classical theory - but at the densities and energies encountered during gravitational collapse, quantum effects become important and cannot be ignored. That’s why the theory is doomed to fail there. This is inevitable and quite independent of coordinate choices, and thus can’t be interpreted away. Geodesic incompleteness is just a part of the topology in this spacetime.

As I mentioned earlier, it is in fact possible to remove the singularity by making a small modification to GR, ie by choosing a different connection on the manifold so that you can have torsion in addition to curvature. It’s called Einstein-Cartan gravity. It’s still a purely classical model, but it doesn’t contain singularities in black holes or at the Big Bang. The trouble is that this necessarily implies a modification to the Dirac equation as well, which to date has not been observed in our world. But the effect would be very small under normal circumstances, so it can’t be definitively ruled out either; ECG is still a contender.

7 hours ago, Mitcher said:

Yes, I understand, but surely it cannot be right. Something is wrong

Yes - we are using a classical model in a situation where quantum effects are not negligible. 

On 9/6/2022 at 7:05 AM, Mitcher said:

I'am looking for theories that avoid it.

See above - have a look at Einstein-Cartan gravity.

One other important point: because the Einstein equations only constitute a local constraint on the metric, the space of all mathematical solutions to these equations is much larger than the space of all physically possible spacetimes. In other words, you can always obtain solutions that formally are valid solutions to the Einstein equations, but which don’t correspond to any physically reasonable spacetime. This is why the issue of boundary conditions and initial values is so important.

Furthermore, even if a solution is both mathematically valid and physically reasonable, it may still be topologically ambiguous in a global sense.

GR really is a very subtle thing.

Edited September 11, 2022 by Markus Hanke

[Mitcher]

16 hours ago, MigL said:

This is a misconception due to the re-orientation of light cones in strongly curved space-time.

I also believe that the light-cone concept is wrong, or rather does not suitably describes what's going on in reality. The light path should be orthogonal to the time translation, not slanted 45 degres in relation to it, to start with. Another reason/consequence is that light should not remain in the plane of simultaneity of the observer who has sent the light ray. Light is like a frozen instant from the past and does not syncronizes with the observer any more, it will always lag in relation to him. I'am probably not being clear enough.

15 hours ago, joigus said:

I'm not sure what you mean here. The linear density?

I'm not familiar with the concept of temporal density either. I'm having problems understanding "the interface between them (BH's?) and the universe." I'm not saying it's meaningless; I'm saying I don't know what it means.

It's just a simple concept I invented, divide the BH mass by its radius, it will always be around 10^27 kg/m, either for the Sun or the Universe, or very close to.

12 hours ago, studiot said:

If you read the wiki article you should note that there are 3 separate angular velocities,   ω1;ω2;ω3      about three mutually perpendicular axes.

You can choose any such set of 3 suitable axes, even not perpendicular ones, but the transformation equations normally work the other way because the 3 principal axes of inertia present the formulae in simplest form (diagonal matrices if you work in matrix format).

Of course all that is moot for a perfect sphere because any axis is a principal axis and the others follow by symmetry.

 

I read the wiki article but unable to see if those 3 rotations descriptions could be simultaneous and how to derive the trajectories of points at the suface from them. I just have the premonition that some points on the suface could be static, or probably only rotating. Might be completely wrong.

10 hours ago, Markus Hanke said:

Ok, so we are in agreement on this point 👍

Yes. But depending on the model geodesics might not continue inside the BH either, that was my point.

11 hours ago, Markus Hanke said:

See above - have a look at Einstein-Cartan gravity.

Yes, thanks, that theory seems really attractive (humour). Another one I read is that due to massive Hawking radiation a star would not have the opportunity to collapse beyond the 2m radius. That would really sort all the disputes isn't ?

[studiot]

30 minutes ago, Mitcher said:

I read the wiki article but unable to see if those 3 rotations descriptions could be simultaneous and how to derive the trajectories of points at the suface from them. I just have the premonition that some points on the suface could be static, or probably only rotating. Might be completely wrong.

Agreed it is hard to picture.

Do you understand resolution of vectors into components ?

And that you can equally achieve the same effect as the one vector  by (re)combining those components ?

Rotations are more complicated motions than linear vectors but the idea is the same.

[Mitcher]

15 hours ago, joigus said:

"the interface between them (BH's?) and the universe."

If accordingly interiors of BH and exteriors are separated by a one way surface we clearly have a definite distinction between two parts of the Universe. Smooth ir not the EH constitutes an interface in some way.

6 minutes ago, studiot said:

Agreed it is hard to picture.

Do you understand resolution of vectors into components ?

And that you can equally achieve the same effect as the one vector  by (re)combining those components ?

Rotations are more complicated motions than linear vectors but the idea is the same.

Yes, vector combination is quite basic. So, having sent a sphere into 3 separate rotations would have the same effect as a simple rotation along a single axis ? This is completely counter-intuitive. How a single axis would be selected in relation to the other 3 ones then ?

15 hours ago, joigus said:

My best guess is that he would see the mirror as a shape obstructing the light from behind, but not reflecting his own image. So it wouldn't be able to work as a mirror for him. But again, for all he knows, the space around him would "feel" quite normal if the BH is big enough that tidal forces are negligible.

I agree, your description makes sense but on the same time leads to contradictons. If mirrors cease to be mirrors inside a BH, then also electronic signals stop to be sent back. Everything is made of electronic signals. Not only space and time would not be quite normal but even matter itself. It cannot be both ways. Even in the absence of measurable tidal forces the inside of a BH would be a very strange place indeed.

[joigus]

1 hour ago, Mitcher said:

It's just a simple concept I invented, divide the BH mass by its radius, it will always be around 10^27 kg/m, either for the Sun or the Universe, or very close to.

Of course, because in natural units (G=1, c=1) M = R_{Schwarzschild}. In natural units M/R_{Schwarzschild} = 1 (I think I'm off by a factor of 2. Little wonder it always gives you the same thing. There's nothing mysterious about it.

42 minutes ago, Mitcher said:

I agree, your description makes sense but on the same time leads to contradictons. If mirrors cease to be mirrors inside a BH, then also electronic signals stop to be sent back. Everything is made of electronic signals. Not only space and time would not be quite normal but even matter itself. It cannot be both ways. Even in the absence of measurable tidal forces the inside of a BH would be a very strange place indeed.

Well, not everything would be completely normal. The equivalence principle (EP) does tell you that the laws of physics are locally the same. Placing things at some distance and expecting they're gonna work the same is not whe the EP tells you. Perhaps mirrors ceasing to work as mirrors is another kind of tidal effect. 

Suppose you have very long arms, and extend them while you're crossing the event horizon. I would expect tidal forces to start manifesting themselves at some point. So not everything is the same.

[Mitcher]

1 hour ago, joigus said:

Of course, because in natural units (G=1, c=1) M = R_{Schwarzschild}. In natural units M/R_{Schwarzschild} = 1 (I think I'm off by a factor of 2. Little wonder it always gives you the same thing. There's nothing mysterious about it.

It's not mysterious but it is a physical constant altogether, with precise measurement and valid for BH of any size. If it is also valid for the Universe then BH's are invisible, I mean, linerarly, BH would go undetected, linear density would be constant everywhere whether there are BH's in the line of sight or not. All this theoretical of course.

1 hour ago, joigus said:

Placing things at some distance and expecting they're gonna work the same is not whe the EP tells you.

I agree but when locally and some distance becomes smaller than the subject of study itself then nothing goes. What could be considered local in those conditions ? 100 meters or the size of a nucleus ? The Planck's scale would be my guess.

[studiot]

3 hours ago, Mitcher said:

3 hours ago, studiot said:

Agreed it is hard to picture.

Do you understand resolution of vectors into components ?

And that you can equally achieve the same effect as the one vector  by (re)combining those components ?

Rotations are more complicated motions than linear vectors but the idea is the same.

Yes, vector combination is quite basic. So, having sent a sphere into 3 separate rotations would have the same effect as a simple rotation along a single axis ? This is completely counter-intuitive. How a single axis would be selected in relation to the other 3 ones then ?

Intuition can be a misleading process.

The relevant theorem is the original version of Poinsot's theorem which was originally stated in statics, long before vectors were invented.

"Every system of of forces  can be reduced to a single force and a single moment (torque) , either or both of which can be zero. "

This gives rise the the two laws of statics viz that

The sum of forces acting in equilibrium is zero.

The sum of moments in equilibrium is zero.

 

This can be widened to refer to vectors in more modern parlance and to dynamics where they become

https://en.wikipedia.org/wiki/Poinsot's_ellipsoid

 

The point being that any number of rotations can be compounded to become a single rotation resultant, just as any number of forces can be compounded to form a single resultant or net force.

 

[joigus]

3 hours ago, Mitcher said:

It's not mysterious but it is a physical constant altogether, with precise measurement and valid for BH of any size. If it is also valid for the Universe then BH's are invisible, I mean, linerarly, BH would go undetected, linear density would be constant everywhere whether there are BH's in the line of sight or not. All this theoretical of course.

Yes. It's a physical constant altogether, and let's see what it's equal to:

\[ R_{S}=\frac{2M_{BH}G}{c^{2}} \]

\[ \frac{M_{BH}}{R_{S}}=\frac{M_{BH}}{2M_{BH}G/c^{2}}=\frac{c^{2}}{2G} \]

It's a universal constant just because it's the quotient of two universal constants. 

It happens to have the dimensions of a linear density. That doesn't mean anything in itself.

3 hours ago, Mitcher said:

I agree but when locally and some distance becomes smaller than the subject of study itself then nothing goes. What could be considered local in those conditions ? 100 meters or the size of a nucleus ? The Planck's scale would be my guess.

Tidal effects begin to be important long before you reach Planck's scale. Planck's scale is more about quantum effects. The h-bar is giving it away, isn't it?

Edited September 11, 2022 by joigus
Latex editing

[NTuft]

7 hours ago, Mitcher said:

Yes, vector combination is quite basic. So, having sent a sphere into 3 separate rotations would have the same effect as a simple rotation along a single axis ? This is completely counter-intuitive. How a single axis would be selected in relation to the other 3 ones then ?

4 hours ago, studiot said:

The point being that any number of rotations can be compounded to become a single rotation resultant, just as any number of forces can be compounded to form a single resultant or net force.

Quote

"It must be stressed that the Rodrigues approach to rotations, by emphasizing their multiplication rules and by regarding the entirely as operators, fully reveals the group properties of the set of all proper rotations, the full orthogonal group SO(3), as it is now called. Rodrigues's analysis is astonishingly penetrating. He recognizes that rotations do not commute and then he goes on to define infinitesimal rotations, which commute and from which, he shows, all other rotations can be generated. He thus anticipates the idea of the infinitesimal group generators later fully developed by Sophus Lie. The whole theory of the rotation group is in fact contained in some form in the Rodrigues paper, as noticed by Gray (1980). [...]
Before we leave this subject, and Rodrigues ([ed:paper from]1840), it should be said that this paper goes further than a mere treatment of the rotation group. For Rodrigues is mainly concerned with the Euclidean group, that is the group of all translations and rotations. He shows that translations commute amongst themselves but not with rotations and has the remarkably ingenious idea of realizing translations as infinitesimal rotations around infinitely distant rotation axes."

Altmann, Simon L., Rotations, Quaternions, and Double Groups, Dover Publications, Inc. 1986.

This is speaking about proper vectors, also known as quaternions, whose combinations as I understand are what translate into rotations as in you don't really select an axis, and they are not so basic to work with and so simple complex number "vectors" were used later instead.

[studiot]

11 minutes ago, NTuft said:

This is speaking about proper vectors, also known as quaternions, whose combinations as I understand are what translate into rotations as in you don't really select an axis, and they are not so basic to work with and so simple complex number "vectors" were used later instead.

This is all perfectly true but nothing to do with the question in hand which concerns something which is spinning with angular velocity omega about an axis.
Your quote concerns the difference between rotation through some finite angle and some infinitesimal angle whereas the angle rotated thorough by spin is unbounded.

[studiot]

OK so here is the simplest form of Maths I know for this subject.

I have marked the starting point page 289, article 217  -  Angular velocites of a body about more than one axis  - on the first attachment.

You will need some simple calculus and trig.

 

Edited September 11, 2022 by studiot

[Markus Hanke]

19 hours ago, Mitcher said:

The light path should be orthogonal to the time translation, not slanted 45 degres in relation to it

Imagine you throw a pebble into a pond - what happens? Wave fronts ripple out at finite speed in all directions on the water. Now plot the position of the leading edge of the wave field against a time axis that runs orthogonally to the water surface - you get a cone.

Light cones work in a similar way. They simply depict regions of causality centered on some event.

19 hours ago, Mitcher said:

Yes. But depending on the model geodesics might not continue inside the BH either, that was my point.

Of course they don’t - that’s the meaning of “singularity”. It’s a region of geodesic incompleteness, past which geodesics cannot be extended. 

19 hours ago, Mitcher said:

Yes, thanks, that theory seems really attractive (humour).

Well, it’s the simplest possible modification of GR, and it avoids the singularity issue even in the classical domain. I wouldn’t dismiss this so readily.

19 hours ago, Mitcher said:

That would really sort all the disputes isn't ?

No, it wouldn’t, because the region bounded by the horizon (the presence of which is required to have Hawking radiation in the first place) would still be geodesically incomplete.