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

EVIDENCE:
- 23.24.1 Application of the null singularity theorem to a Λ-Kerr-Newman black hole Penrose’s(1965)originalsingularitytheoremconsideredhypersurface-orthogonalnullcongruences.ThePretorius and Israel (1998) double-null congruence provides an example of such a congruence in the Λ-KerrNewman geometry.
- If the dog-leg proposition (§18.9.1) held, then
- These 2-surfaces are trapped inside the outer horizon, with negative expansion θ along both outgoing and ingoing congruences.
- the action on the initial hypersurface is S =−t.

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.

23.24.1 Application of the null singularity theorem to a Λ-Kerr-Newman black hole Penrose’s(1965)originalsingularitytheoremconsideredhypersurface-orthogonalnullcongruences.ThePretorius and Israel (1998) double-null congruence provides an example of such a congruence in the Λ-KerrNewman geometry.

If the dog-leg proposition (§18.9.1) held, then

These 2-surfaces are trapped inside the outer horizon, with negative expansion θ along both outgoing and ingoing congruences.

the action on the initial hypersurface is S =−t.

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- "The Search for Unity: Notes for a History of Quantum Field Theory".
- An Introduction to Quantum Field Theory (https://books.google.
- Bibcode:1981PhT....34k..69W (https://ui.adsabs.harvard.edu/abs/1 981PhT....34k..69W).
- Climbing the Mountain: The Scientific Biography of Julian Schwinger (https://books.google.com/books?id=9SmZSN8F164C) (Repr ed.).

"The Search for Unity: Notes for a History of Quantum Field Theory".

An Introduction to Quantum Field Theory (https://books.google.

Climbing the Mountain: The Scientific Biography of Julian Schwinger (https://books.google.com/books?id=9SmZSN8F164C) (Repr ed.).

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

List tests or derivations implied by the evidence.

Bennett
Dirac
Zeilinger
Jian-Wei Pan
De Martini
Feynman
Popper


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

EVIDENCE:
- Then replace the not be confused with the unrelated (r, φ) plane with a surface dimpled in the w direction concept of a gravity well.
- A tachyon could have a spacelike worldline that lies entirely on a single paraboloid.
- according to the equation (Flamm's paraboloid) This surface has the property that distances measured within it match distances in the Schwarzschild metric, because with the definition of w above, Thus, Flamm's paraboloid is useful for visualizing the spatial curvature of the Schwarzschild metric.
- It should not, however, be confused with a gravity well.

Then replace the not be confused with the unrelated (r, φ) plane with a surface dimpled in the w direction concept of a gravity well.

A tachyon could have a spacelike worldline that lies entirely on a single paraboloid.

according to the equation (Flamm's paraboloid) This surface has the property that distances measured within it match distances in the Schwarzschild metric, because with the definition of w above, Thus, Flamm's paraboloid is useful for visualizing the spatial curvature of the Schwarzschild metric.

It should not, however, be confused with a gravity well.

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- The Basics of Chemistry (https://archive.org/details/basicschemistry00mye r).
- National Institute of Standards and Technology.
- 85 (https://archive.org/details/basicschemistry00myer/page/n98).
- "What Do Spectra Tell Us?" (https://i magine.gsfc.nasa.gov/docs/science/how_l1/spectral_what.html).

The Basics of Chemistry (https://archive.org/details/basicschemistry00mye r).

National Institute of Standards and Technology.

"What Do Spectra Tell Us?" (https://i magine.gsfc.nasa.gov/docs/science/how_l1/spectral_what.html).

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 first matrix on the right hand sides of equations (38.109) is the matrix representation of the multivector ε ε · in N dimensions in terms of its representation in N−2 dimensions, A B ( ) ε ε · 0 ε ε ·=(−)(N−2)/2ε ε ·±ε ε ·= A B .
- The signs in equations (38.109) are arranged so that the correct commutation rules of the geometric algebra are respected: γγ and γγ , which are odd, commute/anticommute with ε ε · according as the latter is + − A B even/odd; and γγ γγ and γγ γγ , which are even, always commute with ε ε ·.
- 988 The super geometric algebra B are empty, and ε ε ·=1.
- In terms of scalar and + − − + A B wedge products, the multivectors γγ γγ and γγ γγ in equations (38.111) are + − − + ( ) ( ) 1 0 1 0 γγ γγ =γγ ·γγ ± γγ ∧γγ ,

The first matrix on the right hand sides of equations (38.109) is the matrix representation of the multivector ε ε · in N dimensions in terms of its representation in N−2 dimensions, A B ( ) ε ε · 0 ε ε ·=(−)(N−2)/2ε ε ·±ε ε ·= A B .

The signs in equations (38.109) are arranged so that the correct commutation rules of the geometric algebra are respected: γγ and γγ , which are odd, commute/anticommute with ε ε · according as the latter is + − A B even/odd; and γγ γγ and γγ γγ , which are even, always commute with ε ε ·.

988 The super geometric algebra B are empty, and ε ε ·=1.

In terms of scalar and + − − + A B wedge products, the multivectors γγ γγ and γγ γγ in equations (38.111) are + − − + ( ) ( ) 1 0 1 0 γγ γγ =γγ ·γγ ± γγ ∧γγ ,

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 considered that if a theory cannot, in principle, be falsified by criticism, it is not a scientific theory.[42] Philosophy of science Falsifiability and the problem of demarcation Popper coined the term "critical rationalism" to describe his philosophy.[43] Popper rejected the empiricist view (following from Kant) that basic statements are infallible; rather, according to Popper, they are descriptions in relation to a theoretical framework.[44] Concerning the method of science, the term "critical rationalism" indicates his rejection of classical empiricism, and the classical observationalist-inductivist account of science that had grown out of it.[45] Popper argued strongly against the latter, holding that scientific theories are abstract in nature[46][47][48] and can be tested only indirectly, by reference to their implications.[49] He also held that scientific theory, and human knowledge generally, is irreducibly conjectural or hypothetical, and is generated by the creative imagination to solve problems that have arisen in specific historico-cultural settings.
- The nature of such theories made it impossible for any criticism or experiment—even in principle—to show them to be false.[10] When Popper tackled the problem of demarcation in the philosophy of science, this conclusion led him to posit that the strength of a scientific theory lies in its being susceptible to falsification.
- properly grounded in scientific thought and method, was highly "risky", in the sense that it was possible to deduce consequences from it which differed considerably from those of the then-dominant Newtonian physics; one such prediction, that gravity could deflect light, was verified by Eddington's experiments in 1919.[41] In contrast he thought that nothing could, even in principle, falsify psychoanalytic theories.
- It also inspired him to take falsifiability as his criterion of demarcation between metaphysics and science: a law should be considered scientific if, and only if, it makes predictions, irrespective of their validity or of our capacity

He considered that if a theory cannot, in principle, be falsified by criticism, it is not a scientific theory.[42] Philosophy of science Falsifiability and the problem of demarcation Popper coined the term "critical rationalism" to describe his philosophy.[43] Popper rejected the empiricist view (following from Kant) that basic statements are infallible; rather, according to Popper, they are descriptions in relation to a theoretical framework.[44] Concerning the method of science, the term "critical rationalism" indicates his rejection of classical empiricism, and the classical observationalist-inductivist account of science that had grown out of it.[45] Popper argued strongly against the latter, holding that scientific theories are abstract in nature[46][47][48] and can be tested only indirectly, by reference to their implications.[49] He also held that scientific theory, and human knowledge generally, is irreducibly conjectural or hypothetical, and is generated by the creative imagination to solve problems that have arisen in specific historico-cultural settings.

The nature of such theories made it impossible for any criticism or experiment—even in principle—to show them to be false.[10] When Popper tackled the problem of demarcation in the philosophy of science, this conclusion led him to posit that the strength of a scientific theory lies in its being susceptible to falsification.

properly grounded in scientific thought and method, was highly "risky", in the sense that it was possible to deduce consequences from it which differed considerably from those of the then-dominant Newtonian physics; one such prediction, that gravity could deflect light, was verified by Eddington's experiments in 1919.[41] In contrast he thought that nothing could, even in principle, falsify psychoanalytic theories.

It also inspired him to take falsifiability as his criterion of demarcation between metaphysics and science: a law should be considered scientific if, and only if, it makes predictions, irrespective of their validity or of our capacity

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:
- 017/aug/18/stephen-hawking-blames-tory-poli Retrieved 18 February 2013.
- "Stephen Hawking's political views" (https://w ives-endorsement-general-election-a777401 ww.bbc.co.uk/news/uk-politics-43399019).
- Archived (https:// Trump's climate stance could damage Earth" web.archive.org/web/20180314204857/http:// (https://www.bbc.co.uk/news/science-environ www.bbc.co.uk/news/av/health-40990288/ste ment-40461726).
- Archived (https://web.archive.org/w 2018.

017/aug/18/stephen-hawking-blames-tory-poli Retrieved 18 February 2013.

"Stephen Hawking's political views" (https://w ives-endorsement-general-election-a777401 ww.bbc.co.uk/news/uk-politics-43399019).

Archived (https:// Trump's climate stance could damage Earth" web.archive.org/web/20180314204857/http:// (https://www.bbc.co.uk/news/science-environ www.bbc.co.uk/news/av/health-40990288/ste ment-40461726).

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:
- “It’s not fair to flunk me for being a right-brain thinker.” The urge on the part of pop psychologists to assign all mental abilities to unique left and right compartments probably owes more to politics, social values, and commercial interests than to science.
- Its detractors have dubbed this extreme view “dichotomania” because of pop psychologists’ tendency to dichotomize the two hemispheres’ functions (Corballis, 1999).
- The notion was embraced enthusiastically by New Age proponents of | Chapter 1 Brain Power 27

“It’s not fair to flunk me for being a right-brain thinker.” The urge on the part of pop psychologists to assign all mental abilities to unique left and right compartments probably owes more to politics, social values, and commercial interests than to science.

Its detractors have dubbed this extreme view “dichotomania” because of pop psychologists’ tendency to dichotomize the two hemispheres’ functions (Corballis, 1999).

The notion was embraced enthusiastically by New Age proponents of | Chapter 1 Brain Power 27

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:
- Specific Energy Specific Energy Material energy density energy density Comment (MJ/kg) (MJ/L) (W⋅h/kg) (W⋅h/L) Ethanol 30 24 8333.3 6666.7 DME 31.7 (HHV) 21.24 (HHV) 8805.6 (HHV) 5900.0 (HHV) [36][37] 28.4 (LHV) 19.03 (LHV) 7888.9 (LHV) 5286.1 (LHV) Polyester plastic 26.0[27] 35.6 7222.2 9888.9 Magnesium 24.7 43.0 6861.1 11,944.5 Phosphorus (white) 24.30 44.30 6750 12310 [38] Coal, bituminous 24–35 26–49 6666.7–9722.2 7222.2– [5] 13611.1 PET plastic (impure) 23.5[39] < ~32.4 6527.8 < ~9000 Methanol 19.7 15.6 5472.2 4333.3 Titanium 19.74 88.93 5480 24700 burned to titanium dioxide Hydrazine 19.5 19.3 5416.7 5361.1 burned to nitrogen and water Liquid ammonia 18.6 11.5 5166.7 3194.5 burned to nitrogen and water Potassium 18.6 16.5 5160 4600 burned to dry potassium oxide PVC plastic (improper 18.0[27] 25.2 5000.0 7000.0 combustion toxic) Wood 18.0 5000.0 [40] Peat briquette 17.7 4916.7 [41] Sugars, carbohydrates, 17 26.2 (dextrose) 4722.2 7277.8 metabolism in human body (22% efficiency[42]) and protein Calcium 15.9 24.6 4416.7 6833.3 Glucose 15.55 23.9 4319.5 6638.9 Dry cow dung and 15.5[43] 4305.6 camel dung Coal, lignite 10–20 2777.8–5555.6 Sodium 13.3 12.8 3694.5 3555.6 burned to wet sodium hydroxide Peat 12.8 3,555.6 Nitromethane 11.3 12.85 3138.9 3570 Manganese 9.46 68.2 2630 18900 burned to manganese dioxide Sulfur 9.23 19.11 2563.9 5308.3 burned to sulfur dioxide[44] Sodium 9.1 8.8 2527.8 2444.5 burned to dry sodium oxide Household waste 8.0[45] 2222.2 Iron 7.4 57.7 2052.9 16004.1 burned to iron(III) oxide[46] Iron 6.7 52.2 1858.3 14487.2 burned to Iron(II,III) oxide[46] Zinc 5.3 38.0 1472.2 10555.6 Teflon plastic 5.1 11.2 1416.7 3111.1 combustion toxic, but flame retardant Iron 4.9 38.2 1361.1 10611.1 burned to iron(II) oxide[46] Gunpowder 4.7–11.3[47] 5.9–12.9 1600–3580 TNT 4.184 6.92 1162 1920 Barium 3.99 14.0 1110 3890 burned to barium dioxide ANFO 3.7 1027.8

Specific Energy Specific Energy Material energy density energy density Comment (MJ/kg) (MJ/L) (W⋅h/kg) (W⋅h/L) Ethanol 30 24 8333.3 6666.7 DME 31.7 (HHV) 21.24 (HHV) 8805.6 (HHV) 5900.0 (HHV) [36][37] 28.4 (LHV) 19.03 (LHV) 7888.9 (LHV) 5286.1 (LHV) Polyester plastic 26.0[27] 35.6 7222.2 9888.9 Magnesium 24.7 43.0 6861.1 11,944.5 Phosphorus (white) 24.30 44.30 6750 12310 [38] Coal, bituminous 24–35 26–49 6666.7–9722.2 7222.2– [5] 13611.1 PET plastic (impure) 23.5[39] < ~32.4 6527.8 < ~9000 Methanol 19.7 15.6 5472.2 4333.3 Titanium 19.74 88.93 5480 24700 burned to titanium dioxide Hydrazine 19.5 19.3 5416.7 5361.1 burned to nitrogen and water Liquid ammonia 18.6 11.5 5166.7 3194.5 burned to nitrogen and water Potassium 18.6 16.5 5160 4600 burned to dry potassium oxide PVC plastic (improper 18.0[27] 25.2 5000.0 7000.0 combustion toxic) Wood 18.0 5000.0 [40] Peat briquette 17.7 4916.7 [41] Sugars, carbohydrates, 17 26.2 (dextrose) 4722.2 7277.8 metabolism in human body (22% efficiency[42]) and protein Calcium 15.9 24.6 4416.7 6833.3 Glucose 15.55 23.9 4319.5 6638.9 Dry cow dung and 15.5[43] 4305.6 camel dung Coal, lignite 10–20 2777.8–5555.6 Sodium 13.3 12.8 3694.5 3555.6 burned to wet sodium hydroxide Peat 12.8 3,555.6 Nitromethane 11.3 12.85 3138.9 3570 Manganese 9.46 68.2 2630 18900 burned to manganese dioxide Sulfur 9.23 19.11 2563.9 5308.3 burned to sulfur dioxide[44] Sodium 9.1 8.8 2527.8 2444.5 burned to dry sodium oxide Household waste 8.0[45] 2222.2 Iron 7.4 57.7 2052.9 16004.1 burned to iron(III) oxide[46] Iron 6.7 52.2 1858.3 14487.2 burned to Iron(II,III) oxide[46] Zinc 5.3 38.0 1472.2 10555.6 Teflon plastic 5.1 11.2 1416.7 3111.1 combustion toxic, but flame retardant Iron 4.9 38.2 1361.1 10611.1 burned to iron(II) oxide[46] Gunpowder 4.7–11.3[47] 5.9–12.9 1600–3580 TNT 4.184 6.92 1162 1920 Barium 3.99 14.0 1110 3890 burned to barium dioxide ANFO 3.7 1027.8

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:
- Archived (https://web.archive.org/web/20120606114603/htt p://www.baseball-almanac.com/recbooks/rb_guin.shtml) from the original on 6 June 2012.
- Leibniz's view has two major implications.
- Archived (https://web.archive.org/web/20110514000548/htt p://www.iep.utm.edu/leib-met/#H7) from the original on 14 May 2011.
- Space and time are, rather, ideal.

Archived (https://web.archive.org/web/20120606114603/htt p://www.baseball-almanac.com/recbooks/rb_guin.shtml) from the original on 6 June 2012.

Leibniz's view has two major implications.

Archived (https://web.archive.org/web/20110514000548/htt p://www.iep.utm.edu/leib-met/#H7) from the original on 14 May 2011.

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:
- "Quantum superposition of molecules beyond 25 kDa".
- "Zur Quantenmechanik der Stoßvorgänge" [On the Quantum Mechanics of Collision Processes].
- 14–11 ff), integrated circuits, which are follow-on technology in solid-state physics (vol II, pp.
- Consequently, neither can belong to the particle's Hilbert space.

"Quantum superposition of molecules beyond 25 kDa".

"Zur Quantenmechanik der Stoßvorgänge" [On the Quantum Mechanics of Collision Processes].

14–11 ff), integrated circuits, which are follow-on technology in solid-state physics (vol II, pp.

Consequently, neither can belong to the particle's Hilbert space.

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:
- (38.23) Despite the equality of ε and iγγ in the Pauli representation, ε is defined to transform as a spinor tensor under spatial rotations, not as an element of the geometric algebra.
- The two row spinors (the braces in equation (38.27) signify a set of spinors, not anticommutation) ε ·={ε ·,ε ·} (38.27) a ↑ ↓ provide a basis for row spinors.
- (38.24) a b ab Commuting the spinor metric ε through the orthonormal basis vectors γγ converts them to minus their a transposes, γγ⊤ε=−εγγ .
- 38.7 Row basis spinors 973 In the Pauli representation (13.112), the spinor metric (38.21) is the antisymmetric matrix ( ) 0 1 ε= .

(38.23) Despite the equality of ε and iγγ in the Pauli representation, ε is defined to transform as a spinor tensor under spatial rotations, not as an element of the geometric algebra.

The two row spinors (the braces in equation (38.27) signify a set of spinors, not anticommutation) ε ·={ε ·,ε ·} (38.27) a ↑ ↓ provide a basis for row spinors.

(38.24) a b ab Commuting the spinor metric ε through the orthonormal basis vectors γγ converts them to minus their a transposes, γγ⊤ε=−εγγ .

38.7 Row basis spinors 973 In the Pauli representation (13.112), the spinor metric (38.21) is the antisymmetric matrix ( ) 0 1 ε= .

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:
- Let us be specific: the object S ̃13, for example, transforms into S ̃(cid:5)13=R1kR3lS ̃kl = R11R31S ̃11+R11R32S ̃12+R11R33S ̃13+R12R31S ̃21+R12R32S ̃22+R12R33S ̃23+R13R31S ̃31 +R13R32S ̃32+R13R33S ̃33=(R11R31−R13R33)S ̃11+(R11R32+R12R31)S ̃12+(R11R33+ R13R31)S ̃13+(R12R32−R13R33)S ̃22+(R12R33+R13R32)S ̃23, where in the last equality, weusedS ̃ij =S ̃ji andS ̃33=−(S ̃11+S ̃22).Indeed,S ̃13transformsintoalinearcombinationofS ̃11,S ̃22,S ̃12,S ̃13,S ̃23.
- WedonotcountS ̃33separately,sinceitisequalto−(S ̃11+S ̃22).UnderanSO(3)rotation, these5objectstransformintolinearcombinationsofoneanother,aswejustexplained.
- (Forthereaderswhoknowtheirlinearalgebra,thetechnicalstatementisthatthereexists a similarity transformation that block-diagonalizes D(R) for all R.
- Tosummarize,whatwefoundisthatif,insteadofthebasisconsistingofthe9entities Tij, we use the basis consisting of the 3 entities Aij, the single entity Skk (remember repeatedindexsummation!),andthe5entitiesS ̃ij,the9-by-9matrixD(R)(thatrepresents rotationinthesenseof(4))breaksupintoa3-by-3matrix, a1-by-1matrix, anda5-by-5 matrix“stackedontopofeachother.”Thisisrepresentedschematicallyas ⎡ ⎤ (3-by-3block) 0 0 ⎢ ⎥ ⎢ ⎥ D(R)=(9-by-9matrix)→⎢ 0 (1-by-1block) 0 ⎥ (8) ⎣ ⎦ 0 0 (5-by-5block) Note that once we chose the new basis, this decomposition holds true for all rotations.

Let us be specific: the object S ̃13, for example, transforms into S ̃(cid:5)13=R1kR3lS ̃kl = R11R31S ̃11+R11R32S ̃12+R11R33S ̃13+R12R31S ̃21+R12R32S ̃22+R12R33S ̃23+R13R31S ̃31 +R13R32S ̃32+R13R33S ̃33=(R11R31−R13R33)S ̃11+(R11R32+R12R31)S ̃12+(R11R33+ R13R31)S ̃13+(R12R32−R13R33)S ̃22+(R12R33+R13R32)S ̃23, where in the last equality, weusedS ̃ij =S ̃ji andS ̃33=−(S ̃11+S ̃22).Indeed,S ̃13transformsintoalinearcombinationofS ̃11,S ̃22,S ̃12,S ̃13,S ̃23.

WedonotcountS ̃33separately,sinceitisequalto−(S ̃11+S ̃22).UnderanSO(3)rotation, these5objectstransformintolinearcombinationsofoneanother,aswejustexplained.

(Forthereaderswhoknowtheirlinearalgebra,thetechnicalstatementisthatthereexists a similarity transformation that block-diagonalizes D(R) for all R.

Tosummarize,whatwefoundisthatif,insteadofthebasisconsistingofthe9entities Tij, we use the basis consisting of the 3 entities Aij, the single entity Skk (remember repeatedindexsummation!),andthe5entitiesS ̃ij,the9-by-9matrixD(R)(thatrepresents rotationinthesenseof(4))breaksupintoa3-by-3matrix, a1-by-1matrix, anda5-by-5 matrix“stackedontopofeachother.”Thisisrepresentedschematicallyas ⎡ ⎤ (3-by-3block) 0 0 ⎢ ⎥ ⎢ ⎥ D(R)=(9-by-9matrix)→⎢ 0 (1-by-1block) 0 ⎥ (8) ⎣ ⎦ 0 0 (5-by-5block) Note that once we chose the new basis, this decomposition holds true for all rotations.

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:
- Summary of wormholes These are legal solutions of Einstein’s equations • They probably pop in and out of quantum froth all the time • But to keep them open you need “exotic matter”, which is • matter that has a negative(!!!) energy density There are ways you can do this, and dark energy has • similarities, but we don’t know whether it is possible even in principle to produce a stable wormhole Wormhole!time travel; self-consistency is the key!

Summary of wormholes These are legal solutions of Einstein’s equations • They probably pop in and out of quantum froth all the time • But to keep them open you need “exotic matter”, which is • matter that has a negative(!!!) energy density There are ways you can do this, and dark energy has • similarities, but we don’t know whether it is possible even in principle to produce a stable wormhole Wormhole!time travel; self-consistency is the key!

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:
- Units To ease calculations, a unit of volume is equal to the volume occupied by a unit cube (with a side length of one).
- The largest of such calibrated containers are petroleum storage tanks, some can hold up to 1,000,000 bbl (160,000,000 L) of fluids.
- customary units of volume are also in use, including: cubic inch, cubic foot, cubic yard, acre-foot, cubic mile; minim, drachm, fluid ounce, pint; teaspoon, tablespoon; gill, quart, gallon, barrel; cord, peck, bushel, hogshead.
- There, volume of liquids is measured using graduated cylinders, pipettes and volumetric flasks.

Units To ease calculations, a unit of volume is equal to the volume occupied by a unit cube (with a side length of one).

The largest of such calibrated containers are petroleum storage tanks, some can hold up to 1,000,000 bbl (160,000,000 L) of fluids.

customary units of volume are also in use, including: cubic inch, cubic foot, cubic yard, acre-foot, cubic mile; minim, drachm, fluid ounce, pint; teaspoon, tablespoon; gill, quart, gallon, barrel; cord, peck, bushel, hogshead.

There, volume of liquids is measured using graduated cylinders, pipettes and volumetric flasks.

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- (The distinction between these two notions is generally glossed over in the physics literature, where the term Hermitian is used for either or both classes of operators.
- Substituting the above into Equation (2) we get the Schrödinger uncertainty relation This proof has an issue[19] related to the domains of the operators involved.
- See Chapter 9 of Hall's book[23] for a detailed discussion of this important but technical distinction.) Phase space In the phase space formulation of quantum mechanics, the Robertson–Schrödinger relation follows from a positivity condition on a real star-square function.
- In fact, the Robertson uncertainty relation is false if is an angle variable and is the derivative with respect to this variable.

(The distinction between these two notions is generally glossed over in the physics literature, where the term Hermitian is used for either or both classes of operators.

Substituting the above into Equation (2) we get the Schrödinger uncertainty relation This proof has an issue[19] related to the domains of the operators involved.

See Chapter 9 of Hall's book[23] for a detailed discussion of this important but technical distinction.) Phase space In the phase space formulation of quantum mechanics, the Robertson–Schrödinger relation follows from a positivity condition on a real star-square function.

In fact, the Robertson uncertainty relation is false if is an angle variable and is the derivative with respect to this variable.

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:
- Indeed, beliefs about the “evil eye” (mal ojo) inflicting psychological harm on others have long been widespread in many countries, especially Mexico and those in the Mediterranean, Central America, and the Arab world (Bohigian, 1998; Gross, 1999; Machovec, 1976; Winer, Rader, & Cottrell, 2003).
- Biologist Rupert Sheldrake (2003) even created a stir in the scientific community by conducting research purporting to show that many people can tell they’re being stared at by people they can’t see, but a number of researchers have identified serious flaws in his studies, including the fact that Sheldrake’s subjects may have subtly influenced people to stare back at them (Marks & Colwell, 2000; Shermer, 2005).
- Throughout the ages, poets wrote of the power of the eye to induce profound psychological effects, perhaps indirectly reflecting people’s extramission beliefs (Gross, 1999).
- 9781405131117_4_001.qxd 30/6/09 11:48 AM Page 34 As the great Swiss psychologist Jean Piaget (1929) noted, this belief begins early in life.

Indeed, beliefs about the “evil eye” (mal ojo) inflicting psychological harm on others have long been widespread in many countries, especially Mexico and those in the Mediterranean, Central America, and the Arab world (Bohigian, 1998; Gross, 1999; Machovec, 1976; Winer, Rader, & Cottrell, 2003).

Biologist Rupert Sheldrake (2003) even created a stir in the scientific community by conducting research purporting to show that many people can tell they’re being stared at by people they can’t see, but a number of researchers have identified serious flaws in his studies, including the fact that Sheldrake’s subjects may have subtly influenced people to stare back at them (Marks & Colwell, 2000; Shermer, 2005).

Throughout the ages, poets wrote of the power of the eye to induce profound psychological effects, perhaps indirectly reflecting people’s extramission beliefs (Gross, 1999).

9781405131117_4_001.qxd 30/6/09 11:48 AM Page 34 As the great Swiss psychologist Jean Piaget (1929) noted, this belief begins early in life.

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:
- However, Physics involves modeling the natural further work in the 19th century revealed that these two forces world with theory, usually quantitative.
- Complex problems that seem like they could be solved by a clever application of dynamics and
- This effect was later called magnetism, which was first rigorously studied in the 17th century.
- These experimental results appear to have solved the long-standing solar neutrino problem, and the physics of massive neutrinos remains an area of active theoretical and experimental research.

However, Physics involves modeling the natural further work in the 19th century revealed that these two forces world with theory, usually quantitative.

Complex problems that seem like they could be solved by a clever application of dynamics and

This effect was later called magnetism, which was first rigorously studied in the 17th century.

These experimental results appear to have solved the long-standing solar neutrino problem, and the physics of massive neutrinos remains an area of active theoretical and experimental research.

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:
- Parametric equation In mathematics, a parametric equation expresses several quantities, such as the coordinates of a point, as functions of one or several variables called parameters.[1] In the case of a single parameter, parametric equations are commonly used to express the trajectory of a moving point, in which case, the parameter is often, but not necessarily, time, and the point describes a curve, called a parametric curve.
- Sometimes the parametric equations for the individual scalar output variables are combined into a single parametric equation in vectors: Parametric representations are generally nonunique (see the "Examples in two dimensions" section below), so the same quantities may be expressed by a number of different parameterizations.[1] In addition to curves and surfaces, parametric equations can describe manifolds and algebraic varieties of higher dimension, with the number of parameters being equal to the dimension of the manifold or variety, and the number of equations being equal to the dimension of the space in which the manifold or variety is considered (for curves the dimension is one and one parameter is used, for surfaces dimension two and two parameters, etc.).
- Because of this application, a single parameter is often labeled t; however, parameters can represent other physical quantities (such as geometric variables) or can be selected arbitrarily for convenience.
- Parametric equations are commonly used in kinematics, where the trajectory of an object is represented by equations depending on time as the parameter.

Parametric equation In mathematics, a parametric equation expresses several quantities, such as the coordinates of a point, as functions of one or several variables called parameters.[1] In the case of a single parameter, parametric equations are commonly used to express the trajectory of a moving point, in which case, the parameter is often, but not necessarily, time, and the point describes a curve, called a parametric curve.

Sometimes the parametric equations for the individual scalar output variables are combined into a single parametric equation in vectors: Parametric representations are generally nonunique (see the "Examples in two dimensions" section below), so the same quantities may be expressed by a number of different parameterizations.[1] In addition to curves and surfaces, parametric equations can describe manifolds and algebraic varieties of higher dimension, with the number of parameters being equal to the dimension of the manifold or variety, and the number of equations being equal to the dimension of the space in which the manifold or variety is considered (for curves the dimension is one and one parameter is used, for surfaces dimension two and two parameters, etc.).

Because of this application, a single parameter is often labeled t; however, parameters can represent other physical quantities (such as geometric variables) or can be selected arbitrarily for convenience.

Parametric equations are commonly used in kinematics, where the trajectory of an object is represented by equations depending on time as the parameter.

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:
- five significant figures, whereas GM🜨, the product of Earth's Figure 1 of Cavendish's paper.
- mass and the universal gravitational constant, is known to over nine significant figures.
- Given two objects A and B, of masses M and M , separated by a displacement R , Newton's law of A B AB gravitation states that each object exerts a gravitational force on the other, of magnitude , where G is the universal gravitational constant.
- The above statement may be reformulated in the following way: if g is the magnitude at a given location in a gravitational field, then the gravitational force on an object with gravitational mass M is

five significant figures, whereas GM🜨, the product of Earth's Figure 1 of Cavendish's paper.

mass and the universal gravitational constant, is known to over nine significant figures.

Given two objects A and B, of masses M and M , separated by a displacement R , Newton's law of A B AB gravitation states that each object exerts a gravitational force on the other, of magnitude , where G is the universal gravitational constant.

The above statement may be reformulated in the following way: if g is the magnitude at a given location in a gravitational field, then the gravitational force on an object with gravitational mass M is

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:
- Irrespective of the complexity of a gravitating object collapsing to form a black hole, the object that results (having emitted gravitational waves) is very simple.[158]
- This is stated by the black hole uniqueness theorem: "black holes have no hair", that is, no distinguishing marks like the hairstyles of humans.
- black hole, and introducing interesting features such as the ergosphere).
- With time they become rather simple objects characterized by eleven parameters specifying: electric charge, mass–energy, linear momentum, angular momentum, and location at a specified time.

Irrespective of the complexity of a gravitating object collapsing to form a black hole, the object that results (having emitted gravitational waves) is very simple.[158]

This is stated by the black hole uniqueness theorem: "black holes have no hair", that is, no distinguishing marks like the hairstyles of humans.

black hole, and introducing interesting features such as the ergosphere).

With time they become rather simple objects characterized by eleven parameters specifying: electric charge, mass–energy, linear momentum, angular momentum, and location at a specified time.

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:
- "Lorentz-covariant reduced spin density matrix and Einstein-Podolsky-Rosen–Bohm correlations".
- Retrieved from "https://en.wikipedia.org/w/index.php?title=Dirac_spinor&oldid=1326566096"
- References Caban, Paweł; Rembieliński, Jakub (5 July 2005).
- doi:10.1103/physreva.72.012103 (https://doi.org/10.1103%2Fphysreva.72.012103).

"Lorentz-covariant reduced spin density matrix and Einstein-Podolsky-Rosen–Bohm correlations".

References Caban, Paweł; Rembieliński, Jakub (5 July 2005).

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:
- Wewillstartbystudyingthemotionofmassiveandmasslessparticles, suchasplanets and photons, around a black hole.
- BlackHoles objectandcrossesintotheregionwherer<r ,heorshecannevergetbackouttospatial S infinity.Inshort,forourpurposeshere,ablackholeisdefinedasamassiveobjectwith anaccessiblehorizon.
- Themysteryoftheblackholealsodeepenswiththerealizationthatspacetimeisperfectly smooththere(asindicatedbythebehaviorofR Rμνλρ,forexample).Inthenextchapter, μνλρ wewillconfirmoursuspicionthatthesingularityinthemetricatthehorizonismerely duetoapoorchoiceofcoordinates.
- Acommonmisconceptionaboutblackholes A common misconception is that, around a black hole, an irresistible mysterious force ∗ sucks everything in.

Wewillstartbystudyingthemotionofmassiveandmasslessparticles, suchasplanets and photons, around a black hole.

BlackHoles objectandcrossesintotheregionwherer<r ,heorshecannevergetbackouttospatial S infinity.Inshort,forourpurposeshere,ablackholeisdefinedasamassiveobjectwith anaccessiblehorizon.

Acommonmisconceptionaboutblackholes A common misconception is that, around a black hole, an irresistible mysterious force ∗ sucks everything in.

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- to run the Einstein field equations in the other direction, first specifying a metric and then finding the energy–momentum tensor associated with it, and this is what Alcubierre did in building his metric.
- This practice means that the solution can violate various energy conditions and require exotic matter.
- The need for exotic matter raises questions about whether one can distribute the matter in an initial spacetime that lacks a warp bubble in such a way that the bubble is created at a later time, although some physicists have proposed models of dynamical warp-drive spacetimes in which a warp bubble is formed in a previously flat space.[4] Moreover, according to Serguei Krasnikov,[21] generating a bubble in a previously flat space for a one-way faster-than-light trip requires forcing the exotic matter to move at local faster-than-light speeds, something that would require the existence of tachyons, although Krasnikov also notes that when the spacetime is not flat from the outset, a similar result could be achieved without tachyons by placing in advance some devices along the travel path and programming them to come into operation at preassigned moments and to operate in a preassigned manner.
- Some suggested methods avoid the problem of tachyonic motion, but would probably generate a naked singularity at the front of the bubble.[22][23] Allen Everett and Thomas Roman comment on Krasnikov's finding (Krasnikov tube):

to run the Einstein field equations in the other direction, first specifying a metric and then finding the energy–momentum tensor associated with it, and this is what Alcubierre did in building his metric.

This practice means that the solution can violate various energy conditions and require exotic matter.

The need for exotic matter raises questions about whether one can distribute the matter in an initial spacetime that lacks a warp bubble in such a way that the bubble is created at a later time, although some physicists have proposed models of dynamical warp-drive spacetimes in which a warp bubble is formed in a previously flat space.[4] Moreover, according to Serguei Krasnikov,[21] generating a bubble in a previously flat space for a one-way faster-than-light trip requires forcing the exotic matter to move at local faster-than-light speeds, something that would require the existence of tachyons, although Krasnikov also notes that when the spacetime is not flat from the outset, a similar result could be achieved without tachyons by placing in advance some devices along the travel path and programming them to come into operation at preassigned moments and to operate in a preassigned manner.

Some suggested methods avoid the problem of tachyonic motion, but would probably generate a naked singularity at the front of the bubble.[22][23] Allen Everett and Thomas Roman comment on Krasnikov's finding (Krasnikov tube):

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- the length of Earth's equator 134 ms Ole Rømer first demonstrated that light does not from Moon to Earth 1.3 s travel instantaneously by studying the apparent from Sun to Earth (1 AU) 8.3 min motion of Jupiter's moon Io.
- Much starlight viewed on astronomical units per day 173[Note 1] Earth is from the distant past, allowing humans to study the history of the universe by viewing parsecs per year 0.307[Note 2] distant objects.
- When communicating with Approximate light signal travel times distant space probes, it can take hours for signals Distance Time to travel.
- It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time interval of 1⁄ 299792458 second.

the length of Earth's equator 134 ms Ole Rømer first demonstrated that light does not from Moon to Earth 1.3 s travel instantaneously by studying the apparent from Sun to Earth (1 AU) 8.3 min motion of Jupiter's moon Io.

Much starlight viewed on astronomical units per day 173[Note 1] Earth is from the distant past, allowing humans to study the history of the universe by viewing parsecs per year 0.307[Note 2] distant objects.

When communicating with Approximate light signal travel times distant space probes, it can take hours for signals Distance Time to travel.

It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time interval of 1⁄ 299792458 second.

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:
- 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 1 cosmic scale factor a Φ3 − δ dna Φ3 − δ b c 1.5 a a eq rec 1.0 1 .5 .0 0.1 −.5 −1.0 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 1 cosmic scale factor a Φ2− dna Φ − Θ 0 a a eq rec 0.1 Figure32.3 (Left)Overdensitiesδc−3Φandδ b −3Φofnon-baryonicdarkmatter(brown)andbaryonicmatter(green), and (right) radiation monopole Θ0−3Φ (blue), and minus twice the scalar potential, −2Ψ (black), as a function of cosmic scale factor a in the hydrodynamic approximation.
- Therefore in the hydrodynamic approximation the two potentials are set equal, as in the simple approximation (30.58), Ψ=Φ .
- Curves are labelled with the comoving wavenumber k/(aeqHeq)inunitsoftheHubbledistanceatmatter-radiationequality.Theresultsmaybecomparedtothoseinthe simpleapproximation,Figure30.1,andusingaBoltzmanncomputation,Figure33.3.
- Program the equations in the hydrodynamic approximation.Upgradethecodeyou wrote in Exercise 30.11 to implement the hydrodynamic approximation, equations (32.6)–(32.13).

10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 1 cosmic scale factor a Φ3 − δ dna Φ3 − δ b c 1.5 a a eq rec 1.0 1 .5 .0 0.1 −.5 −1.0 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 1 cosmic scale factor a Φ2− dna Φ − Θ 0 a a eq rec 0.1 Figure32.3 (Left)Overdensitiesδc−3Φandδ b −3Φofnon-baryonicdarkmatter(brown)andbaryonicmatter(green), and (right) radiation monopole Θ0−3Φ (blue), and minus twice the scalar potential, −2Ψ (black), as a function of cosmic scale factor a in the hydrodynamic approximation.

Therefore in the hydrodynamic approximation the two potentials are set equal, as in the simple approximation (30.58), Ψ=Φ .

Curves are labelled with the comoving wavenumber k/(aeqHeq)inunitsoftheHubbledistanceatmatter-radiationequality.Theresultsmaybecomparedtothoseinthe simpleapproximation,Figure30.1,andusingaBoltzmanncomputation,Figure33.3.

Program the equations in the hydrodynamic approximation.Upgradethecodeyou wrote in Exercise 30.11 to implement the hydrodynamic approximation, equations (32.6)–(32.13).

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:
- An equivalent way of saying this is that for infinitesimal θ, the transformation x (cid:5)(cid:16) x+θyandy (cid:5)(cid:16)y−θx(youcouldverifythat(1)indeedreducestothistoleadingorderin
- In particular, a reflection in the x-axis −1 0 ,whichistheproductofP andarotationthrough90 ◦ ,isalsoexcluded.
- FromNewtontoRiemann:CoordinatestoCurvature To focus on rotations, let us exclude reflections by imposing the condition (since detP=−1) detR=1 (5) Matriceswithunitdeterminantarecalledspecial.
- Indeed, suppose we have never even heard of sine and cosine.

An equivalent way of saying this is that for infinitesimal θ, the transformation x (cid:5)(cid:16) x+θyandy (cid:5)(cid:16)y−θx(youcouldverifythat(1)indeedreducestothistoleadingorderin

In particular, a reflection in the x-axis −1 0 ,whichistheproductofP andarotationthrough90 ◦ ,isalsoexcluded.

FromNewtontoRiemann:CoordinatestoCurvature To focus on rotations, let us exclude reflections by imposing the condition (since detP=−1) detR=1 (5) Matriceswithunitdeterminantarecalledspecial.

Indeed, suppose we have never even heard of sine and cosine.

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:
- doi:10.1103/PhysRevLett.88.221602 (https://doi.org/10.1103%2FPhysRevLett.88.221602).
- "On the quantum structure of a black hole".
- In Pisin Chen; Elliott Bloom; Greg Madejski; Vahe Patrosian (eds.).
- PMID 10019680 (https://pubmed.ncbi.nlm.nih.gov/10019680).

"On the quantum structure of a black hole".

In Pisin Chen; Elliott Bloom; Greg Madejski; Vahe Patrosian (eds.).

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:
- (23.10b) y φ y y From the expression (23.4) for the coordinate momenta dxκ/dλ, the trajectory of a freely-falling particle follows from integrating dy/dx=−P /P , equivalent to the implicit equation y x dx dy − = .
- Whereas electric charge is a scalar, magnetic charge is a pseudoscalar.
- (23.12) P Δ P Δ P Δ P Δ x x y y x x y y Againfromexpression(23.4),theaffineparameterλalongthetrajectorysatisfiesdλ/ρ2 =−dx/P =dy/P , x y so similarly reduces to quadratures, ρ2dx ρ2dy dλ=− x + y .
- The Hodge dual of the electromagnetic field (22.11), obtained by multiplying by the pseudoscalar I, equation (13.24), is the same expression with the electric Q and ∙ magnetic Q charges of the black hole exchanged according to ∙ Q →−Q , Q →Q .

(23.10b) y φ y y From the expression (23.4) for the coordinate momenta dxκ/dλ, the trajectory of a freely-falling particle follows from integrating dy/dx=−P /P , equivalent to the implicit equation y x dx dy − = .

Whereas electric charge is a scalar, magnetic charge is a pseudoscalar.

(23.12) P Δ P Δ P Δ P Δ x x y y x x y y Againfromexpression(23.4),theaffineparameterλalongthetrajectorysatisfiesdλ/ρ2 =−dx/P =dy/P , x y so similarly reduces to quadratures, ρ2dx ρ2dy dλ=− x + y .

The Hodge dual of the electromagnetic field (22.11), obtained by multiplying by the pseudoscalar I, equation (13.24), is the same expression with the electric Q and ∙ magnetic Q charges of the black hole exchanged according to ∙ Q →−Q , Q →Q .

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- The Einstein equation (16.250b) is equivalent to the traditional expression G=8πT .
- Notice that if the area element e2 had been taken to be the momentum conjugate to Γ, rather than the line element e, all the components of the area element e2 being considered to be independent degrees of freedom, then the variation (16.241) of the gravitational action with respect to e2 would have yielded an equation of motion for the Riemann tensor R rather than for the Einstein tensor G, and the theory would not be general relativity.
- (16.251) ThecontractedBianchiidentities(16.406)enforceconservationlawsforthetotalspinangular-momentum ** ** Σ and total matter energy-momentum T, §16.14.8 and §16.14.9.
- 458 Action principle for electromagnetism and gravity energy-momentumvector1-formT are(theminussigninthedefinitionofΣconformstotheconventionfor the definition of torsion S, equation (16.209)) Σ≡Σ dxκ =−Σ γγl∧γγmdxκ , (16.249a) κ κlm T ≡T dxκ =T γγmdxκ , (16.249b) κ κm with, for Σ, implicit summation over distinct antisymmetric sets of indices kl.

The Einstein equation (16.250b) is equivalent to the traditional expression G=8πT .

Notice that if the area element e2 had been taken to be the momentum conjugate to Γ, rather than the line element e, all the components of the area element e2 being considered to be independent degrees of freedom, then the variation (16.241) of the gravitational action with respect to e2 would have yielded an equation of motion for the Riemann tensor R rather than for the Einstein tensor G, and the theory would not be general relativity.

(16.251) ThecontractedBianchiidentities(16.406)enforceconservationlawsforthetotalspinangular-momentum ** ** Σ and total matter energy-momentum T, §16.14.8 and §16.14.9.

458 Action principle for electromagnetism and gravity energy-momentumvector1-formT are(theminussigninthedefinitionofΣconformstotheconventionfor the definition of torsion S, equation (16.209)) Σ≡Σ dxκ =−Σ γγl∧γγmdxκ , (16.249a) κ κlm T ≡T dxκ =T γγmdxκ , (16.249b) κ κm with, for Σ, implicit summation over distinct antisymmetric sets of indices kl.

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:
- testing of the land downwind for radioactive material contamination, and this is how it was found.
- Medical tests indicated no harm, but the tests did not include bone marrow.
- Traces of 210Po were found on the hands of Professor Dror Sadeh, a physicist who researched radioactive materials.
- An estimate of 8.8 terabecquerels (240 Ci) of polonium-210 has been made.

testing of the land downwind for radioactive material contamination, and this is how it was found.

Medical tests indicated no harm, but the tests did not include bone marrow.

Traces of 210Po were found on the hands of Professor Dror Sadeh, a physicist who researched radioactive materials.

An estimate of 8.8 terabecquerels (240 Ci) of polonium-210 has been made.

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:
- Just as a two-dimensional space would require a third spatial dimension to bend or fold its plane in order to bring two distant points together, a three-dimensional space must be embedded within a fourth spatial dimension to allow similar manipulation.[70] This analogy underlies many depictions of warp portals, which function by bending or folding three-dimensional space through the fourth spatial axis, allowing distant regions to become adjacent.
- In fiction, wormholes have also served as a method for time travel.
- The existence of such a mechanism would imply that the universe possesses, or is embedded within, a four-dimensional spatial framework, even if that dimension is not directly observable.
- Rosen, "The Particle Problem in the General Theory of Relativity" In fiction Wormholes are a common element in science fiction because they allow interstellar, intergalactic, and sometimes even interuniversal travel within human lifetime scales.

Just as a two-dimensional space would require a third spatial dimension to bend or fold its plane in order to bring two distant points together, a three-dimensional space must be embedded within a fourth spatial dimension to allow similar manipulation.[70] This analogy underlies many depictions of warp portals, which function by bending or folding three-dimensional space through the fourth spatial axis, allowing distant regions to become adjacent.

In fiction, wormholes have also served as a method for time travel.

The existence of such a mechanism would imply that the universe possesses, or is embedded within, a four-dimensional spatial framework, even if that dimension is not directly observable.

Rosen, "The Particle Problem in the General Theory of Relativity" In fiction Wormholes are a common element in science fiction because they allow interstellar, intergalactic, and sometimes even interuniversal travel within human lifetime scales.

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:
- Though this gate set is infinite, it can be replaced with a finite gate set by appealing to the Solovay-Kitaev theorem.
- Adiabatic quantum computing An adiabatic quantum computer, based on quantum annealing, decomposes computation into a slow continuous transformation of an initial Hamiltonian into a final Hamiltonian, whose ground states contain the solution.[40] Neuromorphic quantum computing Neuromorphic quantum computing (abbreviated 'n.quantum computing') is an unconventional process of computing that uses neuromorphic computing to perform quantum operations.
- Implementation of Boolean functions using the few-qubit quantum gates is presented here.[39] Measurement-based quantum computing A measurement-based quantum computer decomposes computation into a sequence of Bell state measurements and single-qubit quantum gates applied to a highly entangled initial state (a cluster state), using a technique called quantum gate teleportation.
- It was suggested that quantum algorithms, which are algorithms that run on a realistic model of quantum computation, can be computed equally efficiently with neuromorphic quantum computing.

Though this gate set is infinite, it can be replaced with a finite gate set by appealing to the Solovay-Kitaev theorem.

Adiabatic quantum computing An adiabatic quantum computer, based on quantum annealing, decomposes computation into a slow continuous transformation of an initial Hamiltonian into a final Hamiltonian, whose ground states contain the solution.[40] Neuromorphic quantum computing Neuromorphic quantum computing (abbreviated 'n.quantum computing') is an unconventional process of computing that uses neuromorphic computing to perform quantum operations.

Implementation of Boolean functions using the few-qubit quantum gates is presented here.[39] Measurement-based quantum computing A measurement-based quantum computer decomposes computation into a sequence of Bell state measurements and single-qubit quantum gates applied to a highly entangled initial state (a cluster state), using a technique called quantum gate teleportation.

It was suggested that quantum algorithms, which are algorithms that run on a realistic model of quantum computation, can be computed equally efficiently with neuromorphic quantum computing.

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:
- Hesaidthepiecewasimpervioustodiamondsawsandtorches,andasmallbody,badly burned,wasrecovered.
- ● https://www.documentcloud.org/documents/20797978-twining-memo(p2) (PUBLICDOMAIN) -1952—CIAProjectMoonstruckallegedlystudieslongrangeuseof frequenciesforthepurposeofstimulationofthebrainsof targets.
- whichwouldbecapableofanapproximaterangeof7,000milesatsubsonicspeeds.” Twiningisspecificallytalkingaboutthe“flyingdisc”phenomenaseenbyciviliansandmilitary personnelthatyear.
- ● RetrievalsoftheThirdKindpresentedatMUFON1978,LeonStringfield,p181

● https://www.documentcloud.org/documents/20797978-twining-memo(p2) (PUBLICDOMAIN) -1952—CIAProjectMoonstruckallegedlystudieslongrangeuseof frequenciesforthepurposeofstimulationofthebrainsof targets.

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 quantity m proves to be the mass of the Higgs boH son, equation (42.153).
- The minimum of the potential V defines the vacuum expectation value ⟨H⟩ of the magnitude of the Higgs field, √ m2 ⟨H⟩= H .
- The Lagrangian L of the 4-component Higgs field H is H L =−1(DmH)·(D H)−V(H ·H) , (42.138) H 2 m whereD H isthegauge-covariantderivative(42.137)oftheHiggsfield,andV(H·H)isapotentialenergy, m afunctionofthescalarproductH2 ≡H·H oftheHiggsfieldH anditsreverseH.ThepotentialenergyV is postulated to have a minimum at a non-zero value of H2, which serves to make it energetically favourable for the Higgs field to acquire a non-zero expectation value.
- A commonly adopted potential, with the virtue of yielding a renormalizable quantum field theory, is a “Mexican hat” quartic, illustrated in Figure 42.3, V(H2)=ρ − 1m2 H2+ 1λH4 .

The quantity m proves to be the mass of the Higgs boH son, equation (42.153).

The minimum of the potential V defines the vacuum expectation value ⟨H⟩ of the magnitude of the Higgs field, √ m2 ⟨H⟩= H .

The Lagrangian L of the 4-component Higgs field H is H L =−1(DmH)·(D H)−V(H ·H) , (42.138) H 2 m whereD H isthegauge-covariantderivative(42.137)oftheHiggsfield,andV(H·H)isapotentialenergy, m afunctionofthescalarproductH2 ≡H·H oftheHiggsfieldH anditsreverseH.ThepotentialenergyV is postulated to have a minimum at a non-zero value of H2, which serves to make it energetically favourable for the Higgs field to acquire a non-zero expectation value.

A commonly adopted potential, with the virtue of yielding a renormalizable quantum field theory, is a “Mexican hat” quartic, illustrated in Figure 42.3, V(H2)=ρ − 1m2 H2+ 1λH4 .

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

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

EVIDENCE:
- Introductory Chemistry: A Foundation (http://college.hmco.com/chemi stry/intro/zumdahl/intro_chemistry/5e/students/protected/periodictables/pt/pt/pt_ar5.html) (5th ed.).
- Introductory Nuclear Physics (https://archive.org/details/introductorynucl00kran).
- "Theoretical calculation of Absolute Radii of Atoms and Ions.
- doi:10.1016/j.nuclphysa.2003.11.003 (https://doi.org/10.1016%2Fj.nuclphysa.2003.11.003).

Introductory Chemistry: A Foundation (http://college.hmco.com/chemi stry/intro/zumdahl/intro_chemistry/5e/students/protected/periodictables/pt/pt/pt_ar5.html) (5th ed.).

Introductory Nuclear Physics (https://archive.org/details/introductorynucl00kran).

"Theoretical calculation of Absolute Radii of Atoms and Ions.

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

EVIDENCE:
- We also calculate the acute angle to obtain the bases that maximize Eq.
- 2.[13] This is important in entanglement-based max quantum key distribution, where the secret key rate depends on the degree of measurement correlations.[14] Let us introduce a 3×3 real matrix with elements , where are the Pauli matrices.
- Then, the expected correlation in bases a, b is The numerical values of the basis vectors, when found, can be directly translated to the configuration of the projective measurements.[16] The optimal set of bases for the state is found by taking the two greatest eigenvalues and the corresponding eigenvectors of , and finding the auxiliary unit vectors where is a free parameter.
- There is a necessary and sufficient condition for a two-qubit density matrix to violate the CHSH inequality, expressed by the maximum attainable polynomial S defined in Eq.

We also calculate the acute angle to obtain the bases that maximize Eq.

2.[13] This is important in entanglement-based max quantum key distribution, where the secret key rate depends on the degree of measurement correlations.[14] Let us introduce a 3×3 real matrix with elements , where are the Pauli matrices.

Then, the expected correlation in bases a, b is The numerical values of the basis vectors, when found, can be directly translated to the configuration of the projective measurements.[16] The optimal set of bases for the state is found by taking the two greatest eigenvalues and the corresponding eigenvectors of , and finding the auxiliary unit vectors where is a free parameter.

There is a necessary and sufficient condition for a two-qubit density matrix to violate the CHSH inequality, expressed by the maximum attainable polynomial S defined in Eq.

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

EVIDENCE:
- man who brutally murdered his wife and three children by stabbing them repeatedly probably “had serious problems controlling his temper.” A better test of whether criminal profiling works is to pit professional profilers against untrained individuals.
- After all, profiling is supposedly a technique that requires specialized training.
- When one puts criminal profiling to this more stringent test, the results are decidedly unimpressive.
- In most studies, professional profilers barely do any better than | 214 Chapter 10 Disorder in the Court

man who brutally murdered his wife and three children by stabbing them repeatedly probably “had serious problems controlling his temper.” A better test of whether criminal profiling works is to pit professional profilers against untrained individuals.

After all, profiling is supposedly a technique that requires specialized training.

When one puts criminal profiling to this more stringent test, the results are decidedly unimpressive.

In most studies, professional profilers barely do any better than | 214 Chapter 10 Disorder in the Court

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

EVIDENCE:
- Regents of the University of California.
- S2CID 122952224 (https://ap i.semanticscholar.org/CorpusID:122952224).
- PMID 10040547 (https://pubmed.ncbi.nlm.nih.gov/10040547).
- "Background-free observation of cold antihydrogen with field ionization analysis of its states" (https://cds.cern.ch/record/977774).

Regents of the University of California.

S2CID 122952224 (https://ap i.semanticscholar.org/CorpusID:122952224).

"Background-free observation of cold antihydrogen with field ionization analysis of its states" (https://cds.cern.ch/record/977774).

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List tests or derivations implied by the evidence.


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

EVIDENCE:
- ● https://www.reddit.com/r/UFOs/comments/zpu9zq/george_knapp_on_ufo_legacy_programs_and whistler/ (PUBLICDOMAIN) -20December2022—Ret.CIAofficialJohnRamirezstatesseveralnewpieces ofinformationinaninterviewincluding:thathewastoldofacompartmentedareaoftheNavyas sensitiveastheManhattanprojectthatmaybeUAPrelated;heheardrumorsofarelationshipbetween ONRandCIAOSWR/WINPACwhereUSOmaterialswererecoveredandtransferredtoCIA;andthat NGAhasthemostUAPevidenceintheformofvideos.
- (PUBLICDOMAIN) -19December2022—JournalistGeorgeKnappstatesheknowsofatleastfour “legacyprograms”thatdealtwithUAPissuesthathebelieveswhistleblowersarespeakingtowhenthey willrefertheirtestimonytoAARO.
- ● https://www.theblackvault.com/documentarchive/highly-classified-nro-system-captures-possible-t ic-tac-object-in-2021/ (PUBLICDOMAIN) -28February2023—JournalistGeorgeKnappsaysAlfredO’Donnell,aformer seniormanageratEG&G,adefensecontractorthatmanagedArea51,toldhim“they”hadaflyingsaucer recoveredinNewMexico.
- Healludestotheideathatifthere isanAmericanantigravityprogramdoneinsecret,hespeculatesitwouldbecenteredinAustin,TXand EastSetauket,NY,thelatterofwhichiswhereRenaissanceislocated.

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

EVIDENCE:
- Guica, Monica; Hartman, Thomas; Song, Wei; Strominger, Andrew (2009).
- arXiv:hep-th/9802150 (https://arxiv.org/abs/hep-th/980215 0).
- "Black hole entropy in M-theory".
- arXiv:0809.4266 (https://arxiv.org/abs/080 9.4266).

Guica, Monica; Hartman, Thomas; Song, Wei; Strominger, Andrew (2009).

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

EVIDENCE:
- As a special treat for those readers whose appetite for unusual psychological findings is still not whetted or who prefer their helpings as baker’s dozens, we close with three “Honorable Mentions”: (11) Although our memories can be quite fallible in some circumstances (see Myths #11–13), they can be astonishingly accurate in others.
- In one study, judges matched the faces of dog owners to their dogs at significantly better than chance levels, although this was true only of purebred, not mixed, breeds (Roy & Christenfeld, 2004).
- One research team showed undergraduate subjects 2,560 photographs of various scenes and objects for only a few seconds each.
- In a recent investigation, two researchers asked subjects to hold either a c up of warm coffee or a cup of iced coffee for a few seconds as a favor for someone, and later asked them to rate a fictitious person on a series of attributes.

As a special treat for those readers whose appetite for unusual psychological findings is still not whetted or who prefer their helpings as baker’s dozens, we close with three “Honorable Mentions”: (11) Although our memories can be quite fallible in some circumstances (see Myths #11–13), they can be astonishingly accurate in others.

In one study, judges matched the faces of dog owners to their dogs at significantly better than chance levels, although this was true only of purebred, not mixed, breeds (Roy & Christenfeld, 2004).

One research team showed undergraduate subjects 2,560 photographs of various scenes and objects for only a few seconds each.

In a recent investigation, two researchers asked subjects to hold either a c up of warm coffee or a cup of iced coffee for a few seconds as a favor for someone, and later asked them to rate a fictitious person on a series of attributes.

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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:
- History The theory was first proposed by Élie Cartan, who was inspired by Cosserat elasticity theory,[3][4] in 1922[5] and expounded in the following few years.[6][7][8] Albert Einstein became affiliated with the theory in 1928 during his unsuccessful attempt to match torsion to the electromagnetic field tensor as part of a unified field theory.
- This line of thought led him to the related but different theory of teleparallelism.[9]
- In particular, regarding the metric and torsion tensors as independent variables gives the correct generalization of the conservation law for the total (orbital plus intrinsic) angular momentum to the presence of the gravitational field.
- That is: "torsion does not propagate".

History The theory was first proposed by Élie Cartan, who was inspired by Cosserat elasticity theory,[3][4] in 1922[5] and expounded in the following few years.[6][7][8] Albert Einstein became affiliated with the theory in 1928 during his unsuccessful attempt to match torsion to the electromagnetic field tensor as part of a unified field theory.

This line of thought led him to the related but different theory of teleparallelism.[9]

In particular, regarding the metric and torsion tensors as independent variables gives the correct generalization of the conservation law for the total (orbital plus intrinsic) angular momentum to the presence of the gravitational field.

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

EVIDENCE:
- "The Uncertainty Principle and Free Will".
- The Logic of Scientific Discovery.
- S2CID 186242764 (https://api.semanticscholar.org/C orpusID:186242764).
- Bibcode:1923Natur.112..540D (https://ui.adsabs.harvard.edu/abs/1923N atur.112..540D).

"The Uncertainty Principle and Free Will".

S2CID 186242764 (https://api.semanticscholar.org/C orpusID:186242764).

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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:
- So, an infrared lamp can warm a large surface, perhaps large enough to keep people comfortable in a cold room, but it cannot give anyone a sunburn.[54] Technological applications Applications of quantum mechanics include the laser, the transistor, the electron microscope, and magnetic resonance imaging.
- "Quantum Mechanics" (https://www.pbs.org/transistor/science/info/quantum.html).
- Such properties of elementary particles are required to take on one of a set of small, discrete allowable values, and since the gap between these values is also small, the discontinuities are only apparent at very tiny (atomic) scales.
- Flash memory chips found in USB drives also use quantum tunneling, to erase their memory cells.[55] See also Einstein's thought experiments Physics portal Macroscopic quantum phenomena Philosophy of physics Quantum computing Virtual particle Teaching quantum mechanics List of textbooks on classical and quantum mechanics References 1.

Introduction to quantum mechanics - Wikipedia.pdf

So, an infrared lamp can warm a large surface, perhaps large enough to keep people comfortable in a cold room, but it cannot give anyone a sunburn.[54] Technological applications Applications of quantum mechanics include the laser, the transistor, the electron microscope, and magnetic resonance imaging.

Such properties of elementary particles are required to take on one of a set of small, discrete allowable values, and since the gap between these values is also small, the discontinuities are only apparent at very tiny (atomic) scales.

Flash memory chips found in USB drives also use quantum tunneling, to erase their memory cells.[55] See also Einstein's thought experiments Physics portal Macroscopic quantum phenomena Philosophy of physics Quantum computing Virtual particle Teaching quantum mechanics List of textbooks on classical and quantum mechanics References 1.

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

EVIDENCE:
- Indeed, all we can do is speculate, for we have no way of telling whether any of the laws of nature that have so painstakingly been worked out by observing the universe around us can hold under the extreme conditions of a black hole.
- It is very difficult to satisfy that curiosity.
- One thing that might happen is that the worst
- ENDINGS AND BEGINNINGS THE END?

Indeed, all we can do is speculate, for we have no way of telling whether any of the laws of nature that have so painstakingly been worked out by observing the universe around us can hold under the extreme conditions of a black hole.

It is very difficult to satisfy that curiosity.

One thing that might happen is that the worst

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List tests or derivations implied by the evidence.


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

EVIDENCE:
- For the squeezed electromagnetic vacuum state, the energy density pE-sqvac is given by (Reference 46): PF."q,~ ~ (2~,ro }inhl;[s inh I; +cosh ~cos (200(1 - z I c) +15)] (J / mO') (5) where L3 is the volume of a large box with sides of length L (i.e., the quantum field is placed in a box with periodic boundary conditions), ~ is the squeezed state amplitude (giving a measure of the mean photon number in a squeezed state), and D is the phase of squeeZing.
- Negative energy can be generated by an array of ultrahigh-intensity lasers using an ultra-fast rotating mirror system (Reference 47).
- It turns out that this is always true for every nonzero value of SI so pE-sqvac becomes negative at some point in the cycle for a general squeezed vacuum state.
- Equation (5) shows that pE-sqvac falls below zero once every cycle when the condition cosh S > sinh ~ is met.

For the squeezed electromagnetic vacuum state, the energy density pE-sqvac is given by (Reference 46): PF."q,~ ~ (2~,ro }inhl;[s inh I; +cosh ~cos (200(1 - z I c) +15)] (J / mO') (5) where L3 is the volume of a large box with sides of length L (i.e., the quantum field is placed in a box with periodic boundary conditions), ~ is the squeezed state amplitude (giving a measure of the mean photon number in a squeezed state), and D is the phase of squeeZing.

Negative energy can be generated by an array of ultrahigh-intensity lasers using an ultra-fast rotating mirror system (Reference 47).

It turns out that this is always true for every nonzero value of SI so pE-sqvac becomes negative at some point in the cycle for a general squeezed vacuum state.

Equation (5) shows that pE-sqvac falls below zero once every cycle when the condition cosh S > sinh ~ is met.

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

EVIDENCE:
- A biography by Galileo's pupil Vincenzo Viviani stated that Galileo had dropped balls of
- In 1609, Johannes Kepler published his three laws of planetary motion, explaining how the planets orbit the Sun.
- The Roman pound and ounce were both defined in terms of different sized collections of the same common mass standard, the carob seed.
- The ratio of a Roman ounce (144 carob seeds) to a Roman pound (1728 carob seeds) was: Planetary motion In 1600 AD, Johannes Kepler sought employment with Tycho Brahe, who had some of the most precise astronomical data available.

A biography by Galileo's pupil Vincenzo Viviani stated that Galileo had dropped balls of

In 1609, Johannes Kepler published his three laws of planetary motion, explaining how the planets orbit the Sun.

The Roman pound and ounce were both defined in terms of different sized collections of the same common mass standard, the carob seed.

The ratio of a Roman ounce (144 carob seeds) to a Roman pound (1728 carob seeds) was: Planetary motion In 1600 AD, Johannes Kepler sought employment with Tycho Brahe, who had some of the most precise astronomical data available.

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List tests or derivations implied by the evidence.


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

EVIDENCE:
- argue that if the vacuum is GUT scale, Λ then it might be expected that ρ ∼ m4 and ρ“∼ m3 in Planck units, in which case ρ“2/ρ ∼ m2 , Λ GUT GUT Λ GUT which is small compared to 1 if the GUT scale is significantly smaller than the Planck scale, m ≪1.
- The energies per unit mass β± and β ̄ are respectively zero at radii r± and r given by 1 1 1 ( M )1/3 ( M )1/3 r± = , r = .
- introduce a dimensionless variable z ≡ r/r+, in terms of which the energy per unit mass β+, 1 1 equation (20.113), is ( 1−z3) β+ = √ , (20.117) 1 −E z2 and the velocity β , equation (20.114), satisfies −Eβ2+V =E , (20.118) where V(z) is a dimensionless effective potential and the constant E is an effective dimensionless energy, ( 1−z3)2 μ V ≡− − , (20.119a) z2 z ( μ2 )2/3 E ≡− , (20.119b) 16πρ“M with the constant μ given by 24πρ“2 μ≡ .
- The radius r− is Λ 1 1 1 positive or negative as ρ is larger or smaller than 6πρ“2.

argue that if the vacuum is GUT scale, Λ then it might be expected that ρ ∼ m4 and ρ“∼ m3 in Planck units, in which case ρ“2/ρ ∼ m2 , Λ GUT GUT Λ GUT which is small compared to 1 if the GUT scale is significantly smaller than the Planck scale, m ≪1.

The energies per unit mass β± and β ̄ are respectively zero at radii r± and r given by 1 1 1 ( M )1/3 ( M )1/3 r± = , r = .

introduce a dimensionless variable z ≡ r/r+, in terms of which the energy per unit mass β+, 1 1 equation (20.113), is ( 1−z3) β+ = √ , (20.117) 1 −E z2 and the velocity β , equation (20.114), satisfies −Eβ2+V =E , (20.118) where V(z) is a dimensionless effective potential and the constant E is an effective dimensionless energy, ( 1−z3)2 μ V ≡− − , (20.119a) z2 z ( μ2 )2/3 E ≡− , (20.119b) 16πρ“M with the constant μ given by 24πρ“2 μ≡ .

The radius r− is Λ 1 1 1 positive or negative as ρ is larger or smaller than 6πρ“2.

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

EVIDENCE:
- ab To determine the energy-momentum in gravitational waves, calculate the Einstein tensor G to second mn order, imposing the vacuum conditions that the unperturbed and linear parts of the Einstein tensor vanish 0 1 G =G =0 .
- Answer: G h (t,x)=− I– ̈ (t−x) .
- (27.123) ab c4x ab tensor 27.14 Energy-momentum carried by gravitational waves The gravitational wave equation (27.27) in empty space appears to describe gravitational waves propagating in a region where the energy-momentum tensor T is zero.
- (27.121) ab x ab tensor Equation (27.121) is valid for long wavelength modes observed at distances x far from the source of gravitational radiation.

ab To determine the energy-momentum in gravitational waves, calculate the Einstein tensor G to second mn order, imposing the vacuum conditions that the unperturbed and linear parts of the Einstein tensor vanish 0 1 G =G =0 .

(27.123) ab c4x ab tensor 27.14 Energy-momentum carried by gravitational waves The gravitational wave equation (27.27) in empty space appears to describe gravitational waves propagating in a region where the energy-momentum tensor T is zero.

(27.121) ab x ab tensor Equation (27.121) is valid for long wavelength modes observed at distances x far from the source of gravitational radiation.

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

EVIDENCE:
- 2 Explain in your own words the importance of the (d) None of those things.
- 6 Who kills Daniel, O’Neil and all the soldiers?
- Think of the beginning, the under the ground?
- Work in pairs or small groups.

2 Explain in your own words the importance of the (d) None of those things.

6 Who kills Daniel, O’Neil and all the soldiers?

Think of the beginning, the under the ground?

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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:
- In most (Type VI with k ̸= 0 is an exception) of the solutions for which R^ is diagonal in the frame where the metric is diagonal, including the BKL ab solutions, the symmetric part n(cd) of the structure coefficients is diagonal in the same frame.
- In this case, the components of the restricted Ricci tensor R^ (17.104) are, in terms of the scale factors a and the ab a parameters n and k of the structure coefficients, equation (17.92), c e ( n n −2k2 2k2 2k2 n2a2 n2a2 n2a2 ) R^ =a2 2 3 1 − 2 − 3 + 1 1 − 2 2 − 3 3 , (17.129a) 11 1 a2 a2 a2 2a2a2 2a2a2 2a2a2 1 2 3 2 3 3 1 1 2 ( n a2−n a3) R^ =k 2 2 3 3 , (17.129b) 23 1 a2 andsimilarlywithpermutedindicesfortheothercomponents.Theoff-diagonalcomponentsR^ andcompany must vanish for the restricted Ricci tensor to be diagonal.
- Equation (17.129b) shows that one possibility, √ √ which covers the majority of cases (Type VII with k ≡ k ̸= 0 and n a = n a is an exception), is 1 2 2 3 3

In most (Type VI with k ̸= 0 is an exception) of the solutions for which R^ is diagonal in the frame where the metric is diagonal, including the BKL ab solutions, the symmetric part n(cd) of the structure coefficients is diagonal in the same frame.

In this case, the components of the restricted Ricci tensor R^ (17.104) are, in terms of the scale factors a and the ab a parameters n and k of the structure coefficients, equation (17.92), c e ( n n −2k2 2k2 2k2 n2a2 n2a2 n2a2 ) R^ =a2 2 3 1 − 2 − 3 + 1 1 − 2 2 − 3 3 , (17.129a) 11 1 a2 a2 a2 2a2a2 2a2a2 2a2a2 1 2 3 2 3 3 1 1 2 ( n a2−n a3) R^ =k 2 2 3 3 , (17.129b) 23 1 a2 andsimilarlywithpermutedindicesfortheothercomponents.Theoff-diagonalcomponentsR^ andcompany must vanish for the restricted Ricci tensor to be diagonal.

Equation (17.129b) shows that one possibility, √ √ which covers the majority of cases (Type VII with k ≡ k ̸= 0 and n a = n a is an exception), is 1 2 2 3 3

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

EVIDENCE:
- Fuller, Wheeler, Friedman, Schleich, Witt, Galloway, Wooglar Science fiction wormholes are forbidden • If the wormhole allows you to go faster than light in the ambient spacetime.
- Then the light ray going through the wormhole is a “fastest” (achronal) light ray.
- • Total focusing is equal to the integral of the null energy.
- They focus as they go into one mouth They defocus as they go out.

Fuller, Wheeler, Friedman, Schleich, Witt, Galloway, Wooglar Science fiction wormholes are forbidden • If the wormhole allows you to go faster than light in the ambient spacetime.

Then the light ray going through the wormhole is a “fastest” (achronal) light ray.

• Total focusing is equal to the integral of the null energy.

They focus as they go into one mouth They defocus as they go out.

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List tests or derivations implied by the evidence.


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

EVIDENCE:
- The limits of family influence: Genes, environment, and behavior.
- Psychological Bulletin, 97, 286–306.
- Skeptical Inquirer, 24(2), 22–26.
- Journal of Child Psychiatry and Psychology, 17, 35–56.

The limits of family influence: Genes, environment, and behavior.

Journal of Child Psychiatry and Psychology, 17, 35–56.

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

EVIDENCE:
- The term S on the right hand side of equation (34.1) embodies
- The terms on the right hand side are sourceterms.ThetermI ontherighthandsideoftheradiativetransferequation(34.1)isamonopoleterm, the Integrated Sachs-Wolfe (ISW) term, I(η,k)≡Ψ ̇(η,k)+Φ ̇(η,k) , (34.2) so-called because, as will be seen in equation (34.17), it contributes a temperature fluctuation that is an integral along the line of sight to the CMB.
- 34.1 Radiative transfer of CMB photons To determine the harmonics Θ (η ,k) of the CMB photon distribution today, return to the Boltzmann l 0 equation (33.80) for the temperature fluctuation Θ(η,k,μ), where μ ≡ k^ · p^ is the cosine of the angle γ between the wavevector k and the photon direction p .
- What gives confidence in the Standard Model is the fact that a broad range of other astronomical observations are consistent with it, including the Hubble diagram of Type I supernovae, the clustering of matter and of galaxies, Big Bang nucleosynthesis, and the age of the oldest stars.

The term S on the right hand side of equation (34.1) embodies

The terms on the right hand side are sourceterms.ThetermI ontherighthandsideoftheradiativetransferequation(34.1)isamonopoleterm, the Integrated Sachs-Wolfe (ISW) term, I(η,k)≡Ψ ̇(η,k)+Φ ̇(η,k) , (34.2) so-called because, as will be seen in equation (34.17), it contributes a temperature fluctuation that is an integral along the line of sight to the CMB.

34.1 Radiative transfer of CMB photons To determine the harmonics Θ (η ,k) of the CMB photon distribution today, return to the Boltzmann l 0 equation (33.80) for the temperature fluctuation Θ(η,k,μ), where μ ≡ k^ · p^ is the cosine of the angle γ between the wavevector k and the photon direction p .

What gives confidence in the Standard Model is the fact that a broad range of other astronomical observations are consistent with it, including the Hubble diagram of Type I supernovae, the clustering of matter and of galaxies, Big Bang nucleosynthesis, and the age of the oldest stars.

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:
- Forcedtoatensorfield Nowthatwehavedealtwithelectromagnetism,weturntogravity,orratheroptionGin chapterIV.1.Recallthatyouobtained (cid:7) (cid:6) OptionGimproved: S=−m −g (x)dxμdxν (1) μν withg =−(1+ 2V),g =g =0,andg =δ asaspecialcase.
- 00 m 0i i0 ij ij Remarkably,ifyouputtheNewtonianpotentialtermVdt outsidethesquareroot,you are led to a vector field A , but if you put it inside, you are forced to a tensor field: two μ indices are needed to match dxμdxν.
- In a sense, we have to thank Pythagoras for this tensorfield.
- Timeandgravity Forthemoment,let’streatthespecialcase " (cid:7) (cid:3) (cid:4) 2V S=−m 1+ dt2−dx(cid:3)2 (2) m ThefractionV/mlooksalittlestrange,butthenyousuddenlyrealizethatiftheparticle, ofmassm,islivinginagravitationalpotentialV =−GMm/r,mwouldcancelout,sothat " (cid:7) (cid:3) (cid:4) 2GM S=−m 1− dt2−dx(cid:3)2 (3) r Thatthemassmcancelsoutdependsofcourseontheprofoundobservationalfactthatthe inertialmassandthegravitationalmassareequal, aswehavealludedtoalreadyseveral times,inchaptersI.1,II.3,andsoforth.

Forcedtoatensorfield Nowthatwehavedealtwithelectromagnetism,weturntogravity,orratheroptionGin chapterIV.1.Recallthatyouobtained (cid:7) (cid:6) OptionGimproved: S=−m −g (x)dxμdxν (1) μν withg =−(1+ 2V),g =g =0,andg =δ asaspecialcase.

00 m 0i i0 ij ij Remarkably,ifyouputtheNewtonianpotentialtermVdt outsidethesquareroot,you are led to a vector field A , but if you put it inside, you are forced to a tensor field: two μ indices are needed to match dxμdxν.

In a sense, we have to thank Pythagoras for this tensorfield.

Timeandgravity Forthemoment,let’streatthespecialcase " (cid:7) (cid:3) (cid:4) 2V S=−m 1+ dt2−dx(cid:3)2 (2) m ThefractionV/mlooksalittlestrange,butthenyousuddenlyrealizethatiftheparticle, ofmassm,islivinginagravitationalpotentialV =−GMm/r,mwouldcancelout,sothat " (cid:7) (cid:3) (cid:4) 2GM S=−m 1− dt2−dx(cid:3)2 (3) r Thatthemassmcancelsoutdependsofcourseontheprofoundobservationalfactthatthe inertialmassandthegravitationalmassareequal, aswehavealludedtoalreadyseveral times,inchaptersI.1,II.3,andsoforth.

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:
- (17.199) z a azb yaz z a 17.11 2+2 split For the particular case of a 2+2 split, equations (17.198) for the Ricci tensor R become km R =R^ −D^ K +(D^ +K )Ka −Kb Ka , (17.200a) vu vu v u a a uv va ub R =−D^ K +(D^ +K )Ka −Kb Ka , (17.200b) vv v v a a vv va vb R =R^ +D^ K −D^ K +K Ky −Ky Kb (17.200c) v+ v++− v v+u u v+v y +v +b vy =R =−R^ −D^ K +D^ K +K Kb −Ky Kb (17.200d) +v +vvu + +v− − +v+ b v+ +b vy =D^ K −D^ K −D^ K +D^ K −2Ky Kb +Ky Kb +Ky Kb , (17.200e) v v+u u v+v + +v− − +v+ +b vy +b yv b+ vy R =(D^ +K )Kz −D^ K −Kz Ky , (17.200f) ++ z z ++ + + +y
- (17.197g) abcd abcd cb zda ca zdb Ifthetetradconnectionsarereplacedbytheirtorsion-freeexpressionsintermsofderivativesofthevierbein, then the various alternative expressions for the Riemann tensor become identities.
- (17.198e) ab ab z z ba a b ay bz Contracting the Ricci tensor yields the Ricci scalar R, R=R^−2D^ Kz−2D^ Ka−KbzaK −KzayK −KzK −KaK .
- 17.11 2+2 split 531 In the M+N split, the expression (17.34) for the Riemann curvature tensor becomes R =R^ +Ka K −Ka K , (17.197a) wxyz wxyz yx azw yw azx R =D^ K −D^ K +(Kc −Kc )K (17.197b) xyaz x azy y azx xy yx azc =R =R^ +Kc K −Kc K , (17.197c) azxy azxy xz cya xa cyz R =D^ K −D^ K +Kx K −Kc K , (17.197d) byaz b azy y azb by azx yb azc R =D^ K −D^ K +(Kx −Kx)K (17.197e) bcaz b azc c azb bc cb azx =R =R^ +KxK −Kx K , (17.197f) azbc azbc bz xca ba xcz R =R^ +Kz K −Kz K .

(17.199) z a azb yaz z a 17.11 2+2 split For the particular case of a 2+2 split, equations (17.198) for the Ricci tensor R become km R =R^ −D^ K +(D^ +K )Ka −Kb Ka , (17.200a) vu vu v u a a uv va ub R =−D^ K +(D^ +K )Ka −Kb Ka , (17.200b) vv v v a a vv va vb R =R^ +D^ K −D^ K +K Ky −Ky Kb (17.200c) v+ v++− v v+u u v+v y +v +b vy =R =−R^ −D^ K +D^ K +K Kb −Ky Kb (17.200d) +v +vvu + +v− − +v+ b v+ +b vy =D^ K −D^ K −D^ K +D^ K −2Ky Kb +Ky Kb +Ky Kb , (17.200e) v v+u u v+v + +v− − +v+ +b vy +b yv b+ vy R =(D^ +K )Kz −D^ K −Kz Ky , (17.200f) ++ z z ++ + + +y

(17.197g) abcd abcd cb zda ca zdb Ifthetetradconnectionsarereplacedbytheirtorsion-freeexpressionsintermsofderivativesofthevierbein, then the various alternative expressions for the Riemann tensor become identities.

(17.198e) ab ab z z ba a b ay bz Contracting the Ricci tensor yields the Ricci scalar R, R=R^−2D^ Kz−2D^ Ka−KbzaK −KzayK −KzK −KaK .

17.11 2+2 split 531 In the M+N split, the expression (17.34) for the Riemann curvature tensor becomes R =R^ +Ka K −Ka K , (17.197a) wxyz wxyz yx azw yw azx R =D^ K −D^ K +(Kc −Kc )K (17.197b) xyaz x azy y azx xy yx azc =R =R^ +Kc K −Kc K , (17.197c) azxy azxy xz cya xa cyz R =D^ K −D^ K +Kx K −Kc K , (17.197d) byaz b azy y azb by azx yb azc R =D^ K −D^ K +(Kx −Kx)K (17.197e) bcaz b azc c azb bc cb azx =R =R^ +KxK −Kx K , (17.197f) azbc azbc bz xca ba xcz R =R^ +Kz K −Kz K .

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:
- "Newton, object 1 (Butlin 306) "Newton"" (https://web.archive.org/web/20130927214741/http://ww w.blakearchive.org/exist/blake/archive/copyinfo.xq?copyid=but306.1).
- John (http://www.gutenber g.org/ebooks/16878) Archived (https://web.archive.org/web/20170120113904/http://www.gutenber g.org/ebooks/16878) 20 January 2017 at the Wayback Machine 1733 238.
- "Isaac Newton, heretic: the strategies of a Nicodemite".
- Archived (https://web.a rchive.org/web/20210414055022/https://books.google.com/books?id=Dz2FzJqaJMUC&q=%22gra vity%20may%20put%20the%20planets%20into%20motion%22&pg=PA436) from the original on 14 April 2021.

"Newton, object 1 (Butlin 306) "Newton"" (https://web.archive.org/web/20130927214741/http://ww w.blakearchive.org/exist/blake/archive/copyinfo.xq?copyid=but306.1).

John (http://www.gutenber g.org/ebooks/16878) Archived (https://web.archive.org/web/20170120113904/http://www.gutenber g.org/ebooks/16878) 20 January 2017 at the Wayback Machine 1733 238.

"Isaac Newton, heretic: the strategies of a Nicodemite".

Archived (https://web.a rchive.org/web/20210414055022/https://books.google.com/books?id=Dz2FzJqaJMUC&q=%22gra vity%20may%20put%20the%20planets%20into%20motion%22&pg=PA436) from the original on 14 April 2021.

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:
- Building 18 Arlington, VA 22203 Wright-Patterson AFB, OH 45433-7251 AFRL/PR Technical Library (2 CD + 1 HC) Dr.
- Boise, ID 83703 Edwards AFB CA 93524-7680 AFRL/PR (1 CD) Dr.
- Saturn Blvd 1041 Los Arabis Ln.
- Box 295 Edwards AFB CA 93524-7130 Harvest, AL 35749 Mr.

Building 18 Arlington, VA 22203 Wright-Patterson AFB, OH 45433-7251 AFRL/PR Technical Library (2 CD + 1 HC) Dr.

Boise, ID 83703 Edwards AFB CA 93524-7680 AFRL/PR (1 CD) Dr.

Box 295 Edwards AFB CA 93524-7130 Harvest, AL 35749 Mr.

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 Section 2.3 we will discuss how or whether we can create such a wormhole in the laboratory.
- 2.2 Engineering the Vacuum Engineering the spacetime vacuum provides a second solution that also satisfies the definition of vmTeleportation.
- Lee stated: “The experimental method to alter the properties of the vacuum may be called vacuum engineering...If indeed we are able to alter the vacuum, then we may encounter some new phenomena, totally unexpected.” This new concept is based on the now-accepted fact that the vacuum is characterized by physical parameters and structure that constitutes an energetic medium which pervades the entire extent of the Approved for public release; distribution unlimited.
- the negative energy configuration constituting the wormhole system).

In Section 2.3 we will discuss how or whether we can create such a wormhole in the laboratory.

2.2 Engineering the Vacuum Engineering the spacetime vacuum provides a second solution that also satisfies the definition of vmTeleportation.

Lee stated: “The experimental method to alter the properties of the vacuum may be called vacuum engineering...If indeed we are able to alter the vacuum, then we may encounter some new phenomena, totally unexpected.” This new concept is based on the now-accepted fact that the vacuum is characterized by physical parameters and structure that constitutes an energetic medium which pervades the entire extent of the Approved for public release; distribution unlimited.

the negative energy configuration constituting the wormhole system).

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- 2.As the astronaut’s body reaches the singularity, it gets stretched out to truly infinite length and squashed transversely to truly zero size.
- 3.The laws of quantum gravity must cut off the fluctuations’ growth.
- 5.Hawking suspects that the growing beam of vacuum fluctuations is nature’s way of enforcing chronology protection: Whenever one tries to make a time machine...
- Just as struggling to understand the evaporation of black holes has helped us to discover a partial marriage of quantum mechanics with general relativity...

2.As the astronaut’s body reaches the singularity, it gets stretched out to truly infinite length and squashed transversely to truly zero size.

3.The laws of quantum gravity must cut off the fluctuations’ growth.

5.Hawking suspects that the growing beam of vacuum fluctuations is nature’s way of enforcing chronology protection: Whenever one tries to make a time machine...

Just as struggling to understand the evaporation of black holes has helped us to discover a partial marriage of quantum mechanics with 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:
- of least action requires that the action is a minimum with respect to variations of the coordinates and momenta along the paths of particles, the coordinates and momenta at the endpoints λ and λ of the integration being held fixed.
- Since the coordinates are fixed at the endpoints, δxμ =0, the i f surface term in equation (4.71) vanishes.
- Minimization of the action with respect to arbitrary independent variations of the coordinates and momenta then yields Hamilton’s equations of motion dxμ ∂H dπ ∂H = , μ =− .

of least action requires that the action is a minimum with respect to variations of the coordinates and momenta along the paths of particles, the coordinates and momenta at the endpoints λ and λ of the integration being held fixed.

Since the coordinates are fixed at the endpoints, δxμ =0, the i f surface term in equation (4.71) vanishes.

Minimization of the action with respect to arbitrary independent variations of the coordinates and momenta then yields Hamilton’s equations of motion dxμ ∂H dπ ∂H = , μ =− .

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

List tests or derivations implied by the evidence.