If Black holes slowly evaporate over time is there a point where they stop being a black hole?

[Mordred]

7 hours ago, Halc said:

The equations are how everybody knows. No links were provided, so I googled the question and the first 8 hits (NASA, Smithsonian, various you-tubes, reddit) all suggest matter is compressed without bound. Much of this list of bad hits is due to my search terms of "black hole infinite density".

 

so you googled a bunch of answers so tell me google what happened to the mass of the collapsing star then.... can you answer that or did it simply disappear. That is directed at google not you lol . Would you like to see equations that directly relate to the density term in regards to the collapsing star?

https://arxiv.org/pdf/2310.17647

see section 4

 

7 hours ago, Halc said:

 

Yes, that’s right. The vacuum. There is no matter. The density is zero everywhere. The Schwarzschild solution is the simplest, spherically symmetric, static vacuum solution of Einstein’s field equations."

 

If Victor Toth stated that then he only looked at the vacuum solutions of the interior  and ignored the coordinate assigned for the singularity R=0. Not surprising though most articles tend to ignore that part as no one feels the singularity condition should exist. I have another related article that suggests that the BH may simply be a neutron star that has collapsed just beyond its EH but is still present. As mentioned no one knows beyond the EH. So its really anyone's game  until we can find a means of indirect evidence to give us more clues. Gravity waves is a viable possibility others are mentioned here. The article that has some suggestive tests via the accretion disk is here

https://arxiv.org/pdf/1104.5499 

there is also a section covering Hawking radiation. However one will be quite surprised at all the processes matter undergoes in that accretion disk. I recall years ago I asked my instructor " How can you possibly have infinite density or near infinite density" his answer was that although fermions cannot share the same space with the same state that restriction does not apply to bosons. " you can in fact have extremely high density and still be a vacuum solution with incredibly high temperature and density.  Vacuum describes a pressure term.  Its not the energy density. You get the energy density by the equations of state for the particles contained within a region. 

Its one thing that often confuses ppl concerning vacuum. It can be positive or negative and can describe any range of energy density. If you think about that you might consider that a star is simply a condensed matter field it can be equally treated as a field. We know in cosmology we get a similar phenomena due to the BB and its mass density terms.

I didnt have time to properly respond to the quoted section earlier as I was at work. Having time now I can provide a better response.

just to add the singularity at r=0 is a true singularity it cannot be removed by a change in metric choice. The event horizon itself is often described as a singularity condition however that is a coordinate singularity and not a true singularity.

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

"Rotating black holes have surfaces where the metric seems to have apparent singularities; the size and shape of these surfaces depends on the black hole's mass and angular momentum. The outer surface encloses the ergosphere and has a shape similar to a flattened sphere. The inner surface marks the event horizon; objects passing into the interior of this horizon can never again communicate with the world outside that horizon. However, neither surface is a true singularity, since their apparent singularity can be eliminated in a different coordinate system. A similar situation obtains when considering the Schwarzschild metric which also appears to result in a singularity at 𝑟=𝑟s dividing the space above and below r_s into two disconnected patches; using a different coordinate transformation one can then relate the extended external patch to the inner patch (see Schwarzschild metric § Singularities and black holes) – such a coordinate transformation eliminates the apparent singularity where the inner and outer surfaces meet. Objects between these two surfaces must co-rotate with the rotating black hole, as noted above; this feature can in principle be used to extract energy from a rotating black hole, up to its invariant mass energy, Mc²."

I suspect this is what Victor Toth was referring to hope that helps Mathius Balu has an excellent coverage of this

http://www.blau.itp.unibe.ch/newlecturesGR.pdf

He will use the BH to help explain: "artifacts of coordinate choice" I would have to dig through it though to relocate the relevant section

lol and just to add flames to the proverbial fire. A rotating BH has more than one event horizon.....

google is useful but unless your aware of other factors getting good answers can often mislead down the wrong google pathway. For example simply googling BH singularity will more likely than not describe the coordinate singularities regarding the EH rather than the R=0 singularity condition.

This wiki has the relevant detail regarding the Gravitational singularity as the link describes it 

 

The case r = 0 is different, however. If one asks that the solution be valid for all r one runs into a true physical singularity, or gravitational singularity, at the origin. To see that this is a true singularity one must look at quantities that are independent of the choice of coordinates. One such important quantity is the Kretschmann invariant, which is given by

𝑅𝛼𝛽𝛾𝛿𝑅𝛼𝛽𝛾𝛿=12𝑟s2𝑟6=48𝐺2𝑀2𝑐4𝑟6.

At r = 0 the curvature becomes infinite, indicating the presence of a singularity. At this point the metric cannot be extended in a smooth manner (the Kretschmann invariant involves second derivatives of the metric), spacetime itself is then no longer well-defined. Furthermore, Sbierski^[21] showed the metric cannot be extended even in a continuous manner. For a long time it was thought that such a solution was non-physical. However, a greater understanding of general relativity led to the realization that such singularities were a generic feature of the theory and not just an exotic special case

https://en.wikipedia.org/wiki/Schwarzschild_metric#Singularities_and_black_holes

 

 

 

Edited May 4 by Mordred

[Halc]

1 hour ago, Mordred said:

so you googled a bunch of answers so tell me google what happened to the mass of the collapsing star then.... can you answer that or did it simply disappear. That is directed at google not you lol . Would you like to see equations that directly relate to the density term in regards to the collapsing star?

The collapsing star is a dynamic state, and quite different from a free-falling object. There's dense stuff, and then not even Pauli-exclusion can support it.  OK, so there's even more squish, at least at first. A spatial dimension rotates and is replaced by time, and that time dimension is bounded. What was the time dimension rotates out to a spatial one, one with nearly infinite extension at that. Lots of new room to spread out, but the causal light cones don't allow arbitrary travel down this big space, so I cannot say the compression ends.

An example of the space available inside a black hole, Sgr-A and the black hole at the center of Andromeda share a common singularity. They're the same black hole, a region of 4D spacetime bounded by a 3D event horizon hypersurface, and in that case, the same (connected) hypersurface. It's only in a slice of coordinate space at a given time that the one object has multiple cross sections, manifesting as a pair of black holes to us, for now.

None of the above is particularly an answer to the question of if there is compressed matter in an established black hole. In coordinate time, yes, it's very dense, but that's more like length contraction than pressure. None of the matter actually reaches the event horizon in coordinate time, and yes, in that state, it (the original collapsing star matter) is very much under compression.

 

1 hour ago, Mordred said:

If Victor Toth stated that then he only looked at the vacuum solutions of the interior  and ignored the coordinate assigned for the singularity R=0. Not surprising though most articles tend to ignore that part as no one feels the singularity condition should exist.

The singularity condition exists, but isn't described, precisely because the physics there is singular, sort of like asking what the perspective of a photon is like.

1 hour ago, Mordred said:

I have another related article that suggests that the BH may simply be a neutron star that has collapsed just beyond its EH but is still present.

Got a link? That sounds like pop nonsense. Is it peer reviewed?

 

1 hour ago, Mordred said:

you can in fact have extremely high density and still be a vacuum solution with incredibly high temperature and density.  Vacuum describes a pressure term.  Its not the energy density

That makes more sense. Still, to be matter, it has to persist, no?

1 hour ago, Mordred said:

just to add the singularity at r=0 is a true singularity it cannot be removed by a change in metric choice. The event horizon itself is often described as a singularity condition however that is a coordinate singularity and not a true singularity.

Agree to all. There's also a naked singularity. You can for instance just keep dropping electrons into a black hole until the charge is more than the gravity and no more (isolated) electrons can be added by any means. Similar issues if the infalling matter adds too much angular momentum. A given mass can only have so much of that. These are examples of frame independent singularities not obscured by the coordinate singularity of the EH.

1 hour ago, Mordred said:

google is useful but unless your aware of other factors getting good answers can often mislead down the wrong google pathway.

Ditto with the LLM answers, which is just massaged google results. Anyway, thanks for the post. Good informative stuff in there.

[Mordred]

49 minutes ago, Halc said:

  The singularity condition exists, but isn't described, precisely because the physics there is singular, sort of like asking what the perspective of a photon is like.

Got a link? That sounds like pop nonsense. Is it peer reviewed?

 It was several peer reviewed article on Arxiv but it was a few years ago that I followed the research and some of the proposed tests.  However I'll take a look and if I can find the papers. If I can I'll post it. 

 Point of detail the singularity conditions regarding the EH involve infinite redshifts and subsequent time dilation relations involving the EH. These conditions are the ones directly involved with regards to Hawking radiation.  This also involves Unruh radiation  

https://arxiv.org/abs/0710.5373

 

Edited May 4 by Mordred

[MigL]

One problem I have with the central singularity.
What happens when you apply Heisenberg's Uncertainty Principle to the singular event ( whether spatial or temporal singularity )?

An exact location would mean indeterminate, even infinite, momentum, while an exact time would mean indeterminate, even infinite, energy.
How would a particle, localized in such a way, remain trapped within the Event Horizon ?
Now make that argument for all particles ingested by the BH and you have a BH which cannot exist.
IOW, the singularity renders the theory inconsistent; either a singularity cannot exist, or Black Holes cannot exist.
Yet we have photographic evidence of Black Holes, so where does that leave us ?

 

2 hours ago, Mordred said:

I have another related article that suggests that the BH may simply be a neutron star that has collapsed just beyond its EH but is still present.

I would be interested in a link to this article also.
I'm  interested in the proposed method ( pressure or force ) used to resist gravitational collapse at this radius.

[Mordred]

20 minutes ago, MigL said:

 

 

I would be interested in a link to this article also.
I'm  interested in the proposed method ( pressure or force ) used to resist gravitational collapse at this radius.

found it I finally recalled it was a series of Black star research

"Irreversible gravitational collapse: black stars or black holes?"

https://arxiv.org/abs/1105.3394

1 hour ago, Halc said:

 

Got a link? That sounds like pop nonsense. Is it peer reviewed?

 

 

 

20 minutes ago, MigL said:

One problem I have with the central singularity.
What happens when you apply Heisenberg's Uncertainty Principle to the singular event ( whether spatial or temporal singularity )?

An exact location would mean indeterminate, even infinite, momentum, while an exact time would mean indeterminate, even infinite, energy.
How would a particle, localized in such a way, remain trapped within the Event Horizon ?
Now make that argument for all particles ingested by the BH and you have a BH which cannot exist.
IOW, the singularity renders the theory inconsistent; either a singularity cannot exist, or Black Holes cannot exist.
Yet we have photographic evidence of Black Holes, so where does that leave us ?

 I certainly  wouldn't argue with this. LOL I know any personal modelling I do I set Planck units as a constraint on anything involving spacetime or other fields. Funny thing is that boundary works well with the FLRW metric for the 10^{-43} second boundary the average estimated temp of  10^19 GeV when you convert is in the same orders of magnitude as Planck temperature. With the temperature scale factor relation. Blackbody temp can be estimated by the inverse of the scale factor. This also corresponds to 1 Planck length.

just an interesting tidbit on last.

Edited May 4 by Mordred

[MSC]

On 5/3/2024 at 2:24 PM, MigL said:

Correct. We know of no force which can resist gravity once Neutron degeneray is exceeded in a neutron star,
This is according to GR and QFT.
Both of which have specific areas of applicability.
When outside those areas they 'fai' by throwing up infinities; like at the center/future of a BH.
IOW, points of infinite density are non-sensical predictions of badly applied models/theories.

Also keep in mind that the central singularity, while being the event where geodesics end ( akin to latitude/longitude at the Earth's poles ), is not a location in space, but an event in time, and an infalling observer would be 'running into other stuff' at the end of time, not at the center.

X-posted with others

What about Quark stars or Strange Stars? 

Thanks everyone for your comments. I'm trying to understand as best I can. There may be more stupid questions to come. 

How do you go about even creating theories that don't end in infinites or end of time stuff? I much prefer the term blackhole core because to me the occams razor suggestion says that it's just simpler for the event horizon to be cloaking some kind of exotic star where the matter density at any single point in the core is finite but unlike matter we have or can observe under other extreme conditions due to the nature of the EH. 

Also is the effect of time dilation upon entering the blackhole exponential? Is there a point where time is experienced at one layer of the core faster than an outer layer and at what point would an observer who can theoretically survive being pulled into a black hole etc, be able to turn, look outward and see the sped up universe and the edges of the very black hole the observer is being pulled into evaporating due to hawking radiation?  

Hmmm I think I can kind of understand what someone said about a vacuum core state at least to an external observer, as in the lifetime of an average human built observer, if you could peel back that EH and peer inside, Time Dilation would not allow you to witness any matter you've seen falling in, to ever reach the internal core.

It would eventually be moving so slow relative to the external observer that there would never be enough time for said observer the witness the full event; from matter or light, you can watch it from the moment it is getting caught in the gravity well. To the external observer, nothing can reach the inner core. Everything is just stuck on its way there. Like our own galaxy being pulled toward a supercluster cosmic expansion will never let ot reach. 

Ngl out of everyone alive on the planet, a group of people like y'all are where I'd lay my bets on finding a theory that unifies or improves upon GR + QM. We should make it a focussed group project lol y'all do the math I'll stay outside the box. This isn't me displaying hubris, it's me asking you guys to have more, because why not?

Quote

Don't desert me Boffins!

- That Mitchell and Webb look

[Markus Hanke]

1 hour ago, MSC said:

occams razor suggestion says that it's just simpler for the event horizon to be cloaking some kind of exotic star where the matter density at any single point in the core is finite

The problem with this is that - at least in ordinary GR - once something is below the horizon, a complete collapse is inevitable, irrespective of what that “something” is made of. This is due to the geometry of spacetime itself - below the horizon, the singularity is in the future of every particle that finds itself there. It is thus not possible to have stable, stationary objects there.

1 hour ago, MSC said:

How do you go about even creating theories that don't end in infinites or end of time stuff?

In the classical realm, it is possible to consider alternative theories of gravity that avoid singularities by postulating different types of geometries, such as for example Einstein-Cartan gravity.

In the quantum realm all bets are off right now, since we don’t have a working model of quantum gravity.

1 hour ago, MSC said:

Also is the effect of time dilation upon entering the blackhole exponential?

Time dilation is a relationship between at least two clocks, so you have to specify which clocks you wish to compare.

Generally speaking, the answer depends on what type of black hole you consider (ie the geometry of spacetime), where the clocks are, and how they move.

1 hour ago, MSC said:

be able to turn, look outward and see the sped up universe and the edges of the very black hole the observer is being pulled into evaporating due to hawking radiation? 

Here’s what you would see if you fell into a Schwarzschild BH (only mass):

And here the same thing for a Kerr-Newman BH (massive, rotating, and electrically charged):

As you can see, what you’d observe depends on the specifics of the black hole.

2 hours ago, MSC said:

To the external observer, nothing can reach the inner core.

For a Schwarzschild BH, an external stationary observer would not even see the test particle reach the horizon in finite time (on his own clock).

[MSC]

8 hours ago, Markus Hanke said:

Time dilation is a relationship between at least two clocks, so you have to specify which clocks you wish to compare.

So lets say a stellar mass black hole about 10 times the mass of our sun. Like Gaia-BH1. 3 clocks. One in a safe stable orbit around the black hole, one fixed just before the event horizon, one in freefall.

Thank you for those videos btw! They were wild. The first one especially. It looks like falling into a water balloon. Will rewatch those more than a few times!

8 hours ago, Markus Hanke said:

For a Schwarzschild BH, an external stationary observer would not even see the test particle reach the horizon in finite time (on his own clock).

That does make it sound like it would be exponentially increasing time dilation between clocks. If I drop multiple clocks in free fall, an hour after the other, would the clocks seem to catch up to the first dropped clock, relative to the external stable orbit clock?

[Halc]

On 5/3/2024 at 11:17 PM, Mordred said:

found it I finally recalled it was a series of Black star research

"Irreversible gravitational collapse: black stars or black holes?"

The paper calls them 'black stars' and seems to posit a complete lack of event horizon, and that the matter all piles up just outside where the EH would be. This is very similar to Schmelzer's alternative generalization of LET (relativity, but denying both premises of SR).  https://arxiv.org/pdf/gr-qc/0001095

He calls the 'frozen stars' since all the matter piles up outside the nonexistent EH. It's a presentist interpretation of relativity, and it equate absolute time with coordinate time.

 

Just my thoughts. The alternative is that the matter gets 'inside' and is somehow prevented from going 'forward' which is the same as positing that if you put enough force on matter, you can push it into the past where there's more room.

[MSC]

8 hours ago, Markus Hanke said:

This is due to the geometry of spacetime itself - below the horizon, the singularity is in the future of every particle that finds itself there. It is thus not possible to have stable, stationary objects there.

See I don't really know how to think about spacetime. Do I perceive to be some kind of material since physicists talk about it in material sounding terms. It has geometry/shape so it's like a sheet and black holes are like bowling balls dragging down the sheet making little pockets in spacetime?

On 5/3/2024 at 11:17 PM, Mordred said:

found it I finally recalled it was a series of Black star research

"Irreversible gravitational collapse: black stars or black holes?"

I've also heard of something called dark star theory which positted something along the lines that some black holes could through some unknown physical processes convert baryonic matter into dark matter. Not sure what to make of that myself. It's interesting watching and listening to you all discuss this stuff!

[Genady]

25 minutes ago, MSC said:

It's interesting watching and listening to you all discuss this stuff!

Free entertainment. Even leaves one with a fuzzy feeling that they learned something.

[Mordred]

1 hour ago, MSC said:

See I don't really know how to think about spacetime. Do I perceive to be some kind of material since physicists talk about it in material sounding terms. It has geometry/shape so it's like a sheet and black holes are like bowling balls dragging down the sheet making little pockets in spacetime?

 

 It's best to regard spacetime as simply the arena where the SM model of particles/fields reside. It really is simply a volume that uses the Interval (ct) to give time dimensionality of a length.

 Spacetime curvature under this describes the particle paths and the easiest way to understand this is to use parallel transport. (The equations of the EFE also uses parallel transport)

If spacetime is flat two parallel beams of light will remain parallel. If you have positve curvature those beams will converge. If you have negative curvature they will diverge.

 For gravity use the same manner. Draw two lines at some distance apart and connect them to the CoM. You will notice those lines converge as you approach the centre of mass.

Another useful tool to understand why particles follow different paths is to realize that all particle motion obeys the principle of least action. (This includes Feymann integrals as well as spacetime geodesics).

Terms such as fabric etc gives false impressions of spacetime being some material or substance. It really is simply a metric that describes a volume and spacetime paths (null geodesics for massless particles such as photons).

What affects the paths is all matter and force  fields of the SM model. The coupling constants collectively give rise to the mass terms (mass is resistance to inertia change).

1 hour ago, MSC said:

 

I've also heard of something called dark star theory which positted something along the lines that some black holes could through some unknown physical processes convert baryonic matter into dark matter. Not sure what to make of that myself. It's interesting watching and listening to you all discuss this stuff!

They also have strange stars that suggest with certain  neutron stars all quarks become strange quarks. Some models really stretch the imagination but surprisingly enough do have enough viability to warrant research.

 As for myself I study some of these as they often include unique ways to mathematically describe fields and states that I find useful for model developments. They also have papers suggesting a dark sector of a wide range of particles such as dark photons etc.

1 hour ago, Halc said:

The paper calls them 'black stars' and seems to posit a complete lack of event horizon, and that the matter all piles up just outside where the EH would be. This is very similar to Schmelzer's alternative generalization of LET (relativity, but denying both premises of SR).  https://arxiv.org/pdf/gr-qc/0001095

He calls the 'frozen stars' since all the matter piles up outside the nonexistent EH. It's a presentist interpretation of relativity, and it equate absolute time with coordinate time.

 

Just my thoughts. The alternative is that the matter gets 'inside' and is somehow prevented from going 'forward' which is the same as positing that if you put enough force on matter, you can push it into the past where there's more room.

Yeah I don't agree with the theory myself I ran across it a few years back.

Edited May 5 by Mordred

[TheVat]

3 hours ago, MSC said:

See I don't really know how to think about spacetime. Do I perceive to be some kind of material since physicists talk about it in material sounding terms. It has geometry/shape so it's like a sheet and black holes are like bowling balls dragging down the sheet making little pockets in spacetime?

I lean towards spacetime as a mathematical entity.  What has increasing curvature near a massive object?  The math.  Curved geometry.  No sheets, no bowling balls, no stuff.  Like going to the bowling alley without equipment and saying, I'm ready, dudes, I brought pi, f=ma,    and a couple other maths with me.

What breaks down at the singularity?  The equations do.

It's math being employed to do things that it's hard for math to do.  

JMAPCO (just my astrophysics-challenged opinion)

Edited May 5 by TheVat

[Genady]

1 hour ago, TheVat said:

What breaks down at the singularity?

Nothing. The 4D-spacetime curvature becomes infinite. So what? The intrinsic curvature at the vertex of a 2D-cone is infinite, too.

[MSC]

4 hours ago, Genady said:

Free entertainment. Even leaves one with a fuzzy feeling that they learned something.

Oh no doubt! Updoots all round.

4 hours ago, Mordred said:

They also have strange stars that suggest with certain  neutron stars all quarks become strange quarks. Some models really stretch the imagination but surprisingly enough do have enough viability to warrant research.

 As for myself I study some of these as they often include unique ways to mathematically describe fields and states that I find useful for model developments. They also have papers suggesting a dark sector of a wide range of particles such as dark photons etc.

How complex is it possible for dark matter objects to get under this premise I wonder? 

While on this note how much of the light spectrum can we actually technologically perceive? All? Most? A little? I sometimes get the feeling that Dark matter is a dog whistle like phenomenon in space beyond our ability to perceive it both organically or technogically. Either that or evidence of a localised dome cloak like technology hiding all the cool alien stuff from us.

[Mordred]

Well the problem with the dark matter spectrum is that indirect evidence gives all the characteristics of either a weakly interactive particle  likely candidate being right hand neutrinos ( this is the main focus of my research at the Professional level) or axioms.

 Being weakly interactive you don't get clumping as you would with matter. So it's rather tricky to get stars made up of dark matter.

As evidence points toward " does not interact with EM force." Clumping is highly unlikely. The same goes with the strong force. A weakly interactive particle may interact with other weakly interactive particles such as other neutrinos . The Higgs field for mass terms and gravity.

For detection using EM any viable detection so far involves x rays due to specific interactions so we can only get indirect evidence using the EM spectrum. Other indirect evidence being the mass terms with gravity.

I don't find anything beyond the above too likely but who knows I could be wrong on that. However from my research I strongly consider right hand neutrinos as the most likely candidate.

Edited May 5 by Mordred

[MigL]

18 hours ago, MSC said:

What about Quark stars or Strange Stars? 

Both are hypothetical, as is much of the other stuff mentioned.
I can, however, show you pictures of a Black Hole.
( or you can google it )

[Carrock]

On 5/3/2024 at 9:30 PM, Mordred said:

On 5/3/2024 at 8:25 PM, Carrock said:

A longer quote to show what I find problematic:

"A photon is its own antiparticle." .... "anti-photons will annihilate with matter photons."

How do photons distinguish between anti-photons and matter photons? Presumably if two high energy matter photons collide it is impossible for them to create a matter/antimatter pair.

 

If you have a reference to there being both anti-photons and matter photons please share it.

How particles scatter, or form new particles etc etc always depends on their cross sections . That uses the Breit Wigner equations along with the Feymann  golden rules.   It not some case of a photon knowing anything. When it encounters another particle the cross sections and Feymann golden rules are used to determine the end results. Granted we also have a table that is helpful . 

https://en.m.wikipedia.org/wiki/Table_of_Clebsch–Gordan_coefficients

Sorry about delay....

I don't think your passive voice alternative for 'distinguish' is any better than my anthropomorphism, which is hard to resist when you claim photons are their own antiparticles but also different from their antiparticles.

 

Saying "Breit Wigner equations .... and Feymann golden rules are used to determine the end results" raises interesting questions. Who or what uses the rules to determine the end results?

You stated it's not the photon. I agree that photons don't know the rules, but I also think that such rules are only (imperfect) descriptions and prescriptive rules are at best only a statistical guide to the end result and certainly don't determine it.

 

As Fermi Golden Rules, Feynman Golden Rules and Feymann Golden Rules don't exist AFAIK, is there an end result?

A reference to anti-photons and matter photons would still be appreciated.

 

[Mordred]

You can easily confirm what I state by simply googling 

Shall I do it for you

"Some particles, such as the photon, are their own antiparticle. Otherwise, for each pair of antiparticle partners, one is designated as the normal particle "

https://en.m.wikipedia.org/wiki/Antiparticle

There that wasn't too hard was it ?

Took me less than 10 seconds....

As far as Breit Wigner and Fermann integrals

https://pdg.lbl.gov/2011/reviews/rpp2011-rev-cross-section-formulae.pdf

For Breit Wigner. 

Feymanns golden rule

https://en.m.wikipedia.org/wiki/Fermi's_golden_rule#:~:text=In quantum physics%2C Fermi's golden,result of a weak perturbation.

Anything else I can do for you?

how about providing you the Feymann integral itself for the photon.

photon

\[i\Delta_{\mu\nu}(k)=\frac{i}{-k^2-i\epsilon}[g_{\mu\nu}-\frac{k_\mu k_\nu}{k^2}(1=\alpha)\]

antiphoton

\[i\Delta_{\mu\nu}(k)=-\frac{i}{-k^2-i\epsilon}[g_{\mu\nu}-\frac{k_\mu k_\nu}{k^2}(1=\alpha)\]

 

notice the only difference is the sign so if the two encounter each other they annihilate. However that is true for every particle antiparticle pair regardless if it is its own antiparticle or has a difference in charge...μ is the polarization index rule 4 of the Feymann rules is you symmetrice between identical bosons as the anti-photon is antisymmetric to begin with you follow the Dirac rules for asymmetric particles in essence fermionic rules.

you Multiply \(\mu\) with \(\epsilon^*_\mu\) for incoming particles and \(\epsilon_\mu \)for outgoing particles. \(\epsilon\) is the dimensional regularization parameter. (renormalization ).The rest of the Feymann rules I suggest you get a decent textbook on the topic.

There is 10 primary rules. However their are also special rules for different theories such as \(\phi^3\) or \(\phi^4\)

here is Fermi's golden rule ( prefer the format)

Fermi's Golden Rule

\[\Gamma=\frac{2\pi}{\hbar}|V_{fi}|^2\frac{dN}{DE_f}\]

density of states

\[\langle x|\psi\rangle\propto exp(ik\cdot x)\]

with periodic boundary condition as "a"\[k_x=2\pi n/a\]

number of momentum states

\[dN=\frac{d^3p}{(2\pi)^2}V\]

decay rate

\[\Gamma\]

Hamilton coupling matrix element between initial and final state

\[V_{fi}\]

density of final state

\[\frac{dN}{dE_f}\]

number of particles remaining at time t (decay law)

\[\frac{dN}{dt}=-\Gamma N\]

average proper lifetime probability

\[p(t)\delta t=-\frac{1}{N}\frac{dN}{dt}\delta t=\Gamma\exp-(\Gamma t)\delta t\]

mean lifetime \[\tau=<t>=\frac{\int_0^\infty tp (t) dt}{\int_0^\infty p (t) dt}=\frac{1}{\Gamma}\]

relativistic decay rate set 

\[L_o=\beta\gamma c\tau\] average number after some distance x

\[N=N_0\exp(-x/l_0)\]

as opposed to here but they are equivalent.

https://en.wikipedia.org/wiki/Fermi's_golden_rule

here is some relevant details on Breit Wigner cross sections.

Breit Wigner cross section

\[\sigma(E)=\frac{2J+1}{2s_1+1)(2S_2+1)}\frac{4\pi}{k^2}[\frac{\Gamma^2/4}{(E-E_0)^2+\Gamma/4)}]B_{in}B_{out}\]

E=c.m energy, J is spin of resonance, (2S_1+1)(2s_2+1) is the #of polarization states of the two incident particles, the c.m., initial momentum k E_0 is the energy c.m. at resonance, \Gamma is full width at half max amplitude, B_[in} B_{out] are the initial and final state for narrow resonance the [] can be replaced by

\[\pi\Gamma\delta(E-E_0)^2/2\]

The production of point-like, spin-1/2 fermions in e+e− annihilation through a virtual photon at c.m.

\[e^+,e^-\longrightarrow\gamma^\ast\longrightarrow f\bar{f}\]

\[\frac{d\sigma}{d\Omega}=N_c{\alpha^2}{4S}\beta[1+\cos^2\theta+(1-\beta^2)\sin^2\theta]Q^2_f\]

where

\[\beta=v/c\]

c/m frame scattering angle

\[\theta\] 

fermion charge

\[Q_f\]

if factor [N_c=1=charged leptons if N_c=3 for quarks.

if v=c then (ultrarelativistic particles)

\[\sigma=N_cQ^2_f\frac{4\pi\alpha^2}{3s}=N_cQ^2_f\frac{86.8 nb}{s (GeV^2)}\]

2 pair quark to 2 pair quark

\[\frac{d\sigma}{d\Omega}(q\bar{q}\rightarrow \acute{q}\acute{\bar{q}})=\frac{\alpha^2_s}{9s}\frac{t^2+u^2}{s^2}\]

cross pair symmetry gives

\[\frac{d\sigma}{d\Omega}(q\bar{q}\rightarrow \acute{q}\acute{\bar{q}})=\frac{\alpha^2_s}{9s}\frac{t^2+u^2}{t^2}\]

The last details I copied from my own research of with regards to Early universe processes including electroweak symmetry breaking. You can find plenty of examples of how to use them here

https://www.scienceforums.net/topic/128332-early-universe-nucleosynthesis/

if you really want to learn Feymannintegrals I suggest

Feynman Integrals Stefan Weinzierl

https://arxiv.org/abs/2201.03593

providing precise step by step on a forum is quite frankly impossible. For example it would take literal pages to explain the Casimer trick for sum of amplitudes for the path integrals. Let alone  using Feymann's trick for the integrals.

thankfully tools such as Mathematica with the Feycalc package can handle those steps.

PS anyone familiar with Breit Wigner may notice that the Fermi golden rule is integrated into the Breit Wigner formula

 

 

Edited May 6 by Mordred

[Mordred]

of course if you want to avoid all that math a simpler way would be to use the Langrangian representation for the photon as  opposed to the antiphoton.

the relativistic Langrangian of a free photon is 

\[\mathcal{L}_0-\frac{\epsilon\hbar}{c}v\cdot \omega\] where \(\epsilon\) in this case is the helicity + for photon - for antiphoton. The conjugate momentum is the linear momentum

\[p-\partial\mathcal{L}_0\partial v=\frac{e\hbar\omega}{c}\]

the energy of the photon and antiphoton is 

\[H-\pm\hbar\omega-\hbar v\]

+ for photon - for antiphoton.

\[H=p\cdot v+s\cdot \omega-L_0-s\cdot \omega\cdot v\]

so now I've shown two methods that describe the same thing....if you like I can skip to the creation annihilation operator method... just let me know or you can pick up quantum field theory Demystified it has an excellent section on it.

for all the previous its the QM methodology and applies their operators (position and momentum) if you have further doubts then please start a new thread so we don't hijack this one.

PS probably the easiest example of how photons interact and how cross sections are applied study Thompson scatterings. Though they likely won't show the anti-photon it will demonstrate the importance of cross sections

 

 

Edited May 6 by Mordred

[Carrock]

Have only a few minutes to correct a handful of the errors you produce at amazing speed:

2 hours ago, Mordred said:

You can easily confirm what I state by simply googling 

Shall I do it for you

"Some particles, such as the photon, are their own antiparticle. Otherwise, for each pair of antiparticle partners, one is designated as the normal particle "

https://en.m.wikipedia.org/wiki/Antiparticle

There that wasn't too hard was it ?

Took me less than 10 seconds....

 

Took longer than 10 seconds to check there was no mention of matter and antimatter photons in the wiki; in your next post you seem to claim that photons and antiphotons have opposite spin (or does helicity have a special meaning to you?) I'm not going to trawl through the rest of your references for a possible source ref.

 

2 hours ago, Mordred said:

Feymanns golden rule

https://en.m.wikipedia.org/wiki/Fermi's_golden_rule#:~:text=In quantum physics%2C Fermi's golden,result of a weak perturbation.

Anything else I can do for you?

I notice you have now gone down from 'Feymanns golden rules' to 'Feymanns golden rule'. It may help you if I point out that while 'Feymann' is not the same as 'Fermi', it is not true that 'Feymann' is the same as 'Fermi'.

If you read through the above quote very carefully you'll notice there isn't even a 'Feymanns golden rule,' let alone, as I wasted my time typing earlier, 'Fermi Golden Rules, Feynman Golden Rules and Feymann Golden Rules don't exist AFAIK.' (You do seem to have corrected some of your references in your penultimate post.)

 

 

Quote

'When it encounters another particle the cross sections and Feymann golden rules are used to determine the end results.'

Of course, with only one 'Feymanns golden rule' instead of at least two, the end results will now be determined differently. I pity whatever has to move the photon around as the determination of the end results keeps changing.*

Seeing you quietly correcting errors you've made without acknowledging I pointed them out is getting tedious.

3 hours ago, Mordred said:

providing precise step by step on a forum is quite frankly impossible.

Quite right.

I've pointed out several times that things which are the same as each other are not also different from each other. I have clearly not made this clear enough for you and to do so is quite frankly impossible.

 

3 hours ago, Mordred said:

photon

 

iΔμν(k)=i−k2−iϵ[gμν−kμkνk2(1=α)

 

antiphoton

 

iΔμν(k)=−i−k2−iϵ[gμν−kμkνk2(1=α)

 

 

I started having a glance through your equations; thanks for the obvious errors in these two which made me realise that was a waste of time.

It seems your equations go from textbook to forum without engaging your brain except to remove explanatory text and symbol definitions.

I'm done wasting my time here.

 

* using your concept that equations determine end results

[Mordred]

whatever you wish to believe I seriously doubt you know how to use any of the equations I mentioned nor have any real understanding of particle physics to  begin with. I won't waste any of my time with you either considering I actually hold a degree in particle physics I don't need to 

.

Go ahead mathematically show me being incorrect with the formulas I provided. Impress me beyond your bland statements of me making errors Prove your case instead of simply declaring errors. If you wish to accuse me of errors you had best back it up.

Edited May 6 by Mordred

[MSC]

5 minutes ago, Mordred said:

whatever you wish to believe I seriously doubt you know how to use any of the equations I mentioned nor have any real understanding of particle physics to  begin with. I won't waste any of my time with you either considering I actually hold a degree in particle physics I don't need to 

You really don't need to waste your time with them. I don't understand your equations at all but same principles of logical debate apply. It's bad faith to say they are incorrect or full of errors without providing corrections and then saying they aren't going to read the rest. Leave them to their delusion. You can't reason with an unreasonable person.

[Mordred]

That is why I don't waste my time trying if they can't take the time to show the error. I don't mind being corrected but I won't change what I've been professionally taught on someone's word without collaborative proof. Certainly not from some bland statement of being wrong without any further detail

[MSC]

17 minutes ago, Carrock said:

Have only a few minutes to correct a handful of the errors you produce at amazing speed:

Took longer than 10 seconds to check there was no mention of matter and antimatter photons in the wiki; in your next post you seem to claim that photons and antiphotons have opposite spin (or does helicity have a special meaning to you?) I'm not going to trawl through the rest of your references for a possible source ref.

 

I notice you have now gone down from 'Feymanns golden rules' to 'Feymanns golden rule'. It may help you if I point out that while 'Feymann' is not the same as 'Fermi', it is not true that 'Feymann' is the same as 'Fermi'.

If you read through the above quote very carefully you'll notice there isn't even a 'Feymanns golden rule,' let alone, as I wasted my time typing earlier, 'Fermi Golden Rules, Feynman Golden Rules and Feymann Golden Rules don't exist AFAIK.' (You do seem to have corrected some of your references in your penultimate post.)

 

 

Of course, with only one 'Feymanns golden rule' instead of at least two, the end results will now be determined differently. I pity whatever has to move the photon around as the determination of the end results keeps changing.*

Seeing you quietly correcting errors you've made without acknowledging I pointed them out is getting tedious.

Quite right.

I've pointed out several times that things which are the same as each other are not also different from each other. I have clearly not made this clear enough for you and to do so is quite frankly impossible.

 

I started having a glance through your equations; thanks for the obvious errors in these two which made me realise that was a waste of time.

It seems your equations go from textbook to forum without engaging your brain except to remove explanatory text and symbol definitions.

I'm done wasting my time here.

 

* using your concept that equations determine end results

Mordreds better at explaining his math in plain English than you are at refuting it in plain English. If he's made errors then for the sake of the people that understand the math, provide corrections.

Also chill with the personal attacks. Nobody gets graded for anything said here and cutting the attacks out of sincere debate is entirely within your capabilities. 

2 minutes ago, Mordred said:

That is why I don't waste my time trying if they can't take the time to show the error. I don't mind being corrected but I won't change what I've been professionally taught on someone's word without collaborative proof. Certainly not from some bland statement of being wrong without any further detail

That's pretty much how I go about debating philosophical and ethical topics. Where I often stand corrected because it's not a precision science like particle physics is but these are pretty uniform academic standards that you don't even have to attend university to infer from just observing debates in different venues. 

I appreciate you taking the time on this topic. Maybe one day I'll understand the math!