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

EVIDENCE:
- a timelike vector and a spacelike vector are multiplied by i, so that rotations in a time-space plane take the exponential form eθ/2, rather than being rotations by a phase, e−iθ/2.
- The orthonormal basis bivectors remain traceless and unitary, but whereas time-time and space-space bivectors remain skew-Hermitian, the time-space bivectors become Hermitian.
- The rotor group in K spatial dimensions and M time dimensions is called Spin(K,M).
- The construction (38.109) described in Exercise 38.3 embeds Spin(K,M) as a subgroup of the group SL(2[N/2],C), where N ≡ K+M, of complex 2[N/2]×2[N/2] matrices of unit determinant, Spin(K,M) ⊆ SL(2[N/2],C) , N ≡K+M .

a timelike vector and a spacelike vector are multiplied by i, so that rotations in a time-space plane take the exponential form eθ/2, rather than being rotations by a phase, e−iθ/2.

The orthonormal basis bivectors remain traceless and unitary, but whereas time-time and space-space bivectors remain skew-Hermitian, the time-space bivectors become Hermitian.

The rotor group in K spatial dimensions and M time dimensions is called Spin(K,M).

The construction (38.109) described in Exercise 38.3 embeds Spin(K,M) as a subgroup of the group SL(2[N/2],C), where N ≡ K+M, of complex 2[N/2]×2[N/2] matrices of unit determinant, Spin(K,M) ⊆ SL(2[N/2],C) , N ≡K+M .

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:
- Earth (5.972 x 1024 kg), M is the mass of the astronomical body, and r.~ is the Schwarzschild radius of the astronomical body.8 Note that rs is only a convenient radial distance parameter for any object under examination and so there is no black hole collapse involved in this analysis.
- UNCLASSlfIEDllrSR errI@IAL Ita!
- Any radial distance from the body in question can be chosen to perform this analysis, but using rs makes the equation simpler in form.
- Also note that Equation (6) contains an extra factor of two (compared to the j derived in Reference 21) in order to account for the photon spin.

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.

Earth (5.972 x 1024 kg), M is the mass of the astronomical body, and r.~ is the Schwarzschild radius of the astronomical body.8 Note that rs is only a convenient radial distance parameter for any object under examination and so there is no black hole collapse involved in this analysis.

Any radial distance from the body in question can be chosen to perform this analysis, but using rs makes the equation simpler in form.

Also note that Equation (6) contains an extra factor of two (compared to the j derived in Reference 21) in order to account for the photon spin.

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:
- rather than a connected particle aggregation.
- These are usually abstracted as point-like particles, even though they have volumes, shapes, structures, etc.
- Particle In the physical sciences, a particle (or corpuscle in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass.[1][2] They vary greatly in size or quantity, from subatomic particles like the electron, to microscopic particles like atoms and molecules, to macroscopic particles like powders and other granular materials.
- They gradually strip the baseball of most of its properties, by first idealizing it as a rigid Particles are often smooth sphere, then by neglecting rotation, buoyancy and friction, represented as dots.

rather than a connected particle aggregation.

These are usually abstracted as point-like particles, even though they have volumes, shapes, structures, etc.

Particle In the physical sciences, a particle (or corpuscle in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass.[1][2] They vary greatly in size or quantity, from subatomic particles like the electron, to microscopic particles like atoms and molecules, to macroscopic particles like powders and other granular materials.

They gradually strip the baseball of most of its properties, by first idealizing it as a rigid Particles are often smooth sphere, then by neglecting rotation, buoyancy and friction, represented as dots.

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:
- Considerthebottomface3ofthecube.Whenweturnedonc −1wewentfromNewtonian mechanicstospecialrelativity.Whenweturnedon(cid:2),wewentfromNewtonianmechanics toquantummechanics.Whenweturnedonbothc −1and(cid:2),wearrivedatquantumfield theory,inmyopinionthegreatestmonumentof20thcenturyphysics.
- NewtonhimselfhadalreadymoveduptheverticalaxisfromNewtonianmechanicsto NewtoniangravitybyturningonG.Turningonc −1,Einsteintookusfromthatcornerto Einsteingravity,themainsubjectofthisbook.‡AlltheSt ̈urmundDrangofthepastfew decadesistheattempttocrossfromthatcornertotheHolyGrailofquantumgravity,when (glorygloryhallelujah!)allthreefundamentalconstantsareturnedon.§ ∗ These days, fundamental principles are posted on the physics archive with abandon.
- Twofascinatingexperimentsinthisareaare:(1)dribblingneutronslikebasketballs,and(2)interferinganeutron beamwithitselfinagravitationalfield.4 §Thisstatementcarriesaslightcaveat,whichwewillcometoinchapterVII.3.
- †BythisImeanthethreelaws,F=maandsoon,notincludingthelawofuniversalgravitation.

NewtonhimselfhadalreadymoveduptheverticalaxisfromNewtonianmechanicsto NewtoniangravitybyturningonG.Turningonc −1,Einsteintookusfromthatcornerto Einsteingravity,themainsubjectofthisbook.‡AlltheSt ̈urmundDrangofthepastfew decadesistheattempttocrossfromthatcornertotheHolyGrailofquantumgravity,when (glorygloryhallelujah!)allthreefundamentalconstantsareturnedon.§ ∗ These days, fundamental principles are posted on the physics archive with abandon.

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:
- Leonard Susskind The result was widely advertised by Murray Gell-Mann, leading Gabriele Veneziano to construct a scattering amplitude that had the property of Dolen–Horn–Schmid duality, later renamed world-sheet duality.
- In the t-channel, the particles exchange intermediate states by emission and absorption.
- Schmid developed some sum rules for hadron exchange.
- Over the next years, hundreds of physicists worked to complete the bootstrap program for this model, with many surprises.

Leonard Susskind The result was widely advertised by Murray Gell-Mann, leading Gabriele Veneziano to construct a scattering amplitude that had the property of Dolen–Horn–Schmid duality, later renamed world-sheet duality.

In the t-channel, the particles exchange intermediate states by emission and absorption.

Schmid developed some sum rules for hadron exchange.

Over the next years, hundreds of physicists worked to complete the bootstrap program for this model, with many surprises.

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 boy stood below it, lifting his arms to the light.
- S(I;IIthatshelencu; andDanielS,ll!' inuncdiatclywhatitIms.
- 'How did you find us?' Daniel said.
- They saw writingalloverthe walls inside,and assoon asDaniel saw it he was sure: Sha'uri's language was almost the same as Egyptian.

A boy stood below it, lifting his arms to the light.

They saw writingalloverthe walls inside,and assoon asDaniel saw it he was sure: Sha'uri's language was almost the same as Egyptian.

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:
- doi:10.1119/1.16243 (https://doi.org/1 0.1119%2F1.16243).
- "Yale researchers 'teleport' a quantum gate" (https://news.yale.
- "Multiparticle Interferometry and the Superposition Principle" (https://physicstoday.scitation.org/doi/10.1063/1.881360).
- Archived (https://web.archive.org/we b/20210423204829/https://physicstoday.scitation.org/doi/10.1063/1.881360) from the original on 23 April 2021.

"Yale researchers 'teleport' a quantum gate" (https://news.yale.

"Multiparticle Interferometry and the Superposition Principle" (https://physicstoday.scitation.org/doi/10.1063/1.881360).

Archived (https://web.archive.org/we b/20210423204829/https://physicstoday.scitation.org/doi/10.1063/1.881360) from the original on 23 April 2021.

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

EVIDENCE:
- μν Inotherwords,weaskwhetherg μν (x) ∂ ∂ x x (cid:5) μ ρ∂ ∂ x x (cid:5) ν σ =(cid:24)2(x (cid:5) )g ρσ (x (cid:5) ),aneasier-goingversion of (IX.6.1), has any solutions.
- As before, we content ourselves with an infinitesimal transformationx (cid:5)μ=xμ+εξμ(x).Inthesmallεlimit,weexpand(cid:24)2(x (cid:5) )(cid:16)1+εκ(x (cid:5) )= 1+εκ(x)+O(ε2).
- Suppose we feel more relaxed, and, instead, impose the more ρσ ρσ forgivingconditionthatg (cid:5) (x (cid:5) )=(cid:24)2(x (cid:5) )g (x (cid:5) )forsomeunknownfunction(cid:24).Tousea ρσ ρσ (cid:5) terminology first introduced in chapter I.6, we do not demand that the two metrics g μν andg areequal,butmerelythattheyareconformallyrelated.
- RecallthatinchapterIX.6,anisometryisdefinedasatransformationx→x (cid:5) (x)under which g (cid:5) (x (cid:5) )=g (x (cid:5) ).

μν Inotherwords,weaskwhetherg μν (x) ∂ ∂ x x (cid:5) μ ρ∂ ∂ x x (cid:5) ν σ =(cid:24)2(x (cid:5) )g ρσ (x (cid:5) ),aneasier-goingversion of (IX.6.1), has any solutions.

As before, we content ourselves with an infinitesimal transformationx (cid:5)μ=xμ+εξμ(x).Inthesmallεlimit,weexpand(cid:24)2(x (cid:5) )(cid:16)1+εκ(x (cid:5) )= 1+εκ(x)+O(ε2).

Suppose we feel more relaxed, and, instead, impose the more ρσ ρσ forgivingconditionthatg (cid:5) (x (cid:5) )=(cid:24)2(x (cid:5) )g (x (cid:5) )forsomeunknownfunction(cid:24).Tousea ρσ ρσ (cid:5) terminology first introduced in chapter I.6, we do not demand that the two metrics g μν andg areequal,butmerelythattheyareconformallyrelated.

RecallthatinchapterIX.6,anisometryisdefinedasatransformationx→x (cid:5) (x)under which g (cid:5) (x (cid:5) )=g (x (cid:5) ).

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:
- transfer where there is a task of translating research knowledge to be application in practice, and ensuring that practitioners are made aware of it.
- Academics have been criticized for not attempting to transfer the knowledge they produce to practitioners.[15]: 804 [16] Another framing supposes that theory and knowledge seek to understand different problems and model the world in different words (using different ontologies and epistemologies).
- Another framing says that research does not produce theory that is relevant to practice.[15]: 803 In the context of management, Van de Van and Johnson propose a form of engaged scholarship where scholars examine problems that occur in practice, in an interdisciplinary fashion, producing results that create both new practical results as well as new theoretical models, but targeting theoretical results shared in an academic fashion.[15]: 815 They use a metaphor of "arbitrage" of ideas between disciplines, distinguishing it from collaboration.[15]: 803

transfer where there is a task of translating research knowledge to be application in practice, and ensuring that practitioners are made aware of it.

Academics have been criticized for not attempting to transfer the knowledge they produce to practitioners.[15]: 804 [16] Another framing supposes that theory and knowledge seek to understand different problems and model the world in different words (using different ontologies and epistemologies).

Another framing says that research does not produce theory that is relevant to practice.[15]: 803 In the context of management, Van de Van and Johnson propose a form of engaged scholarship where scholars examine problems that occur in practice, in an interdisciplinary fashion, producing results that create both new practical results as well as new theoretical models, but targeting theoretical results shared in an academic fashion.[15]: 815 They use a metaphor of "arbitrage" of ideas between disciplines, distinguishing it from collaboration.[15]: 803

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:
- "Cosmological perturbations: Myths and facts".
- Bibcode:2016A&A...594A..20P (https://ui.adsabs.harvard.edu/abs/2016A&A...594A..20P).
- Bibcode:2006PhRvL..96k1301B (https://ui.adsabs.harvard.edu/abs/2006PhRvL..96k1301B).
- "The Road to Precision Cosmology" (https://www.annual reviews.org/doi/10.1146/annurev-nucl-111119-041046).

"Cosmological perturbations: Myths and facts".

"The Road to Precision Cosmology" (https://www.annual reviews.org/doi/10.1146/annurev-nucl-111119-041046).

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:
- According to Newton's law of gravity, and independently verified by experiments such as that of Eötvös and its successors (see Eötvös experiment), there is a universality of free fall (also According to general relativity, objects in known as the weak equivalence principle, or the universal a gravitational field behave similarly to equality of inertial and passive-gravitational mass): the objects within an accelerating enclosure.
- As can be shown using simple thought experiments following the free-fall trajectories of different test particles, the result of transporting spacetime vectors that can denote a particle's velocity (time-like vectors) will vary with the particle's trajectory; mathematically speaking, the Newtonian connection is not integrable.
- impossible to decide, by mapping the trajectory of bodies such as a dropped ball, whether the room is stationary in a gravitational field and the ball accelerating, or in free space aboard a rocket that is accelerating at a rate equal to that of the gravitational field versus the ball which upon release has nil acceleration.[29] Given the universality of free fall, there is no observable distinction between inertial motion and motion under the influence of the gravitational force.
- However, spacetime as a whole is more complicated.

According to Newton's law of gravity, and independently verified by experiments such as that of Eötvös and its successors (see Eötvös experiment), there is a universality of free fall (also According to general relativity, objects in known as the weak equivalence principle, or the universal a gravitational field behave similarly to equality of inertial and passive-gravitational mass): the objects within an accelerating enclosure.

As can be shown using simple thought experiments following the free-fall trajectories of different test particles, the result of transporting spacetime vectors that can denote a particle's velocity (time-like vectors) will vary with the particle's trajectory; mathematically speaking, the Newtonian connection is not integrable.

impossible to decide, by mapping the trajectory of bodies such as a dropped ball, whether the room is stationary in a gravitational field and the ball accelerating, or in free space aboard a rocket that is accelerating at a rate equal to that of the gravitational field versus the ball which upon release has nil acceleration.[29] Given the universality of free fall, there is no observable distinction between inertial motion and motion under the influence of the gravitational force.

However, spacetime as a whole is more complicated.

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 Universe passes through the Big Crunch and emerges in a hot Big Bang phase.
- The minimal coupling between torsion and Dirac spinors generates a spin-spin interaction that is significant in fermionic matter at extremely high densities.
- Inflation and loop quantum gravity When investigating the effects the theory of loop quantum gravity would have on cosmology, a loop quantum cosmology model has evolved that provides a possible mechanism for cosmological inflation.
- As the density of the Universe decreases, the effects of torsion weaken and the Universe smoothly enters the radiation-dominated era.

The Universe passes through the Big Crunch and emerges in a hot Big Bang phase.

The minimal coupling between torsion and Dirac spinors generates a spin-spin interaction that is significant in fermionic matter at extremely high densities.

Inflation and loop quantum gravity When investigating the effects the theory of loop quantum gravity would have on cosmology, a loop quantum cosmology model has evolved that provides a possible mechanism for cosmological inflation.

As the density of the Universe decreases, the effects of torsion weaken and the Universe smoothly enters the radiation-dominated era.

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:
- Kinematics Kinematics is a subfield of physics and a branch of geometry.
- Overview Kinematics is a subfield of physics and mathematics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move.[1][2][3] Kinematics differs from dynamics (also known as kinetics) which studies the effect of forces on bodies.
- Kinematics is concerned with systems of specification of objects' positions and velocities and mathematical transformations between such systems.
- To quantify the properties of space, he compared the dimensions of a body when it was in motion versus when it was at rest.[7] Another way to describe kinematics is as the specification of the possible states of a physical system.

Kinematics Kinematics is a subfield of physics and a branch of geometry.

Overview Kinematics is a subfield of physics and mathematics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move.[1][2][3] Kinematics differs from dynamics (also known as kinetics) which studies the effect of forces on bodies.

Kinematics is concerned with systems of specification of objects' positions and velocities and mathematical transformations between such systems.

To quantify the properties of space, he compared the dimensions of a body when it was in motion versus when it was at rest.[7] Another way to describe kinematics is as the specification of the possible states of a physical system.

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:
- As all known stars rotate and realistic collisions have nonzero angular momentum, it is expected that all black holes in nature are rotating black holes.[1][2] Since observed astronomical objects do not possess an appreciable net electric charge, only the Kerr solution has astrophysical relevance.
- Note that astrophysical black holes are expected to have non-zero angular momentum, due to their formation via collapse of rotating stellar objects, but effectively zero charge, since any net charge will quickly attract the opposite charge and neutralize.
- For this reason the term "astrophysical" black hole is usually reserved for the Kerr black hole.[5] While from an infalling observer's perspective the plunge into a rotating black hole occurs in a finite proper time and with very high rapidity (left), from Formation the perspective of a coordinate observer at infinity they slow down, approaching zero velocity at the horizon relative to a stationary probe on site while Rotating black holes are formed in the gravitational being whirled around forever by the black hole's collapse of a massive spinning star or from the collapse frame-dragging effect (right).
- Relation with gamma ray bursts The formation of a rotating black hole by a collapsar is thought to be observed as the emission of gamma ray bursts.

As all known stars rotate and realistic collisions have nonzero angular momentum, it is expected that all black holes in nature are rotating black holes.[1][2] Since observed astronomical objects do not possess an appreciable net electric charge, only the Kerr solution has astrophysical relevance.

Note that astrophysical black holes are expected to have non-zero angular momentum, due to their formation via collapse of rotating stellar objects, but effectively zero charge, since any net charge will quickly attract the opposite charge and neutralize.

For this reason the term "astrophysical" black hole is usually reserved for the Kerr black hole.[5] While from an infalling observer's perspective the plunge into a rotating black hole occurs in a finite proper time and with very high rapidity (left), from Formation the perspective of a coordinate observer at infinity they slow down, approaching zero velocity at the horizon relative to a stationary probe on site while Rotating black holes are formed in the gravitational being whirled around forever by the black hole's collapse of a massive spinning star or from the collapse frame-dragging effect (right).

Relation with gamma ray bursts The formation of a rotating black hole by a collapsar is thought to be observed as the emission of gamma ray bursts.

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 decay energy is initially released as the energy of emitted photons plus the kinetic energy of massive emitted particles (that is, particles that have rest mass).
- This is a restatement of the equivalent laws of conservation of energy and conservation of mass.
- Types Radioactive decay results in a reduction of summed rest mass, once the released energy (the disintegration energy) has escaped in some way.
- This is true because the decay energy must always carry mass with it, wherever it appears (see mass in special relativity) according to the formula E = mc2.

The decay energy is initially released as the energy of emitted photons plus the kinetic energy of massive emitted particles (that is, particles that have rest mass).

This is a restatement of the equivalent laws of conservation of energy and conservation of mass.

Types Radioactive decay results in a reduction of summed rest mass, once the released energy (the disintegration energy) has escaped in some way.

This is true because the decay energy must always carry mass with it, wherever it appears (see mass in special relativity) according to the formula E = mc2.

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 heavier mass m must be much + + larger than the lighter mass m , − m ≫m .
- The electron e and its positron partner e ̄ , and the heavy electron states e − − + and e ̄ orthogonal to them, must be + −e −ie e −ie e ≈ rR√ lL , e ≈ lR√ rL , (42.178a) − + 2 2 ie +e −ie +e e ̄ ≈ rR√ lL , e ̄ ≈ lR√ rL .
- (42.178b) − + 2 2 Therelations(42.178)arewrittenasapproximations,notequalities,toallowthepossibilitythattherecould besomesmalldeparturefromexactequalityofweakandDiracchiralityoflightelectrons.Theheavyelectron eigenstates (42.178b) have the same SM charges as the light electron eigenstates (42.178a), but the heavy electrons have Dirac chirality opposite to their weak chirality.
- Experiment establishes that the weak chirality of observed electrons (and other fundamental fermions) coincides with their Dirac chirality; that is, only left-handed (L) electrons (and right-handed (R) positrons) experience the weak force.

The heavier mass m must be much + + larger than the lighter mass m , − m ≫m .

The electron e and its positron partner e ̄ , and the heavy electron states e − − + and e ̄ orthogonal to them, must be + −e −ie e −ie e ≈ rR√ lL , e ≈ lR√ rL , (42.178a) − + 2 2 ie +e −ie +e e ̄ ≈ rR√ lL , e ̄ ≈ lR√ rL .

(42.178b) − + 2 2 Therelations(42.178)arewrittenasapproximations,notequalities,toallowthepossibilitythattherecould besomesmalldeparturefromexactequalityofweakandDiracchiralityoflightelectrons.Theheavyelectron eigenstates (42.178b) have the same SM charges as the light electron eigenstates (42.178a), but the heavy electrons have Dirac chirality opposite to their weak chirality.

Experiment establishes that the weak chirality of observed electrons (and other fundamental fermions) coincides with their Dirac chirality; that is, only left-handed (L) electrons (and right-handed (R) positrons) experience the weak force.

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:
- Thenextsceneshowedtheinteriorofthecraft,apanelwithleversallcoloredinmutedpasteland white;thenextsceneshowedtwodeadbodiesontablesinsideatent,smallwithlargeheadsthatlooked “mongoloid”withsmallnoses,mouthsandshuteyes,eachworeatightsuitpastelincolor;whenthefilm endedtherewasnocreditandtheofficerinchargeinstructedtheviewersto“thinkaboutthemovie”and “don’trelateitscontentstoanyone.” TwoweekslaterMr.Twasapproachedbyanintelligenceofficeronthebaseandtoldto“forget themovieyousaw,itwasahoax.” Mr.Tstatesthefilmwasshotbyaninexperiencedcameramanwith poorcoloringandtexture.
- ● RetrievalsoftheThirdKindpresentedatMUFON1978,LeonStringfield,p10 (PUBLICDOMAIN) -1953—Asource“H.J.”sayshewasintheArmyatFortPolk,LAin1953when theysawaneggshapedUAPcrashlandintosoftsandysoil.
- Heheardthesurvivorswere senttoahospitalandalldied,andheheardtheircadaversweresenttoamedicalcenternearWashington, DC.
- H.J.sayshewastoldnottodiscusstheincidentandafterhetalkedtoLeonardStringfield,hisphone wasdisconnected.

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:
- Wernerstateshisjobwastodeterminefromtheangleoftheimpacthowfastthevehiclewas goingatthetimeofimpact;hestatesthevehiclehadnolandinggearbutalsohadnodentsoreven scratches;afterwardshewastoldbyaUSAFCol.Towritehisreportinlonghand,nottotypeor reproduceit,andnottorevealwhathesaw;Wernerstateshewasgivenatelephonenumbertocallwhen thereportwascompleteanditwashandedofftoanairman.
- Wernerstatesithadoneoccupant, nowdead,darkbrowncomplexion,4feettall,wearingasilverymetallicsuitandaskullcapwithnoface coveringorhelmet.
- (PUBLICDOMAIN) -21May1953—USAFprojectengineeroncontractwiththeAECfor OperationUpshot-Knothole“FritzWerner”(pseudonym)stateshewasplacedonspecialassignmentto investigateacrashedUAPnearKingman,AZon21May1953,inasignedaffidavit.
- AllconfirmedUAPreportsmustnowbesentto AirForceIntelligence,andreportsbyUSAFpersonneldonotgotoProjectBlueBook.

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:
- Using a different coordinate transformation one can then relate s the extended external patch to the inner patch.[20] The case r = 0 is different, however.
- The singularity at r = r divides the Schwarzschild coordinates in two disconnected patches.
- Since the Schwarzschild metric is expected to be valid only for those radii larger than the radius R of the gravitating body, there is no problem as long as R > r .
- The interior Schwarzschild solution with 0 ≤ r < r , which contains the singularity at s r = 0, is completely separated from the outer patch by the singularity at r = r .

Using a different coordinate transformation one can then relate s the extended external patch to the inner patch.[20] The case r = 0 is different, however.

The singularity at r = r divides the Schwarzschild coordinates in two disconnected patches.

Since the Schwarzschild metric is expected to be valid only for those radii larger than the radius R of the gravitating body, there is no problem as long as R > r .

The interior Schwarzschild solution with 0 ≤ r < r , which contains the singularity at s r = 0, is completely separated from the outer patch by the singularity at r = r .

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:
- More modern systems include the Global Positioning System, other satellite systems, Coordinated Universal Time and mean solar time.
- Although these systems differ from one another, with careful measurements they can be synchronized.
- In physics, time is a fundamental concept to define other quantities, such as velocity.
- To avoid a circular definition,[15] time in physics is operationally defined as "what a clock reads", specifically a count of repeating events such as the SI second.[6][16][17] Although this aids in practical

More modern systems include the Global Positioning System, other satellite systems, Coordinated Universal Time and mean solar time.

Although these systems differ from one another, with careful measurements they can be synchronized.

In physics, time is a fundamental concept to define other quantities, such as velocity.

To avoid a circular definition,[15] time in physics is operationally defined as "what a clock reads", specifically a count of repeating events such as the SI second.[6][16][17] Although this aids in practical

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:
- Although tetrad-frame 0ab connections are generically not tetrad-frame tensors, the unperturbed connection Γ 0 ≡−(a ̇/a2)δ , equa0ab ab tion (29.23), is a tetrad-frame tensor, because the spatial unit matrix δ can be expressed as the tensor ab u u +η ,whereu isthetetrad-frame4-velocityoftheLorentz-transformedtetradframerelativetothe m n mn m rest tetrad frame.
- 29.5.3 Tetrad-frame Einstein tensor The tetrad-frame Einstein tensor G follows from the usual formulae (11.61), (11.78), and (11.80).
- According to the rule established in §26.7, the change in a quantity under an infinitesimal coordinate gauge transformation equals minus its Lie derivative L with respect to the infinitesimal coorε dinate shift ε.
- (29.26) 0dηa2 1 1 1 1 Thus perturbations Γ , Γ with a̸=b, Γ , and Γ are coordinate gauge-invariant, while the transfor0a0 0ab ab0 abc mation (29.26) of F implies that Γ with a = b transforms under an infinitesimal coordinate transforma0ab tion (26.9) as Γ 1 →Γ 1 − ε 0 d a ̇ δ .

Although tetrad-frame 0ab connections are generically not tetrad-frame tensors, the unperturbed connection Γ 0 ≡−(a ̇/a2)δ , equa0ab ab tion (29.23), is a tetrad-frame tensor, because the spatial unit matrix δ can be expressed as the tensor ab u u +η ,whereu isthetetrad-frame4-velocityoftheLorentz-transformedtetradframerelativetothe m n mn m rest tetrad frame.

29.5.3 Tetrad-frame Einstein tensor The tetrad-frame Einstein tensor G follows from the usual formulae (11.61), (11.78), and (11.80).

According to the rule established in §26.7, the change in a quantity under an infinitesimal coordinate gauge transformation equals minus its Lie derivative L with respect to the infinitesimal coorε dinate shift ε.

(29.26) 0dηa2 1 1 1 1 Thus perturbations Γ , Γ with a̸=b, Γ , and Γ are coordinate gauge-invariant, while the transfor0a0 0ab ab0 abc mation (29.26) of F implies that Γ with a = b transforms under an infinitesimal coordinate transforma0ab tion (26.9) as Γ 1 →Γ 1 − ε 0 d a ̇ δ .

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:
- “I have no idea, but you’re not well.” And she looks at me with the compassion that has always defined her.
- I wrap the covers tightly around me, but I can’t stop shaking, and a full-body ache is growing more painful by the second, my legs restless, my head throbbing.
- I would never lie to you.” This is possibly the most surreal moment I’ve experienced since coming to consciousness in that lab—sitting in bed in the guest room of the apartment of the woman who is my wife but isn’t, talking about the son we apparently never had, about the life that wasn’t ours.
- I say, “Look, maybe you believe what I’m telling you, maybe you don’t, but I need you to know that I believe it.

“I have no idea, but you’re not well.” And she looks at me with the compassion that has always defined her.

I wrap the covers tightly around me, but I can’t stop shaking, and a full-body ache is growing more painful by the second, my legs restless, my head throbbing.

I would never lie to you.” This is possibly the most surreal moment I’ve experienced since coming to consciousness in that lab—sitting in bed in the guest room of the apartment of the woman who is my wife but isn’t, talking about the son we apparently never had, about the life that wasn’t ours.

I say, “Look, maybe you believe what I’m telling you, maybe you don’t, but I need you to know that I believe it.

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 relation between cosmic time t and affine parameter λ is dt =pt =e tp0 =E , (31.34) dλ 0 where E = p0 is the proper energy of the particle in the tetrad rest-frame.
- Since f is dimensionless while the affine parameter dλ≡dτ/m has units of time/mass, the units of the collision term C[f] are mass/time.
- (31.32) dλ Equation (31.32) is the collisional Boltzmann equation.
- The collisionless Boltzmann equation holds without modification for particles that do not collide, such as neutrinos or non-baryonic dark matter particles, but it fails for particles whose trajectories are substantially modifiedbycollisionswithotherparticles,suchasphotonsorbaryons.Collisionsarebothasinkandasource of particles, destroying particles of momentum p and creating others of momentum p′ in the single-particle distribution f.

The relation between cosmic time t and affine parameter λ is dt =pt =e tp0 =E , (31.34) dλ 0 where E = p0 is the proper energy of the particle in the tetrad rest-frame.

Since f is dimensionless while the affine parameter dλ≡dτ/m has units of time/mass, the units of the collision term C[f] are mass/time.

(31.32) dλ Equation (31.32) is the collisional Boltzmann equation.

The collisionless Boltzmann equation holds without modification for particles that do not collide, such as neutrinos or non-baryonic dark matter particles, but it fails for particles whose trajectories are substantially modifiedbycollisionswithotherparticles,suchasphotonsorbaryons.Collisionsarebothasinkandasource of particles, destroying particles of momentum p and creating others of momentum p′ in the single-particle distribution f.

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:
- KAstatesagroundcrewinfatigueswasalreadythereandpreventedthemfromgettingcloser, andbegantaggingtheobjectsandthebodies.
- (PUBLICDOMAIN) -12April1954—“KA”statesheservedintheUSAFin1954-55atRoswellAir ForceBasewherehebeganhelicoptertraining.
- Moredeadbodieswereinsideasmallhatchwayinthe crasheddisc.
- KAstateshewassentbacktoSampson AFBwherehissuperioraccusedhimofgoingAWOLandpunishedhim,towhichKAtoldhissuperior abouttheUAPcrashandhissuperiordidnotbelievehim.

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

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

EVIDENCE:
- Riemanncurvatureandparalleltransport Thisdiscussionsuggestsyetanotherwayforthemitegeometers(namelyus)tomeasure curvature.ConsideraclosedcurveC(ingeneralnotageodesic)startingandendingatthe pointP.Let’sparalleltransportavectorS alongCstartingatP.
- ∗Youareprobablyfamiliarwiththissortofargumentfromavarietyofcontextsinphysics(mostlikelyina courseonelectromagnetism)andmathematics.
- μ Inflatspace,itwill.Thus,theextenttowhichitdoesnotprovidesuswithameasureof thecurvature.Inotherwords,wewanttocalculate(cid:10)S aswegoaroundC.
- Clearly, the curved segment shared by C and C are traversed 1 2 twice in opposite directions, producing canceling contributions to (cid:10)S .

Clearly, the curved segment shared by C and C are traversed 1 2 twice in opposite directions, producing canceling contributions to (cid:10)S .

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

EVIDENCE:
- The averaged strong energy condition (ASEC) - this condition is true for a timelike curve Γ if: (cid:90) (cid:18) (cid:19) T VμVν + T ds ≥ 0 μν Γ Attheendofsection3westatedMorris’sandThorne’sconditionthatinatraversable wormhole’s throat the tension must be larger than the total density of mass-energy τ > ρ - the so-called exotic material property.
- RAYCHAUDHURI-LANDAU EQUATION Section 4.
- 4.2 Raychaudhuri-Landau equation A fundamental result for the motion of adjacent particles was discovered independentlybytheIndianphysicistAmalKumarRaychaudhuri[4]andbytheSovietphysicistLevLandau[15].
- ThisresultisreferredtoastheRaychaudhuri-Landauequation, or sometimes only as Raychaudhuri’s equation.

The averaged strong energy condition (ASEC) - this condition is true for a timelike curve Γ if: (cid:90) (cid:18) (cid:19) T VμVν + T ds ≥ 0 μν Γ Attheendofsection3westatedMorris’sandThorne’sconditionthatinatraversable wormhole’s throat the tension must be larger than the total density of mass-energy τ > ρ - the so-called exotic material property.

4.2 Raychaudhuri-Landau equation A fundamental result for the motion of adjacent particles was discovered independentlybytheIndianphysicistAmalKumarRaychaudhuri[4]andbytheSovietphysicistLevLandau[15].

ThisresultisreferredtoastheRaychaudhuri-Landauequation, or sometimes only as Raychaudhuri’s equation.

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 pitch-black inside the shed and redolent of gasoline and old grass clippings.
- Does the idea that I was drugged and kidnapped by a secret, experimental group for the purposes of mind control or God-knows-what fit that bill?
- What else could have crashed through me with such debilitating speed?
- Or—is it more plausible that a tumor in my brain has turned my world upside down?

It’s pitch-black inside the shed and redolent of gasoline and old grass clippings.

Does the idea that I was drugged and kidnapped by a secret, experimental group for the purposes of mind control or God-knows-what fit that bill?

What else could have crashed through me with such debilitating speed?

Or—is it more plausible that a tumor in my brain has turned my world upside down?

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:
- Considersolutionsdependingonbotht andythatarehomogeneousandisotropicinthethreetransverse directions,andwritethemetricintheform (cid:12) (cid:14) (cid:12) (cid:14) ds2=e2A(t,y) −dt2+dy2 +e2B(t,y) dx2+dx2+dx2 1 2 3 (cid:12) (cid:14) =−e2A(u,v)dudv+e2B(u,v) dx2+dx2+dx2 (23) 1 2 3 withthelightconecoordinatesu≡t−y andv≡t+y.Themetricforthe(3+1)-dimensionaluniverseon the brane can be written in the standard(cid:8)form of a Friedmann-Lemaˆıtre-Robertson-Walker universe ds2= −dτ2+a(τ)2(dx2+dx2+dx2),withτ= dteA(t,0)anda(τ)=eB(t(τ),0).
- 1 2 3 NowsolveEinstein’sequation(20)andthedilatonequation(22)inthebulk,thatis,awayfromthebrane,to obtainA(t,y),B(t,y),andφ(t,y)fory>0andy<0separately,andthenmatchthesolutionsacrossthebrane.
- ∗The 4 isconventionalandcanbeabsorbedinb.
- GravityPast,Present,andFuture Appendix:Outgoingbranewavemodel Aspromised,fromthevastliteratureonbraneworlds,hereIbrieflydescribetheoutgoingbranewavemodel13 togiveyouaflavorofthesortofcalculationsonecanindulgein.Also,thefollowingdiscussioninvolvessolving Einstein’sequationsinlightconecoordinates,somethingthatmightbeofmethodologicalinterest.Letmefirst setupthemodelandthendescribeitsphysicalproperties.

Considersolutionsdependingonbotht andythatarehomogeneousandisotropicinthethreetransverse directions,andwritethemetricintheform (cid:12) (cid:14) (cid:12) (cid:14) ds2=e2A(t,y) −dt2+dy2 +e2B(t,y) dx2+dx2+dx2 1 2 3 (cid:12) (cid:14) =−e2A(u,v)dudv+e2B(u,v) dx2+dx2+dx2 (23) 1 2 3 withthelightconecoordinatesu≡t−y andv≡t+y.Themetricforthe(3+1)-dimensionaluniverseon the brane can be written in the standard(cid:8)form of a Friedmann-Lemaˆıtre-Robertson-Walker universe ds2= −dτ2+a(τ)2(dx2+dx2+dx2),withτ= dteA(t,0)anda(τ)=eB(t(τ),0).

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

EVIDENCE:
- Relativistic kinematics applies the special theory of relativity to the geometry of object motion.
- It encompasses time dilation, length contraction and the Lorentz transformation.[11]: 12.8 The kinematics of relativity operates in a spacetime geometry where spatial points are augmented with a time coordinate to form 4-vectors.[12]: 221 Werner Heisenberg reinterpreted classical kinematics for quantum systems in his 1925 paper "On the quantum-theoretical reinterpretation of kinematical and mechanical relationships".[13] Dirac noted the similarity in structure between Heisenberg's formulations and classical Poisson brackets.[14]: 143 In a follow up paper in 1927 Heisenberg showed that classical kinematic notions like velocity and energy are valid in quantum mechanics, but pairs of conjugate kinematic and dynamic quantities cannot be simultaneously measured, a result he called indeterminacy, but which became known as the uncertainty principle.[15] Etymology The term kinematic is the English version of A.M.
- Ampère's cinématique,[16] which he constructed from the Greek κίνημα kinema ("movement, motion"), itself derived from κινεῖν kinein ("to move").[17][18] Kinematic and cinématique are related to the French word cinéma, but neither are directly derived from it.
- The position of a particle is defined as the coordinate vector from the origin of a coordinate frame to the particle.

Relativistic kinematics applies the special theory of relativity to the geometry of object motion.

It encompasses time dilation, length contraction and the Lorentz transformation.[11]: 12.8 The kinematics of relativity operates in a spacetime geometry where spatial points are augmented with a time coordinate to form 4-vectors.[12]: 221 Werner Heisenberg reinterpreted classical kinematics for quantum systems in his 1925 paper "On the quantum-theoretical reinterpretation of kinematical and mechanical relationships".[13] Dirac noted the similarity in structure between Heisenberg's formulations and classical Poisson brackets.[14]: 143 In a follow up paper in 1927 Heisenberg showed that classical kinematic notions like velocity and energy are valid in quantum mechanics, but pairs of conjugate kinematic and dynamic quantities cannot be simultaneously measured, a result he called indeterminacy, but which became known as the uncertainty principle.[15] Etymology The term kinematic is the English version of A.M.

Ampère's cinématique,[16] which he constructed from the Greek κίνημα kinema ("movement, motion"), itself derived from κινεῖν kinein ("to move").[17][18] Kinematic and cinématique are related to the French word cinéma, but neither are directly derived from it.

The position of a particle is defined as the coordinate vector from the origin of a coordinate frame to the particle.

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

EVIDENCE:
- Any two Riemannian manifolds of the same constant curvature are locally isometric, and so it follows that any Riemannian manifold of constant curvature κ can be covered by coordinate charts relative to which the metric has the above form.[33] A Riemannian space form is a Riemannian manifold with constant curvature which is additionally connected and geodesically complete.
- A Riemannian space form is said to be a spherical space form if the curvature is positive, a Euclidean space form if the curvature is zero, and a hyperbolic space form or hyperbolic manifold if the curvature is negative.
- In any dimension, the sphere with its standard Riemannian metric, Euclidean space, and hyperbolic space are Riemannian space forms of constant curvature 1, 0, and −1 respectively.
- Given any finite subgroup G thereof in which only the identity matrix possesses 1 as an eigenvalue, the natural group action of the orthogonal group on the n-sphere restricts to a group action of G, with the quotient n manifold S / G inheriting a geodesically complete Riemannian metric of constant curvature 1.

Any two Riemannian manifolds of the same constant curvature are locally isometric, and so it follows that any Riemannian manifold of constant curvature κ can be covered by coordinate charts relative to which the metric has the above form.[33] A Riemannian space form is a Riemannian manifold with constant curvature which is additionally connected and geodesically complete.

A Riemannian space form is said to be a spherical space form if the curvature is positive, a Euclidean space form if the curvature is zero, and a hyperbolic space form or hyperbolic manifold if the curvature is negative.

In any dimension, the sphere with its standard Riemannian metric, Euclidean space, and hyperbolic space are Riemannian space forms of constant curvature 1, 0, and −1 respectively.

Given any finite subgroup G thereof in which only the identity matrix possesses 1 as an eigenvalue, the natural group action of the orthogonal group on the n-sphere restricts to a group action of G, with the quotient n manifold S / G inheriting a geodesically complete Riemannian metric of constant curvature 1.

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

EVIDENCE:
- 1942; died 1945) Feynman Lectures on Physics (1961–1964).
- Tolman professorship in theoretical physics Los Angeles, California, U.S.
- 1965 Commission, the panel that investigated the Space Shuttle Challenger disaster.
- In a 1999 poll Resting place Mountain View Cemetery and of 130 leading physicists worldwide by the British Mausoleum journal Physics World, he was ranked the seventhEducation Massachusetts Institute of greatest physicist of all time.[3] Technology (SB) Feynman was a keen physics popularizer through books Princeton University (PhD) and lectures, including a talk on top-down Known for nanotechnology, "There's Plenty of Room at the Bottom" (1959) and his undergraduate lectures, The Spouses Arline Greenbaum (m.

1942; died 1945) Feynman Lectures on Physics (1961–1964).

Tolman professorship in theoretical physics Los Angeles, California, U.S.

1965 Commission, the panel that investigated the Space Shuttle Challenger disaster.

In a 1999 poll Resting place Mountain View Cemetery and of 130 leading physicists worldwide by the British Mausoleum journal Physics World, he was ranked the seventhEducation Massachusetts Institute of greatest physicist of all time.[3] Technology (SB) Feynman was a keen physics popularizer through books Princeton University (PhD) and lectures, including a talk on top-down Known for nanotechnology, "There's Plenty of Room at the Bottom" (1959) and his undergraduate lectures, The Spouses Arline Greenbaum (m.

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:
- One minute in, some of the panels begin to show looped imagery.
- I choose a path, and as I enter the labyrinth, the transparency vanishes.
- A low-level sound that has been there all along catches my attention—it’s not music so much as white noise, like television static, hissing over a deep, sustained tone.
- Birth—child screaming, mother weeping with joy.

One minute in, some of the panels begin to show looped imagery.

I choose a path, and as I enter the labyrinth, the transparency vanishes.

A low-level sound that has been there all along catches my attention—it’s not music so much as white noise, like television static, hissing over a deep, sustained tone.

Birth—child screaming, mother weeping with joy.

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:
- Indeed, back in chapter VI.2, we studied the universe described by ((cid:24) ,(cid:24) )= m,0 (cid:27),0 (0,1),aflatuniversewithnomatter,onlyacosmologicalconstant.Wefoundthereandin
- Abovethestraightline(cid:24) = 1(cid:24) ,cosmicexpansionaccelerates:theuniversewill (cid:27),0 2 m,0 expand faster and faster.
- Below this line, cosmic expansiondecelerates.
- The two axes are labeled by (cid:24) and (cid:24) , two parameters that we will define in this m,0 (cid:27),0 chapter and that measure how much matter and how much dark energy, respectively, theuniversecontainsatpresent.Heretheterm“matter”includesbothdarkandbaryonic matter,withthebulk(morethan80%)ofitindarkmatter,aswewillsee.AsinchapterVI.2 andtheprecedingchapter,darkenergyispresumedtobethemanifestationofEinstein’s cosmological constant.

Indeed, back in chapter VI.2, we studied the universe described by ((cid:24) ,(cid:24) )= m,0 (cid:27),0 (0,1),aflatuniversewithnomatter,onlyacosmologicalconstant.Wefoundthereandin

Abovethestraightline(cid:24) = 1(cid:24) ,cosmicexpansionaccelerates:theuniversewill (cid:27),0 2 m,0 expand faster and faster.

Below this line, cosmic expansiondecelerates.

The two axes are labeled by (cid:24) and (cid:24) , two parameters that we will define in this m,0 (cid:27),0 chapter and that measure how much matter and how much dark energy, respectively, theuniversecontainsatpresent.Heretheterm“matter”includesbothdarkandbaryonic matter,withthebulk(morethan80%)ofitindarkmatter,aswewillsee.AsinchapterVI.2 andtheprecedingchapter,darkenergyispresumedtobethemanifestationofEinstein’s cosmological constant.

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:
- μ μν ρσ μ ν TherolesplayedbyS andS −1areinterchanged:anupperindextransformswithS −1and alowerindexwithS.
- For a rotation matrix R, RTR=RRT =I,sothatR −1=RT.Thus,ifShappenstobearotationmatrixR,thatis,if thetransformationfromunprimedtoprimedcoordinatesisamererotation,theninthis specialcase(andonlyinthisspecialcase)(S −1)T =(ST)T =S.Thetransformationlaw(17) ofthemetriccollapsestoI =SIS −1,sinceinthiscasethemetricisjusttheidentitymatrix I inboththeunprimedandprimedcoordinates.Inotherwords,rotationsaredefinedas linearcoordinatetransformationsthatleavetheEuclideanmetricinvariant,aswelearned inchapterI.3.
- Let us see how the formalism just developed works for rotations.
- In the two preceding paragraphs, we talked about going from unprimed to primed coordinates.Goingfromprimedtounprimedcoordinates,wehaveofcourse,asalready remarked,theinversetransformations:dxν=(S −1)ν (x (cid:5) )dx (cid:5)μandg (x)=g (cid:5) (x (cid:5) )Sρ Sσ.

μ μν ρσ μ ν TherolesplayedbyS andS −1areinterchanged:anupperindextransformswithS −1and alowerindexwithS.

For a rotation matrix R, RTR=RRT =I,sothatR −1=RT.Thus,ifShappenstobearotationmatrixR,thatis,if thetransformationfromunprimedtoprimedcoordinatesisamererotation,theninthis specialcase(andonlyinthisspecialcase)(S −1)T =(ST)T =S.Thetransformationlaw(17) ofthemetriccollapsestoI =SIS −1,sinceinthiscasethemetricisjusttheidentitymatrix I inboththeunprimedandprimedcoordinates.Inotherwords,rotationsaredefinedas linearcoordinatetransformationsthatleavetheEuclideanmetricinvariant,aswelearned inchapterI.3.

Let us see how the formalism just developed works for rotations.

In the two preceding paragraphs, we talked about going from unprimed to primed coordinates.Goingfromprimedtounprimedcoordinates,wehaveofcourse,asalready remarked,theinversetransformations:dxν=(S −1)ν (x (cid:5) )dx (cid:5)μandg (x)=g (cid:5) (x (cid:5) )Sρ Sσ.

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

EVIDENCE:
- Two world lines starting at the same event in spacetime, each following its own path afterwards, may represent e.g.
- In the limit , this difference divided by defines a vector, the tangent vector of the world line at the point .
- The sum of two vectors is again a tangent vector to some other curve and the same holds for multiplying by a scalar.
- It is a four-dimensional vector, defined in the point .

Two world lines starting at the same event in spacetime, each following its own path afterwards, may represent e.g.

In the limit , this difference divided by defines a vector, the tangent vector of the world line at the point .

The sum of two vectors is again a tangent vector to some other curve and the same holds for multiplying by a scalar.

It is a four-dimensional vector, defined in the point .

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List tests or derivations implied by the evidence.


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

EVIDENCE:
- deuterium, to the mass of the proton in free space (which is greater by about 0.239% —this is due to the binding energy of deuterium).
- Thus, for example, if the reference weight m is taken to be the mass of the neutron in free space, and the relative accelerations for the proton and neutron in deuterium are computed, then the above formula over-estimates the mass m (by 0.239%) for the proton in deuterium.
- At best, Mach's formula can only be used to obtain ratios of masses, that is, as m / m = |a | / |a |.
- An additional difficulty was pointed out by Henri Poincaré, which is that 1 2 2 1

deuterium, to the mass of the proton in free space (which is greater by about 0.239% —this is due to the binding energy of deuterium).

Thus, for example, if the reference weight m is taken to be the mass of the neutron in free space, and the relative accelerations for the proton and neutron in deuterium are computed, then the above formula over-estimates the mass m (by 0.239%) for the proton in deuterium.

At best, Mach's formula can only be used to obtain ratios of masses, that is, as m / m = |a | / |a |.

An additional difficulty was pointed out by Henri Poincaré, which is that 1 2 2 1

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

EVIDENCE:
- Relaxing the conditions for convexity generates a further 10 nonconvex regular 4polytopes.
- The Cartesian product of two circles may be taken to obtain a duocylinder.
- Hypersphere The set of points in Euclidean 4-space having the same distance R from a fixed point P forms a hypersurface known as a 3-sphere.
- dimensional world, can, without special practice, make spatial judgments about line segments embedded in four-dimensional space, based on their length (one-dimensional) and the angle (two-dimensional) between them.[17] The researchers noted that "the participants in our study had minimal practice in these tasks, and it remains an open question whether it is possible to obtain more sustainable, definitive, and richer 4D representations with increased perceptual experience

Relaxing the conditions for convexity generates a further 10 nonconvex regular 4polytopes.

The Cartesian product of two circles may be taken to obtain a duocylinder.

Hypersphere The set of points in Euclidean 4-space having the same distance R from a fixed point P forms a hypersurface known as a 3-sphere.

dimensional world, can, without special practice, make spatial judgments about line segments embedded in four-dimensional space, based on their length (one-dimensional) and the angle (two-dimensional) between them.[17] The researchers noted that "the participants in our study had minimal practice in these tasks, and it remains an open question whether it is possible to obtain more sustainable, definitive, and richer 4D representations with increased perceptual experience

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

EVIDENCE:
- These signs give you an ‘address’in the sky.
- There is also another stone, which has some signs on it.
- There are signs around it, and seven stones.
- 3 Open answers © Pearson Education 2000

These signs give you an ‘address’in the sky.

There is also another stone, which has some signs on it.

There are signs around it, and seven stones.

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

EVIDENCE:
- Considering the second term in equation (3.32), one has F Fρσ = gρρ(cid:48) gσσ(cid:48) F F , ρσ ρσ ρ(cid:48)σ(cid:48) = 2g00g11F F +2g22g33F F , 01 01 23 23 (cid:16) (cid:17) = 2 −e−2(α+β)E2/c2+B2 .
- Therefore, the function B representing the magnetic (cid:28)eld of a point magnetic monopole can be written as μ P 1 B = .
- Therefore, equation (3.37) becomes ∇ F +∇ F +∇ F = 0, ρ μν μ νρ ν ρμ ∂ F +∂ F +∂ F = 0, ρ μν μ νρ ν ρμ ∂ F = 0, r θφ C ∂ (r2sinθB) = 0, ⇒ B = , (3.42) r r2 where C is a constant.
- In order to (cid:28)nd the form of B, let us consider the second equation of motion with component (ρ,μ,ν) = (r,θ,φ).

Considering the second term in equation (3.32), one has F Fρσ = gρρ(cid:48) gσσ(cid:48) F F , ρσ ρσ ρ(cid:48)σ(cid:48) = 2g00g11F F +2g22g33F F , 01 01 23 23 (cid:16) (cid:17) = 2 −e−2(α+β)E2/c2+B2 .

Therefore, the function B representing the magnetic (cid:28)eld of a point magnetic monopole can be written as μ P 1 B = .

Therefore, equation (3.37) becomes ∇ F +∇ F +∇ F = 0, ρ μν μ νρ ν ρμ ∂ F +∂ F +∂ F = 0, ρ μν μ νρ ν ρμ ∂ F = 0, r θφ C ∂ (r2sinθB) = 0, ⇒ B = , (3.42) r r2 where C is a constant.

In order to (cid:28)nd the form of B, let us consider the second equation of motion with component (ρ,μ,ν) = (r,θ,φ).

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

EVIDENCE:
- Steinhardt was probably the first job Feynman had, and during this Stephen Wolfram time he showed early signs of an aptitude for his later career in theoretical physics, when he would analyze the Signature issues theoretically and arrive at the solutions.[12] When he was in grade school, he created a home burglar alarm system while his parents were out for the day running errands.[13]

Steinhardt was probably the first job Feynman had, and during this Stephen Wolfram time he showed early signs of an aptitude for his later career in theoretical physics, when he would analyze the Signature issues theoretically and arrive at the solutions.[12] When he was in grade school, he created a home burglar alarm system while his parents were out for the day running errands.[13]

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:
- Answer:The acceptance of the term 'black hole' represents a significant shift in the understanding of celestial phenomena and a bridging of concepts in astrophysics.
- Meanwhile, Oppenheimer's cautious approach ultimately led him to significant insights but also reflected his hesitation to engage in speculation.
- The Search| Q&A 1.Question What are black holes and how have they evolved in scientific understanding since the early 1960s?
- It encapsulates the realization of a definitive and quantifiable entity in the universe, fostering deeper investigations into the fundamental theories of physics, particularly the interfaces of general relativity and quantum mechanics.

Answer:The acceptance of the term 'black hole' represents a significant shift in the understanding of celestial phenomena and a bridging of concepts in astrophysics.

Meanwhile, Oppenheimer's cautious approach ultimately led him to significant insights but also reflected his hesitation to engage in speculation.

The Search| Q&A 1.Question What are black holes and how have they evolved in scientific understanding since the early 1960s?

It encapsulates the realization of a definitive and quantifiable entity in the universe, fostering deeper investigations into the fundamental theories of physics, particularly the interfaces of general relativity and quantum mechanics.

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:
- Isotopy When two given continuous functions f and g from the topological space X to the topological space Y are embeddings, one can ask whether they can be connected 'through embeddings'.
- For example, the map from the interval [−1, 1] into the real numbers defined by f(x) = −x is not isotopic to the identity g(x) = x.
- Moreover, f has changed the orientation of the interval and g has not, which is impossible under an isotopy.
- Formally: if f and g are continuous maps from X to Y and K is a subset of X, then we say that f and g are homotopic relative to K if there exists a homotopy H : X × [0, 1] → Y between f and g such that H(k, t) = f(k) = g(k) for all k ∈ K and t ∈ [0, 1].

Isotopy When two given continuous functions f and g from the topological space X to the topological space Y are embeddings, one can ask whether they can be connected 'through embeddings'.

For example, the map from the interval [−1, 1] into the real numbers defined by f(x) = −x is not isotopic to the identity g(x) = x.

Moreover, f has changed the orientation of the interval and g has not, which is impossible under an isotopy.

Formally: if f and g are continuous maps from X to Y and K is a subset of X, then we say that f and g are homotopic relative to K if there exists a homotopy H : X × [0, 1] → Y between f and g such that H(k, t) = f(k) = g(k) for all k ∈ K and t ∈ [0, 1].

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:
- AspectsofGravity Maximallysymmetric3-spaces Let’sfoliate3-spacewithaseriesofspheresseparatedbythedistanceF(r)dr: ds2=F(r)2dr2+r2dθ2+r2sin2θdφ2 (25) sothat e1=F(r)dr, e2=rdθ, e3=rsinθdφ (26) Weobtaineasilyde1=0,de2=drdθ,de3=sinθdrdφ+r cosθdθdφ.
- We can formally go to the maximally symmetric negatively curved space by setting L2→−L2,sothatF2= 1 .Withr =Lsinhχ,themetricisgivenby 1+r2 L2 (cid:12) (cid:14) ds2=L2 dχ2+sinh2χdθ2+sinh2χsin2θdφ2 (31)
- (cid:29) (cid:30) From(28), wesimplyreadoffR1 = F(cid:5) , R1 = F(cid:5) , andR2 = 1 1− 1 .(Con212 rF3 313 rF3 323 r2 F2 fusedaboutthefactorsof2,orabsencethereof?Seetheprecedingchapter.)Contracting Riemann,wearriveatthenicelysymmetricRiccitensor 2F(cid:5) R = δ (29) ab rF3 ab Ifwerequirethatthespacebemaximallysymmetricandpositivelycurvedwithsome lengthscaleL,sothatR = 2 δ ,weobtainthedifferentialequation F(cid:5) = 1 ,withthe ab L2 ab rF3 L2 solutionF2= 1 .Asacheck,pluggingthisbackinto(28),wefindR1 = 1 e1e2,R1 = 1−r2 2 L2 3 L2 1 e1e3,andR2 = 1 e2e3.Withthechangeofvariabler =Lsinχ,themetricbecomes L2 3 L2 (cid:12) (cid:14) ds2=L2 dχ2+sin2χdθ2+sin2χsin2θdφ2 (30) HerewehavenothingotherthanS3,ofcourse,asyoumightrecallfromexerciseI.5.9.
- b bμ bc PlugthisintoCartan’sfirststructuralequation(1)andmatchterms.Weobtain 1 1 1 1 cosθ ω1 =− e2=− dθ, ω1 =− e3=− sinθdφ, ω2 =− e3=−cosθdφ (27) 2 rF F 3 rF F 3 rsinθ Cartan’ssecondequationgivesus (cid:3) (cid:4) F(cid:5) F(cid:5) 1 1 R1 = e1e2, R1 = e1e3, R2 = 1− e2e3 (28) 2 rF3 3 rF3 3 F2 r2 Using differential forms, you have to exercise some judgment.

AspectsofGravity Maximallysymmetric3-spaces Let’sfoliate3-spacewithaseriesofspheresseparatedbythedistanceF(r)dr: ds2=F(r)2dr2+r2dθ2+r2sin2θdφ2 (25) sothat e1=F(r)dr, e2=rdθ, e3=rsinθdφ (26) Weobtaineasilyde1=0,de2=drdθ,de3=sinθdrdφ+r cosθdθdφ.

We can formally go to the maximally symmetric negatively curved space by setting L2→−L2,sothatF2= 1 .Withr =Lsinhχ,themetricisgivenby 1+r2 L2 (cid:12) (cid:14) ds2=L2 dχ2+sinh2χdθ2+sinh2χsin2θdφ2 (31)

(cid:29) (cid:30) From(28), wesimplyreadoffR1 = F(cid:5) , R1 = F(cid:5) , andR2 = 1 1− 1 .(Con212 rF3 313 rF3 323 r2 F2 fusedaboutthefactorsof2,orabsencethereof?Seetheprecedingchapter.)Contracting Riemann,wearriveatthenicelysymmetricRiccitensor 2F(cid:5) R = δ (29) ab rF3 ab Ifwerequirethatthespacebemaximallysymmetricandpositivelycurvedwithsome lengthscaleL,sothatR = 2 δ ,weobtainthedifferentialequation F(cid:5) = 1 ,withthe ab L2 ab rF3 L2 solutionF2= 1 .Asacheck,pluggingthisbackinto(28),wefindR1 = 1 e1e2,R1 = 1−r2 2 L2 3 L2 1 e1e3,andR2 = 1 e2e3.Withthechangeofvariabler =Lsinχ,themetricbecomes L2 3 L2 (cid:12) (cid:14) ds2=L2 dχ2+sin2χdθ2+sin2χsin2θdφ2 (30) HerewehavenothingotherthanS3,ofcourse,asyoumightrecallfromexerciseI.5.9.

b bμ bc PlugthisintoCartan’sfirststructuralequation(1)andmatchterms.Weobtain 1 1 1 1 cosθ ω1 =− e2=− dθ, ω1 =− e3=− sinθdφ, ω2 =− e3=−cosθdφ (27) 2 rF F 3 rF F 3 rsinθ Cartan’ssecondequationgivesus (cid:3) (cid:4) F(cid:5) F(cid:5) 1 1 R1 = e1e2, R1 = e1e3, R2 = 1− e2e3 (28) 2 rF3 3 rF3 3 F2 r2 Using differential forms, you have to exercise some judgment.

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:
- I got you covered.” “They’ll know you helped me.” “By the time they realize, I’ll be out of here.” The computerized female voice says, Name, please.
- “I thought there was always someone monitoring—” “I’m on duty tonight.
- “From everything I’ve heard, stepping inside that thing is like jumping out of an airplane and not knowing if your chute is going to open.” “If the plane’s going down anyway, does it really matter?” “What about the camera?” “The one in here?
- “You coming?” I use the bed frame to drag myself onto my feet.

I got you covered.” “They’ll know you helped me.” “By the time they realize, I’ll be out of here.” The computerized female voice says, Name, please.

“I thought there was always someone monitoring—” “I’m on duty tonight.

“From everything I’ve heard, stepping inside that thing is like jumping out of an airplane and not knowing if your chute is going to open.” “If the plane’s going down anyway, does it really matter?” “What about the camera?” “The one in here?

“You coming?” I use the bed frame to drag myself onto my feet.

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 photon sphere equivalent occurs where the denominator of the integrand of equation (20.156) not only vanishes, vx =0, but is an extremum, which happens where the horizon function Δ is an extremum, dΔ =0 at photon sphere equivalent .
- 604 General spherically symmetric spacetimes uμ ≡ dxμ/dτ that follows from equations (20.152) takes its simplest form in the conformal coordinates {t ,x,θ,φ} of the ray-tracing metric (20.147), × ut × = vτ , ux =± 1 [v2τ2−(R2+L2)Δ ]1/2 , uφ = L .
- IntheReissner-Nordströmgeometrythereisaradius,thephotonsphere,wherephotonscanorbitincircles forever.Innon-stationaryself-similarsolutionsthereisnoconformalradiuswherephotonscanorbitforever (to remain at fixed conformal radius r, the photon angular momentum would have to increase in proportion to the conformal factor R).
- (20.155) R2Δ R2 R2 Equations (20.155) yield the shape of a null geodesic by quadrature, ∫ Jdx φ= .

The photon sphere equivalent occurs where the denominator of the integrand of equation (20.156) not only vanishes, vx =0, but is an extremum, which happens where the horizon function Δ is an extremum, dΔ =0 at photon sphere equivalent .

604 General spherically symmetric spacetimes uμ ≡ dxμ/dτ that follows from equations (20.152) takes its simplest form in the conformal coordinates {t ,x,θ,φ} of the ray-tracing metric (20.147), × ut × = vτ , ux =± 1 [v2τ2−(R2+L2)Δ ]1/2 , uφ = L .

IntheReissner-Nordströmgeometrythereisaradius,thephotonsphere,wherephotonscanorbitincircles forever.Innon-stationaryself-similarsolutionsthereisnoconformalradiuswherephotonscanorbitforever (to remain at fixed conformal radius r, the photon angular momentum would have to increase in proportion to the conformal factor R).

(20.155) R2Δ R2 R2 Equations (20.155) yield the shape of a null geodesic by quadrature, ∫ Jdx φ= .

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:
- To μν 5 P fixitspowerin(6),weinvokedimensionalanalysisasfollows.
- This can only be Maxwell’s action (recall (V.6.18)).
- That S must reduce correctly follows KK fromgeneralinvarianceconsiderations:theformofG justgivenismaintainedunder MN the4-dimensionalgeneralcoordinatetransformationxμ=xμ(x (cid:5)ν), x5=x (cid:5)5.
- ThescalarcurvatureR(inanyspacetimedimension)containstwoderivativesactingon themetricandhencehasdimensionsofaninverselengthsquared.Thus,asexplainedat (cid:8) √ lengthinchapterVI.1,wehavetomultiply d4x −gR(g)bysomethingwithdimension ofinverselengthsquaredtomaketheEinstein-Hilbertactiondimensionless,andwedefine thatsomethingasthesquareofthePlanckmassM .(Innaturalunitswith(cid:2)=1andc=1, P masshasdimensionsofaninverselength,aswasexplainedwaybackintheintroduction.) (cid:8) √ In5-dimensionalspacetime,wehavetomultiply d4xdy −GR(G)bysomethingwith dimensions of inverse length cubed, that is, by some mass cubed.

To μν 5 P fixitspowerin(6),weinvokedimensionalanalysisasfollows.

This can only be Maxwell’s action (recall (V.6.18)).

That S must reduce correctly follows KK fromgeneralinvarianceconsiderations:theformofG justgivenismaintainedunder MN the4-dimensionalgeneralcoordinatetransformationxμ=xμ(x (cid:5)ν), x5=x (cid:5)5.

ThescalarcurvatureR(inanyspacetimedimension)containstwoderivativesactingon themetricandhencehasdimensionsofaninverselengthsquared.Thus,asexplainedat (cid:8) √ lengthinchapterVI.1,wehavetomultiply d4x −gR(g)bysomethingwithdimension ofinverselengthsquaredtomaketheEinstein-Hilbertactiondimensionless,andwedefine thatsomethingasthesquareofthePlanckmassM .(Innaturalunitswith(cid:2)=1andc=1, P masshasdimensionsofaninverselength,aswasexplainedwaybackintheintroduction.) (cid:8) √ In5-dimensionalspacetime,wehavetomultiply d4xdy −GR(G)bysomethingwith dimensions of inverse length cubed, that is, by some mass cubed.

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List tests or derivations implied by the evidence.


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

EVIDENCE:
- The parameters of the model are the same as those for Figure 10.11.
- The Figure illustrates that regions that are in causal contact prior to inflation can fly out of causal contact during the accelerated expansion of inflation.
- Figure 10.14 shows the mass-energy density ρ as a function of time t for the same flat ΛCDM model as shown in Figure 10.13.
- Since the Universe here is taken to be flat, the density equals the critical density at all times, and is proportional to the inverse square of the Hubble distance c/H plotted in Figure 10.13.

The parameters of the model are the same as those for Figure 10.11.

The Figure illustrates that regions that are in causal contact prior to inflation can fly out of causal contact during the accelerated expansion of inflation.

Figure 10.14 shows the mass-energy density ρ as a function of time t for the same flat ΛCDM model as shown in Figure 10.13.

Since the Universe here is taken to be flat, the density equals the critical density at all times, and is proportional to the inverse square of the Hubble distance c/H plotted in Figure 10.13.

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:
- In the simplified, one-dimensional scenarios that we have been considering, only one additional constraint is necessary before the outgoing momenta of the particles can be determined—an energy condition.
- In the one-dimensional case of a completely elastic collision with no loss of kinetic energy, the outgoing velocities of the rebounding particles in the COM frame will be precisely equal and opposite to their incoming velocities.
- Einstein was faced with either having to give up conservation of momentum, or to change the definition of momentum.
- Mass is no longer conserved independently, because it has been subsumed into the total relativistic energy.

In the simplified, one-dimensional scenarios that we have been considering, only one additional constraint is necessary before the outgoing momenta of the particles can be determined—an energy condition.

In the one-dimensional case of a completely elastic collision with no loss of kinetic energy, the outgoing velocities of the rebounding particles in the COM frame will be precisely equal and opposite to their incoming velocities.

Einstein was faced with either having to give up conservation of momentum, or to change the definition of momentum.

Mass is no longer conserved independently, because it has been subsumed into the total relativistic energy.

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:
- Confusio:“Igetit!So,anappleandastonedroppedfromtheLeaningTowerofPisadid notfall,butweresuspendedmotionlessinspace.Itwasactuallythegroundwhichrushed uptomeetthem!Thatwouldexplainwhytheappleandthestonehitthegroundatexactly thesameinstant.Therelativityofmotion!” Indeed,amiteonthegroundlookedupattheenormousapplecomingdownandsawhis entirelifeflashbyhiminaninstant,butamiteontheapple,equallyterrorized,watched thegroundrushinguptocrushher.
- “Asif”isgoodenough But yet, this sounds like total nonsense.
- The earth carrying the Leaning Tower and the entire town of Pisa rushing up toward the apple and the stone?
- 270 | ProloguetoBookTwo Figure3 Thefloorrushinguptomeettheapple.

“Asif”isgoodenough But yet, this sounds like total nonsense.

The earth carrying the Leaning Tower and the entire town of Pisa rushing up toward the apple and the stone?

270 | ProloguetoBookTwo Figure3 Thefloorrushinguptomeettheapple.

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:
- It is possible to do without tetrads by working with coordinate tangent axes e and the associated coordinate connections, but it is not possible to do without coordinates.
- In general relativity, tetrad axes γγ (xμ) are defined at each point xμ of spacetime.
- The m coordinatesxμ ofthespacetimemanifoldprovidethecanvasuponwhichtetradscanbeerected,andthrough which tetrads can be transported.
- It is possible — indeed, this is the traditional Hilbert approach — to work entirely with μνκ coordinate-frame expressions, the coordinate metric and the coordinate connections, without introducing tetrads.

It is possible to do without tetrads by working with coordinate tangent axes e and the associated coordinate connections, but it is not possible to do without coordinates.

In general relativity, tetrad axes γγ (xμ) are defined at each point xμ of spacetime.

The m coordinatesxμ ofthespacetimemanifoldprovidethecanvasuponwhichtetradscanbeerected,andthrough which tetrads can be transported.

It is possible — indeed, this is the traditional Hilbert approach — to work entirely with μνκ coordinate-frame expressions, the coordinate metric and the coordinate connections, without introducing tetrads.

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 night after I escaped from this place, I went to a hospital.
- When I saw you at Daniela’s art reception, you asked me if the ‘compound’ had worked out.
- They ran a tox screen that returned traces of a mysterious psychoactive compound.
- I had no idea what the application was.” “You understand the concept behind Schrödinger’s cat?” “Sure.” “And how observation determines reality?” “Yes.” “This other version of me was trying to put a human being into superposition.

The night after I escaped from this place, I went to a hospital.

When I saw you at Daniela’s art reception, you asked me if the ‘compound’ had worked out.

They ran a tox screen that returned traces of a mysterious psychoactive compound.

I had no idea what the application was.” “You understand the concept behind Schrödinger’s cat?” “Sure.” “And how observation determines reality?” “Yes.” “This other version of me was trying to put a human being into superposition.

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:
- 3.Einstein's principles of special relativity state that the laws of physics are different for observers in different inertial frames.
- Black Holes Discovered and Rejected| Quiz and Test 1.Hermann Minkowski introduced the concept of spacetime, asserting it should be understood as a unified four-dimensional fabric rather than separate entities.
- 2.Einstein's principle of equivalence suggests that freely falling frames of reference do not behave similarly to inertial frames in a gravity-free environment.
- 3.Einstein fully embraced the prediction of black holes as a physical reality from the outset of his theories.

3.Einstein's principles of special relativity state that the laws of physics are different for observers in different inertial frames.

Black Holes Discovered and Rejected| Quiz and Test 1.Hermann Minkowski introduced the concept of spacetime, asserting it should be understood as a unified four-dimensional fabric rather than separate entities.

2.Einstein's principle of equivalence suggests that freely falling frames of reference do not behave similarly to inertial frames in a gravity-free environment.

3.Einstein fully embraced the prediction of black holes as a physical reality from the outset of his theories.

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List tests or derivations implied by the evidence.


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

EVIDENCE:
- On May 30, 2008 in Hartford, Connecticut, a 78-year-old man on his way home from buying milk in a grocery store, Angel Arce Torres, was hit by a car in the middle of a busy street during rush hour.
- 9781405131117_4_007.qxd 30/6/09 11:46 AM Page 140 held Deletha up in the air and asked bystanders whether anyone wanted a piece of “the bitch.” About 40 people drove by in their cars, but none intervened or called the police.
- Rather than blaming the causes of the Genovese murder and similar events on apathetic city-goers, Darley and Latane suspected that the causes of bystander nonintervention lay far more in commonplace psychological processes than in the interpersonal nature of the large urban environment.
- Not a single person paused to assist Torres before a police officer arrived on the scene.

On May 30, 2008 in Hartford, Connecticut, a 78-year-old man on his way home from buying milk in a grocery store, Angel Arce Torres, was hit by a car in the middle of a busy street during rush hour.

9781405131117_4_007.qxd 30/6/09 11:46 AM Page 140 held Deletha up in the air and asked bystanders whether anyone wanted a piece of “the bitch.” About 40 people drove by in their cars, but none intervened or called the police.

Rather than blaming the causes of the Genovese murder and similar events on apathetic city-goers, Darley and Latane suspected that the causes of bystander nonintervention lay far more in commonplace psychological processes than in the interpersonal nature of the large urban environment.

Not a single person paused to assist Torres before a police officer arrived on the scene.

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:
- We also see from the lightcone at the line r = 2μ in the upper region that the particle never escaped from black hole while, in the lower region, particle never moved into the white hole.
- Thus it is impossible to move from region I to region I(cid:48) or inversely from region I(cid:48) to region I.
- We can see that the particle at the origin of the diagram are restricted to go the region II which is the black hole.
- This leads to the fact that there exists another world in region I(cid:48) which cannot be in(cid:29)uenced by our world.

We also see from the lightcone at the line r = 2μ in the upper region that the particle never escaped from black hole while, in the lower region, particle never moved into the white hole.

Thus it is impossible to move from region I to region I(cid:48) or inversely from region I(cid:48) to region I.

We can see that the particle at the origin of the diagram are restricted to go the region II which is the black hole.

This leads to the fact that there exists another world in region I(cid:48) which cannot be in(cid:29)uenced by our world.

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:
- "Space and Time Warps" (https://web.archive.org/web/20120210233225/http://www.hawking.org.u k/space-and-time-warps.html).
- doi:10.1103/PhysRevLett.61.1446 (https://doi.org/10.1103%2FPh ysRevLett.61.1446).
- arXiv:0908.2651 (https://arxiv.org/abs/0908.2651).
- arXiv:gr-qc/9309017 (https://arxiv.org/abs/gr-qc/930901 7).

"Space and Time Warps" (https://web.archive.org/web/20120210233225/http://www.hawking.org.u k/space-and-time-warps.html).

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List tests or derivations implied by the evidence.


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

EVIDENCE:
- If there is a coincidence between detectors f1 and f2, then photon 3 is predicted to be polarized at a 45° angle.
- They then used a two-photon interferometry for analyzing the entanglement so that the quantum property could be recognized when it is transferred from one photon to the other.[3] Photon 1 was polarized at 45° in the first experiment conducted by Zeilinger's group.
- Quantum teleportation is verified when both photons are detected in the state, which has a probability of 25%.
- Two detectors, f1 and f2, are placed behind the beam splitter, and recording the coincidence will identify the state.

If there is a coincidence between detectors f1 and f2, then photon 3 is predicted to be polarized at a 45° angle.

They then used a two-photon interferometry for analyzing the entanglement so that the quantum property could be recognized when it is transferred from one photon to the other.[3] Photon 1 was polarized at 45° in the first experiment conducted by Zeilinger's group.

Quantum teleportation is verified when both photons are detected in the state, which has a probability of 25%.

Two detectors, f1 and f2, are placed behind the beam splitter, and recording the coincidence will identify the state.

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:
- Wewillnotaddressfurtherthisdifficultyofthetheorynotcontainingchargedparticles thatarealsomassless(thatis,withmassesmuchlessthanM ).
- In quantum mechanics, de Broglie tells us that frequency and wave number become energy E=(cid:2)ω and momentum p(cid:3)=(cid:2)k (cid:3) , respectively.
- The solution is (cid:26)(t,x(cid:3),y)=e −iωt+ik (cid:3).x(cid:3)+iκy, with ω2=(k (cid:3)2+κ2).
- P Note also that we have not inquired about the “mechanism” that breaks the 5-dimensionalspacetimeinto4cosmologicallylargecoordinatesandonesmallcoordinate.

In quantum mechanics, de Broglie tells us that frequency and wave number become energy E=(cid:2)ω and momentum p(cid:3)=(cid:2)k (cid:3) , respectively.

The solution is (cid:26)(t,x(cid:3),y)=e −iωt+ik (cid:3).x(cid:3)+iκy, with ω2=(k (cid:3)2+κ2).

P Note also that we have not inquired about the “mechanism” that breaks the 5-dimensionalspacetimeinto4cosmologicallylargecoordinatesandonesmallcoordinate.

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:
- of 210Po being released from the coolant.[63][64] The longer-lived isotopes of polonium, 208Po and 209Po, can be formed by proton or deuteron bombardment of bismuth using a cyclotron.
- Other more neutron-deficient and more unstable isotopes can be formed by the irradiation of platinum with carbon nuclei.[65] Applications Polonium-based sources of alpha particles were produced in the former Soviet Union.[66] Such sources were applied for measuring the thickness of industrial coatings via attenuation of alpha radiation.[67] Because of intense alpha radiation, a one-gram sample of 210Po will spontaneously heat up to above 500 °C (932 °F) generating about 140 watts of power.
- Therefore, 210Po is used as an atomic heat source to power radioisotope thermoelectric generators via thermoelectric materials.[6][18][68][69] For example, 210Po heat sources were used in the Lunokhod 1 (1970) and Lunokhod 2 (1973) Moon rovers to keep their internal components warm during the lunar nights, as well as the Kosmos 84 and 90 satellites (1965).[66][70] The alpha particles emitted by polonium can be converted to neutrons using beryllium oxide, at a rate of 93 neutrons per million alpha particles.[68] Po-BeO mixtures are used as passive neutron sources with a gamma-rayto-neutron production ratio of 1.13 ± 0.05, lower than for nuclear fission-based neutron sources.[71] Examples of

of 210Po being released from the coolant.[63][64] The longer-lived isotopes of polonium, 208Po and 209Po, can be formed by proton or deuteron bombardment of bismuth using a cyclotron.

Other more neutron-deficient and more unstable isotopes can be formed by the irradiation of platinum with carbon nuclei.[65] Applications Polonium-based sources of alpha particles were produced in the former Soviet Union.[66] Such sources were applied for measuring the thickness of industrial coatings via attenuation of alpha radiation.[67] Because of intense alpha radiation, a one-gram sample of 210Po will spontaneously heat up to above 500 °C (932 °F) generating about 140 watts of power.

Therefore, 210Po is used as an atomic heat source to power radioisotope thermoelectric generators via thermoelectric materials.[6][18][68][69] For example, 210Po heat sources were used in the Lunokhod 1 (1970) and Lunokhod 2 (1973) Moon rovers to keep their internal components warm during the lunar nights, as well as the Kosmos 84 and 90 satellites (1965).[66][70] The alpha particles emitted by polonium can be converted to neutrons using beryllium oxide, at a rate of 93 neutrons per million alpha particles.[68] Po-BeO mixtures are used as passive neutron sources with a gamma-rayto-neutron production ratio of 1.13 ± 0.05, lower than for nuclear fission-based neutron sources.[71] Examples of

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 has dimension of length with SI unit of metre (m).
- Definition and formulation Deformation is the change in the metric properties of a continuous body, meaning that a curve drawn in the initial body placement changes its length when displaced to a curve in the final placement.
- It is quantified as the residual displacement of particles in a non-rigid body, from an initial configuration to a final configuration, excluding the body's average translation and rotation (its rigid transformation).[1] A configuration is a set containing the positions of all particles of the body.
- In this (such as gravity or electromagnetic forces), or changes particular case of bending, displacements in temperature, moisture content, or chemical associated with rigid translations and rotations reactions, etc.

It has dimension of length with SI unit of metre (m).

Definition and formulation Deformation is the change in the metric properties of a continuous body, meaning that a curve drawn in the initial body placement changes its length when displaced to a curve in the final placement.

It is quantified as the residual displacement of particles in a non-rigid body, from an initial configuration to a final configuration, excluding the body's average translation and rotation (its rigid transformation).[1] A configuration is a set containing the positions of all particles of the body.

In this (such as gravity or electromagnetic forces), or changes particular case of bending, displacements in temperature, moisture content, or chemical associated with rigid translations and rotations reactions, etc.

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:
- Thedurationforwhichthewormholeistraversabledependsonthemechanismsemployed to create it.
- In the examples discussed above, ANECviolationwasobtainedeitherthroughaCasimir-likenegativeenergyproduced by fermions moving along magnetic field lines (sections 5.1 and 5.2.5), through an interactionbetweenCFTslivingontheboundariesofaneternalBTZblackhole(sections 5.2.1 and 5.2.2) or through backreaction of quantum fields (sections 5.2.3 and 5.2.4).
- Section 6 Conclusion This dissertation is a review of the most important concepts in wormhole physics and of the most recent recipes to construct these extreme objects.
- There could exist eternally traversable wormholes (as discussed in section 5.2.2) or long-lived wormholes which are only traversable for a brief period of their existence (like the example provided in section 5.2.1).

Thedurationforwhichthewormholeistraversabledependsonthemechanismsemployed to create it.

In the examples discussed above, ANECviolationwasobtainedeitherthroughaCasimir-likenegativeenergyproduced by fermions moving along magnetic field lines (sections 5.1 and 5.2.5), through an interactionbetweenCFTslivingontheboundariesofaneternalBTZblackhole(sections 5.2.1 and 5.2.2) or through backreaction of quantum fields (sections 5.2.3 and 5.2.4).

Section 6 Conclusion This dissertation is a review of the most important concepts in wormhole physics and of the most recent recipes to construct these extreme objects.

There could exist eternally traversable wormholes (as discussed in section 5.2.2) or long-lived wormholes which are only traversable for a brief period of their existence (like the example provided in section 5.2.1).

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:
- 13.4.1 Pseudoscalar Define the pseudoscalar I in N dimensions to be N I ≡γγ ∧γγ ∧...∧γγ (13.19) N 1 2 N
- Reversion leaves unchanged all multivectors whose grade is 0 or 1, modulo 4, and changes the sign of all multivectors whose grade is 2 or 3, modulo 4.
- For example, the Hodge dual of a bivector in 2 dimensions is a 0-dimensional object, a pseudoscalar.
- For example, scalars and vectors are unchanged by reversion, but bivectors and trivectors change sign.

13.4.1 Pseudoscalar Define the pseudoscalar I in N dimensions to be N I ≡γγ ∧γγ ∧...∧γγ (13.19) N 1 2 N

Reversion leaves unchanged all multivectors whose grade is 0 or 1, modulo 4, and changes the sign of all multivectors whose grade is 2 or 3, modulo 4.

For example, the Hodge dual of a bivector in 2 dimensions is a 0-dimensional object, a pseudoscalar.

For example, scalars and vectors are unchanged by reversion, but bivectors and trivectors change sign.

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:
- Singapore, New Jersey, London: World Scientific Publishing.
- Rheology: principles, measurement and applications.
- Elastoplasticity Theory (https://books.google.com/books?id=1P0LybL4o AgC).
- Inelastic Analysis of Structures (https://books.google.com/books?i d=8mz-xPdvH00C).

Singapore, New Jersey, London: World Scientific Publishing.

Rheology: principles, measurement and applications.

Elastoplasticity Theory (https://books.google.com/books?id=1P0LybL4o AgC).

Inelastic Analysis of Structures (https://books.google.com/books?i d=8mz-xPdvH00C).

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:
- Normallyinquantummechanics,aspin1particlehas3spinstates.Foradiscussionofwhyamassless spin1particlehasonly2spinstates,see,forexample,QFTNut,chapterIII.4.
- WhethertheyareequationsoridentitiesdependsonwhetheryouregardA μorF μνasfundamental.Inthe quantumworld,youareforcedtotreatA μasfundamental.
- Thissectionisadaptedfrompp.95–100ofmypopularbookFearful,writtenfortheeducatedpublic.
- See,forexample,H.Goldstein,ClassicalMechanics.

WhethertheyareequationsoridentitiesdependsonwhetheryouregardA μorF μνasfundamental.Inthe quantumworld,youareforcedtotreatA μasfundamental.

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:
- This means that the bottom area is .
- This equation is applicable when the final velocity v is unknown.
- We also know that or is the area under a velocity–time graph.[21] We can take by adding the top area and the bottom area.
- Adding and results in the equation results in the equation [23] .

This equation is applicable when the final velocity v is unknown.

We also know that or is the area under a velocity–time graph.[21] We can take by adding the top area and the bottom area.

Adding and results in the equation results in the equation [23] .

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

List tests or derivations implied by the evidence.