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

EVIDENCE:
- As a result of that review, 45 SRI originated project documents were reviewed and all references regarding DoD involvement were excised and the documents - declassified.
- (S/NF) REPRODUCTION REQUIRES WARNING NOTICE-INTELLIGENCE APPROVAL OF ORIGINATOR SOURCES OR METHODS INVOLVED OR HIGHER DOD AUTHORITY FURTHER DISSEMINATION LIMITED DISSEMINATION ONLY AS DIRECTED BY PAG OR HIGHER DOD AUTHORITY SECRET NOT RELEASABLE TO FOREIGN NATIONALS STAR GATE
- DIA’s General Counsel and Public Affairs Office concurred with the declassification action in September 1994.
- This DIA initiative should establish the precedent for the DCI’s declassification committee to continue this effort once the program is officially transferred to CIA.

As a result of that review, 45 SRI originated project documents were reviewed and all references regarding DoD involvement were excised and the documents - declassified.

(S/NF) REPRODUCTION REQUIRES WARNING NOTICE-INTELLIGENCE APPROVAL OF ORIGINATOR SOURCES OR METHODS INVOLVED OR HIGHER DOD AUTHORITY FURTHER DISSEMINATION LIMITED DISSEMINATION ONLY AS DIRECTED BY PAG OR HIGHER DOD AUTHORITY SECRET NOT RELEASABLE TO FOREIGN NATIONALS STAR GATE

DIA’s General Counsel and Public Affairs Office concurred with the declassification action in September 1994.

This DIA initiative should establish the precedent for the DCI’s declassification committee to continue this effort once the program is officially transferred to CIA.

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:
- Newton: Challenges Newton's view of absolute space and time, presenting them as relative and interconnected.
- The Relativity of Space and Time Einstein vs.
- Preface Thorne shares his quest to understand Einstein’s theories and make complex scientific ideas accessible to both laypersons and scientists.
- Chapter 1 Summary : Prologue: A Voyage among the Holes Section Summary Foreword Stephen Hawking introduces the revolutionary concepts of space and time from Einstein's theories, detailing the journey toward understanding black holes.

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.

Newton: Challenges Newton's view of absolute space and time, presenting them as relative and interconnected.

The Relativity of Space and Time Einstein vs.

Preface Thorne shares his quest to understand Einstein’s theories and make complex scientific ideas accessible to both laypersons and scientists.

Chapter 1 Summary : Prologue: A Voyage among the Holes Section Summary Foreword Stephen Hawking introduces the revolutionary concepts of space and time from Einstein's theories, detailing the journey toward understanding black holes.

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 result is that we see two images of the same quasar, one arriving a short time after the other (about 417.1 days later).
- the sky, leading to speculation that a cosmic string had been found.[7] However, observations by the Hubble Space Telescope in January 2005 showed them to be a pair of similar galaxies, not two images of the same galaxy.[8][9] A cosmic string would produce a similar duplicate image of fluctuations in the cosmic microwave background, which it was thought might have been detectable by the Planck Surveyor mission.[10] However, a 2013 analysis of data from the Planck mission failed to find any evidence of cosmic strings.[11] A piece of evidence supporting cosmic string theory is a phenomenon noticed in observations of the "double quasar" called Q0957+561A,B.
- However, a team of astronomers at the Harvard-Smithsonian Center for Astrophysics led by Rudolph Schild studied the quasar and found that during the period between September 1994 and July 1995 the two images
- Originally discovered by Dennis Walsh, Bob Carswell, and Ray Weymann in 1979, the double image of this quasar is caused by a galaxy positioned between it and the Earth.

The result is that we see two images of the same quasar, one arriving a short time after the other (about 417.1 days later).

the sky, leading to speculation that a cosmic string had been found.[7] However, observations by the Hubble Space Telescope in January 2005 showed them to be a pair of similar galaxies, not two images of the same galaxy.[8][9] A cosmic string would produce a similar duplicate image of fluctuations in the cosmic microwave background, which it was thought might have been detectable by the Planck Surveyor mission.[10] However, a 2013 analysis of data from the Planck mission failed to find any evidence of cosmic strings.[11] A piece of evidence supporting cosmic string theory is a phenomenon noticed in observations of the "double quasar" called Q0957+561A,B.

However, a team of astronomers at the Harvard-Smithsonian Center for Astrophysics led by Rudolph Schild studied the quasar and found that during the period between September 1994 and July 1995 the two images

Originally discovered by Dennis Walsh, Bob Carswell, and Ray Weymann in 1979, the double image of this quasar is caused by a galaxy positioned between it and the Earth.

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:
- However, some experts argue that because a particle’s or a photon’s quantum state is its defining characteristic, teleporting its quantum state is completely equivalent to teleporting the particle/photon even though the original particle’s/photon’s quantum state (and defining characteristic) was completely destroyed in the process (more on this in Section 3.3).
- Research is continuing on whether decoherence can be reduced, circumvented, or otherwise be (partially or totally) eliminated.
- The quantum link (or EPR interaction) between a pair of systems is subject to noise or decoherence through photon loss or heating of the phonons.
- Decoherence degrades the fidelity of the quantum link (i.e., the set of pure EPR entangled pairs) between two quantum systems, thus introducing a certain level of error in the exchange of quantum information between the systems.

However, some experts argue that because a particle’s or a photon’s quantum state is its defining characteristic, teleporting its quantum state is completely equivalent to teleporting the particle/photon even though the original particle’s/photon’s quantum state (and defining characteristic) was completely destroyed in the process (more on this in Section 3.3).

Research is continuing on whether decoherence can be reduced, circumvented, or otherwise be (partially or totally) eliminated.

The quantum link (or EPR interaction) between a pair of systems is subject to noise or decoherence through photon loss or heating of the phonons.

Decoherence degrades the fidelity of the quantum link (i.e., the set of pure EPR entangled pairs) between two quantum systems, thus introducing a certain level of error in the exchange of quantum information between the systems.

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

EVIDENCE:
- If we extend unlimited tolerance even to those who are intolerant, if we are not prepared to defend a tolerant society against the onslaught of the intolerant, then the tolerant will be destroyed, and tolerance with them.
- This view cannot be right, Popper argued, because "nothing ever comes off exactly as intended."[71] According to philosopher David Coady, "Popper has often been cited by critics of conspiracy theories, and his views on the topic continue to constitute an orthodoxy in some circles."[72] However, philosopher Charles Pigden has pointed out that Popper's argument only applies to a very extreme kind of conspiracy theory, not to conspiracy theories generally.[73] Metaphysics Truth As early as 1934, Popper wrote of the search for truth as "one of the strongest motives for scientific discovery."[74] Still, he describes in Objective Knowledge (1972) early concerns about the muchcriticised notion of truth as correspondence.
- In this formulation, I do not imply, for instance, that we should always suppress the utterance of intolerant philosophies; as long as we can counter them by rational argument and keep them in check by public opinion, suppression would certainly be most unwise.
- and that emphasis should be put on a hierarchy of humanitarian values rather than advocacy of a free market as envisioned by classical liberalism.[66] The paradox of tolerance Although Popper was an advocate of toleration, he also warned against unlimited tolerance.

If we extend unlimited tolerance even to those who are intolerant, if we are not prepared to defend a tolerant society against the onslaught of the intolerant, then the tolerant will be destroyed, and tolerance with them.

This view cannot be right, Popper argued, because "nothing ever comes off exactly as intended."[71] According to philosopher David Coady, "Popper has often been cited by critics of conspiracy theories, and his views on the topic continue to constitute an orthodoxy in some circles."[72] However, philosopher Charles Pigden has pointed out that Popper's argument only applies to a very extreme kind of conspiracy theory, not to conspiracy theories generally.[73] Metaphysics Truth As early as 1934, Popper wrote of the search for truth as "one of the strongest motives for scientific discovery."[74] Still, he describes in Objective Knowledge (1972) early concerns about the muchcriticised notion of truth as correspondence.

In this formulation, I do not imply, for instance, that we should always suppress the utterance of intolerant philosophies; as long as we can counter them by rational argument and keep them in check by public opinion, suppression would certainly be most unwise.

and that emphasis should be put on a hierarchy of humanitarian values rather than advocacy of a free market as envisioned by classical liberalism.[66] The paradox of tolerance Although Popper was an advocate of toleration, he also warned against unlimited tolerance.

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- You could verify as a trivial exercise that S (cid:5)ij =RikRjlSkl.
- Butwearenotdoneyet.The6-dimensionalrepresentationisalsoreducible.Toseethis, note S (cid:5)ii=RikRilSkl=(RT)kiRilSkl=(R −1)kiRilSkl=δklSkl=Skk (6) where we have used the O in SO(D).
- The 6-by-6 matrix describing the linear transformation of the 6 objects Sij breaks up into a 1-by-1 block and a 5-by-5 block.
- For D=3, the number of components in A and S are 1 .3.2=3 and 1 .3.4=6, 2 2 respectively.(ForD=4,thenumberofcomponentsinAandSare6and10,respectively.) Thus, inasuitablebasis, the9-by-9matrixreferredtoaboveactuallybreaksupintoa3by-3blockanda6-by-6block.Wesaythatthe9-dimensionalrepresentationisreducible: itcouldbereducedtosmallerrepresentations.

You could verify as a trivial exercise that S (cid:5)ij =RikRjlSkl.

Butwearenotdoneyet.The6-dimensionalrepresentationisalsoreducible.Toseethis, note S (cid:5)ii=RikRilSkl=(RT)kiRilSkl=(R −1)kiRilSkl=δklSkl=Skk (6) where we have used the O in SO(D).

The 6-by-6 matrix describing the linear transformation of the 6 objects Sij breaks up into a 1-by-1 block and a 5-by-5 block.

For D=3, the number of components in A and S are 1 .3.2=3 and 1 .3.4=6, 2 2 respectively.(ForD=4,thenumberofcomponentsinAandSare6and10,respectively.) Thus, inasuitablebasis, the9-by-9matrixreferredtoaboveactuallybreaksupintoa3by-3blockanda6-by-6block.Wesaythatthe9-dimensionalrepresentationisreducible: itcouldbereducedtosmallerrepresentations.

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:
- uni-goettingen.de/no_cache/dms/load/img/?IDDOC=63971) on 8 March 2013 ——— (1926b), "Ableitung der Elementarteilertheorie aus der Gruppentheorie" (https://web.archiv e.org/web/20130308102932/http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=24 8861) [Derivation of the Theory of Elementary Divisor from Group Theory], Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), 34 (Abt.
- ——— (1923), "Zur Theorie der Polynomideale und Resultanten" (http://www.digizeitschriften.de/d ownload/PPN235181684_0088/log7.pdf) (PDF), Mathematische Annalen (in German), 88 (1–2): 53–79, doi:10.1007/BF01448441 (https://doi.org/10.1007%2FBF01448441), S2CID 122226025 (h ttps://api.semanticscholar.org/CorpusID:122226025) ——— (1923b), "Eliminationstheorie und allgemeine Idealtheorie" (http://www.digizeitschriften.de/ download/PPN235181684_0090/log25.pdf) (PDF), Mathematische Annalen (in German), 90 (3– 4), Germany: 229–261, doi:10.1007/BF01455443 (https://doi.org/10.1007%2FBF01455443), S2CID 121239880 (https://api.semanticscholar.org/CorpusID:121239880) ——— (1924), "Eliminationstheorie und Idealtheorie" (https://web.archive.org/web/201303081029 26/http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=248880), Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), 33: 116–120, archived from the original (http:// gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=248880) on 8 March 2013 ——— (1926), "Der Endlichkeitsatz der Invarianten endlicher linearer Gruppen der Charakteristik p" (https://web.archive.org/web/20130308102929/http://gdz.sub.uni-goettingen.de/no_cache/dms/l oad/img/?IDDOC=63971) [Proof of the Finiteness of the Invariants of Finite Linear Groups of Characteristic p], Nachr.
- viii, 777, ISBN 978-3-540-11504-5, MR 0703862 (https://mathscinet.ams.org/mathscinet-getitem?mr=0703862)
- Wiss (in German): 28–35, archived from the original (http://gdz.sub.

uni-goettingen.de/no_cache/dms/load/img/?IDDOC=63971) on 8 March 2013 ——— (1926b), "Ableitung der Elementarteilertheorie aus der Gruppentheorie" (https://web.archiv e.org/web/20130308102932/http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=24 8861) [Derivation of the Theory of Elementary Divisor from Group Theory], Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), 34 (Abt.

——— (1923), "Zur Theorie der Polynomideale und Resultanten" (http://www.digizeitschriften.de/d ownload/PPN235181684_0088/log7.pdf) (PDF), Mathematische Annalen (in German), 88 (1–2): 53–79, doi:10.1007/BF01448441 (https://doi.org/10.1007%2FBF01448441), S2CID 122226025 (h ttps://api.semanticscholar.org/CorpusID:122226025) ——— (1923b), "Eliminationstheorie und allgemeine Idealtheorie" (http://www.digizeitschriften.de/ download/PPN235181684_0090/log25.pdf) (PDF), Mathematische Annalen (in German), 90 (3– 4), Germany: 229–261, doi:10.1007/BF01455443 (https://doi.org/10.1007%2FBF01455443), S2CID 121239880 (https://api.semanticscholar.org/CorpusID:121239880) ——— (1924), "Eliminationstheorie und Idealtheorie" (https://web.archive.org/web/201303081029 26/http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=248880), Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), 33: 116–120, archived from the original (http:// gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=248880) on 8 March 2013 ——— (1926), "Der Endlichkeitsatz der Invarianten endlicher linearer Gruppen der Charakteristik p" (https://web.archive.org/web/20130308102929/http://gdz.sub.uni-goettingen.de/no_cache/dms/l oad/img/?IDDOC=63971) [Proof of the Finiteness of the Invariants of Finite Linear Groups of Characteristic p], Nachr.

viii, 777, ISBN 978-3-540-11504-5, MR 0703862 (https://mathscinet.ams.org/mathscinet-getitem?mr=0703862)

Wiss (in German): 28–35, archived from the original (http://gdz.sub.

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:
- Consideragameoftennis.Compareaharddrivedownthelineandasoftlobhighoverthenet.Inboth cases,wearetosolveNewton’slawd2x =0,d2y =−g,withtheboundaryconditionsx(0)=0,x(T)=L,and dt2 dt2 y(0)=y(T)=0.(Theproblemissoelementarythatwewon’tbothertoexplainthenotation,thatydenotes theverticaldirection,thaty=0istheground,thatT isthetimeofflightbeforetheballhitstheground, thatListhelengthofthetenniscourt,andsoonandsoforth.Youmightwanttodrawyourownfigure.) Thesolutionisx=Lt/T, y=1g(T −t)t.Notethatthetwotypesoftennisshotsaregovernedbythesame equationandthesameL.Henceweobtainthesamesolution,butkeepinmindthatT issmallinthecase oftheharddriveandthatT islargeinthecaseofthesoftlob.Noweliminatet toobtainyasafunction ofx,namelyy(x)= 1gT2(1− x)x,aparabolainbothcases(ofcourse).Butcomparethecurvatureofthe 2 L L twoparabolas:wehave d2y =−g(T/L)2,verysmallinthecaseoftheharddrive(smallT)andverylargein dx2 thecaseofthelob(largeT).Theharddrivedownthelinebarelyskimmingoverthenet,andthesoftlob climbinglazilyhighupintothesky,lookandfeeltotallydifferentpicturedinspace.Incontrast,considery asafunctionoft.Wealsohavetwoparabolas(ofcourse),namelyy(t)= 1g(T −t)t,asgivenearlier.Now comparethecurvatureofthetwoparabolas:wehaved2y =−g,thesameinbothcases.Thecurvatureofthe dt2 ball’strajectoryinspacetimeisuniversal(universalgravity,getit?).Butwetendtoseeinourmind’seyethe twoparabolasy(x)inspace,onefortheharddriveandoneforthelob,whichlookquitedifferent,ratherthan theparabolasy(t)inspacetime,whichhavethesamecurvature.IlearnedthislongagofromJohnWheeler.
- Currentlytoonepartin1013.ThemodernroundofexperimentsstartedwithLor ́andE ̈otv ̈osin1885and continueswiththeE ̈ot-WashexperimentledbyE.Adelbergerinourdays.
- Rigorousmathematiciansgoberserkatphysicists’useoftheword“function”here;theyprefertocallita distribution,definedasthelimitofafunction.Butworkingphysicistsdonotgiveaflyingbarfaboutsuch niceties.Inanycase,Idonotpersonallyknowatheoreticalphysicistsufferinganyharmbycallingδ(t)a function.
- 5 ProveNewton’ssecondsuperbtheorem.

Currentlytoonepartin1013.ThemodernroundofexperimentsstartedwithLor ́andE ̈otv ̈osin1885and continueswiththeE ̈ot-WashexperimentledbyE.Adelbergerinourdays.

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:
- Superdense Coding over Optical Fiber Links with Complete Bell-State Measurements.
- doi:10.1103/PhysRevA.71.044305 (https://doi.org/10.1103%2FPhysRevA.71.044305).
- doi:10.1126/sciadv.aat9304 (https://doi.org/10.1126%2Fsciadv.aat9304).
- PMID 28211745 (https://pubmed.ncbi.nlm.nih.gov/2821174 5).Williams, B.

Superdense Coding over Optical Fiber Links with Complete Bell-State Measurements.

PMID 28211745 (https://pubmed.ncbi.nlm.nih.gov/2821174 5).Williams, B.

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:
- During this preliminary test, it was claimed that the Eldridge was rendered nearly invisible, with some witnesses describing a greenish fog enveloping the ship.
- Detailed descriptions of the events center on two main occurrences.
- The more widely cited and dramatic experiment allegedly happened on October 28, 1943.
- However, the accounts become even more extraordinary with the details of the second test.

During this preliminary test, it was claimed that the Eldridge was rendered nearly invisible, with some witnesses describing a greenish fog enveloping the ship.

Detailed descriptions of the events center on two main occurrences.

The more widely cited and dramatic experiment allegedly happened on October 28, 1943.

However, the accounts become even more extraordinary with the details of the second test.

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:
- Fundamentals Definitions Non-relativistic classical mechanics treats time as a universal quantity of measurement that is uniform throughout, is separate from space, and is agreed on by all observers.
- Spacetime diagrams are useful in visualizing and understanding relativistic effects, such as how different observers perceive where and when events occur.
- Until the turn of the 20th century, the assumption had been that the three-dimensional geometry of the universe (its description in terms of locations, shapes, distances, and directions) was distinct from time (the measurement of when events occur within the universe).
- This interpretation proved vital to the general theory of relativity, wherein spacetime is curved by mass and energy.

Fundamentals Definitions Non-relativistic classical mechanics treats time as a universal quantity of measurement that is uniform throughout, is separate from space, and is agreed on by all observers.

Spacetime diagrams are useful in visualizing and understanding relativistic effects, such as how different observers perceive where and when events occur.

Until the turn of the 20th century, the assumption had been that the three-dimensional geometry of the universe (its description in terms of locations, shapes, distances, and directions) was distinct from time (the measurement of when events occur within the universe).

This interpretation proved vital to the general theory of relativity, wherein spacetime is curved by mass and 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:
- debated by physicists and by philosophers.
- While the significance of Bell's theorem is not in doubt, different interpretations of quantum mechanics disagree about what exactly it implies.
- Theorem There are many variations on the basic idea, some employing stronger mathematical assumptions than others.[6] Significantly, Bell-type theorems do not refer to any particular theory of local hidden variables, but instead show that quantum physics violates general assumptions behind classical

debated by physicists and by philosophers.

While the significance of Bell's theorem is not in doubt, different interpretations of quantum mechanics disagree about what exactly it implies.

Theorem There are many variations on the basic idea, some employing stronger mathematical assumptions than others.[6] Significantly, Bell-type theorems do not refer to any particular theory of local hidden variables, but instead show that quantum physics violates general assumptions behind classical

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:
- For example, they reflect the fact that observers moving at different velocities may measure different distances, elapsed times, and even different orderings of events, but always such that the speed of light is the same in all inertial reference frames.
- features that do not appear in Galilean transformations.
- Historically, the transformations were the result of attempts by Lorentz and others to explain how the speed of light was observed to be independent of the reference frame, and to understand the symmetries of the laws of electromagnetism.
- The invariance of light speed is one of the postulates of special relativity.

For example, they reflect the fact that observers moving at different velocities may measure different distances, elapsed times, and even different orderings of events, but always such that the speed of light is the same in all inertial reference frames.

features that do not appear in Galilean transformations.

Historically, the transformations were the result of attempts by Lorentz and others to explain how the speed of light was observed to be independent of the reference frame, and to understand the symmetries of the laws of electromagnetism.

The invariance of light speed is one of the postulates of special relativity.

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:
- GravityGoesLive | 397 Hereisacodatothestory.NowthatLorentzknewwhatLwas,hestudied∗theactionprincipleforgravityin aseriesofpapers,usingamoregeneralS thanHilbertdid.AsforEinstein,ayearlater,onNovember26, matter 1916,hepresentedapapertitled“Hamilton’sPrincipleandtheGeneralTheoryofRelativity”inwhichhewrote pointedly,“Weshallmakeasfewspecializingassumptionsaspossible,inmarkedcontrasttoHilbert’streatment.” NowIcomebacktothePalatiniformalism.ItisnowheretobefoundinthepaperPalatinipresentedto theCircoloMatematicodiPalermoonAugust10,1919!HeimprovedHilbert’scalculationandinfacthadthe Palatiniidentityintroducedinthischapter.WhatthetextbooksnowcallthePalatiniformalismwasactually inventedin1925byEinstein!Astheyearspassed, apparentlypeoplemixedupthePalatiniidentityandthe Palatiniformalism,andvariouspeople,includingEinstein,†referredto(24)asthePalatiniformalism.Afterall, boththePalatiniidentityandthePalatiniformalismabsolveusofhavingtovarytheRiccitensor,soitiseasyto mixupthetwo.Thisconfusionhasbeenperpetuatedunwittinglyinmanytextbooks(andwittinglyinthisone).
- Appendix7:AnalternativeformoftheEinstein-Hilbertaction WecanrewritetheEinstein-HilbertactioninanalternativeformthatIdonotlikeforreasonsthatwillbecome clearbutthatImentionhereforthesakeofcompleteness.LookinguptheexpressionfortheRiccitensorgiven in(25),wewritetheactionin(1)(suppressingtheirrelevantoverallconstant)as (cid:7) (cid:7) √ √ S= d4x −ggρνR = d4x −ggρν(∂ (cid:25)σ −∂ (cid:25)σ +(cid:25)σ (cid:25)κ −(cid:25)σ (cid:25)κ ) ρν σ νρ ν σρ σκ νρ νκ σρ =S +S +S +S (27) I II III IV WenextintegratebypartstogetridofthederivativesontheChristoffelsymbols,assumingthatallsurfaceterms canbethrownaway.Firstwehave (cid:7) (cid:7) √ √ S = d4x −ggρν∂ (cid:25)σ =− d4x∂ ( −ggρν)(cid:25)σ I σ νρ σ νρ √ √ Differentiating, we write ∂ σ ( −ggρν)= − (cid:8) g((cid:25) σ η √η gρν+∂ σ gρν).
- ∗ LorentzmentionedthattheBelgianTh ́eophileErnestdeDonder(1872–1957)alsocontributed.History hasnotbeenkindtodeDonder.(See,however,thefootnoteonp.21ofQFTNut.)Theharmonicgaugetobe introducedinchapterIX.4isalsoknownasthedeDondergauge.Perhapsyouhaveseenthefamousphotograph ofthepost–quantummechanics1927SolvayConference?DeDonderstoodbehindDirac,whointurnwasseated behindLorentzandEinstein.
- Invoking the identity D σ gρν=0=∂ σ gρν+ (cid:25)ρ gλν+(cid:25)ν gρλ,wecanthenwriteS =− d4x −g((cid:25)η gρν−(cid:25)ρ gλν−(cid:25)ν gρλ)(cid:25)σ .Similarly,weobtain σλ σλ I ση σλ σλ νρ (cid:7) (cid:7) √ √ S =− d4x −ggρν∂ (cid:25)σ = d4x∂ ( −ggρν)(cid:25)σ II ν σρ ν σρ (cid:7) √ = d4x −g((cid:25)η gρν−(cid:25)ρ gλν−(cid:25)ν gρλ)(cid:25)σ νη νλ νλ σρ (cid:8) √ Adding,wefindS +S =2 d4x −g[(cid:25)ρ (cid:25)σ gλν−((cid:25)ρ gλν)(cid:25)σ ].WenextobservethatS +S ,aswritten I II σλ νρ νλ σρ III IV in(27)withnofurthermassagingneeded,isjust,interestingly,−1timesthisexpression.Thus,weendupwith analternativeformfortheEinstein-Hilbertaction (cid:7) √ S= d4x −g[(cid:25)ρ (cid:25)σ gλν−((cid:25)ρ gλν)(cid:25)η ] (28) σλ ρν νλ ρη ThereasonthatIdon’tlikethisformoftheactionisnowclear:theintegrandisnotascalarundercoordinate transformation,asisthecasefortheEinstein-Hilbertaction.Throwingboundarytermsawayatwillhasdone violencetotheunderlyinginvariancepropertiesoftheaction.Nevertheless,thisformoftheactionisusefulin somesituations.

Appendix7:AnalternativeformoftheEinstein-Hilbertaction WecanrewritetheEinstein-HilbertactioninanalternativeformthatIdonotlikeforreasonsthatwillbecome clearbutthatImentionhereforthesakeofcompleteness.LookinguptheexpressionfortheRiccitensorgiven in(25),wewritetheactionin(1)(suppressingtheirrelevantoverallconstant)as (cid:7) (cid:7) √ √ S= d4x −ggρνR = d4x −ggρν(∂ (cid:25)σ −∂ (cid:25)σ +(cid:25)σ (cid:25)κ −(cid:25)σ (cid:25)κ ) ρν σ νρ ν σρ σκ νρ νκ σρ =S +S +S +S (27) I II III IV WenextintegratebypartstogetridofthederivativesontheChristoffelsymbols,assumingthatallsurfaceterms canbethrownaway.Firstwehave (cid:7) (cid:7) √ √ S = d4x −ggρν∂ (cid:25)σ =− d4x∂ ( −ggρν)(cid:25)σ I σ νρ σ νρ √ √ Differentiating, we write ∂ σ ( −ggρν)= − (cid:8) g((cid:25) σ η √η gρν+∂ σ gρν).

Invoking the identity D σ gρν=0=∂ σ gρν+ (cid:25)ρ gλν+(cid:25)ν gρλ,wecanthenwriteS =− d4x −g((cid:25)η gρν−(cid:25)ρ gλν−(cid:25)ν gρλ)(cid:25)σ .Similarly,weobtain σλ σλ I ση σλ σλ νρ (cid:7) (cid:7) √ √ S =− d4x −ggρν∂ (cid:25)σ = d4x∂ ( −ggρν)(cid:25)σ II ν σρ ν σρ (cid:7) √ = d4x −g((cid:25)η gρν−(cid:25)ρ gλν−(cid:25)ν gρλ)(cid:25)σ νη νλ νλ σρ (cid:8) √ Adding,wefindS +S =2 d4x −g[(cid:25)ρ (cid:25)σ gλν−((cid:25)ρ gλν)(cid:25)σ ].WenextobservethatS +S ,aswritten I II σλ νρ νλ σρ III IV in(27)withnofurthermassagingneeded,isjust,interestingly,−1timesthisexpression.Thus,weendupwith analternativeformfortheEinstein-Hilbertaction (cid:7) √ S= d4x −g[(cid:25)ρ (cid:25)σ gλν−((cid:25)ρ gλν)(cid:25)η ] (28) σλ ρν νλ ρη ThereasonthatIdon’tlikethisformoftheactionisnowclear:theintegrandisnotascalarundercoordinate transformation,asisthecasefortheEinstein-Hilbertaction.Throwingboundarytermsawayatwillhasdone violencetotheunderlyinginvariancepropertiesoftheaction.Nevertheless,thisformoftheactionisusefulin somesituations.

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

EVIDENCE:
- The energy of photons, the kinetic energy of emitted particles, and, later, the thermal energy of the surrounding matter, all contribute to the invariant mass of the system.
- This is a restatement of the equivalent laws of conservation of energy and conservation of mass.
- Lead, atomic number 82, is the heaviest element to have any isotopes stable (to the limit of measurement) to radioactive decay.
- Gamma rays are emitted from excited states of nuclei as a side effect of alpha or beta decay.[27] Beta decay is the only type to be observed in all the elements.

The energy of photons, the kinetic energy of emitted particles, and, later, the thermal energy of the surrounding matter, all contribute to the invariant mass of the system.

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

Lead, atomic number 82, is the heaviest element to have any isotopes stable (to the limit of measurement) to radioactive decay.

Gamma rays are emitted from excited states of nuclei as a side effect of alpha or beta decay.[27] Beta decay is the only type to be observed in all the elements.

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

EVIDENCE:
- 17.1 ADM formalism 497 Withtherestrictedcovariantdivergenceconvertedtoatotalcovariantdivergence,theRicciscalar(17.37)is R=R^−2D Km−KpmnK +KpK .
- At this point however no equations of motion have yet been obtained: equations are obtained later, §17.2, from variation of the action.
- In the ADM formalism, equation (17.40c) for R is discarded as redundant.
- (17.40d) abcd abcd ca db cb da Equations (17.40a), (17.40b), (17.40c), and (17.40d) are called respectively the Ricci, Codazzi-Mainardi, BSSN, and Gauss equations.

17.1 ADM formalism 497 Withtherestrictedcovariantdivergenceconvertedtoatotalcovariantdivergence,theRicciscalar(17.37)is R=R^−2D Km−KpmnK +KpK .

At this point however no equations of motion have yet been obtained: equations are obtained later, §17.2, from variation of the action.

In the ADM formalism, equation (17.40c) for R is discarded as redundant.

(17.40d) abcd abcd ca db cb da Equations (17.40a), (17.40b), (17.40c), and (17.40d) are called respectively the Ricci, Codazzi-Mainardi, BSSN, and Gauss equations.

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

EVIDENCE:
- Archived (https://web.archive.org/web/20191228124040/https://books.google.com/books?
- Archived (https://web.archi ve.org/web/20191227145317/https://books.google.com/books?id=HoGDAwAAQBAJ) from the original on 27 December 2019.
- An Introduction to Measure Theory (https://books.google.com/books?id=HoG DAwAAQBAJ).
- id=mHwcCAAAQBAJ&pg=PA111) from the original on 28 December 2019.

Archived (https://web.archi ve.org/web/20191227145317/https://books.google.com/books?id=HoGDAwAAQBAJ) from the original on 27 December 2019.

An Introduction to Measure Theory (https://books.google.com/books?id=HoG DAwAAQBAJ).

id=mHwcCAAAQBAJ&pg=PA111) from the original on 28 December 2019.

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

EVIDENCE:
- Numerically, it is better to characterize an orbit by L/ 1−α rather than by L itself, since the former remains finite as α → 1, whereas L and 1 − α both tend to zero at α → 1.
- Substituting the two (±) expressions (23.66) for L into dP2/dr, and setting the product of the resulting two expressions for dP2/dr x x equal to zero, equation (23.60), yields a quartic equation p +p P +p P2+p P3+p P4 =0 , (23.67) 0 1 2 3 4

Numerically, it is better to characterize an orbit by L/ 1−α rather than by L itself, since the former remains finite as α → 1, whereas L and 1 − α both tend to zero at α → 1.

Substituting the two (±) expressions (23.66) for L into dP2/dr, and setting the product of the resulting two expressions for dP2/dr x x equal to zero, equation (23.60), yields a quartic equation p +p P +p P2+p P3+p P4 =0 , (23.67) 0 1 2 3 4

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

EVIDENCE:
- The teleportation protocol begins with a quantum state or qubit , in Alice's possession, that she wants to convey to Bob.
- , In the following, assume that Alice and Bob share the state Alice obtains one of the particles in the pair, with the other going to Bob.
- Next, the protocol requires that Alice and Bob share a maximally entangled state.
- At this point, Alice has two particles (C, the one she wants to teleport, and A, one of the entangled pair), and Bob has one particle, B.

The teleportation protocol begins with a quantum state or qubit , in Alice's possession, that she wants to convey to Bob.

, In the following, assume that Alice and Bob share the state Alice obtains one of the particles in the pair, with the other going to Bob.

Next, the protocol requires that Alice and Bob share a maximally entangled state.

At this point, Alice has two particles (C, the one she wants to teleport, and A, one of the entangled pair), and Bob has one particle, B.

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

EVIDENCE:
- These are the directions with respect to which the Weyl tensor and the electric field vector align.
- Geodesics along the principal null congruences satisfy dθ =dφ−ωdt=0 , (9.28) where ω = a/R2 is the azimuthal angular velocity of the geodesics through the coordinates.
- After tetrads, it will emerge that, in the Boyer-Lindquist tetrad, the electromagnetic field is purely radial, and the electromagnetic potential has only a time component.
- (9.27) μ ρ2 R Δ 9.16 Principal null congruences The Kerr-Newman geometry admits a special set of space-filling, non-overlapping null geodesics called the principal outgoing and ingoing null congruences.

These are the directions with respect to which the Weyl tensor and the electric field vector align.

Geodesics along the principal null congruences satisfy dθ =dφ−ωdt=0 , (9.28) where ω = a/R2 is the azimuthal angular velocity of the geodesics through the coordinates.

After tetrads, it will emerge that, in the Boyer-Lindquist tetrad, the electromagnetic field is purely radial, and the electromagnetic potential has only a time component.

(9.27) μ ρ2 R Δ 9.16 Principal null congruences The Kerr-Newman geometry admits a special set of space-filling, non-overlapping null geodesics called the principal outgoing and ingoing null congruences.

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

EVIDENCE:
- Einstein’sFieldEquationDerivedandPuttoWork antisymmetry of Fμν and the fact that ∂ and ∂ commute.
- To verify this, simply differentiatethequantity∇(cid:3) .E (cid:3)−ρwithrespecttotime:∂ (∇(cid:3) .E (cid:3)−ρ)=∇(cid:3) .∂ E (cid:3)−∂ ρ= 0 0 0 ∇(cid:3) .(∇(cid:3) ×B (cid:3)−J (cid:3) )+∇(cid:3) .J (cid:3)=0,whereweusedaMaxwellequationandcurrentconservation.
- Indeed, the invariance of physics under general coordinate transformation xμ→x (cid:5)μ(x)allowsustoeliminate4ofthe10unknownfunctionsg (x).Inotherwords, μν Einstein’s field equations should not determine g (x) completely.
- ∗ Note that this identity together with Maxwell’s equation implies that the current must be conserved: ∂ ν (∂ μ Fμν−Jν)=0gives∂ ν Jν=0.

Einstein’sFieldEquationDerivedandPuttoWork antisymmetry of Fμν and the fact that ∂ and ∂ commute.

To verify this, simply differentiatethequantity∇(cid:3) .E (cid:3)−ρwithrespecttotime:∂ (∇(cid:3) .E (cid:3)−ρ)=∇(cid:3) .∂ E (cid:3)−∂ ρ= 0 0 0 ∇(cid:3) .(∇(cid:3) ×B (cid:3)−J (cid:3) )+∇(cid:3) .J (cid:3)=0,whereweusedaMaxwellequationandcurrentconservation.

Indeed, the invariance of physics under general coordinate transformation xμ→x (cid:5)μ(x)allowsustoeliminate4ofthe10unknownfunctionsg (x).Inotherwords, μν Einstein’s field equations should not determine g (x) completely.

∗ Note that this identity together with Maxwell’s equation implies that the current must be conserved: ∂ ν (∂ μ Fμν−Jν)=0gives∂ ν Jν=0.

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

EVIDENCE:
- Futamase & Yoshida Possible measurement of vacuum energy (https://web.archive.org/web/2018 0104132548/https://arxiv.org/abs/gr-qc/0011083v1).
- Study of Vacuum Energy Physics for Breakthrough Propulsion (https://ntrs.nasa.gov/archive/nasa/ casi.ntrs.nasa.gov/20040171927_2004178265.pdf) 2004, NASA Glenn Technical Reports Server (PDF, 57 pages, Retrieved 2013-09-18).
- Retrieved from "https://en.wikipedia.org/w/index.php?title=Vacuum_energy&oldid=1329374446"

Futamase & Yoshida Possible measurement of vacuum energy (https://web.archive.org/web/2018 0104132548/https://arxiv.org/abs/gr-qc/0011083v1).

Study of Vacuum Energy Physics for Breakthrough Propulsion (https://ntrs.nasa.gov/archive/nasa/ casi.ntrs.nasa.gov/20040171927_2004178265.pdf) 2004, NASA Glenn Technical Reports Server (PDF, 57 pages, Retrieved 2013-09-18).

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List tests or derivations implied by the evidence.


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

EVIDENCE:
- A 2D sphere (which forms a 1D circle intheembeddingdiagramofFigure10.5)atfixeddistancesurroundingtheobserverhasgeodesic distance r defined by ‖ r ≡proper distance to sphere measured along a radial geodesic , (10.5) ‖ and circumferential radius r defined by r ≡ ( x2+y2+z2)1/2 , (10.6) w φ r //= r= R Rsin χ χ Rsinχdφ R R d χ χ y x Figure10.5 EmbeddingdiagramoftheFLRWgeometry.
- (10.4) An observer is sitting at the north pole of the diagram, at {0,0,0,1}.
- Figure 10.5 shows an embedding diagram of a 3D hypersphere in 4D Euclidean space.
- The 3D hypersphere is a set of points {x,y,z,w} satisfying ( x2+y2+z2+w2)1/2 =R=constant .

A 2D sphere (which forms a 1D circle intheembeddingdiagramofFigure10.5)atfixeddistancesurroundingtheobserverhasgeodesic distance r defined by ‖ r ≡proper distance to sphere measured along a radial geodesic , (10.5) ‖ and circumferential radius r defined by r ≡ ( x2+y2+z2)1/2 , (10.6) w φ r //= r= R Rsin χ χ Rsinχdφ R R d χ χ y x Figure10.5 EmbeddingdiagramoftheFLRWgeometry.

(10.4) An observer is sitting at the north pole of the diagram, at {0,0,0,1}.

Figure 10.5 shows an embedding diagram of a 3D hypersphere in 4D Euclidean space.

The 3D hypersphere is a set of points {x,y,z,w} satisfying ( x2+y2+z2+w2)1/2 =R=constant .

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

EVIDENCE:
- remained full even as the first-year students dropped away.
- Accounts vary about how successful the original lectures were.
- At the end of the year, Feynman was invited to give a lecture on his teaching experiences, and he agreed to do so, provided he could speak frankly, which he did.[169][170]
- Feynman's own preface, Feynman's vision of a medical use for written just after an exam on which the students did poorly, nanotechnology by swallowing the doctor was somewhat pessimistic.

remained full even as the first-year students dropped away.

Accounts vary about how successful the original lectures were.

At the end of the year, Feynman was invited to give a lecture on his teaching experiences, and he agreed to do so, provided he could speak frankly, which he did.[169][170]

Feynman's own preface, Feynman's vision of a medical use for written just after an exam on which the students did poorly, nanotechnology by swallowing the doctor was somewhat pessimistic.

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List tests or derivations implied by the evidence.


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

EVIDENCE:
- This joint has three degrees of freedom.
- A spherical joint, or ball joint, requires that a point in the moving body maintain contact with a point in the fixed body.
- A cylindrical joint requires that a line, or axis, in the moving body remain co-linear with a line in the fixed body.
- Kinematic chains Rigid bodies ("links") connected by kinematic pairs ("joints") are known as kinematic chains.

This joint has three degrees of freedom.

A spherical joint, or ball joint, requires that a point in the moving body maintain contact with a point in the fixed body.

A cylindrical joint requires that a line, or axis, in the moving body remain co-linear with a line in the fixed body.

Kinematic chains Rigid bodies ("links") connected by kinematic pairs ("joints") are known as kinematic chains.

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

EVIDENCE:
- " " [Cosmological model of the Universe with a time vector inversion].
- doi:10.1007/BF01375497 (https://doi.org/10.1007%2FBF01375497).
- S2CID 122830231 (https://ap i.semanticscholar.org/CorpusID:122830231).
- doi:10.1103/PhysRevD.90.101502 (https://doi.org/10.1103%2FPhysRevD.90.101502).

" " [Cosmological model of the Universe with a time vector inversion].

S2CID 122830231 (https://ap i.semanticscholar.org/CorpusID:122830231).

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List tests or derivations implied by the evidence.


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

EVIDENCE:
- Washington, DC: National Institute of Justice.
- Segal of the Minnesota Study of Twins Reared Apart re-unite to discuss behavior genetics and evolutionary psychology.
- Annals of Dyslexia, 34, 121–139.
- Intergenerational daycare and preschoolers’ attitudes about aging.

Washington, DC: National Institute of Justice.

Segal of the Minnesota Study of Twins Reared Apart re-unite to discuss behavior genetics and evolutionary psychology.

Intergenerational daycare and preschoolers’ attitudes about aging.

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

EVIDENCE:
- 1.13; for an elementary account, see Wheeler 1990, ch.
- Einstein's condemnation would prove to be premature, cf.
- 17ff; a derivation can be found in Mermin 2005, ch.
- Relativity – The Special and General Theory (https://books.google.com/boo ks?id=yhN9CgAAQBAJ).

1.13; for an elementary account, see Wheeler 1990, ch.

Einstein's condemnation would prove to be premature, cf.

17ff; a derivation can be found in Mermin 2005, ch.

Relativity – The Special and General Theory (https://books.google.com/boo ks?id=yhN9CgAAQBAJ).

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

EVIDENCE:
- The first complete quantum field theory, quantum electrodynamics, provides a fully quantum description of the electromagnetic interaction.
- Quantum field theories for the strong nuclear force and the weak nuclear force have also been developed.
- [7]: 285 Likewise, in a Stern– Gerlach experiment, a charged particle is modeled as a quantum system, while the background magnetic field is described classically.[42]: 26 This "semi-classical" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by charged particles.
- For example, the stability of bulk matter (consisting of atoms and molecules which would quickly collapse under electric forces alone), the rigidity of solids, and the mechanical, thermal, chemical, optical and magnetic properties of matter are all results of the interaction of electric charges under the rules of quantum mechanics.[44] Special relativity and electrodynamics Early attempts to merge quantum mechanics with special relativity involved the replacement of the Schrödinger equation with a covariant equation such as the Klein–Gordon equation or the Dirac equation.

The first complete quantum field theory, quantum electrodynamics, provides a fully quantum description of the electromagnetic interaction.

Quantum field theories for the strong nuclear force and the weak nuclear force have also been developed.

[7]: 285 Likewise, in a Stern– Gerlach experiment, a charged particle is modeled as a quantum system, while the background magnetic field is described classically.[42]: 26 This "semi-classical" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by charged particles.

For example, the stability of bulk matter (consisting of atoms and molecules which would quickly collapse under electric forces alone), the rigidity of solids, and the mechanical, thermal, chemical, optical and magnetic properties of matter are all results of the interaction of electric charges under the rules of quantum mechanics.[44] Special relativity and electrodynamics Early attempts to merge quantum mechanics with special relativity involved the replacement of the Schrödinger equation with a covariant equation such as the Klein–Gordon equation or the Dirac equation.

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List tests or derivations implied by the evidence.


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

EVIDENCE:
- Inflation: Alcubierre Metric1 Warp Metric: Space expansion ds2 dt2 (dx v f (r )dt)2 dy2 dz2 Location of behind ship s s ship proper Apparent speed Shaping Function: Shell thickness Shell size parameter parameter tanh( (r R)) tanh( (r R)) f (r ) s s s 2tanh( R) York Time: x df (r ) York Time is measure of v s s expansion/contraction s r dr s s of space Energy Density: 1 1 v2(y2 z2) df (r ) G00 s s Space contraction 8 8 4r 2 dr in front of ship s s 1.
- Alcubierre, M., “The warp drive: hyper-fast travel within general relativity,” Class.

Inflation: Alcubierre Metric1 Warp Metric: Space expansion ds2 dt2 (dx v f (r )dt)2 dy2 dz2 Location of behind ship s s ship proper Apparent speed Shaping Function: Shell thickness Shell size parameter parameter tanh( (r R)) tanh( (r R)) f (r ) s s s 2tanh( R) York Time: x df (r ) York Time is measure of v s s expansion/contraction s r dr s s of space Energy Density: 1 1 v2(y2 z2) df (r ) G00 s s Space contraction 8 8 4r 2 dr in front of ship s s 1.

Alcubierre, M., “The warp drive: hyper-fast travel within general relativity,” Class.

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

EVIDENCE:
- 23.6 Principal null congruences The middle expression of equation (23.9) shows that the Carter constant K is necessarily positive.
- (23.28) Theline-element(22.1)definesatetrad(theBoyer-Lindquisttetrad)thatisalignedwiththeprincipalnull
- The vanishing of the Carter constant, K=0 , (23.23) defines a special set of geodesics, called the principal outgoing and ingoing null congruences.
- Table 23.1 lists the constraints on the Hamilton-Jacobi parameters P and P in each of the regions of the t x Penrose diagram of Figure 9.6.

23.6 Principal null congruences The middle expression of equation (23.9) shows that the Carter constant K is necessarily positive.

The vanishing of the Carter constant, K=0 , (23.23) defines a special set of geodesics, called the principal outgoing and ingoing null congruences.

Table 23.1 lists the constraints on the Hamilton-Jacobi parameters P and P in each of the regions of the t x Penrose diagram of Figure 9.6.

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List tests or derivations implied by the evidence.


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

EVIDENCE:
- universe and might bridge the gap between general relativity and quantum mechanics.
- 5.Question What are the implications of vacuum fluctuations near a black hole's horizon?
- 4.Question What happens to time within a black hole's interior according to physicist John Wheeler?
- Answer:According to Wheeler, time within a black hole's interior flows inexorably towards the singularity, making it impossible for anything that crosses the event horizon to escape or influence the outside universe.

universe and might bridge the gap between general relativity and quantum mechanics.

5.Question What are the implications of vacuum fluctuations near a black hole's horizon?

4.Question What happens to time within a black hole's interior according to physicist John Wheeler?

Answer:According to Wheeler, time within a black hole's interior flows inexorably towards the singularity, making it impossible for anything that crosses the event horizon to escape or influence the outside universe.

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

EVIDENCE:
- We learned in (V.6.14) that the secondcovariantderivativeisgivenby D D S =∂ (D S )−(cid:25)σ D S −(cid:25)σ D S (6) μ ν ρ μ ν ρ μν σ ρ μρ ν σ Tocalculate[D ,D ]S ,wearetosubtractfrom(6)theexpressionweobtainbyinterchangμ ν ρ ingμandν in(6).Wenoticethatthesecondtermissymmetricinμandν andsocanbe dropped.Next,forD S inthefirstandthirdterms,weinserttheexpressionin(1).Thus, ν ρ [D ,D ]S =∂ (∂ S −(cid:25)σ S )−(cid:25)σ (∂ S −(cid:25)κ S )−(μ↔ν) μ ν ρ μ ν ρ νρ σ μρ ν σ νσ κ =∂ ∂ S −(∂ (cid:25)σ )S −(cid:25)σ ∂ S −(cid:25)σ ∂ S +(cid:25)σ (cid:25)κ S −(μ↔ν) (7) μ ν ρ μ νρ σ νρ μ σ μρ ν σ μρ νσ κ Indeed,thecalculationisevensimplerthanshownhereifwetrusttheprecedingargument anddonotevenbothertowritedownanyterminvolving∂.S.and∂.∂.S..Butlet’sbecareful.
- involves‡ two derivatives acting on the metric, as we anticipated,hereandasfarbackasinchapterI.6.
- istheinverse ofg..andrecalltheidentity(V.6.7).
- Happily,lotsoftermsknockeachotheroffwhenweantisymmetrizeinμandν.Thefirst termgoesaway,andthethirdandfourthtermsgoawaytogether,inaccordancewithour earlier“sloppier”argument.Thusweareleftwith(forconvenience,wehaveinterchanged thedummyindicesκ andσ inthefifthtermin(7)) [D ,D ]S =−(∂ (cid:25)σ )S +(cid:25)κ (cid:25)σ S −(μ↔ν) μ ν ρ μ νρ σ μρ νκ σ (cid:12) (cid:14) =− ∂ (cid:25)σ −(cid:25)κ (cid:25)σ −(μ↔ν) S (8) μ νρ μρ νκ σ in agreement with (4).

We learned in (V.6.14) that the secondcovariantderivativeisgivenby D D S =∂ (D S )−(cid:25)σ D S −(cid:25)σ D S (6) μ ν ρ μ ν ρ μν σ ρ μρ ν σ Tocalculate[D ,D ]S ,wearetosubtractfrom(6)theexpressionweobtainbyinterchangμ ν ρ ingμandν in(6).Wenoticethatthesecondtermissymmetricinμandν andsocanbe dropped.Next,forD S inthefirstandthirdterms,weinserttheexpressionin(1).Thus, ν ρ [D ,D ]S =∂ (∂ S −(cid:25)σ S )−(cid:25)σ (∂ S −(cid:25)κ S )−(μ↔ν) μ ν ρ μ ν ρ νρ σ μρ ν σ νσ κ =∂ ∂ S −(∂ (cid:25)σ )S −(cid:25)σ ∂ S −(cid:25)σ ∂ S +(cid:25)σ (cid:25)κ S −(μ↔ν) (7) μ ν ρ μ νρ σ νρ μ σ μρ ν σ μρ νσ κ Indeed,thecalculationisevensimplerthanshownhereifwetrusttheprecedingargument anddonotevenbothertowritedownanyterminvolving∂.S.and∂.∂.S..Butlet’sbecareful.

involves‡ two derivatives acting on the metric, as we anticipated,hereandasfarbackasinchapterI.6.

Happily,lotsoftermsknockeachotheroffwhenweantisymmetrizeinμandν.Thefirst termgoesaway,andthethirdandfourthtermsgoawaytogether,inaccordancewithour earlier“sloppier”argument.Thusweareleftwith(forconvenience,wehaveinterchanged thedummyindicesκ andσ inthefifthtermin(7)) [D ,D ]S =−(∂ (cid:25)σ )S +(cid:25)κ (cid:25)σ S −(μ↔ν) μ ν ρ μ νρ σ μρ νκ σ (cid:12) (cid:14) =− ∂ (cid:25)σ −(cid:25)κ (cid:25)σ −(μ↔ν) S (8) μ νρ μρ νκ σ in agreement with (4).

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:
- Thesystemthenrestarts spontaneouslyandthemissileagaingoesintolaunchmode;theofficer’snext“inhibit”orderdoesn’t work.
- ● http://www.noufors.com/Documents/Books,%20Manuals%20and%20Published%20Pap ers/Specialty%20UFO%20Publications/Personal%20Collection%20of%20UFO%20Pub lications/MUFON%20Journals/1999/07/July%201999.pdf(p17) (PUBLICDOMAIN) -1974—ASgt.XclaimshewastoldbyaretiredUSArmyofficialatCampSan LuisObispothatinformationofacrashedsaucerandrecoveredoccupantswouldbereleasedbytheUSG withinthreemonths;LeonardStringfieldclaimsheheardothersourcestellhimthatanannouncement wasexpectedinlate1974,possiblybyPresidentGeraldFordandthePresidentofFranceGiscard d’Estaing;WalterAndrusofMUFONstatedhewastoldtojoinAllenHynekofCUFOSinChicagoina pressconferenceifofficialwordwasmade;itneveroccurred.
- ● https://ufos-scientificresearch.blogspot.com/2018/04/kit-green-in-phenomena-book-by-annie.html (PUBLICDOMAIN) -11February1974—JacquesValleestateshemetKitGreenanddiscussed Green’sconnectionsintheExecutivebranchandICthatwereinterestedinUAP;Valleenotedhecould nottellJ.AllenHynekbecausePuthoffaskedValleenottoshareanything.
- Amissile launchofficerwiththe564thStrategicMissileSquadronreportsanuclearmissilecountdownisstarted, andtheofficerflipsthe“inhibit”switchtoturnthesystemoffline.

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:
- Constraints amplifies the NEC-violating portions of a sequence of excited states.
- The second violation was constructed purely from the geometric anomalous terms in the stress tensor.
- Both of these violations can become arbitrarily large [29].
- However, the addition of the self-consistency criteria could be one possibility to exclude exotic phenomena from general relativity.

Constraints amplifies the NEC-violating portions of a sequence of excited states.

The second violation was constructed purely from the geometric anomalous terms in the stress tensor.

Both of these violations can become arbitrarily large [29].

However, the addition of the self-consistency criteria could be one possibility to exclude exotic phenomena from general relativity.

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:
- He made Feynman an offer in May 1945, but Feynman turned it down.
- had the good sense to fire me."[85] As early as October 30, 1943, Bethe had written to the chairman of the physics department of his university, Cornell, to recommend that Feynman be hired.
- Cornell matched its salary offer of $3,900 (equivalent to $68,000 in 2024) per annum.[86] Feynman became one of the first of the Los Alamos Laboratory's group leaders to depart, leaving for Ithaca, New York, in October 1945.[89] Because Feynman was no longer working at the Los Alamos Laboratory, he was no longer exempt from the draft.
- Oppenheimer had hoped to recruit Feynman to the University of California, but Birge was reluctant.

He made Feynman an offer in May 1945, but Feynman turned it down.

had the good sense to fire me."[85] As early as October 30, 1943, Bethe had written to the chairman of the physics department of his university, Cornell, to recommend that Feynman be hired.

Cornell matched its salary offer of $3,900 (equivalent to $68,000 in 2024) per annum.[86] Feynman became one of the first of the Los Alamos Laboratory's group leaders to depart, leaving for Ithaca, New York, in October 1945.[89] Because Feynman was no longer working at the Los Alamos Laboratory, he was no longer exempt from the draft.

Oppenheimer had hoped to recruit Feynman to the University of California, but Birge was reluctant.

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 Lie derivative is also a covariant derivative: the Lie derivative of a coordinate tensor is a coordinate tensor.
- The Lie derivative of a coordinate tensor, whose mathematical form is derived in §§7.34.2–7.34.6, is physically minus the rate of change of the coordinate tensor with respect to a prescribed change in the coordinates, equation (7.124).
- 184 Schwarzschild Black Hole 7.34 Lie derivative It was remarked above that Killing’s equation (7.116) can be recast as the statement that the Lie derivative of the metric along the Killing vector vanishes, equation (7.117).
- ε→0 ε The reason for the minus sign in the definition (7.125) of the Lie derivative is that, as will be seen below, equation(7.151),theprincipaltermintheexpansionoftheLiederivativeofatensorA..

The Lie derivative is also a covariant derivative: the Lie derivative of a coordinate tensor is a coordinate tensor.

The Lie derivative of a coordinate tensor, whose mathematical form is derived in §§7.34.2–7.34.6, is physically minus the rate of change of the coordinate tensor with respect to a prescribed change in the coordinates, equation (7.124).

184 Schwarzschild Black Hole 7.34 Lie derivative It was remarked above that Killing’s equation (7.116) can be recast as the statement that the Lie derivative of the metric along the Killing vector vanishes, equation (7.117).

ε→0 ε The reason for the minus sign in the definition (7.125) of the Lie derivative is that, as will be seen below, equation(7.151),theprincipaltermintheexpansionoftheLiederivativeofatensorA..

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:
- Spinor axes are defined by 2[N/2] basis spinors ε , a a ε ≡ε (38.85) a a1...a[N/2] where a ...a denotes not a set of indices, but rather a bitcode specifying the single index a.
- Thus the condition (38.89) is determined by the commutation properties of ε with the orthonormal bivectors of the geometric algebra (an orthonormal bivector is defined here to be a wedge product of orthonormal vectors; the square of an orthonormal bivector is thus −1).
- A rotor R is a real linear combination of even elements of the geometric algebra in an orthonormal basis.
- Thetransformation(38.86)showsthateachbasisspinorε hasi’thspinweighteither+1 or−1 ineach a 2 2 of its [N/2] bits.

Spinor axes are defined by 2[N/2] basis spinors ε , a a ε ≡ε (38.85) a a1...a[N/2] where a ...a denotes not a set of indices, but rather a bitcode specifying the single index a.

Thus the condition (38.89) is determined by the commutation properties of ε with the orthonormal bivectors of the geometric algebra (an orthonormal bivector is defined here to be a wedge product of orthonormal vectors; the square of an orthonormal bivector is thus −1).

A rotor R is a real linear combination of even elements of the geometric algebra in an orthonormal basis.

Thetransformation(38.86)showsthateachbasisspinorε hasi’thspinweighteither+1 or−1 ineach a 2 2 of its [N/2] bits.

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- But, as illustrated in the middle panel of Figure 21.6, the gravitational force acts in opposite directions for outgoing and ingoing streams, tending to acceleratethestreamsfasterthrougheachother.Anintuitivewaytounderstandthisisthatthegravitational force is always inwards, meaning in the direction of smaller radius, but the inward direction is towards the black hole for ingoing streams, and away from the black hole for outgoing streams.
- It follows that, as illustrated in the middle panel of Figure 21.6, the change in β slightly exceeds the change in β , which drives the 4-vector β , already nearly lightlike, to be even more 1 0 m nearly lightlike.
- This follows from the fact that for mass inflation to begin,thegravitationalforceproducedbythecounter-streamingpressurepmustbecomecomparabletothat produced by the mass M, which for streams of low proper density requires a hyper-relativistic streaming velocity.Foracoldstreamofproperdensityρmovingat4-velocityum ≡{u0,u1,0,0},thestreamingenergy flux would be f ∼ ρu0u1, while the streaming pressure would be p ∼ ρ(u1)2.
- 21.6.3 Collapse phase Itmightseemthatinflationislockedintoanexponentialgrowthfromwhichthereisnoexit.ButtheEinstein equations (20.62) have one more trick up their sleave.

But, as illustrated in the middle panel of Figure 21.6, the gravitational force acts in opposite directions for outgoing and ingoing streams, tending to acceleratethestreamsfasterthrougheachother.Anintuitivewaytounderstandthisisthatthegravitational force is always inwards, meaning in the direction of smaller radius, but the inward direction is towards the black hole for ingoing streams, and away from the black hole for outgoing streams.

It follows that, as illustrated in the middle panel of Figure 21.6, the change in β slightly exceeds the change in β , which drives the 4-vector β , already nearly lightlike, to be even more 1 0 m nearly lightlike.

This follows from the fact that for mass inflation to begin,thegravitationalforceproducedbythecounter-streamingpressurepmustbecomecomparabletothat produced by the mass M, which for streams of low proper density requires a hyper-relativistic streaming velocity.Foracoldstreamofproperdensityρmovingat4-velocityum ≡{u0,u1,0,0},thestreamingenergy flux would be f ∼ ρu0u1, while the streaming pressure would be p ∼ ρ(u1)2.

21.6.3 Collapse phase Itmightseemthatinflationislockedintoanexponentialgrowthfromwhichthereisnoexit.ButtheEinstein equations (20.62) have one more trick up their sleave.

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:
- Absolutely continuous univariate distributions A probability density function is most commonly associated with absolutely continuous univariate distributions.
- This is the probability that the bacterium dies within an infinitesimal window of time around 5 hours, where dt is the duration of this window.
- For example, the probability that it lives longer than 5 hours, but shorter than (5 hours + 1 nanosecond), is (2 hour−1)×(1 nanosecond) ≈ 6 ×10−13 (using the unit conversion 3.6 ×1012 nanoseconds = 1 hour).
- The integral of f over any window of time (not only density functions.

Probability density function - Wikipedia.pdf

Absolutely continuous univariate distributions A probability density function is most commonly associated with absolutely continuous univariate distributions.

This is the probability that the bacterium dies within an infinitesimal window of time around 5 hours, where dt is the duration of this window.

For example, the probability that it lives longer than 5 hours, but shorter than (5 hours + 1 nanosecond), is (2 hour−1)×(1 nanosecond) ≈ 6 ×10−13 (using the unit conversion 3.6 ×1012 nanoseconds = 1 hour).

The integral of f over any window of time (not only density functions.

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:
- Heisenberg used such an observer effect at the quantum level (see Heisenberg's gamma-ray below) as a physical "explanation" of quantum uncertainty.[95] It has microscope for locating an since become clearer, however, that the uncertainty principle is electron (shown in blue).
- Intrinsic quantum uncertainty Historically, the uncertainty principle has been confused[92][66] with a related effect in physics, called the observer effect, which notes that measurements of certain systems cannot be made without affecting the system,[93][94] that is, without changing something in a system.
- There is no way to say what the state of a system fundamentally is, only what the result of observations might be.
- statement about the observational success of current technology.[97] Classical optics shows that the electron position can be resolved only up to an uncertainty Δx that Critical reactions depends on θ and the wavelength λ of the incoming light.

Heisenberg used such an observer effect at the quantum level (see Heisenberg's gamma-ray below) as a physical "explanation" of quantum uncertainty.[95] It has microscope for locating an since become clearer, however, that the uncertainty principle is electron (shown in blue).

Intrinsic quantum uncertainty Historically, the uncertainty principle has been confused[92][66] with a related effect in physics, called the observer effect, which notes that measurements of certain systems cannot be made without affecting the system,[93][94] that is, without changing something in a system.

There is no way to say what the state of a system fundamentally is, only what the result of observations might be.

statement about the observational success of current technology.[97] Classical optics shows that the electron position can be resolved only up to an uncertainty Δx that Critical reactions depends on θ and the wavelength λ of the incoming light.

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 past, in very large telescopes, an observer would sit inside the telescope in an A prime focus telescope design.
- Cassegrain focus For telescopes built to the Cassegrain design or other related designs, the image is formed behind the primary mirror, at the focal point of the secondary mirror.
- The Solano telescope doesn't contain any toric surfaces.
- The Yolo design eliminates coma, but leaves significant astigmatism, which is reduced by deformation of the secondary mirror by some form of warping harness, or alternatively, polishing a toroidal figure into the secondary.

In the past, in very large telescopes, an observer would sit inside the telescope in an A prime focus telescope design.

Cassegrain focus For telescopes built to the Cassegrain design or other related designs, the image is formed behind the primary mirror, at the focal point of the secondary mirror.

The Solano telescope doesn't contain any toric surfaces.

The Yolo design eliminates coma, but leaves significant astigmatism, which is reduced by deformation of the secondary mirror by some form of warping harness, or alternatively, polishing a toroidal figure into the secondary.

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:
- Non-relativistic mechanics also assumes that forces act instantaneously (see also Action at a distance).
- Kinematics The position of a point particle is defined in relation to a coordinate system centered on an arbitrary fixed reference point in space called the origin O.
- Description of objects and their motion Classical mechanics models real-world objects as point particles, that is, objects with negligible size.
- The rocket equation extends the notion of rate of change of an object's The analysis of projectile motion is a part momentum to include the effects of an object "losing mass".

Non-relativistic mechanics also assumes that forces act instantaneously (see also Action at a distance).

Kinematics The position of a point particle is defined in relation to a coordinate system centered on an arbitrary fixed reference point in space called the origin O.

Description of objects and their motion Classical mechanics models real-world objects as point particles, that is, objects with negligible size.

The rocket equation extends the notion of rate of change of an object's The analysis of projectile motion is a part momentum to include the effects of an object "losing mass".

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:
- This solution is called the Schwazchild solution It predicted the existence of Black holes a very dense object having an enormous gravity even spacetime will not make sense because of it!
- Later, in 1935 Einstein and Rosen [4] studied the extension of Schwarzchild solution to predict an even stranger object, knows as Einstein-Rosen Bridge.
- Or what is known as a Wormhole.
- WormholePhysics QuantumWormholePhysics Summary Introduction Many solutions to (1) were found, a lot of them were very interesting and related to observations.

This solution is called the Schwazchild solution It predicted the existence of Black holes a very dense object having an enormous gravity even spacetime will not make sense because of it!

Later, in 1935 Einstein and Rosen [4] studied the extension of Schwarzchild solution to predict an even stranger object, knows as Einstein-Rosen Bridge.

WormholePhysics QuantumWormholePhysics Summary Introduction Many solutions to (1) were found, a lot of them were very interesting and related to observations.

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 system dynamically identifies these layers based on the model architecture (e.g., Llama-3, Mistral, Qwen) loaded via the GGUF backend, ensuring compatibility across "any LLM" as requested.
- weights, satisfying the Genesis requirement for a "permanent injection of LLM core".
- This surgical approach maximizes the retention of new knowledge while minimizing the impact on the model's linguistic coherence.
- Knowledge Localization and Layer Specificity, Not all layers in a transformer are equal targets for injection.

4db7f3e0-af6c-4bcc-9197-eae84d21c19d_Genesis X - GUI.pdf

The system dynamically identifies these layers based on the model architecture (e.g., Llama-3, Mistral, Qwen) loaded via the GGUF backend, ensuring compatibility across "any LLM" as requested.

weights, satisfying the Genesis requirement for a "permanent injection of LLM core".

This surgical approach maximizes the retention of new knowledge while minimizing the impact on the model's linguistic coherence.

Knowledge Localization and Layer Specificity, Not all layers in a transformer are equal targets for injection.

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:
- Before I go further, let me warn the reader that as this book goes to press, there is considerabledebateregardingwhetherinflationarycosmologyisstillviable.7Youwillhave toformyourownopinionoverthecomingyears.
- Inflationaryepoch Withtheblindingclarityofhindsight, wenowseethatalltheseproblemscanbesolved if the early universe went through an inflationary epoch,5 during which it expanded exponentially.Aswasjustshown,d (t)wouldthengrowexponentially,thussolving6 horizon thehorizonandhomogeneityproblems,andaswewillseebelow,alsotheflatnessproblem.
- As you have known since chapter V.3, the desired exponential expansion can be producedreadilybyaconstantenergydensitycorrespondingtosomeeffectivecosmological constant.Withacosmologicalconstant,(1)isreplacedby (cid:24) ̇ =H(cid:24) (2(cid:24) +(cid:24) −2(cid:24) ) (5) k k r m (cid:27) With(2(cid:24) +(cid:24) −2(cid:24) )<0,theflowaroundsmall(cid:24) goesfrombeingunstabletostable, r m (cid:27) k andtheflatnessproblemissolved.Anyinitialvalue(cid:24) getsdrivento0.
- InflationaryCosmology | 533 For(1−γ)<0,thatis,ifγ >1,thesituationisobviouslyreversed:theregimearound t contributes the most to J(t,t ).

Before I go further, let me warn the reader that as this book goes to press, there is considerabledebateregardingwhetherinflationarycosmologyisstillviable.7Youwillhave toformyourownopinionoverthecomingyears.

Inflationaryepoch Withtheblindingclarityofhindsight, wenowseethatalltheseproblemscanbesolved if the early universe went through an inflationary epoch,5 during which it expanded exponentially.Aswasjustshown,d (t)wouldthengrowexponentially,thussolving6 horizon thehorizonandhomogeneityproblems,andaswewillseebelow,alsotheflatnessproblem.

As you have known since chapter V.3, the desired exponential expansion can be producedreadilybyaconstantenergydensitycorrespondingtosomeeffectivecosmological constant.Withacosmologicalconstant,(1)isreplacedby (cid:24) ̇ =H(cid:24) (2(cid:24) +(cid:24) −2(cid:24) ) (5) k k r m (cid:27) With(2(cid:24) +(cid:24) −2(cid:24) )<0,theflowaroundsmall(cid:24) goesfrombeingunstabletostable, r m (cid:27) k andtheflatnessproblemissolved.Anyinitialvalue(cid:24) getsdrivento0.

InflationaryCosmology | 533 For(1−γ)<0,thatis,ifγ >1,thesituationisobviouslyreversed:theregimearound t contributes the most to J(t,t ).

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:
- TheCosmologicalConstantParadox | 753 cosmologicalconstantparadox,however,indicatesthatwewillultimatelyhavetofaceup tothequestion,“Whatisvacuumenergy?” Thequestionremindsmeofanearlierquestion:“Whatisheat?”(Orperhapsalso,“What istheether?”)Everybodybutphysicistsknewwhatheatwas,butitprovedtobeextremely elusivetodefine.Itwasnotasubstanceorafluid(thecaloric),aswasoncethought.Thetrue answerhadtowaituntilthenatureofmatterwasunderstood.AsyouandIknow,physics progresses through asking first the what, then the how, and finally the why questions, for example, “What is matter made of?” “How do these atoms behave?” “Why are there atoms?”Nowweknowtheanswerstoallthreequestions,butthewhyquestionhadtowait untiltheadventofbaryogenesisandleptogenesis(aswasdiscussedinchapterVIII.3),and skepticscertainlystillabound.Attheleast,wedon’tknowthedetailedanswer.
- It would be a bit disappointing if dark energy or the cosmological constant proves to be merely due to some mundane mechanism, such as the presence of dime-a-dozen scalarfields(seechapterVIII.4),whichultimatelyhavetobefine-tuned.Ithinkthatmost theoretical physicists would hope that the cosmological constant paradox, like the great paradoxesofthelate19thcentury,willleadustoadeeperunderstandingofphysics.
- Theuniversesaystothequantumfieldtheorist,“Iamdoingjustfine,thankyou,but somethingiswrongwithyourunderstandingofthevacuumenergy,oryourunderstanding ofhowthegravitationalfieldrespondstothevacuumenergy.” A distinguished colleague said to me recently, “The cosmological constant paradox is morethanaparadox;it’saprofoundpublichumiliationoftheoreticalphysicists.” Appendix1:Scalingatcosmologicaldistances Thehistoryofphysicsisfullofexamplesofreasoningbyanalogythatturnouttobefruitful.Asexplainedinthe text,thecosmologicalconstantparadoxcanbesummarizedasfollows: expectedvalue enormous (1) decreedvalue mathematically0 observedvalue tinybutnot0 Haveweeverencounteredsomethingsimilar?Iproposedlongagothatthestoryofprotondecaymayprovide suchananalogy.11 Iwillnotgointotheparticlephysicsbehindprotondecayhere.Sufficeittosaythat,atonetime,theexpected valueoftheprotondecayratewasenormous,thenitwasdecreedtobemathematically0,whiletheobserved value∗turnsouttobeextremelytinybutnonzero.Theimportantquestionishowtheoristsmanagedtoreduce theenormousexpectedvaluedowntotheextremelytinyobservedvalue.
- ∗Iamfudgingslightlyhere:atthemoment,weonlyhaveanupperboundfortheobservedvalue.Experimentalistshaveyettoobserveprotondecay,butthatunfortunatefactmightmerelybeduetothefactthatthe detectorsconstructedthusfararetoosmall.AsIexplainedinchapterVIII.3,theoristshavecompellingreason tobelievethattheprotondoesdecay,sowecaneasilyimaginethatexperimentalistsinsomeothercivilization werenotasunluckyandhadobservedprotondecaysoonaftergrandunifiedtheorywasproposed.Inanycase, theparticulardetailsofhowparticlephysicsevolvedinourcivilizationdonotconcernushere.

TheCosmologicalConstantParadox | 753 cosmologicalconstantparadox,however,indicatesthatwewillultimatelyhavetofaceup tothequestion,“Whatisvacuumenergy?” Thequestionremindsmeofanearlierquestion:“Whatisheat?”(Orperhapsalso,“What istheether?”)Everybodybutphysicistsknewwhatheatwas,butitprovedtobeextremely elusivetodefine.Itwasnotasubstanceorafluid(thecaloric),aswasoncethought.Thetrue answerhadtowaituntilthenatureofmatterwasunderstood.AsyouandIknow,physics progresses through asking first the what, then the how, and finally the why questions, for example, “What is matter made of?” “How do these atoms behave?” “Why are there atoms?”Nowweknowtheanswerstoallthreequestions,butthewhyquestionhadtowait untiltheadventofbaryogenesisandleptogenesis(aswasdiscussedinchapterVIII.3),and skepticscertainlystillabound.Attheleast,wedon’tknowthedetailedanswer.

It would be a bit disappointing if dark energy or the cosmological constant proves to be merely due to some mundane mechanism, such as the presence of dime-a-dozen scalarfields(seechapterVIII.4),whichultimatelyhavetobefine-tuned.Ithinkthatmost theoretical physicists would hope that the cosmological constant paradox, like the great paradoxesofthelate19thcentury,willleadustoadeeperunderstandingofphysics.

Theuniversesaystothequantumfieldtheorist,“Iamdoingjustfine,thankyou,but somethingiswrongwithyourunderstandingofthevacuumenergy,oryourunderstanding ofhowthegravitationalfieldrespondstothevacuumenergy.” A distinguished colleague said to me recently, “The cosmological constant paradox is morethanaparadox;it’saprofoundpublichumiliationoftheoreticalphysicists.” Appendix1:Scalingatcosmologicaldistances Thehistoryofphysicsisfullofexamplesofreasoningbyanalogythatturnouttobefruitful.Asexplainedinthe text,thecosmologicalconstantparadoxcanbesummarizedasfollows: expectedvalue enormous (1) decreedvalue mathematically0 observedvalue tinybutnot0 Haveweeverencounteredsomethingsimilar?Iproposedlongagothatthestoryofprotondecaymayprovide suchananalogy.11 Iwillnotgointotheparticlephysicsbehindprotondecayhere.Sufficeittosaythat,atonetime,theexpected valueoftheprotondecayratewasenormous,thenitwasdecreedtobemathematically0,whiletheobserved value∗turnsouttobeextremelytinybutnonzero.Theimportantquestionishowtheoristsmanagedtoreduce theenormousexpectedvaluedowntotheextremelytinyobservedvalue.

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:
- Sincewearehuntingfor∂2,wecouldcarelessaboutthe(cid:25)(cid:25)terms.Sofarsogood,butitwouldappear thatwestillhavetoslaveawaytoobtainthe∂∂g..terms.NowweappealtoProfessorFlatforhelp.Goto alocallyflatcoordinatesystem.BackinchapterVI.1,weobtainedR....locallyintermsofthe∂∂g...The generalexpressionforR....mustreducetotheexpressioninchapterVI.1;henceweconcludethat R τρμν = 1(g τν,μρ −g ρν,μτ −g τμ,νρ +g ρμ,ντ )+(cid:25)(cid:25)terms wherewehaveswitchedtothecommanotationforpartialderivatives:g τν,μρ ≡∂ μ ∂ ρ g τν.Keepinginmind theantisymmetricpropertiesofR τρμν,weseethatwewon’tencounterg 00,00 andg i0,00 .Wearedown toR i0j0 = 2 1(g i0,j0 −g 00,ji −g ij,00 +g 0j,0i )+(cid:25)(cid:25)terms,leadingtoR i0j0 “=”− 2 1g ij,00 .Tosavewriting, weintroducethesymbol“=”tomeanequaluptotermsnotcontaining∂2.
- Now he cries out, “But I get E =R − 1g R“=”−1(gij+g g0ig0j− 00 00 2 00 2 00 g g00gij)g ij,00 andthiscontains∂ 2!” PerhapsyoucouldhelpConfusioout.Youchidehim,“Bynow,youshouldknowtheimportanceof distinguishingupperandlowerindices!Inthetext,thestatementisthatE0μdoesnotcontain∂2.” Wewillraisetheindicesintwosteps.First, R 0 0=g00R 00 +g0iR 0i“=” 2 (g0ig0j−g00gij)g ij,00 Indeed, E0=R0− R“=”0 0 0 2 Similarly, E i 0=R i 0=g00R 0i +g0jR ji“=”0 Nowwearealmostdone: E00=g00E0+g0iE0“=”0 and E0i=gi0E0+gijE0“=”0 0 i 0 j Phew!It’snicetohavetheBianchiidentityproof!
- VI.6 InitialValueProblemsandNumericalRelativity 1 To obtain E0μ, we have to calculate the Riemann curvature tensor.
- (which,ifyouwant,youcouldexpressintermsof(cid:25)andgusingthedefinitionof(cid:25),but youdon’tevenhavetobotherforthepurposesathand).Thus,R....∼∂∂g..+(cid:25)(cid:25).

Sincewearehuntingfor∂2,wecouldcarelessaboutthe(cid:25)(cid:25)terms.Sofarsogood,butitwouldappear thatwestillhavetoslaveawaytoobtainthe∂∂g..terms.NowweappealtoProfessorFlatforhelp.Goto alocallyflatcoordinatesystem.BackinchapterVI.1,weobtainedR....locallyintermsofthe∂∂g...The generalexpressionforR....mustreducetotheexpressioninchapterVI.1;henceweconcludethat R τρμν = 1(g τν,μρ −g ρν,μτ −g τμ,νρ +g ρμ,ντ )+(cid:25)(cid:25)terms wherewehaveswitchedtothecommanotationforpartialderivatives:g τν,μρ ≡∂ μ ∂ ρ g τν.Keepinginmind theantisymmetricpropertiesofR τρμν,weseethatwewon’tencounterg 00,00 andg i0,00 .Wearedown toR i0j0 = 2 1(g i0,j0 −g 00,ji −g ij,00 +g 0j,0i )+(cid:25)(cid:25)terms,leadingtoR i0j0 “=”− 2 1g ij,00 .Tosavewriting, weintroducethesymbol“=”tomeanequaluptotermsnotcontaining∂2.

Now he cries out, “But I get E =R − 1g R“=”−1(gij+g g0ig0j− 00 00 2 00 2 00 g g00gij)g ij,00 andthiscontains∂ 2!” PerhapsyoucouldhelpConfusioout.Youchidehim,“Bynow,youshouldknowtheimportanceof distinguishingupperandlowerindices!Inthetext,thestatementisthatE0μdoesnotcontain∂2.” Wewillraisetheindicesintwosteps.First, R 0 0=g00R 00 +g0iR 0i“=” 2 (g0ig0j−g00gij)g ij,00 Indeed, E0=R0− R“=”0 0 0 2 Similarly, E i 0=R i 0=g00R 0i +g0jR ji“=”0 Nowwearealmostdone: E00=g00E0+g0iE0“=”0 and E0i=gi0E0+gijE0“=”0 0 i 0 j Phew!It’snicetohavetheBianchiidentityproof!

VI.6 InitialValueProblemsandNumericalRelativity 1 To obtain E0μ, we have to calculate the Riemann curvature tensor.

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:
- (DFA) which is defined as an abstract mathematical concept, but due to the deterministic nature of a DFA, it is implementable in hardware and software for solving various specific problems.
- For example, the following is a DFA M with a binary alphabet, which requires that the input contains an even number of 0s: where and is defined by the following state-transition table:

(DFA) which is defined as an abstract mathematical concept, but due to the deterministic nature of a DFA, it is implementable in hardware and software for solving various specific problems.

For example, the following is a DFA M with a binary alphabet, which requires that the input contains an even number of 0s: where and is defined by the following state-transition table:

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:
- So the test seems virtually infallible: If the person fails and admits he lied, the test “worked,” and if the person fails and doesn’t admit he lied, the test also “worked.” Of course, if the person passes, he’ll essentially always agree that he was telling the truth, so the test again “worked.” This “heads I win, tails you lose” reasoning renders the rationale underlying the polygraph test difficult or impossible to falsify.
- Given these limitations, the National Research Council (2003) was reluctant to estimate the CQT’s accuracy.
- David Lykken (1998) characterized an accuracy of 85% for guilty individuals and 60% for innocent individuals as charitable.
- That 40% of honest examinees appear deceptive provides exceedingly poor protection for innocent suspects, and this problem is compounded when polygraphers administer tests to many suspects.

So the test seems virtually infallible: If the person fails and admits he lied, the test “worked,” and if the person fails and doesn’t admit he lied, the test also “worked.” Of course, if the person passes, he’ll essentially always agree that he was telling the truth, so the test again “worked.” This “heads I win, tails you lose” reasoning renders the rationale underlying the polygraph test difficult or impossible to falsify.

Given these limitations, the National Research Council (2003) was reluctant to estimate the CQT’s accuracy.

David Lykken (1998) characterized an accuracy of 85% for guilty individuals and 60% for innocent individuals as charitable.

That 40% of honest examinees appear deceptive provides exceedingly poor protection for innocent suspects, and this problem is compounded when polygraphers administer tests to many suspects.

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:
- He based the whole description of planetary motions on a theoretical law of dynamic interactions.
- These theories were supported by the results of the Michelson–Morley experiment in which evidence for the motion of aether was conclusively absent.[23] The results of the experiment influenced many physicists of the time and contributed to the eventual development of Einstein's theory of special relativity.[24] Aether and gravitation In 1682, Jakob Bernoulli formulated the theory that the hardness of the bodies depended on the pressure of the aether.[25] Aether has been used in various gravitational theories as a medium to help explain gravitation and what causes it.
- In his aether model, Newton describes aether as a medium that "flows" continually downward toward the
- He renounced standing attempts at accounting for this particular form of interaction between distant bodies by introducing a mechanism of propagation through an intervening medium.[26] He calls this intervening medium aether.

Aether (classical element) - Wikipedia.pdf

He based the whole description of planetary motions on a theoretical law of dynamic interactions.

These theories were supported by the results of the Michelson–Morley experiment in which evidence for the motion of aether was conclusively absent.[23] The results of the experiment influenced many physicists of the time and contributed to the eventual development of Einstein's theory of special relativity.[24] Aether and gravitation In 1682, Jakob Bernoulli formulated the theory that the hardness of the bodies depended on the pressure of the aether.[25] Aether has been used in various gravitational theories as a medium to help explain gravitation and what causes it.

In his aether model, Newton describes aether as a medium that "flows" continually downward toward the

He renounced standing attempts at accounting for this particular form of interaction between distant bodies by introducing a mechanism of propagation through an intervening medium.[26] He calls this intervening medium aether.

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 one study, 48% of undergraduates agreed that “Autism is caused by immunization shots” (Lenz et al., 2009).
- | 196 Chapter 9 Sad, Mad, and Bad
- Understandably, many people have attempted to pinpoint the sources of this baffling upsurge.
- One of thimerosal’s breakdown products is mercury, which can produce neurological damage at high doses (Figure 9.1).

In one study, 48% of undergraduates agreed that “Autism is caused by immunization shots” (Lenz et al., 2009).

Understandably, many people have attempted to pinpoint the sources of this baffling upsurge.

One of thimerosal’s breakdown products is mercury, which can produce neurological damage at high doses (Figure 9.1).

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:
- There are reflecting telescope designs that use modified mirror surfaces (such as the Ritchey–Chrétien telescope) or some form of correcting lens (such as catadioptric telescopes) that correct some of these aberrations.
- usually involves a combination of two or three aperture sized lenses (see achromat and apochromat for more details).
- There are structural problems involved in manufacturing and manipulating large-aperture lenses.
- Astigmatism is not usually a problem in a narrow field of view, but in a wide field image it gets rapidly worse and varies quadratically with field angle.

There are reflecting telescope designs that use modified mirror surfaces (such as the Ritchey–Chrétien telescope) or some form of correcting lens (such as catadioptric telescopes) that correct some of these aberrations.

usually involves a combination of two or three aperture sized lenses (see achromat and apochromat for more details).

There are structural problems involved in manufacturing and manipulating large-aperture lenses.

Astigmatism is not usually a problem in a narrow field of view, but in a wide field image it gets rapidly worse and varies quadratically with field angle.

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:
- Like some other myths in this book, the one you’re about to encounter has a grain of truth to it.
- Nevertheless, this grain can be a bit hard to find amidst the mounds of misinformation that bury it.
- In this rarely performed procedure, surgeons sever the nerve tracts connecting opposite points in the brain’s left and right hemispheres in a last-ditch attempt to control severe epilepsy.
- Then clasp it firmly to your chest and run as fast as you can.

Like some other myths in this book, the one you’re about to encounter has a grain of truth to it.

Nevertheless, this grain can be a bit hard to find amidst the mounds of misinformation that bury it.

In this rarely performed procedure, surgeons sever the nerve tracts connecting opposite points in the brain’s left and right hemispheres in a last-ditch attempt to control severe epilepsy.

Then clasp it firmly to your chest and run as fast as you can.

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:
- different rates of fall are tied by a string, does the combined system fall faster because it is now more massive, or does the lighter body in its slower fall hold back the heavier body?
- The groove was lined with "parchment, also smooth and polished as possible".
- One of Galileo's fictional characters, Salviati, describes an experiment using a bronze ball and a wooden ramp.
- The time was measured using a water clock described as follows: a large vessel of water placed in an elevated position; to the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which we collected in a small glass during the time of each descent, whether for the whole length of the channel or for a part of its length; the water thus collected was weighed, after each descent, on a very accurate balance; the differences and ratios of these weights gave us the differences and ratios of the times, and this with such accuracy that although the operation was repeated many, many times, there was no appreciable discrepancy in the results.[13] Galileo found that for an object in free fall, the distance that the object has fallen is always proportional to the square of the elapsed time: Distance traveled by a freely Galileo had shown that objects in free fall under the influence of the falling ball is proportional to the Earth's gravitational field have a constant acceleration, and Galileo's square of the elapsed time.

different rates of fall are tied by a string, does the combined system fall faster because it is now more massive, or does the lighter body in its slower fall hold back the heavier body?

The groove was lined with "parchment, also smooth and polished as possible".

One of Galileo's fictional characters, Salviati, describes an experiment using a bronze ball and a wooden ramp.

The time was measured using a water clock described as follows: a large vessel of water placed in an elevated position; to the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which we collected in a small glass during the time of each descent, whether for the whole length of the channel or for a part of its length; the water thus collected was weighed, after each descent, on a very accurate balance; the differences and ratios of these weights gave us the differences and ratios of the times, and this with such accuracy that although the operation was repeated many, many times, there was no appreciable discrepancy in the results.[13] Galileo found that for an object in free fall, the distance that the object has fallen is always proportional to the square of the elapsed time: Distance traveled by a freely Galileo had shown that objects in free fall under the influence of the falling ball is proportional to the Earth's gravitational field have a constant acceleration, and Galileo's square of the elapsed time.

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:
- ratio (g /g ) : (g /g ) = 1 : 3 of their statistical weights.
- The value of the spon2s 2 2p 2 4 4 taneous decay coefficient A itself is not actually needed since it cancels in the Sobolev approximation, 2p−1s equation (31.102), g 1 g 8πH 2pA P = 1 , (31.64) g 2p−1s S X n g λ3 2 1 + 2 2p−1s where H is the Hubble parameter.
- The statistical weight factor is g /g = 1.
- 1 2 4 DetailedbalancerequiresthatdX /dt,equation(31.61),mustvanishinthermodynamicequilibrium(TE), p so the ratio of photoionization to recombination rate coefficients must be β ( n n ) g g ( m T )3/2 2 = p e = p e e e−E2/T , (31.65) α n g 2π(cid:126)2 2 2 TE 2

ratio (g /g ) : (g /g ) = 1 : 3 of their statistical weights.

The value of the spon2s 2 2p 2 4 4 taneous decay coefficient A itself is not actually needed since it cancels in the Sobolev approximation, 2p−1s equation (31.102), g 1 g 8πH 2pA P = 1 , (31.64) g 2p−1s S X n g λ3 2 1 + 2 2p−1s where H is the Hubble parameter.

The statistical weight factor is g /g = 1.

1 2 4 DetailedbalancerequiresthatdX /dt,equation(31.61),mustvanishinthermodynamicequilibrium(TE), p so the ratio of photoionization to recombination rate coefficients must be β ( n n ) g g ( m T )3/2 2 = p e = p e e e−E2/T , (31.65) α n g 2π(cid:126)2 2 2 TE 2

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

List tests or derivations implied by the evidence.