Prompt:
Design a retrieval plan to answer a missing detail without guessing.

EVIDENCE:
- m,0 (cid:27),0 NotethattheuniverseconsideredinchapterVI.2sitsatthepoint{(cid:24) =0,(cid:24) =1}.
- Keepinmindthatourindexj runsoverr,m,and(cid:27),whiletheindexnrunsoverthe rangeofj plusthecurvaturetermk.
- IntroductiontoOurUniverse Hence,settinga =1,wehave (cid:3) (cid:4) H 2 1 (cid:24) = 0 (cid:24) (13) j H aγj j,0 (As usual, the subscript 0 indicates the present value of various quantities.) It is also convenient to define the analogous expression for (cid:24) with γ =2.
- We can then rewrite k k (10)as (cid:3) (cid:4) H 2(cid:10) (cid:24) 1= 0 n,0 (14) H aγn n or (cid:3) (cid:4) (cid:10) (cid:24) (cid:24) (cid:24) (cid:24) H2=H2 n,0 =H2 m,0 + r,0 +(cid:24) + k,0 (15) 0 aγn 0 a3 a4 (cid:27),0 a2 n Theobservedvaluesforthesecosmologicalparametersare (cid:24) ∼0.7, (cid:24) ∼0.3, (cid:24) ∼5×10 −5 (16) (cid:27),0 m,0 r,0 andH ∼70km/sec/Mpc∼2×10 −18sec,where1Mpc=106parsecor∼3×1019km.

m,0 (cid:27),0 NotethattheuniverseconsideredinchapterVI.2sitsatthepoint{(cid:24) =0,(cid:24) =1}.

Keepinmindthatourindexj runsoverr,m,and(cid:27),whiletheindexnrunsoverthe rangeofj plusthecurvaturetermk.

IntroductiontoOurUniverse Hence,settinga =1,wehave (cid:3) (cid:4) H 2 1 (cid:24) = 0 (cid:24) (13) j H aγj j,0 (As usual, the subscript 0 indicates the present value of various quantities.) It is also convenient to define the analogous expression for (cid:24) with γ =2.

We can then rewrite k k (10)as (cid:3) (cid:4) H 2(cid:10) (cid:24) 1= 0 n,0 (14) H aγn n or (cid:3) (cid:4) (cid:10) (cid:24) (cid:24) (cid:24) (cid:24) H2=H2 n,0 =H2 m,0 + r,0 +(cid:24) + k,0 (15) 0 aγn 0 a3 a4 (cid:27),0 a2 n Theobservedvaluesforthesecosmologicalparametersare (cid:24) ∼0.7, (cid:24) ∼0.3, (cid:24) ∼5×10 −5 (16) (cid:27),0 m,0 r,0 andH ∼70km/sec/Mpc∼2×10 −18sec,where1Mpc=106parsecor∼3×1019km.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Design a retrieval plan to answer a missing detail without guessing.

EVIDENCE:
- Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning (https://archive.org/details/wondersofnumbers0000pick_g6a7/page/78).
- Einstein voted 'greatest physicist ever' by leading physicists; Newton runner-up" (htt p://news.bbc.co.uk/2/hi/science/nature/541840.stm).
- Mathematical Association of America.
- The Queen of Mathematics: A Historically Motivated Guide to Number Theory (https://books.google.com/books?id=A0FZDwAAQBAJ&pg=PA88).

Locate highest-signal evidence sentences from chunks for the question.

Check consistency, units, conservation constraints, and whether evidence supports claims.

Compose bounded answer and (if applicable) a proposal/hypothesis clearly labeled.

Generate falsification tests and identify unknowns or missing evidence.

Merge outputs, remove speculation, keep only evidence-backed facts.

Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning (https://archive.org/details/wondersofnumbers0000pick_g6a7/page/78).

Einstein voted 'greatest physicist ever' by leading physicists; Newton runner-up" (htt p://news.bbc.co.uk/2/hi/science/nature/541840.stm).

The Queen of Mathematics: A Historically Motivated Guide to Number Theory (https://books.google.com/books?id=A0FZDwAAQBAJ&pg=PA88).

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.

Bennett
Dirac
Zeilinger
Jian-Wei Pan
De Martini
Feynman
Popper


Prompt:
Design a retrieval plan to answer a missing detail without guessing.

EVIDENCE:
- doi:10.1086/339437 (https://doi.org/10.1086%2F339437).
- arXiv:astro-ph/0506468 (https://arxiv.org/abs/astro-ph/0506468).
- "Vesta: Spin Pole, Size, and Shape from HST Images" (https://doi.or g/10.1006%2Ficar.1997.5736).
- doi:10.1016/j.icarus.2009.08.007 (https://doi.org/10.101 6%2Fj.icarus.2009.08.007).

"Vesta: Spin Pole, Size, and Shape from HST Images" (https://doi.or g/10.1006%2Ficar.1997.5736).

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
What does the evidence say about necessary conditions and constraints?

EVIDENCE:
- Inyourrestframe(figure3b)asyoufallalongwiththeballs,however,youseeatidal force acting on the circular ring (or a spherical shell if you prefer) of balls.
- Note also for future reference that the tidal force Rij(x(cid:3))≡∂i∂jV(x(cid:3)) involves two derivativesactingonthegravitationalpotentialV(x(cid:3)).
- and ε12...D=1 (11) Inotherwords, theantisymmetricsymbolε flipssignupontheinterchangeofanypair of indices.
- It follows that ε vanishes when two indices are equal.

Inyourrestframe(figure3b)asyoufallalongwiththeballs,however,youseeatidal force acting on the circular ring (or a spherical shell if you prefer) of balls.

Note also for future reference that the tidal force Rij(x(cid:3))≡∂i∂jV(x(cid:3)) involves two derivativesactingonthegravitationalpotentialV(x(cid:3)).

and ε12...D=1 (11) Inotherwords, theantisymmetricsymbolε flipssignupontheinterchangeofanypair of indices.

It follows that ε vanishes when two indices are equal.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Summarize the most important claims and how to verify them.

EVIDENCE:
- History From ancient times, people were familiar with four types of phenomena that today would all be explained using the concept of electric charge: (a) lightning, (b) the torpedo fish (or electric ray), (c) St Elmo's Fire, and (d) that amber rubbed with fur would attract small, light objects.[9] The first account of the amber effect is often attributed to the ancient Greek mathematician Thales of Miletus, who lived from c.
- The Coulomb's torsion balance Greeks observed that the charged amber buttons could attract light objects such as hair.
- One faraday is the magnitude of the charge of one mole of elementary charges,[8] i.e.
- They also found that if they rubbed the amber for long enough, they could even get an electric spark to jump, but there is also a claim that no mention of electric sparks appeared until late 17th century.[12] This property derives from the triboelectric effect.

History From ancient times, people were familiar with four types of phenomena that today would all be explained using the concept of electric charge: (a) lightning, (b) the torpedo fish (or electric ray), (c) St Elmo's Fire, and (d) that amber rubbed with fur would attract small, light objects.[9] The first account of the amber effect is often attributed to the ancient Greek mathematician Thales of Miletus, who lived from c.

The Coulomb's torsion balance Greeks observed that the charged amber buttons could attract light objects such as hair.

One faraday is the magnitude of the charge of one mole of elementary charges,[8] i.e.

They also found that if they rubbed the amber for long enough, they could even get an electric spark to jump, but there is also a claim that no mention of electric sparks appeared until late 17th century.[12] This property derives from the triboelectric effect.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
What does the evidence say about necessary conditions and constraints?

EVIDENCE:
- soluble among the alkaline earth hydroxides and a stronger base than its barium congener, barium hydroxide.[26] It is also more soluble than actinium hydroxide and thorium hydroxide: these three adjacent hydroxides may be separated by precipitating them with ammonia.[26] Radium chloride (RaCl ) is a colorless, luminescent compound.
- It becomes yellow after some time due to self-damage by the alpha radiation given off by radium when it decays.
- Small amounts of barium impurities give the compound a rose color.[26] It is soluble in water, though less so than barium chloride, and its solubility decreases with increasing concentration of hydrochloric acid.
- Crystallization from aqueous solution gives the dihydrate RaCl ·2H O, isomorphous with its barium 2 2 analog.[26]

soluble among the alkaline earth hydroxides and a stronger base than its barium congener, barium hydroxide.[26] It is also more soluble than actinium hydroxide and thorium hydroxide: these three adjacent hydroxides may be separated by precipitating them with ammonia.[26] Radium chloride (RaCl ) is a colorless, luminescent compound.

It becomes yellow after some time due to self-damage by the alpha radiation given off by radium when it decays.

Small amounts of barium impurities give the compound a rose color.[26] It is soluble in water, though less so than barium chloride, and its solubility decreases with increasing concentration of hydrochloric acid.

Crystallization from aqueous solution gives the dihydrate RaCl ·2H O, isomorphous with its barium 2 2 analog.[26]

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Design a retrieval plan to answer a missing detail without guessing.

EVIDENCE:
- Bibcode:1978GReGr...9..511K (https://ui.adsabs.harvar d.edu/abs/1978GReGr...9..511K).
- "On the Gauge Aspects of Gravity".
- "Nonholonomic Mapping Principle for Classical and Quantum Mechanics in Spaces with Curvature and Torsion".
- Bibcode:2019EPJC...79..697C (https://ui.adsabs.harvard.edu/abs/2019EPJC...79..697 C).

"Nonholonomic Mapping Principle for Classical and Quantum Mechanics in Spaces with Curvature and Torsion".

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
What does the evidence say about necessary conditions and constraints?

EVIDENCE:
- Whycouldn’tthebookmakeupitsmind?Isitfugal ∗ orisitpetal†?
- dt2 WhatyoucallforceIcouldcallapieceoftheacceleration, andwhatyoucallcentripetal Icouldcallcentrifugal.Einstein’sinsightwasthatthemostcommonlyexperiencedforce ofall,thegravitationalforce,maybeanexampleofa“fictitiousforce.” Exercise 1 Aheliumballoonisattachedtoachild’sseatinthebackofacar.Whenthespeedingcarsuddenlybrakesto astop,howdoestheballoonmove?
- You know the resolution of all this confusing talk: the book was just moving bits and piecesof d2x(cid:3) backandforthbetweentherighthandsideandthelefthandsideofma=F.
- Thenextlineis“Sictransitgloriamundi”.

You know the resolution of all this confusing talk: the book was just moving bits and piecesof d2x(cid:3) backandforthbetweentherighthandsideandthelefthandsideofma=F.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Summarize the most important claims and how to verify them.

EVIDENCE:
- the Milky Way, spiral nebulae, and the dimensions of the universe.
- To support his claim that the Great Andromeda Nebula is an external galaxy, Curtis noted the appearance of dark lanes resembling the dust clouds in the Milky Way, as well as the significant Doppler shift.[48] In 1922, the Estonian astronomer Ernst Öpik gave a distance determination that supported the theory that the Andromeda Nebula is indeed a distant extra-galactic object.[49] Using the new 100-inch Mount Wilson telescope, Edwin Hubble was able to resolve the outer parts of some spiral nebulae as collections of individual stars and identified some Cepheid variables, thus allowing him to estimate the distance to the nebulae: they were far too distant to be part of the Milky Way.[50] In 1926 Hubble produced a classification of galactic morphology that is used to this day.[51][52]

the Milky Way, spiral nebulae, and the dimensions of the universe.

To support his claim that the Great Andromeda Nebula is an external galaxy, Curtis noted the appearance of dark lanes resembling the dust clouds in the Milky Way, as well as the significant Doppler shift.[48] In 1922, the Estonian astronomer Ernst Öpik gave a distance determination that supported the theory that the Andromeda Nebula is indeed a distant extra-galactic object.[49] Using the new 100-inch Mount Wilson telescope, Edwin Hubble was able to resolve the outer parts of some spiral nebulae as collections of individual stars and identified some Cepheid variables, thus allowing him to estimate the distance to the nebulae: they were far too distant to be part of the Milky Way.[50] In 1926 Hubble produced a classification of galactic morphology that is used to this day.[51][52]

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Design a retrieval plan to answer a missing detail without guessing.

EVIDENCE:
- imaginary n-sphere defined by the equation is a real hypersurface without any real point, which is defined over the rational numbers.
- It has no rational point, but has many points that are rational over the Gaussian rationals.
- Projective algebraic hypersurface A projective (algebraic) hypersurface of dimension n – 1 in a projective space of dimension n over a field k is defined by a homogeneous polynomial in n + 1 indeterminates.
- As usual, homogeneous polynomial means that all monomials of P have the same degree, or, equivalently that

imaginary n-sphere defined by the equation is a real hypersurface without any real point, which is defined over the rational numbers.

It has no rational point, but has many points that are rational over the Gaussian rationals.

Projective algebraic hypersurface A projective (algebraic) hypersurface of dimension n – 1 in a projective space of dimension n over a field k is defined by a homogeneous polynomial in n + 1 indeterminates.

As usual, homogeneous polynomial means that all monomials of P have the same degree, or, equivalently that

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
What does the evidence say about necessary conditions and constraints?

EVIDENCE:
- It constitutes the definition of Riemann curvature R in terms of the Lorentz connection Γ.
- The commutator 1[Γ,Γ] of the 1-form Γ with itself is the bivector 2-form 1[Γ,Γ]= 1[Γ ,Γ ]d2xκλ =(Γp Γ −Γp Γ ) γγm∧γγnd2xκλ , (16.207) 4 2 κ λ mλ pnκ mκ pnλ again implicitly summed over distinct antisymmetric indices mn and κλ.
- The d in the rightmost expression of mκλ equations (16.210) are the vierbein derivatives defined by equation (11.32).
- 452 Action principle for electromagnetism and gravity 16.14.2 Curvature and torsion forms The Riemann bivector 2-form R is defined by R≡R d2xκλ =R γγm∧γγnd2xκλ , (16.205) κλ κλmn again implicitly summed over distinct antisymmetric indices mn and κλ.

It constitutes the definition of Riemann curvature R in terms of the Lorentz connection Γ.

The commutator 1[Γ,Γ] of the 1-form Γ with itself is the bivector 2-form 1[Γ,Γ]= 1[Γ ,Γ ]d2xκλ =(Γp Γ −Γp Γ ) γγm∧γγnd2xκλ , (16.207) 4 2 κ λ mλ pnκ mκ pnλ again implicitly summed over distinct antisymmetric indices mn and κλ.

The d in the rightmost expression of mκλ equations (16.210) are the vierbein derivatives defined by equation (11.32).

452 Action principle for electromagnetism and gravity 16.14.2 Curvature and torsion forms The Riemann bivector 2-form R is defined by R≡R d2xκλ =R γγm∧γγnd2xκλ , (16.205) κλ κλmn again implicitly summed over distinct antisymmetric indices mn and κλ.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Design a retrieval plan to answer a missing detail without guessing.

EVIDENCE:
- (4.28) κν dλ κ κν dτ This requires that the constant in equation (4.23) must equal the rest mass m, dτ =m .
- The correct m normalizationissuchthatthegeneralizedmomentumofthetestparticle,definedbyequation(4.26a),equal its ordinary momentum p , in agreement with equation (4.11), μ dxν dxν g =p ≡g m .
- (4.29) dλ This is just the definition of the affine parameter λ, equation (2.93).
- (4.27) dλ κν dλ μνκ dλ dλ Equations (4.27) are valid subject to the condition (4.23), which asserts that dλ ∝ dτ.

(4.28) κν dλ κ κν dτ This requires that the constant in equation (4.23) must equal the rest mass m, dτ =m .

The correct m normalizationissuchthatthegeneralizedmomentumofthetestparticle,definedbyequation(4.26a),equal its ordinary momentum p , in agreement with equation (4.11), μ dxν dxν g =p ≡g m .

(4.29) dλ This is just the definition of the affine parameter λ, equation (2.93).

(4.27) dλ κν dλ μνκ dλ dλ Equations (4.27) are valid subject to the condition (4.23), which asserts that dλ ∝ dτ.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Summarize the most important claims and how to verify them.

EVIDENCE:
- Spatial rotations correspond to real unimodular quaternions, and account for 3 of the 6 degrees of freedom of Lorentz transformations.
- A spatial rotation by angle θ right-handedly about the 1-axis (the x-axis) is the real Lorentz rotor R=e−ı1θ/2 =cos(θ/2)−ı sin(θ/2) , (14.57)
- Compared to standard rotation matrices, quaternions offer increased speed and require less storage, and theiralgebraicpropertiessimplifyinterpolationandsplining.Complexquaternionsretainsimilaradvantages for implementing Lorentz transformations.
- 14.5 How to implement Lorentz transformations on a computer The advantages of quaternions for implementing spatial rotations are well-known to 3D game programmers.

Spatial rotations correspond to real unimodular quaternions, and account for 3 of the 6 degrees of freedom of Lorentz transformations.

A spatial rotation by angle θ right-handedly about the 1-axis (the x-axis) is the real Lorentz rotor R=e−ı1θ/2 =cos(θ/2)−ı sin(θ/2) , (14.57)

Compared to standard rotation matrices, quaternions offer increased speed and require less storage, and theiralgebraicpropertiessimplifyinterpolationandsplining.Complexquaternionsretainsimilaradvantages for implementing Lorentz transformations.

14.5 How to implement Lorentz transformations on a computer The advantages of quaternions for implementing spatial rotations are well-known to 3D game programmers.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Which parts are established vs speculative, and what evidence supports each?

EVIDENCE:
- 9–11; use of these models is justified by the fact that, at large scales of around hundred million light-years and more, our universe indeed appears to be isotropic and homogeneous, cf.
- that at least some of the compact objects in question appear to have no solid surface, which can be deduced from the examination of X-ray bursts for which the central compact object is either a neutron star or a black hole; cf.
- 2006 for an overview, Narayan 2006, sec.
- Falcke, Melia & Agol 2000 130.

9–11; use of these models is justified by the fact that, at large scales of around hundred million light-years and more, our universe indeed appears to be isotropic and homogeneous, cf.

that at least some of the compact objects in question appear to have no solid surface, which can be deduced from the examination of X-ray bursts for which the central compact object is either a neutron star or a black hole; cf.

2006 for an overview, Narayan 2006, sec.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Which parts are established vs speculative, and what evidence supports each?

EVIDENCE:
- to involve the production of electron–positron pairs, as ordinary matter gains kinetic energy while falling into a stellar remnant.[31][32] Antimatter may exist in relatively large amounts in far-away galaxies due to cosmic inflation in the primordial time of the universe.
- Antimatter galaxies, if they exist, are expected to have the same chemistry and absorption and emission spectra as normal-matter galaxies, and their astronomical objects would be observationally identical, making them difficult to distinguish.[33] NASA is trying to determine if such galaxies exist by looking for X-ray and gamma ray signatures of annihilation events in colliding superclusters.[34]

to involve the production of electron–positron pairs, as ordinary matter gains kinetic energy while falling into a stellar remnant.[31][32] Antimatter may exist in relatively large amounts in far-away galaxies due to cosmic inflation in the primordial time of the universe.

Antimatter galaxies, if they exist, are expected to have the same chemistry and absorption and emission spectra as normal-matter galaxies, and their astronomical objects would be observationally identical, making them difficult to distinguish.[33] NASA is trying to determine if such galaxies exist by looking for X-ray and gamma ray signatures of annihilation events in colliding superclusters.[34]

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Which parts are established vs speculative, and what evidence supports each?

EVIDENCE:
- ToCosmologyasQuicklyasPossible | 361 afundamentalprinciple,quantumfluctuationswouldforcethattermonyou.Youmust haveareasonforwhyagiventermshouldnotbethere.Roughly,thatisbecausequantum physicsisprobabilistic.Physicistscanonlydeterminetheprobabilitiesofvariousprocesses ∗ occurring.Anyprocessnotexplicitlyforbidden willoccur,eventhoughtheprobabilityof theprocessactuallyoccurringmaybeverysmall.
- 3 Youmighthavenoticedthatthenumbers3and2in(8)and(9),respectively,dependonthedimensionof space.Determinethesenumbersforarbitrarydimensions.
- In part II, we touched upon the notion of a classical field theory, and in part IV we studied electromagnetic field theory.
- ∗InT.H.White’sTheOnceandFutureKing,theboyArthurdreamsofvisitingakingdomgovernedonthe principlethatwhateverisnotforbiddenismandatory.ThestoryinspiredthephysicistMurrayGell-Manntoquip thatinquantumphysicswhatisnottabooisacommandment.

In part II, we touched upon the notion of a classical field theory, and in part IV we studied electromagnetic field theory.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Summarize the most important claims and how to verify them.

EVIDENCE:
- the mathematical formalism of quantum theory as given, rather than trying to understand how it might arise from something more fundamental.
- Jaynes described the mathematical formalism of quantum physics as "a peculiar mixture describing in part realities of Nature, in part incomplete human information about Nature—all scrambled up together by Heisenberg and Bohr into an omelette that nobody has seen how to unscramble".[99]) This dissatisfaction has motivated new interpretative variants as well as technical work in quantum foundations.[65][100] See also Bohr–Einstein debates Einstein's thought experiments Fifth Solvay Conference

Copenhagen interpretation - Wikipedia.pdf

the mathematical formalism of quantum theory as given, rather than trying to understand how it might arise from something more fundamental.

Jaynes described the mathematical formalism of quantum physics as "a peculiar mixture describing in part realities of Nature, in part incomplete human information about Nature—all scrambled up together by Heisenberg and Bohr into an omelette that nobody has seen how to unscramble".[99]) This dissatisfaction has motivated new interpretative variants as well as technical work in quantum foundations.[65][100] See also Bohr–Einstein debates Einstein's thought experiments Fifth Solvay Conference

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Summarize the most important claims and how to verify them.

EVIDENCE:
- Understanding the Properties of Matter (https://books.google.com/books?id =h8BNvnR050cC&pg=PA8) (2nd ed.).
- International Journal of Modern Physics E.
- Particles and Nuclei: An Introduction to the Physical Concepts (https://books.google.com/books?id=rJe4k8tkq7sC&pg=PA103).
- Particles and Nuclei: An Introduction to the Physical Concepts (4th ed.).

Understanding the Properties of Matter (https://books.google.com/books?id =h8BNvnR050cC&pg=PA8) (2nd ed.).

International Journal of Modern Physics E.

Particles and Nuclei: An Introduction to the Physical Concepts (https://books.google.com/books?id=rJe4k8tkq7sC&pg=PA103).

Particles and Nuclei: An Introduction to the Physical Concepts (4th ed.).

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Design a retrieval plan to answer a missing detail without guessing.

EVIDENCE:
- where the prime symbol denotes a scattered electron.
- The energy of an electron beam can be more easily tuned than pion or kaon beams, making it easier to measure and calibrate hypernuclear energy levels.[45] Initially theoretically predicted in the 1980s, this method was first used experimentally in the early 2000s.[46] Hyperon capture The capture of a Ξ− baryon by a nucleus can make a Ξ− exotic atom or hypernucleus.[33] Upon capture, it changes to a ΛΛ hypernucleus or two Λ hypernuclei.[47] The disadvantage is that the Ξ− baryon is harder to make into a beam than singly strange hadrons.[48] However, an experiment at JPARC begun in 2020 will compile data on Ξ and ΛΛ hypernuclei using a similar, non-beam setup where scattered Ξ− baryons rain onto an emulsion target.[33] Similar species Kaonic nuclei The K– meson can orbit a nucleus in an exotic atom, such as in kaonic hydrogen.[49] Although the K–- proton strong interaction in kaonic hydrogen is repulsive,[50] the K––nucleus interaction is attractive for larger systems, so this meson can enter a strongly bound state closely related to a hypernucleus;[6] in particular, the K––proton–proton system is experimentally known and more tightly bound than a normal nucleus.[51] Charmed hypernuclei Nuclei containing a charm quark have been predicted theoretically since 1977,[52] and are described as charmed hypernuclei despite the possible absence of strange quarks.[53] In particular, the lightest charmed baryons, the Λ and Σ baryons,[c] are predicted to exist in bound states in charmed c c hypernuclei, and could be created in processes analogous to those used to make hypernuclei.[53] The depth of the Λ potential in nuclear matter is predicted to be 58 MeV,[53] but unlike Λ hypernuclei, c larger hypernuclei containing the positively charged Λ would be less stable than the corresponding Λ c hypernuclei due to Coulomb repulsion.
- [54] The mass difference between the Λ and the Σ + is too large c c for appreciable mixing of these baryons to occur in hypernuclei.[55] Weak decays of charmed hypernuclei have strong relativistic corrections compared to those in ordinary hypernuclei, as the energy released in the decay process is comparable to the mass of the Λ baryon.[56] Antihypernuclei In August 2024 the STAR Collaboration reported the observation of the heaviest antimatter nucleus known, antihyperhydrogen-4 consisting of one antiproton, two antineutrons and an antihyperon.[57][58][59] The anti-lambda hyperon [60] and the antihypertriton [61] have also been previously observed.

where the prime symbol denotes a scattered electron.

The energy of an electron beam can be more easily tuned than pion or kaon beams, making it easier to measure and calibrate hypernuclear energy levels.[45] Initially theoretically predicted in the 1980s, this method was first used experimentally in the early 2000s.[46] Hyperon capture The capture of a Ξ− baryon by a nucleus can make a Ξ− exotic atom or hypernucleus.[33] Upon capture, it changes to a ΛΛ hypernucleus or two Λ hypernuclei.[47] The disadvantage is that the Ξ− baryon is harder to make into a beam than singly strange hadrons.[48] However, an experiment at JPARC begun in 2020 will compile data on Ξ and ΛΛ hypernuclei using a similar, non-beam setup where scattered Ξ− baryons rain onto an emulsion target.[33] Similar species Kaonic nuclei The K– meson can orbit a nucleus in an exotic atom, such as in kaonic hydrogen.[49] Although the K–- proton strong interaction in kaonic hydrogen is repulsive,[50] the K––nucleus interaction is attractive for larger systems, so this meson can enter a strongly bound state closely related to a hypernucleus;[6] in particular, the K––proton–proton system is experimentally known and more tightly bound than a normal nucleus.[51] Charmed hypernuclei Nuclei containing a charm quark have been predicted theoretically since 1977,[52] and are described as charmed hypernuclei despite the possible absence of strange quarks.[53] In particular, the lightest charmed baryons, the Λ and Σ baryons,[c] are predicted to exist in bound states in charmed c c hypernuclei, and could be created in processes analogous to those used to make hypernuclei.[53] The depth of the Λ potential in nuclear matter is predicted to be 58 MeV,[53] but unlike Λ hypernuclei, c larger hypernuclei containing the positively charged Λ would be less stable than the corresponding Λ c hypernuclei due to Coulomb repulsion.

[54] The mass difference between the Λ and the Σ + is too large c c for appreciable mixing of these baryons to occur in hypernuclei.[55] Weak decays of charmed hypernuclei have strong relativistic corrections compared to those in ordinary hypernuclei, as the energy released in the decay process is comparable to the mass of the Λ baryon.[56] Antihypernuclei In August 2024 the STAR Collaboration reported the observation of the heaviest antimatter nucleus known, antihyperhydrogen-4 consisting of one antiproton, two antineutrons and an antihyperon.[57][58][59] The anti-lambda hyperon [60] and the antihypertriton [61] have also been previously observed.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Which parts are established vs speculative, and what evidence supports each?

EVIDENCE:
- Nonlinear gravitational clusteringcausesthepotentialΦtochangeintime,becomingdeeper(morenegative)inmorehighlyclustered regions.Photonsthattravelthroughaclusterseeaslightlydeeperpotentialwhentheyexittheclusterthan whentheyenteredit,causingthephotontobeslightlyredshifted.Figure34.3doesnotincludethenonlinear ISW effect.
- In the tight-coupling approximation, the baryon velocity on the third line approximates the photon velocity, v ≈3Θ .
- In practice, there are “early” and “late” ISW effects that arise respectively from the contributions of radiation to the background density shortly after recombination, and of dark energy (and possibly curvature) near the present time.
- (34.19) dη dη g g dη ISW early ISW late ISW The early and late time contributions to the ISW term are illustrated in Figure 34.3.

In the tight-coupling approximation, the baryon velocity on the third line approximates the photon velocity, v ≈3Θ .

In practice, there are “early” and “late” ISW effects that arise respectively from the contributions of radiation to the background density shortly after recombination, and of dark energy (and possibly curvature) near the present time.

(34.19) dη dη g g dη ISW early ISW late ISW The early and late time contributions to the ISW term are illustrated in Figure 34.3.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Summarize the most important claims and how to verify them.

EVIDENCE:
- Instead of making straight up-and-down motions, observer B sees the pulses moving along a zig-zag line.
- A time interval measured using a single clock that is motionless in a particular reference frame is called a proper time interval.[43] Fig.
- Among synchronized clocks associated with an his numerous contributions to the foundations of special observer's reference frame relativity were independent work on the mass–energy relationship, a thorough examination of the twin paradox, and investigations into rotating coordinate systems.
- The concept of time dilation is frequently light-clock to explain time dilation taught using a light-clock that is traveling in uniform inertial motion perpendicular to a line connecting the two mirrors.[39][40][41][42] (Langevin himself made use of a light-clock oriented parallel to its line of motion.[38]) Consider the scenario illustrated in Fig.

Instead of making straight up-and-down motions, observer B sees the pulses moving along a zig-zag line.

A time interval measured using a single clock that is motionless in a particular reference frame is called a proper time interval.[43] Fig.

Among synchronized clocks associated with an his numerous contributions to the foundations of special observer's reference frame relativity were independent work on the mass–energy relationship, a thorough examination of the twin paradox, and investigations into rotating coordinate systems.

The concept of time dilation is frequently light-clock to explain time dilation taught using a light-clock that is traveling in uniform inertial motion perpendicular to a line connecting the two mirrors.[39][40][41][42] (Langevin himself made use of a light-clock oriented parallel to its line of motion.[38]) Consider the scenario illustrated in Fig.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
What does the evidence say about necessary conditions and constraints?

EVIDENCE:
- isocurvature initial conditions, §30.12.
- The line increasing diagonally from bottom left to top rightisthecomovinghorizondistanceη.Abovethislinearesuperhorizonfluctuations,whosecomovingwavelengths exceed the horizon distance, while below the line are subhorizon fluctuations, whose comoving wavelengths are less thanthehorizondistance.Thedashedverticallineatcosmicscalefactoraeq≈a0/3400marksthemomentofmatterradiation equality.
- adiabatic initial conditions, §30.11; b.
- Once a fluctuation enters the horizon, the non-baryonic matter fluctuation tends to grow, whereas the radiation fluctuation tend to decay, so there is an epoch prior to matter-radiation equality where gravitational perturbations aredominatedbymatterratherthanradiationfluctuations,eventhoughradiationdominatesthebackgroundenergy density.Thedashedverticallineatarec≈a0/1100marksrecombination,wherethetemperaturecooledtothepoint that baryons changed from being mostly ionized to mostly neutral, and the Universe changed from being opaque to transparent.TheobservedCMBcomesfromthetimeofrecombination.

isocurvature initial conditions, §30.12.

The line increasing diagonally from bottom left to top rightisthecomovinghorizondistanceη.Abovethislinearesuperhorizonfluctuations,whosecomovingwavelengths exceed the horizon distance, while below the line are subhorizon fluctuations, whose comoving wavelengths are less thanthehorizondistance.Thedashedverticallineatcosmicscalefactoraeq≈a0/3400marksthemomentofmatterradiation equality.

adiabatic initial conditions, §30.11; b.

Once a fluctuation enters the horizon, the non-baryonic matter fluctuation tends to grow, whereas the radiation fluctuation tend to decay, so there is an epoch prior to matter-radiation equality where gravitational perturbations aredominatedbymatterratherthanradiationfluctuations,eventhoughradiationdominatesthebackgroundenergy density.Thedashedverticallineatarec≈a0/1100marksrecombination,wherethetemperaturecooledtothepoint that baryons changed from being mostly ionized to mostly neutral, and the Universe changed from being opaque to transparent.TheobservedCMBcomesfromthetimeofrecombination.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
What does the evidence say about necessary conditions and constraints?

EVIDENCE:
- Professor Flat ambles by again, just in time to save us a lot of work.
- ProfessorFlatpopsup,declaring,“That’seasytoprove!Supposethereisanothertensor with the same properties as the Riemann curvature tensor.
- (11) τμ τμ τμ,λσ wheretheTaylorcoefficientB is,byconstruction,symmetricundertheinterchange τμ,λσ of τ and μ or of λ and σ.
- He says: “Since youarelookingforsymmetrypropertiesofatensor,youcouldsimplygotoalocallyflat coordinatesystemaroundsomegenericpointP,justasinchapterI.6.” So,translateourcoordinatesystemsothatPisattheoriginx=0.Expanding,wewrite g (x)=η +B xλxσ +...

Professor Flat ambles by again, just in time to save us a lot of work.

ProfessorFlatpopsup,declaring,“That’seasytoprove!Supposethereisanothertensor with the same properties as the Riemann curvature tensor.

(11) τμ τμ τμ,λσ wheretheTaylorcoefficientB is,byconstruction,symmetricundertheinterchange τμ,λσ of τ and μ or of λ and σ.

He says: “Since youarelookingforsymmetrypropertiesofatensor,youcouldsimplygotoalocallyflat coordinatesystemaroundsomegenericpointP,justasinchapterI.6.” So,translateourcoordinatesystemsothatPisattheoriginx=0.Expanding,wewrite g (x)=η +B xλxσ +...

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Summarize the most important claims and how to verify them.

EVIDENCE:
- (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol.
- Sophus Lie (1893) Vorlesungen über Continuierliche Gruppen (http://neo-classical-physics.info/upl oads/3/0/6/5/3065888/lie-_infinite_continuous_groups_-_i.pdf), English translation by D.H.
- (1995) Applied Functional Analysis: Main Principles and Their Applications SpringerVerlag 5.
- Paul Cohn (1957) Lie Groups, page 58, Cambridge Tracts in Mathematics and Mathematical Physics #46 4.

(2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol.

Sophus Lie (1893) Vorlesungen über Continuierliche Gruppen (http://neo-classical-physics.info/upl oads/3/0/6/5/3065888/lie-_infinite_continuous_groups_-_i.pdf), English translation by D.H.

(1995) Applied Functional Analysis: Main Principles and Their Applications SpringerVerlag 5.

Paul Cohn (1957) Lie Groups, page 58, Cambridge Tracts in Mathematics and Mathematical Physics #46 4.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
What does the evidence say about necessary conditions and constraints?

EVIDENCE:
- that he is Ra, and he stays (b) Ra keeps all the white stone that the miners on their world as their king.
- 2 How does O’Neil find Daniel?
- How do Daniel and O’Neil use it for work in a group.
- Work in pairs or small groups.

that he is Ra, and he stays (b) Ra keeps all the white stone that the miners on their world as their king.

How do Daniel and O’Neil use it for work in a group.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Summarize the most important claims and how to verify them.

EVIDENCE:
- The Bohr model is a relatively primitive model of the hydrogen atom, compared to the valence shell model.
- However, because of its simplicity, and its correct results for selected systems (see below for application), the Bohr model is still commonly taught to introduce students to quantum mechanics or energy level diagrams before moving on to the more accurate, but more complex, valence shell atom.
- A related quantum model was proposed by Arthur Erich Haas in 1910 but was
- the fundamental physical constants that make up the formula's empirical results.

The Bohr model is a relatively primitive model of the hydrogen atom, compared to the valence shell model.

However, because of its simplicity, and its correct results for selected systems (see below for application), the Bohr model is still commonly taught to introduce students to quantum mechanics or energy level diagrams before moving on to the more accurate, but more complex, valence shell atom.

A related quantum model was proposed by Arthur Erich Haas in 1910 but was

the fundamental physical constants that make up the formula's empirical results.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
What does the evidence say about necessary conditions and constraints?

EVIDENCE:
- Observational evidence is consistent with the predictions of the theory of inflation in its simplest form —theexpansionoftheUniverse,thespatialflatnessoftheUniverse,thenearuniformityoftemperature fluctuationsoftheCMB(thehorizonproblem),thepresenceofacousticpeaksandtroughsinthepower spectrum of fluctuations of the CMB, the near power law shape of the power spectrum at large scales, its spectral index (tilt), the gaussian distribution of fluctuations at large scales.
- What is non-baryonic dark matter?
- The power spectrum of fluctuations of the CMB has enabled precise measurements of cosmological parameters.
- It is this simplicity that makes it possible to model the early Universe with some degree of confidence.

Observational evidence is consistent with the predictions of the theory of inflation in its simplest form —theexpansionoftheUniverse,thespatialflatnessoftheUniverse,thenearuniformityoftemperature fluctuationsoftheCMB(thehorizonproblem),thepresenceofacousticpeaksandtroughsinthepower spectrum of fluctuations of the CMB, the near power law shape of the power spectrum at large scales, its spectral index (tilt), the gaussian distribution of fluctuations at large scales.

The power spectrum of fluctuations of the CMB has enabled precise measurements of cosmological parameters.

It is this simplicity that makes it possible to model the early Universe with some degree of confidence.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Summarize the most important claims and how to verify them.

EVIDENCE:
- possible ends to the universe.
- After inflation ended, the universe continued to expand.
- This heating effect led to the universe being repopulated with a dense, hot mixture of Standard Model particles.
- One of the theoretical products of this phase transition was a scalar field called the inflaton field.[19] As this field settled into its lowest-energy state throughout the universe, it generated an enormous repulsive force that led to a rapid expansion of the universe.

Chronology of the universe - Wikipedia.pdf

After inflation ended, the universe continued to expand.

This heating effect led to the universe being repopulated with a dense, hot mixture of Standard Model particles.

One of the theoretical products of this phase transition was a scalar field called the inflaton field.[19] As this field settled into its lowest-energy state throughout the universe, it generated an enormous repulsive force that led to a rapid expansion of the universe.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Which parts are established vs speculative, and what evidence supports each?

EVIDENCE:
- A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity.
- A property that is invariant under such deformations is a topological property.
- The ideas underlying topology go back to Gottfried Wilhelm Leibniz, who in the 17th century envisioned the geometria situs and analysis situs.
- The term topology was introduced by Johann Benedict Listing in the 19th century, although, it was not until the first decades of the 20th century that the idea of a topological space was developed.

A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity.

A property that is invariant under such deformations is a topological property.

The ideas underlying topology go back to Gottfried Wilhelm Leibniz, who in the 17th century envisioned the geometria situs and analysis situs.

The term topology was introduced by Johann Benedict Listing in the 19th century, although, it was not until the first decades of the 20th century that the idea of a topological space was developed.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Design a retrieval plan to answer a missing detail without guessing.

EVIDENCE:
- K , (17.36) km km k m n mk kn mp mk p with R^ ≡ γlnR^ the restricted Ricci tensor.
- Contracting the Ricci tensor (17.36) yields the Ricci km klmn scalar R, R=R^−2D^ Km−KpmnK −KpK , (17.37) m nmp p with R^ ≡γkmR^ the restricted Ricci scalar.
- km A restricted covariant divergence D^ Am can be converted to a total covariant divergence D Am through m m D Am =D^ Am+Km Ap =D^ Am+K Ap .

K , (17.36) km km k m n mk kn mp mk p with R^ ≡ γlnR^ the restricted Ricci tensor.

Contracting the Ricci tensor (17.36) yields the Ricci km klmn scalar R, R=R^−2D^ Km−KpmnK −KpK , (17.37) m nmp p with R^ ≡γkmR^ the restricted Ricci scalar.

km A restricted covariant divergence D^ Am can be converted to a total covariant divergence D Am through m m D Am =D^ Am+Km Ap =D^ Am+K Ap .

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Summarize the most important claims and how to verify them.

EVIDENCE:
- The deformation gradient is given by
- The equations that govern the mechanics of such materials include the balance laws for mass, momentum, and energy.
- there is a source of the physical quantity inside the volume.
- In the continuum mechanics of solids, the second law of thermodynamics is satisfied if the Clausius–Duhem form of the entropy inequality is satisfied.

The equations that govern the mechanics of such materials include the balance laws for mass, momentum, and energy.

there is a source of the physical quantity inside the volume.

In the continuum mechanics of solids, the second law of thermodynamics is satisfied if the Clausius–Duhem form of the entropy inequality is satisfied.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Design a retrieval plan to answer a missing detail without guessing.

EVIDENCE:
- Each bookhas an introductionand extensiveactivitymaterial.Theyare published at seven levelsfrom Easystarts (200 words) to Advanced (3000 words).
- • ISBN 0-997-B0579-X 1111111 Published and distributed by Pearson EducationLimited 9 780997 805796
- PenguinReadersare simplified texts designedin association with Longman, the world famouseducational publisher.to provide a step-by-step approach to the joysof reading for pleasure.
- Will they find a way to get back to Earth?

Each bookhas an introductionand extensiveactivitymaterial.Theyare published at seven levelsfrom Easystarts (200 words) to Advanced (3000 words).

• ISBN 0-997-B0579-X 1111111 Published and distributed by Pearson EducationLimited 9 780997 805796

PenguinReadersare simplified texts designedin association with Longman, the world famouseducational publisher.to provide a step-by-step approach to the joysof reading for pleasure.

Will they find a way to get back to Earth?

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Design a retrieval plan to answer a missing detail without guessing.

EVIDENCE:
- a Bell state) is prepared using a Bell circuit or gate by Charlie, a third person.
- It is worth stressing that if Alice and Bob do not pre-share entanglement, then the superdense protocol is impossible, as this would violate Holevo's theorem.
- Superdense coding is the underlying principle of secure quantum secret coding.
- The necessity of having both qubits to decode the information being sent eliminates the risk of eavesdroppers intercepting messages.[4] Overview Suppose Alice wants to send two classical bits of information (00, 01, 10, or 11) to Bob using qubits (instead of classical bits).

a Bell state) is prepared using a Bell circuit or gate by Charlie, a third person.

It is worth stressing that if Alice and Bob do not pre-share entanglement, then the superdense protocol is impossible, as this would violate Holevo's theorem.

Superdense coding is the underlying principle of secure quantum secret coding.

The necessity of having both qubits to decode the information being sent eliminates the risk of eavesdroppers intercepting messages.[4] Overview Suppose Alice wants to send two classical bits of information (00, 01, 10, or 11) to Bob using qubits (instead of classical bits).

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Which parts are established vs speculative, and what evidence supports each?

EVIDENCE:
- McCormmach, Russell (1 January 1966).
- Bohr, Niels (7 November 1962).
- S2CID 120797894 (ht tps://api.semanticscholar.org/CorpusID:120797894).
- OCLC 557599205 (https://search.worldcat.or g/oclc/557599205).

S2CID 120797894 (ht tps://api.semanticscholar.org/CorpusID:120797894).

OCLC 557599205 (https://search.worldcat.or g/oclc/557599205).

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
What does the evidence say about necessary conditions and constraints?

EVIDENCE:
- Wright, Laura (November 6, 2003).
- Vaidman, Lev (January 17, 2014).
- "Many-Worlds Interpretation of Quantum Mechanics" (http://plat o.stanford.edu/entries/qm-manyworlds/).
- "The night the planets were aligned with Baltimore lunacy" (http s://web.archive.org/web/20081206170526/http://select.nytimes.com/gst/abstract.html?res=F70E1 3FD395F0C728DDDAA0894DA484D81).

"Many-Worlds Interpretation of Quantum Mechanics" (http://plat o.stanford.edu/entries/qm-manyworlds/).

"The night the planets were aligned with Baltimore lunacy" (http s://web.archive.org/web/20081206170526/http://select.nytimes.com/gst/abstract.html?res=F70E1 3FD395F0C728DDDAA0894DA484D81).

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Design a retrieval plan to answer a missing detail without guessing.

EVIDENCE:
- Compute the CMB power spectrum C (η ) today in your choice of the instantaneous and rapid l 0 recombination approximations, equations (34.24) or (34.25), with source functions calculated in your choice of level of detail, simple, §30.7, hydrodynamic, §32.2, or full Boltzmann, §33.1).
- If you’d like to write a code that competes with CAMB in speed, expect to invest a substantial time developing it.
- 0 rec 34.3.3 Rapid recombination approximation to the CMB power spectrum Modern, publicly available codes such as CAMB compute an entire model CMB power spectrum C (η ) in l 0 just a few seconds, which is amazingly fast.
- (34.35) l 0 l 0 ζ (2π)3 l 0 ζ k If the primordial power spectrum Δ2 is a power-law with tilt n, equation (30.137), then the CMB power ζ spectrum today is ∫ ∞ ( k )n−1 dk C (η )=4πΔ2(k ) |T (η ,k)|2 .

Compute the CMB power spectrum C (η ) today in your choice of the instantaneous and rapid l 0 recombination approximations, equations (34.24) or (34.25), with source functions calculated in your choice of level of detail, simple, §30.7, hydrodynamic, §32.2, or full Boltzmann, §33.1).

If you’d like to write a code that competes with CAMB in speed, expect to invest a substantial time developing it.

0 rec 34.3.3 Rapid recombination approximation to the CMB power spectrum Modern, publicly available codes such as CAMB compute an entire model CMB power spectrum C (η ) in l 0 just a few seconds, which is amazingly fast.

(34.35) l 0 l 0 ζ (2π)3 l 0 ζ k If the primordial power spectrum Δ2 is a power-law with tilt n, equation (30.137), then the CMB power ζ spectrum today is ∫ ∞ ( k )n−1 dk C (η )=4πΔ2(k ) |T (η ,k)|2 .

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
What does the evidence say about necessary conditions and constraints?

EVIDENCE:
- Riemannian geometry, the study of Riemannian manifolds, has deep connections to other areas of mathematics, including geometric topology, complex geometry, and algebraic geometry.
- A Riemannian manifold is a smooth manifold together with a Riemannian The dot product of two vectors tangent to metric.
- The dot products on every and parallel transport.
- Riemannian manifolds take their name from German mathematician Bernhard Riemann, who first conceptualized them in 1854.

Riemannian geometry, the study of Riemannian manifolds, has deep connections to other areas of mathematics, including geometric topology, complex geometry, and algebraic geometry.

A Riemannian manifold is a smooth manifold together with a Riemannian The dot product of two vectors tangent to metric.

The dot products on every and parallel transport.

Riemannian manifolds take their name from German mathematician Bernhard Riemann, who first conceptualized them in 1854.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Design a retrieval plan to answer a missing detail without guessing.

EVIDENCE:
- Continuum (measurement) Continuum (pl.: continua or continuums) theories or models explain variation as involving gradual quantitative transitions without abrupt changes or discontinuities.
- Bell isolates two distinct historical conceptions of infinitesimal, one by Leibniz and one by Nieuwentijdt, and argues that Leibniz's conception was implemented in Robinson's hyperreal continuum, whereas Nieuwentijdt's, in Lawvere's smooth infinitesimal analysis, characterized by the presence of nilsquare infinitesimals: "It may be said that Leibniz recognized the need for the first, but not the second type of infinitesimal and Nieuwentijdt, vice versa.
- Those levels include dichotomous (a person either has a personality trait or not) and non-dichotomous approaches.
- A spectrum in physics, such as the electromagnetic spectrum, is often termed as either continuous (with energy at all wavelengths) or discrete (energy at only certain wavelengths).

Continuum (measurement) Continuum (pl.: continua or continuums) theories or models explain variation as involving gradual quantitative transitions without abrupt changes or discontinuities.

Bell isolates two distinct historical conceptions of infinitesimal, one by Leibniz and one by Nieuwentijdt, and argues that Leibniz's conception was implemented in Robinson's hyperreal continuum, whereas Nieuwentijdt's, in Lawvere's smooth infinitesimal analysis, characterized by the presence of nilsquare infinitesimals: "It may be said that Leibniz recognized the need for the first, but not the second type of infinitesimal and Nieuwentijdt, vice versa.

Those levels include dichotomous (a person either has a personality trait or not) and non-dichotomous approaches.

A spectrum in physics, such as the electromagnetic spectrum, is often termed as either continuous (with energy at all wavelengths) or discrete (energy at only certain wavelengths).

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
What does the evidence say about necessary conditions and constraints?

EVIDENCE:
- However, gravitational stripping of the interstellar gas and dust that makes up the spiral arms produces a long train of stars known as tidal tails.
- Examples of these formations can be seen
- For example, a high redshift (z ~ 16) photometric observation by James Webb Space Telescope (JWST) was later corrected to be closer to z ~ 5.[189] Nevertheless, confirmed observations from the JWST and other observatories are accumulating, allowing systematic comparison of early galaxies to predictions of theory.[190] Evidence for individual Population III stars in early galaxies is even more challenging.
- Even seemingly confirmed spectroscopic evidence may turn out to have other origins.

However, gravitational stripping of the interstellar gas and dust that makes up the spiral arms produces a long train of stars known as tidal tails.

Examples of these formations can be seen

For example, a high redshift (z ~ 16) photometric observation by James Webb Space Telescope (JWST) was later corrected to be closer to z ~ 5.[189] Nevertheless, confirmed observations from the JWST and other observatories are accumulating, allowing systematic comparison of early galaxies to predictions of theory.[190] Evidence for individual Population III stars in early galaxies is even more challenging.

Even seemingly confirmed spectroscopic evidence may turn out to have other origins.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
What does the evidence say about necessary conditions and constraints?

EVIDENCE:
- Similarly, in the present construction the time axisγγ =iγγ− preservesthestandard-modelchargesofmassivespinorsintheirrestframes,whileitspartner 0 T γγ+ transforms massive spinors to their anti-spinor partners of opposite standard-model charge.
- E However, the experimental evidence constrains the number of neutrino types at energies below electroweak symmetry breaking, equation (42.78), so the arrangement of neutrino mass couplings could differ from the diagram (42.193) before electroweak symmetry breaking.
- WheredoestheMajoranamasstermcomefrom,inthepresentconstruction?InDiractheory,thetimeaxis γγ is diagonal acting on massive spinors in their rest frame, while the time axis’ Newman-Penrose partner γγ transforms massive spinors to their anti-spinor partners.
- Dirac mass terms, which are generated by the electroweak Higgs field, cease to operate before electroweak symmetry breaking.

Similarly, in the present construction the time axisγγ =iγγ− preservesthestandard-modelchargesofmassivespinorsintheirrestframes,whileitspartner 0 T γγ+ transforms massive spinors to their anti-spinor partners of opposite standard-model charge.

E However, the experimental evidence constrains the number of neutrino types at energies below electroweak symmetry breaking, equation (42.78), so the arrangement of neutrino mass couplings could differ from the diagram (42.193) before electroweak symmetry breaking.

WheredoestheMajoranamasstermcomefrom,inthepresentconstruction?InDiractheory,thetimeaxis γγ is diagonal acting on massive spinors in their rest frame, while the time axis’ Newman-Penrose partner γγ transforms massive spinors to their anti-spinor partners.

Dirac mass terms, which are generated by the electroweak Higgs field, cease to operate before electroweak symmetry breaking.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
What does the evidence say about necessary conditions and constraints?

EVIDENCE:
- The 1 constraint (16.77a) arises α ̄ α ̄ from minimizing the action with respect to the 1 time component A t ̄ of the coordinates, and the 3 identities (16.77b) from minimizing with respect to the 3 time components *F t ̄ of the momenta.
- Other gauge-fixing choices can be made; see §27.6 for further discussion of this issue.
- The magnetic field *F t ̄, or equivalently its dual F α ̄ , is not a gauge field (that is, it cannot be adjusted by a gauge transformation), but rather an auxiliary field that arises when the electromagnetic field is treated as a generally covariant 4-dimensional object.
- 16.6.7 3+1 split of the variation of the electromagnetic action Theequationsofmotion(16.76)andconstraintandidentities(16.77)followdirectlyfromsplittingHamilton’s equations (16.46) into time and space parts; but they can also be derived more fundamentally from splitting the variation (16.62) of the action into time and space parts, [ 1 ∮ ]tf δS = *F ∧δA (16.81) 4π ti 1 ∫ tf + 4π −(d*F −4π*j) t ̄∧δA α ̄ −(d*F −4π*j) α ̄ ∧δA t ̄+δ*F α ̄ ∧(dA−F) t ̄+δ*F t ̄∧(dA−F) α ̄ .

The 1 constraint (16.77a) arises α ̄ α ̄ from minimizing the action with respect to the 1 time component A t ̄ of the coordinates, and the 3 identities (16.77b) from minimizing with respect to the 3 time components *F t ̄ of the momenta.

Other gauge-fixing choices can be made; see §27.6 for further discussion of this issue.

The magnetic field *F t ̄, or equivalently its dual F α ̄ , is not a gauge field (that is, it cannot be adjusted by a gauge transformation), but rather an auxiliary field that arises when the electromagnetic field is treated as a generally covariant 4-dimensional object.

16.6.7 3+1 split of the variation of the electromagnetic action Theequationsofmotion(16.76)andconstraintandidentities(16.77)followdirectlyfromsplittingHamilton’s equations (16.46) into time and space parts; but they can also be derived more fundamentally from splitting the variation (16.62) of the action into time and space parts, [ 1 ∮ ]tf δS = *F ∧δA (16.81) 4π ti 1 ∫ tf + 4π −(d*F −4π*j) t ̄∧δA α ̄ −(d*F −4π*j) α ̄ ∧δA t ̄+δ*F α ̄ ∧(dA−F) t ̄+δ*F t ̄∧(dA−F) α ̄ .

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Which parts are established vs speculative, and what evidence supports each?

EVIDENCE:
- It’s Saturday afternoon, the last home game of the regular season, and she’s with Jason and Charlie, watching the Cubs get their asses kicked in their sold-out ballpark.
- The air redolent of— Roasted peanuts.
- Daniela finds the roar of the crowd strangely comforting, and they’re far enough back from home plate to notice a delay between swing and bat-crack—speed of light versus speed of sound—when a player sends a pitch sailing beyond the wall.
- Plastic cups filled to the brim with beer.

It’s Saturday afternoon, the last home game of the regular season, and she’s with Jason and Charlie, watching the Cubs get their asses kicked in their sold-out ballpark.

Daniela finds the roar of the crowd strangely comforting, and they’re far enough back from home plate to notice a delay between swing and bat-crack—speed of light versus speed of sound—when a player sends a pitch sailing beyond the wall.

Plastic cups filled to the brim with beer.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Design a retrieval plan to answer a missing detail without guessing.

EVIDENCE:
- Wood Prize Recipient | Optica" (https:// www.optica.org/about/newsroom/news_releases/2019/the_optical_society_names_jian-wei_pan_t he_2019_r_w_wood_prize_recipient/).
- "Jian-Wei Pan to Receive ZEISS Research Award" (https://www.zeiss.com/corporate/int/newsroo m/press-releases/zeiss-pressemitteilung/jian-wei-pan-to-receive-zeiss-research-award.html).
- "Jian-Wei Pan: The World's 100 Most Influential People" (https://time.com/collection/most-influenti al-people-2018/5238153/jian-wei-pan/).
- "Jian-Wei Pan Appointed Lead Editor of Physical Review Research" (https://www.aps.org/publicati ons/apsnews/updates/prr-pan.cfm).

Wood Prize Recipient | Optica" (https:// www.optica.org/about/newsroom/news_releases/2019/the_optical_society_names_jian-wei_pan_t he_2019_r_w_wood_prize_recipient/).

"Jian-Wei Pan to Receive ZEISS Research Award" (https://www.zeiss.com/corporate/int/newsroo m/press-releases/zeiss-pressemitteilung/jian-wei-pan-to-receive-zeiss-research-award.html).

"Jian-Wei Pan: The World's 100 Most Influential People" (https://time.com/collection/most-influenti al-people-2018/5238153/jian-wei-pan/).

"Jian-Wei Pan Appointed Lead Editor of Physical Review Research" (https://www.aps.org/publicati ons/apsnews/updates/prr-pan.cfm).

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Which parts are established vs speculative, and what evidence supports each?

EVIDENCE:
- 12 THE COLLAPSING UNIVERSE gained a negative charge, and the one that loses electrons can no longer balance the charge of its nucleus completely and has gained a positive charge.
- In some cases millions of atoms can form a molecule.
- Long chains and complicated rings of carbon atoms can therefore form.
- In either case, electron transfer or electron sharing, it takes considerable energy to pull the atoms apart, and under ordinary circumstances, they remain together.

12 THE COLLAPSING UNIVERSE gained a negative charge, and the one that loses electrons can no longer balance the charge of its nucleus completely and has gained a positive charge.

In some cases millions of atoms can form a molecule.

Long chains and complicated rings of carbon atoms can therefore form.

In either case, electron transfer or electron sharing, it takes considerable energy to pull the atoms apart, and under ordinary circumstances, they remain together.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Design a retrieval plan to answer a missing detail without guessing.

EVIDENCE:
- You may board now for Downtown Station.
- It shoves me deep into the cushioned seat for two seconds, and then we’re floating along a single rail at an inconceivable speed, no sense of friction beneath us
- I glance up and down the line, but I don’t see anything approaching.
- As I lean back, padded restraints emerge from the chair and gently secure my shoulders and waist.

It shoves me deep into the cushioned seat for two seconds, and then we’re floating along a single rail at an inconceivable speed, no sense of friction beneath us

I glance up and down the line, but I don’t see anything approaching.

As I lean back, padded restraints emerge from the chair and gently secure my shoulders and waist.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
What does the evidence say about necessary conditions and constraints?

EVIDENCE:
- I retreat to the sidewalk and watch the flow of traffic resume.
- I mean that.” “Jason.” “Thanks for the drink.” Back outside, I stalk down the sidewalk.
- “Well, I’m so glad I popped in for this.” I step down off the barstool.
- And I’m not even sure at whom.

I retreat to the sidewalk and watch the flow of traffic resume.

I mean that.” “Jason.” “Thanks for the drink.” Back outside, I stalk down the sidewalk.

“Well, I’m so glad I popped in for this.” I step down off the barstool.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Summarize the most important claims and how to verify them.

EVIDENCE:
- Archived (https://web.archive.org/web/20191225141623/https://books.google.com/books?id=F1Tl 9ok-7_IC) from the original on 25 December 2019.
- Paul Marriott; Mark Salmon (2000).
- Archived (https://web.archive.org/web/ 20191231233050/https://books.google.com/books?id=1Mz-BAAAQBAJ) from the original on 31 December 2019.
- S2CID 118964353 (https://api.semanticscholar.org/CorpusID:118964353).

Archived (https://web.archive.org/web/20191225141623/https://books.google.com/books?id=F1Tl 9ok-7_IC) from the original on 25 December 2019.

Archived (https://web.archive.org/web/ 20191231233050/https://books.google.com/books?id=1Mz-BAAAQBAJ) from the original on 31 December 2019.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Design a retrieval plan to answer a missing detail without guessing.

EVIDENCE:
- Flipping any axis changes the signs of all the n , so the number of positive eigenvalues can always be c chosen to be greater than or equal to the number of negative eigenvalues.
- Finally, the axes can be reordered arbitrarily.Thustheinvariantpropertiesoftheeigenvaluesn arethenumbersofnegative,zero,andpositive c eigenvalues.
- If the parameter k is non-zero, and if n and n are non-zero (Bianchi Types VI and VII), then 2 3 k cannotberescaledindependently,sincek ∝1/a ∝|n n |1/2 isfixedbythescalingofn andn .Ifonthe 1 2 3 2 3

Flipping any axis changes the signs of all the n , so the number of positive eigenvalues can always be c chosen to be greater than or equal to the number of negative eigenvalues.

Finally, the axes can be reordered arbitrarily.Thustheinvariantpropertiesoftheeigenvaluesn arethenumbersofnegative,zero,andpositive c eigenvalues.

If the parameter k is non-zero, and if n and n are non-zero (Bianchi Types VI and VII), then 2 3 k cannotberescaledindependently,sincek ∝1/a ∝|n n |1/2 isfixedbythescalingofn andn .Ifonthe 1 2 3 2 3

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Summarize the most important claims and how to verify them.

EVIDENCE:
- Using (4), we evaluate V(r =r ,l=l )=−1 + 3 − 3 =−1.
- Through dissipative processes, such as collisions between particles and electromagnetic radiation, particles in the disk gradually lose energy and angularmomentumandmoveinwarduntiltheyreachr .Fromthere,furtherlossof ISCO angularmomentumwillcausethemtofallin.Byangularmomentumconservation,this processwillcausetheblackholetorotate,sothateventuallytheSchwarzschildsolutionis nolongeradequate.InchapterVII.5wewilldiscussrotatingblackholes.
- What is the energy of a particle in the innermost stable circular orbit?
- Accretiondisks:Mightierthannuclearfusion (cid:3) (cid:5) (cid:4) √ As l decreases toward l = 3r , the stable orbit with radius r = l2 1+ 1− 3r S 2 c S min r S l2 keepsshrinkinguntilr reachesitsminimumvalueof min (cid:31) (cid:31) r =r (cid:31) =3r (8) ISCO min l=lc S Theorbitwithradiusr isknownastheinnermoststablecircularorbitinrelativistic ISCO astrophysics.NotethatitsitswelloutsidetheSchwarzschildradius.

Using (4), we evaluate V(r =r ,l=l )=−1 + 3 − 3 =−1.

Through dissipative processes, such as collisions between particles and electromagnetic radiation, particles in the disk gradually lose energy and angularmomentumandmoveinwarduntiltheyreachr .Fromthere,furtherlossof ISCO angularmomentumwillcausethemtofallin.Byangularmomentumconservation,this processwillcausetheblackholetorotate,sothateventuallytheSchwarzschildsolutionis nolongeradequate.InchapterVII.5wewilldiscussrotatingblackholes.

What is the energy of a particle in the innermost stable circular orbit?

Accretiondisks:Mightierthannuclearfusion (cid:3) (cid:5) (cid:4) √ As l decreases toward l = 3r , the stable orbit with radius r = l2 1+ 1− 3r S 2 c S min r S l2 keepsshrinkinguntilr reachesitsminimumvalueof min (cid:31) (cid:31) r =r (cid:31) =3r (8) ISCO min l=lc S Theorbitwithradiusr isknownastheinnermoststablecircularorbitinrelativistic ISCO astrophysics.NotethatitsitswelloutsidetheSchwarzschildradius.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Summarize the most important claims and how to verify them.

EVIDENCE:
- In addition, Fμν contains 6 independent components (it is true only in 4 dimensional spacetime).
- Hence, it is possible to assume that (cid:3) ≡ Fμν, (3.4) then, fμ = qFμνu .
- (3.5) ν Considering the case of pure force (a force which preserves the rest mass) as dm fμu = 0 ∝ .
- Moreover, it must be a 2-rank tensor in order to contract with a vector (u(4)) then obtain another vector (f(4)).

In addition, Fμν contains 6 independent components (it is true only in 4 dimensional spacetime).

Hence, it is possible to assume that (cid:3) ≡ Fμν, (3.4) then, fμ = qFμνu .

(3.5) ν Considering the case of pure force (a force which preserves the rest mass) as dm fμu = 0 ∝ .

Moreover, it must be a 2-rank tensor in order to contract with a vector (u(4)) then obtain another vector (f(4)).

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Design a retrieval plan to answer a missing detail without guessing.

EVIDENCE:
- 952 Cosmological perturbations including polarization The spherical harmonic components of the rotation operator are correspondingly (no sum over m′, m) D lm′m (χ′,ψ,χ)=e−im′χ′ d lm′m (ψ)e−imχ .
- (35.125) Thematrixd lm′m (ψ)isthepolarpartofthefullrotationmatrixD lm′m (χ′,ψ,χ).Thepolarrotationmatrix d lm′m (ψ) is a real matrix, orthogonal with respect to m′m, with matrix inverse d lm′m (ψ)−1 =d lm′m (−ψ)=d lmm′ (ψ)=d l,−m′,−m (ψ)=(−)m′−md lm′m (ψ) .
- (35.126) A parity transformation ψ →π−ψ flips the sign of one of the indices m′ or m and multiplies by (−)l−m or (−)l+m′ , d lm′m (π−ψ)=(−)l−md l,−m′,m (ψ)=(−)l+m′ d l,m′,−m (ψ) .
- The explicit form of the Wigner rotation matrix elements D lm′m (χ′,ψ,χ) is derived most elegantly from the Newman-Penrose components L ,L of the total angular momentum operator L, which are (Newman z ± and Penrose, 1962; Goldberg et al., 1967; Geroch, Held, and Penrose, 1973) ∂ e±iχ ( ∂ 1 ∂ cosψ ∂ ) L ≡−i , L ≡ √ ± +i +i .

952 Cosmological perturbations including polarization The spherical harmonic components of the rotation operator are correspondingly (no sum over m′, m) D lm′m (χ′,ψ,χ)=e−im′χ′ d lm′m (ψ)e−imχ .

(35.125) Thematrixd lm′m (ψ)isthepolarpartofthefullrotationmatrixD lm′m (χ′,ψ,χ).Thepolarrotationmatrix d lm′m (ψ) is a real matrix, orthogonal with respect to m′m, with matrix inverse d lm′m (ψ)−1 =d lm′m (−ψ)=d lmm′ (ψ)=d l,−m′,−m (ψ)=(−)m′−md lm′m (ψ) .

(35.126) A parity transformation ψ →π−ψ flips the sign of one of the indices m′ or m and multiplies by (−)l−m or (−)l+m′ , d lm′m (π−ψ)=(−)l−md l,−m′,m (ψ)=(−)l+m′ d l,m′,−m (ψ) .

The explicit form of the Wigner rotation matrix elements D lm′m (χ′,ψ,χ) is derived most elegantly from the Newman-Penrose components L ,L of the total angular momentum operator L, which are (Newman z ± and Penrose, 1962; Goldberg et al., 1967; Geroch, Held, and Penrose, 1973) ∂ e±iχ ( ∂ 1 ∂ cosψ ∂ ) L ≡−i , L ≡ √ ± +i +i .

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Summarize the most important claims and how to verify them.

EVIDENCE:
- Descartes's mechanics, in which interactions happened exclusively by contact.[92] Though Newton's gravity would seem to be a power of bodies, Newton himself did not admit it to be an essential property of matter.
- Carrying the logic forward more consistently, Joseph Priestley (1733– 1804) argued that corporeal properties transcend contact mechanics: chemical properties require the capacity for attraction.[92] He argued matter has other inherent powers besides the so-called primary qualities of Descartes, et al.[93] 19th and 20th centuries Since Priestley's time, there has been a massive expansion in knowledge of the constituents of the material world (viz., molecules, atoms, subatomic particles).
- In the 19th century, following the development of the periodic table, and of atomic theory, atoms were seen as being the fundamental

Descartes's mechanics, in which interactions happened exclusively by contact.[92] Though Newton's gravity would seem to be a power of bodies, Newton himself did not admit it to be an essential property of matter.

Carrying the logic forward more consistently, Joseph Priestley (1733– 1804) argued that corporeal properties transcend contact mechanics: chemical properties require the capacity for attraction.[92] He argued matter has other inherent powers besides the so-called primary qualities of Descartes, et al.[93] 19th and 20th centuries Since Priestley's time, there has been a massive expansion in knowledge of the constituents of the material world (viz., molecules, atoms, subatomic particles).

In the 19th century, following the development of the periodic table, and of atomic theory, atoms were seen as being the fundamental

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Design a retrieval plan to answer a missing detail without guessing.

EVIDENCE:
- MinkowskiandtheGeometryofSpacetime | 185 Surfaces generated by light rays, not surprisingly, form another important class of surfaces known as null surfaces.
- Wecouldutterthefollowingpeculiar-soundingstatement:thenormaltoanullsurface is a null vector that lies in the surface.
- Thenullsurfaceofthelightconehasanotherpeculiarproperty:it’sa“oneway”surface.
- μν tt rr Theothertwotangentvectorsaregivenby hμ=(0,0,r −1,0) and kμ=(0,0,0,(rsinθ) −1) (Wehavenormalizedthemtog hμhν=1andg kμkν=1.)Evidently,theyarespacelike μν μν andorthogonaltolμ,sinceg lμhν=0andg lμkν=0.Theyarealsomutuallyorthogμν μν onal,sinceg hμkν=0.Thesethree4-vectorsl,h,k furnishthethreetangentvectorsto μν thesurface.

MinkowskiandtheGeometryofSpacetime | 185 Surfaces generated by light rays, not surprisingly, form another important class of surfaces known as null surfaces.

Wecouldutterthefollowingpeculiar-soundingstatement:thenormaltoanullsurface is a null vector that lies in the surface.

μν tt rr Theothertwotangentvectorsaregivenby hμ=(0,0,r −1,0) and kμ=(0,0,0,(rsinθ) −1) (Wehavenormalizedthemtog hμhν=1andg kμkν=1.)Evidently,theyarespacelike μν μν andorthogonaltolμ,sinceg lμhν=0andg lμkν=0.Theyarealsomutuallyorthogμν μν onal,sinceg hμkν=0.Thesethree4-vectorsl,h,k furnishthethreetangentvectorsto μν thesurface.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Which parts are established vs speculative, and what evidence supports each?

EVIDENCE:
- The Mead and Nachamkin (1996) device is actually designed to extract electrical energy from the higher frequency/higher energy density ZPE modes.
- The dimensions of the optical bench used by the University of CaliforniaSan Diego is ≈ 5 m × 12 m (or ≈ 60 m2; see Mourou et al., 1998).
- vacuum ZPE can be exploited (via the Casimir Effect, for example) to extract electrical energy or actuate microelectromechanical devices (see for example, Ambjφrn and Wolfram, 1983; Forward, 1984, 1996, 1998; Puthoff, 1990, 1993; Cole and Puthoff, 1993; Milonni, 1994; Mead and Nachamkin, 1996; Lamoreaux, 1997; Chan et al., 2001, and the references cited therein).
- The power intensity of these lasers has reached the point to where they actually probe QED vacuum physics and general relativistic physics, and they have even modified the vacuum itself.

The Mead and Nachamkin (1996) device is actually designed to extract electrical energy from the higher frequency/higher energy density ZPE modes.

The dimensions of the optical bench used by the University of CaliforniaSan Diego is ≈ 5 m × 12 m (or ≈ 60 m2; see Mourou et al., 1998).

vacuum ZPE can be exploited (via the Casimir Effect, for example) to extract electrical energy or actuate microelectromechanical devices (see for example, Ambjφrn and Wolfram, 1983; Forward, 1984, 1996, 1998; Puthoff, 1990, 1993; Cole and Puthoff, 1993; Milonni, 1994; Mead and Nachamkin, 1996; Lamoreaux, 1997; Chan et al., 2001, and the references cited therein).

The power intensity of these lasers has reached the point to where they actually probe QED vacuum physics and general relativistic physics, and they have even modified the vacuum itself.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Which parts are established vs speculative, and what evidence supports each?

EVIDENCE:
- This asymmetry is due to unavoidable distortions in how spacetime coordinates map onto a Cartesian plane.
- Since the speed of light is an invariant, the worldlines of two photons passing through the origin at time still plot as 45° diagonal lines.
- The speed of light is so much larger than anything most humans encounter that some of the effects predicted by relativity are initially counterintuitive.
- In other words, given two events that are timelike separated, it is possible to find a frame in which the two events happen at the same place.

This asymmetry is due to unavoidable distortions in how spacetime coordinates map onto a Cartesian plane.

Since the speed of light is an invariant, the worldlines of two photons passing through the origin at time still plot as 45° diagonal lines.

The speed of light is so much larger than anything most humans encounter that some of the effects predicted by relativity are initially counterintuitive.

In other words, given two events that are timelike separated, it is possible to find a frame in which the two events happen at the same place.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Design a retrieval plan to answer a missing detail without guessing.

EVIDENCE:
- A collage of automated cargo spacecraft Space probes used in the past or present to resupply Space probes are robotic spacecraft that are sent to explore the International Space Station deep space, or astronomical bodies other than Earth.
- that spot and do not have to move to track the satellite.
- To avoid signal interference, international organizations have regulations for which frequency ranges or "bands" certain organizations are allowed to use.
- Being robotic eliminates the need for expensive, heavy life-support systems (the Apollo crewed Moon landings required the use of the Saturn V rocket that cost over a billion dollars per launch, adjusted for inflation) and so allows for lighter, less expensive rockets.

A collage of automated cargo spacecraft Space probes used in the past or present to resupply Space probes are robotic spacecraft that are sent to explore the International Space Station deep space, or astronomical bodies other than Earth.

that spot and do not have to move to track the satellite.

To avoid signal interference, international organizations have regulations for which frequency ranges or "bands" certain organizations are allowed to use.

Being robotic eliminates the need for expensive, heavy life-support systems (the Apollo crewed Moon landings required the use of the Saturn V rocket that cost over a billion dollars per launch, adjusted for inflation) and so allows for lighter, less expensive rockets.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Design a retrieval plan to answer a missing detail without guessing.

EVIDENCE:
- The p^ ·vvv term in the γ′ b integrand of (33.75) is odd, and vanishes on angular integration: ∫ [ 1+ 1P (p^ ·p^ ) ] p^ do γ′ =0 .
- (33.74) 16πm2 γ γ′ b γ γ′ 4π e Finally,thecollisionintegral(33.74)mustbeintegratedoverthedirectionp^ ofthescatteredphoton.The γ′ integrationisfacilitatediftheangulardependenceoftheamplitudesquared⟨|M|2⟩givenbyequation(33.55) is expanded in Legendre polynomials, 1(1+μ2) = 2[ 1+ 1P (μ) ] .
- 892 Cosmological perturbations: Boltzmann treatment Integration over the electron momentum brings the collision integral (33.71) to C[Θ]= n ̄ e a ∫ ⟨|M|2⟩ [ (p^ −p^ )·vvv −Θ(p^ )+Θ(p^ ) ]do γ′ .
- To perform the remaining angular integral over the photon direction b b p^ , expand the Legendre polynomial P (p^ ·p^ ) in the integrand in spherical harmonics using the addition γ′ 2 γ γ′ theorem (33.103), expand the temperature fluctuation Θ(k,p^ ) in scalar multipole moments according to γ′ equation (33.47), and invoke orthogonality of the spherical harmonics.

The p^ ·vvv term in the γ′ b integrand of (33.75) is odd, and vanishes on angular integration: ∫ [ 1+ 1P (p^ ·p^ ) ] p^ do γ′ =0 .

(33.74) 16πm2 γ γ′ b γ γ′ 4π e Finally,thecollisionintegral(33.74)mustbeintegratedoverthedirectionp^ ofthescatteredphoton.The γ′ integrationisfacilitatediftheangulardependenceoftheamplitudesquared⟨|M|2⟩givenbyequation(33.55) is expanded in Legendre polynomials, 1(1+μ2) = 2[ 1+ 1P (μ) ] .

892 Cosmological perturbations: Boltzmann treatment Integration over the electron momentum brings the collision integral (33.71) to C[Θ]= n ̄ e a ∫ ⟨|M|2⟩ [ (p^ −p^ )·vvv −Θ(p^ )+Θ(p^ ) ]do γ′ .

To perform the remaining angular integral over the photon direction b b p^ , expand the Legendre polynomial P (p^ ·p^ ) in the integrand in spherical harmonics using the addition γ′ 2 γ γ′ theorem (33.103), expand the temperature fluctuation Θ(k,p^ ) in scalar multipole moments according to γ′ equation (33.47), and invoke orthogonality of the spherical harmonics.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Summarize the most important claims and how to verify them.

EVIDENCE:
- 862 Cosmological perturbations: the hydrodynamic approximation primarilyinthescaleoftheBAO.InFigure10.4theyallowedforpossiblescale-dependentbiasandincipient non-linearity by applying a more or less arbitrary polynomial correction to the model power spectrum.
- Computetheeffectof1ormorespeciesofmassiveneutrinoonthematterpowerspectrum.Forsimplicity, assume an abrupt transition of the neutrino evolution equations from relativistic to non-relativistic.
- Since decoupling, neutrinos free-streamed, with particle momenta p and temperature T redshifting as p ∝ T ∝ a−1.
- (32.18) 3 ∂lnm An approximation good to 1% for the density ρ(m,T), and which yields an approximation good to 4% for the pressure p(m,T) given in terms of ρ(m,T) by formula (32.18), is √ 1+βμ2+γα2μ4 R(μ)≈ , (32.19) 1+γμ2 where the constants α (equation (32.16)), β, γ are chosen such that both density ρ and pressure p have the correct asymptotic behaviour at both μ→0 and μ→∞, 10 10 [ −7π2+3240ζ(3)2] β = +γ =0.2902 , γ = =0.1454 .

862 Cosmological perturbations: the hydrodynamic approximation primarilyinthescaleoftheBAO.InFigure10.4theyallowedforpossiblescale-dependentbiasandincipient non-linearity by applying a more or less arbitrary polynomial correction to the model power spectrum.

Computetheeffectof1ormorespeciesofmassiveneutrinoonthematterpowerspectrum.Forsimplicity, assume an abrupt transition of the neutrino evolution equations from relativistic to non-relativistic.

Since decoupling, neutrinos free-streamed, with particle momenta p and temperature T redshifting as p ∝ T ∝ a−1.

(32.18) 3 ∂lnm An approximation good to 1% for the density ρ(m,T), and which yields an approximation good to 4% for the pressure p(m,T) given in terms of ρ(m,T) by formula (32.18), is √ 1+βμ2+γα2μ4 R(μ)≈ , (32.19) 1+γμ2 where the constants α (equation (32.16)), β, γ are chosen such that both density ρ and pressure p have the correct asymptotic behaviour at both μ→0 and μ→∞, 10 10 [ −7π2+3240ζ(3)2] β = +γ =0.2902 , γ = =0.1454 .

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Summarize the most important claims and how to verify them.

EVIDENCE:
- 18.6 Hypersurface-orthogonality for a timelike congruence Singularity theorems consider special congruences that are both geodesic and vorticity-free.
- The Sachs coefficients characterize how the shape of the congruence of light rays evolves as it propagates, as illustrated in Figure 18.1.
- The Raychaudhuriequation(18.18b)guaranteesthatifthevorticityπ vanishesontheinitial3-dimensionalhypersurface ab ofatimelikegeodesiccongruence,thenthevorticitywillvanishidenticallyeverywherealongthecongruence.
- Can vorticity be non-zero while shear vanishes?Answer.Yes.TheprincipalnullcongruencesoftheΛ-Kerr-Newmangeometryprovideanexampleofcongruencesthathavenon-zero vorticity but are shear-free, Exercise 23.11.

18.6 Hypersurface-orthogonality for a timelike congruence Singularity theorems consider special congruences that are both geodesic and vorticity-free.

The Sachs coefficients characterize how the shape of the congruence of light rays evolves as it propagates, as illustrated in Figure 18.1.

Can vorticity be non-zero while shear vanishes?Answer.Yes.TheprincipalnullcongruencesoftheΛ-Kerr-Newmangeometryprovideanexampleofcongruencesthathavenon-zero vorticity but are shear-free, Exercise 23.11.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
What does the evidence say about necessary conditions and constraints?

EVIDENCE:
- Some researchers have proposed that when a black hole forms, a Big Bang may occur at the core/singularity, which would create a new universe that expands outside of the parent universe.[14][15][16] The Einstein–Cartan–Sciama–Kibble theory of gravity extends general relativity by removing a constraint of the symmetry of the affine connection and regarding its antisymmetric part, the torsion tensor, as a dynamical variable.
- To be able to exist, a white hole must either arise from a physical process leading to its formation, or be present from the creation of the universe.
- Torsion naturally accounts for the quantum-mechanical, intrinsic angular momentum (spin) of matter.
- another example of a non-deterministic configuration, but does not have the status of a white hole, however, because there is no region inaccessible from a given region.

Some researchers have proposed that when a black hole forms, a Big Bang may occur at the core/singularity, which would create a new universe that expands outside of the parent universe.[14][15][16] The Einstein–Cartan–Sciama–Kibble theory of gravity extends general relativity by removing a constraint of the symmetry of the affine connection and regarding its antisymmetric part, the torsion tensor, as a dynamical variable.

To be able to exist, a white hole must either arise from a physical process leading to its formation, or be present from the creation of the universe.

Torsion naturally accounts for the quantum-mechanical, intrinsic angular momentum (spin) of matter.

another example of a non-deterministic configuration, but does not have the status of a white hole, however, because there is no region inaccessible from a given region.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Summarize the most important claims and how to verify them.

EVIDENCE:
- (23.102) Of course, there are trajectories with negative energy E in the outer ergosphere, but these trajectories are not circular.
- (Concept question: p Why?)Butthesamelargecharge-to-massratiomeansthatqQcouldbeappreciableinspiteofthesmallness of the black hole charge Q.
- Areprincipalnullgeodesicscircularorbits?Outgoingprincipalnullgeodesics hold steady on the outer horizon, remaining at constant r = r as time t goes by.
- If a massive particle is put on the outer horizon on a relativistic geodesic, then the massive particle necessarily falls off the horizon into the black hole in a finite proper time: it is impossible for the geodesic to hold steady on the horizon.

(23.102) Of course, there are trajectories with negative energy E in the outer ergosphere, but these trajectories are not circular.

(Concept question: p Why?)Butthesamelargecharge-to-massratiomeansthatqQcouldbeappreciableinspiteofthesmallness of the black hole charge Q.

Areprincipalnullgeodesicscircularorbits?Outgoingprincipalnullgeodesics hold steady on the outer horizon, remaining at constant r = r as time t goes by.

If a massive particle is put on the outer horizon on a relativistic geodesic, then the massive particle necessarily falls off the horizon into the black hole in a finite proper time: it is impossible for the geodesic to hold steady on the horizon.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Summarize the most important claims and how to verify them.

EVIDENCE:
- Answer:Young researchers played a pivotal role in advancing black hole research during the golden age.
- They brought fresh perspectives, engaged in collaborative work, and led many theoretical advancements that established the foundational framework of black hole physics, including active contributions to understanding black hole properties and related phenomena.
- 5.Question What was the role of young researchers in the golden age of black hole research?
- those undergoing mergers or pulsations.

Answer:Young researchers played a pivotal role in advancing black hole research during the golden age.

They brought fresh perspectives, engaged in collaborative work, and led many theoretical advancements that established the foundational framework of black hole physics, including active contributions to understanding black hole properties and related phenomena.

5.Question What was the role of young researchers in the golden age of black hole research?

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
What does the evidence say about necessary conditions and constraints?

EVIDENCE:
- SpaceandTimeUnified t 2 C B (1, 1) x A Figure1 Astraightlinebetweentwopointsin spacetimeisnotnecessarilythepathofshortest distance.Inthetriangleshown,thesumofthe lengthofthetwosides,ABandBC,islessthan thelengthofthesideAC.
- ∗Likeobjectsrepelinelectromagnetismandattractingravity.Thisamazingfact,whichtoalargeextentis responsibleforthephysicalworldasweknowit,canbeexplainedintermsofthissign.SeeQFTNut,p.37.
- “Astubbornlypersistentillusion”(?) With this most valiant piece of chalk I might project upon the blackboard four world-axes.
- Thewholeuniverseisseentoresolveitselfinto similarworld-lines,andIwouldfainanticipatemyselfbysaying thatinmyopinionphysicallawsmightfindtheirmostperfect expressionasreciprocalrelationsbetweentheseworld-lines.

SpaceandTimeUnified t 2 C B (1, 1) x A Figure1 Astraightlinebetweentwopointsin spacetimeisnotnecessarilythepathofshortest distance.Inthetriangleshown,thesumofthe lengthofthetwosides,ABandBC,islessthan thelengthofthesideAC.

“Astubbornlypersistentillusion”(?) With this most valiant piece of chalk I might project upon the blackboard four world-axes.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Design a retrieval plan to answer a missing detail without guessing.

EVIDENCE:
- Does the result agree with what you would expect from equation (2.149)?
- Equation of motion of a massive particle.
- Confirmthatthenon-zeroconnectioncoefficientsare(coefficientsasbelowbutwiththelasttwoindices swapped are the same by the no-torsion condition Γκ =Γκ ) μν νμ ∂Φ Γt =Γα =Γα =−Γβ =−Γα = (α̸=β =x,y,z) .
- [Hint: Work to linear t order in Φ.

Does the result agree with what you would expect from equation (2.149)?

Equation of motion of a massive particle.

Confirmthatthenon-zeroconnectioncoefficientsare(coefficientsasbelowbutwiththelasttwoindices swapped are the same by the no-torsion condition Γκ =Γκ ) μν νμ ∂Φ Γt =Γα =Γα =−Γβ =−Γα = (α̸=β =x,y,z) .

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Design a retrieval plan to answer a missing detail without guessing.

EVIDENCE:
- Determinant of a Lorentz transformation.
- This section collects three famous paradoxes: time dilation, Lorentz contraction, and the Twin paradox.
- The proper time along the interval is √ √ t τ = t2−r2 =t 1−v2 = .
- 1.7.2 Fitzgerald-Lorentz contraction Considerarocketofproperlengthl,sothatintherocket’sownrestframe(primed)thebackandfrontends of the rocket move through time t′ with coordinates {t′,x′}={t′,0} and {t′,l} .

Determinant of a Lorentz transformation.

This section collects three famous paradoxes: time dilation, Lorentz contraction, and the Twin paradox.

The proper time along the interval is √ √ t τ = t2−r2 =t 1−v2 = .

1.7.2 Fitzgerald-Lorentz contraction Considerarocketofproperlengthl,sothatintherocket’sownrestframe(primed)thebackandfrontends of the rocket move through time t′ with coordinates {t′,x′}={t′,0} and {t′,l} .

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Summarize the most important claims and how to verify them.

EVIDENCE:
- The distance travelled by light from the planet (or its moon) to Earth is shorter when the Earth is at the point in its orbit that is closest to its planet than when the Earth is at the farthest point in its orbit, the difference in distance being the diameter of the Earth's orbit around the Sun.
- Ole Rømer used an astronomical measurement to make the first quantitative estimate of the speed of light in the year 1676.[101][102] When measured from Earth, the periods of moons orbiting a distant planet are shorter when the Earth is approaching the planet than when the Earth is receding from it.
- distance in the Solar System, such as the radius of the Earth's orbit.
- Historically, such measurements could be made fairly accurately, compared to how accurately the length of the reference distance is known in Earth-based units.

The distance travelled by light from the planet (or its moon) to Earth is shorter when the Earth is at the point in its orbit that is closest to its planet than when the Earth is at the farthest point in its orbit, the difference in distance being the diameter of the Earth's orbit around the Sun.

Ole Rømer used an astronomical measurement to make the first quantitative estimate of the speed of light in the year 1676.[101][102] When measured from Earth, the periods of moons orbiting a distant planet are shorter when the Earth is approaching the planet than when the Earth is receding from it.

distance in the Solar System, such as the radius of the Earth's orbit.

Historically, such measurements could be made fairly accurately, compared to how accurately the length of the reference distance is known in Earth-based units.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
What does the evidence say about necessary conditions and constraints?

EVIDENCE:
- Given a fixed connection on and a smooth curve , there is a unique operator , called the covariant derivative along , such that:[19] 1.
- Levi-Civita connection Two Riemannian manifolds with different connections have different geometry.
- Once a Riemannian metric is fixed, there exists a unique Levi-Civita connection.[17] Note that the definition of preserving the metric uses the regularity of .
- Thankfully, there is a natural connection associated to a Riemannian manifold called the Levi-Civita connection.

Given a fixed connection on and a smooth curve , there is a unique operator , called the covariant derivative along , such that:[19] 1.

Levi-Civita connection Two Riemannian manifolds with different connections have different geometry.

Once a Riemannian metric is fixed, there exists a unique Levi-Civita connection.[17] Note that the definition of preserving the metric uses the regularity of .

Thankfully, there is a natural connection associated to a Riemannian manifold called the Levi-Civita connection.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.