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

EVIDENCE:
- DifferentialFormsandVielbein | 597 We call the object A≡A dxμ a 1-form.
- TheimportantpointhereisthattheformAdoesnotdependonanyspecificcoordinate system:itiscoordinatefree.Butyouarewelcometoexpressitasalinearcombinationof dxμinacoordinatesystemofyourchoice.
- (If you confuse the d here with the d in the precedingparagraph,youareintrouble!) TheformAdoesnotcarryanyindicesandsodoesnottransform.Indeed,weinsertthe transformationofdxμjustmentionedtoobtainA≡A μ dxμ=A μ ∂ ∂ x x (cid:5) μ ν dx (cid:5)ν≡A (cid:5) ν dx (cid:5)ν.The (cid:5) laststepdefinesA andthusreproducesthestandardtransformationlawofvectorswitha ν lowerindexundercoordinatetransformation:A (cid:5) ν =A μ ∂ ∂ x x (cid:5) μ ν .HenceA μ (x)isavectorfield.
- Thereasonforanticommuting Wenowfacethequestionofhowtothinkabouttheproductsofdifferentialsdxμdxν.Inan elementarycourseoncalculus,welearnedthatdxdyrepresentstheareaofaninfinitesimal rectanglewithlengthdxandwidthdy.Atthatlevel,wemoreorlessautomaticallyregard dydx asthesameasdxdy.Theorderofwritingthedifferentialsdoesnotmatter.

DifferentialFormsandVielbein | 597 We call the object A≡A dxμ a 1-form.

(If you confuse the d here with the d in the precedingparagraph,youareintrouble!) TheformAdoesnotcarryanyindicesandsodoesnottransform.Indeed,weinsertthe transformationofdxμjustmentionedtoobtainA≡A μ dxμ=A μ ∂ ∂ x x (cid:5) μ ν dx (cid:5)ν≡A (cid:5) ν dx (cid:5)ν.The (cid:5) laststepdefinesA andthusreproducesthestandardtransformationlawofvectorswitha ν lowerindexundercoordinatetransformation:A (cid:5) ν =A μ ∂ ∂ x x (cid:5) μ ν .HenceA μ (x)isavectorfield.

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:
- WormholePhysics QuantumWormholePhysics Summary Summary • The general theory of relativity had changed how we view space and time Perhaps, wormholes could allow interstellar or even intergalactic travel one day!
- In our current work, we are trying to study the implications of this conjecture such as Bekenstein-Hawking entropy counting assuming multiply-connected spacetime.
- • Outlook • We need a complete theory of quantum gravity to be able to fully understand spacetime.
- • Paradoxes push theoretical physics forward, it makes scientists think and discover solution that advances our understanding of nature.

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.

WormholePhysics QuantumWormholePhysics Summary Summary • The general theory of relativity had changed how we view space and time Perhaps, wormholes could allow interstellar or even intergalactic travel one day!

In our current work, we are trying to study the implications of this conjecture such as Bekenstein-Hawking entropy counting assuming multiply-connected spacetime.

• Outlook • We need a complete theory of quantum gravity to be able to fully understand spacetime.

• Paradoxes push theoretical physics forward, it makes scientists think and discover solution that advances our understanding of nature.

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:
- R = 6 r = 4 d = 1 R = 7 r = 4 d = 1 R = 8 r = 3 d = 2 R = 7 r = 4 d = 2 R = 15 r = 14 d = 1 Parametric space curves Helix Parametric equations are convenient for describing curves in higher-dimensional spaces.
- For example: 0:06 describes a three-dimensional Animated Parametric helix curve, the helix, with a radius of a and rising by 2πb units per turn.
- The equations are identical in the plane to those for a circle.
- Such expressions as the one above are commonly written as where r is a three-dimensional vector.

R = 6 r = 4 d = 1 R = 7 r = 4 d = 1 R = 8 r = 3 d = 2 R = 7 r = 4 d = 2 R = 15 r = 14 d = 1 Parametric space curves Helix Parametric equations are convenient for describing curves in higher-dimensional spaces.

For example: 0:06 describes a three-dimensional Animated Parametric helix curve, the helix, with a radius of a and rising by 2πb units per turn.

The equations are identical in the plane to those for a circle.

Such expressions as the one above are commonly written as where r is a three-dimensional vector.

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:
- Jason9: DANIELA AND CHARLIE ARE WITH ME.
- Jason9: How doesn’t matter, but yes, they’re safe.
- I’ve been giving this a lot of thought.
- I assume we all share the same basic desire, that no matter what, Daniela and Charlie can’t be harmed?

Jason9: DANIELA AND CHARLIE ARE WITH ME.

Jason9: How doesn’t matter, but yes, they’re safe.

I assume we all share the same basic desire, that no matter what, Daniela and Charlie can’t be harmed?

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:
- (21.42) dτ ξ0 T ⃒ b b b rb=r+ Meanwhile the horizon radius R + expands in proportion to the conformal factor, R + ∝evtb, and dt b /dτ b = ∂ t =1/(R ξ0), so the Bekenstein-Hawking entropy S =πR2 increases as 0 b b b BH + dS 2πR2v BH = + .
- b b ThustheentropyS ,equation(21.13),accretedthroughthehorizon,atconformalradiusr ,perunitproper b + time of the fluid is dS b = 4πR b 2ξ b 1(1+w b )ρ b ⃒ ⃒ ⃒ .
- 21.10 Black hole accreting a conducting charged plasma 637 For a spherical black hole of horizon radius R , the area is A = 4πR2.
- The proper velocity of the baryonic fluid through the similarity frame equals ξ1/ξ0, equation (20.145).

(21.42) dτ ξ0 T ⃒ b b b rb=r+ Meanwhile the horizon radius R + expands in proportion to the conformal factor, R + ∝evtb, and dt b /dτ b = ∂ t =1/(R ξ0), so the Bekenstein-Hawking entropy S =πR2 increases as 0 b b b BH + dS 2πR2v BH = + .

b b ThustheentropyS ,equation(21.13),accretedthroughthehorizon,atconformalradiusr ,perunitproper b + time of the fluid is dS b = 4πR b 2ξ b 1(1+w b )ρ b ⃒ ⃒ ⃒ .

21.10 Black hole accreting a conducting charged plasma 637 For a spherical black hole of horizon radius R , the area is A = 4πR2.

The proper velocity of the baryonic fluid through the similarity frame equals ξ1/ξ0, equation (20.145).

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:
- Theargumentisasfollows.Connectthediagramby,say,twophotonlinestothepropagatorofsomeatomic nucleus,say,aluminumoriron.Thisdiagramthuscontributestothegravitationalmassofthenucleus.Onthe otherhand,considerthesamediagramwiththeatomicnucleusbutwiththegravitonremoved,adiagramthat presumablyhasnothingtodowithgravity.Butthisdiagramcontributestotheinertialmassofthenucleus.
- (cid:8) Instead of nailing g down, we might consider a softer constraint by including in the action a term like d4xV(g),wherethefunctionV(g)hasadeepminimumatg=−1(forexample,V(g)=A(g+1)2,withA large).Thiswouldservetoencouragegtostaycloseto−1.Again,wewouldnolongerbeallowedtomakejust anycoordinatetransformationwefeellike.
- However,anorderofmagnitudeestimatewouldseemtosuggestthattheeffectisfartoosmall.
- Appendix5:Decayingcosmologicalconstant Overtheyears,manyphysicistshavehadmany(“crazy”)thoughtsaboutgravity.Onepossibility,onceconsidered highlyspeculative,wastoentertainadecayingcosmologicalconstant19 d(cid:27)(cid:7)=0.Butthesedays,withamultitude dt ofscalarfields20around,rollingdownthishillorthat,thispossibilitywouldbeconsideredcommonplacerather thanoutrageous.

(cid:8) Instead of nailing g down, we might consider a softer constraint by including in the action a term like d4xV(g),wherethefunctionV(g)hasadeepminimumatg=−1(forexample,V(g)=A(g+1)2,withA large).Thiswouldservetoencouragegtostaycloseto−1.Again,wewouldnolongerbeallowedtomakejust anycoordinatetransformationwefeellike.

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:
- • if something is entangled with other objects, then a measurement of it simultaneously provides information about its partners • some quantum systems can have a little entanglement while others will have a lot • the more entanglement available, the better suited a system is to quantum information processing • decoherence degrades the fidelity of the quantum link (i.e., the set of pure EPR entangled pairs) between two quantum systems, thus introducing a certain level of error in the exchange of quantum information between the systems; thus limiting our ability to perform quantum information processing (see more on this issue in the next paragraph below) • mixed entangled states may be measured, distilled, concentrated, diluted, and manipulated • the basic resource of classical information is the bit (i.e., the two values 0 and 1), while quantum information comes in quantum bits (i.e., qubits) that are described by their quantum state; qubits can exist in superpositions that simultaneously involve 0 and 1, thus giving them an infinite range of values; groups of qubits can be entangled; qubits must be insulated against decoherence, so that the coherent state of the quantum system in a quantum computer is preserved for a time that is long enough to set up a calculation, perform it, and read out the results • quantum computers processing qubits or entangled qubits can outperform classical computers; functional requirements of quantum computers: (cid:153) they must have the ability to initialize any qubit in a specified state, and to measure the state of a specific qubit (cid:153) they must have universal quantum gates, which are logical elements capable of arranging any desired logical relationship between the states of qubits (cid:153) they must also have a processor capable of interlinking quantum gates to establish rules and boundary conditions for their inter-relationships – in a quantum computation, the arrangement of quantum gates connects the qubits in a logical pattern, according to a program or algorithm, and after an interval the qubits assigned to the result are read out • quantum error correction codes exist, whereby qubits are passed through a circuit (the quantum analogue of logic gates) that will successfully fix an error in any one of the qubits without actually reading what all the individual qubit states are; no qubit cloning is required • a completely secure quantum key can be generated and distributed (for communication and decoding of encrypted messages) using entangled photons has been demonstrated (Tittel et al., 2000; Jennewein et al., 2000; Naik et al., 2000); any eavesdropper’s attempt

• if something is entangled with other objects, then a measurement of it simultaneously provides information about its partners • some quantum systems can have a little entanglement while others will have a lot • the more entanglement available, the better suited a system is to quantum information processing • decoherence degrades the fidelity of the quantum link (i.e., the set of pure EPR entangled pairs) between two quantum systems, thus introducing a certain level of error in the exchange of quantum information between the systems; thus limiting our ability to perform quantum information processing (see more on this issue in the next paragraph below) • mixed entangled states may be measured, distilled, concentrated, diluted, and manipulated • the basic resource of classical information is the bit (i.e., the two values 0 and 1), while quantum information comes in quantum bits (i.e., qubits) that are described by their quantum state; qubits can exist in superpositions that simultaneously involve 0 and 1, thus giving them an infinite range of values; groups of qubits can be entangled; qubits must be insulated against decoherence, so that the coherent state of the quantum system in a quantum computer is preserved for a time that is long enough to set up a calculation, perform it, and read out the results • quantum computers processing qubits or entangled qubits can outperform classical computers; functional requirements of quantum computers: (cid:153) they must have the ability to initialize any qubit in a specified state, and to measure the state of a specific qubit (cid:153) they must have universal quantum gates, which are logical elements capable of arranging any desired logical relationship between the states of qubits (cid:153) they must also have a processor capable of interlinking quantum gates to establish rules and boundary conditions for their inter-relationships – in a quantum computation, the arrangement of quantum gates connects the qubits in a logical pattern, according to a program or algorithm, and after an interval the qubits assigned to the result are read out • quantum error correction codes exist, whereby qubits are passed through a circuit (the quantum analogue of logic gates) that will successfully fix an error in any one of the qubits without actually reading what all the individual qubit states are; no qubit cloning is required • a completely secure quantum key can be generated and distributed (for communication and decoding of encrypted messages) using entangled photons has been demonstrated (Tittel et al., 2000; Jennewein et al., 2000; Naik et al., 2000); any eavesdropper’s attempt

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

List tests or derivations implied by the evidence.

Bennett
Dirac
Zeilinger
Jian-Wei Pan
De Martini
Feynman
Popper


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

EVIDENCE:
- On the other hand, the existence of antiparticles leads to the conclusion that relativistic quantum mechanics is not enough for a more accurate and complete theory of particle interactions.
- This equation not only described the intrinsic angular momentum of the electrons called spin, it also led to the prediction of the antiparticle of the electron (the positron),[p 19][p 20] and fine structure could only be fully explained with special relativity.
- But at macroscopic scales and in the absence of strong gravitational fields, special relativity is experimentally tested to extremely high degree of accuracy (10−20)[97] and thus accepted by the physics community.
- Experimental results that appear to contradict it are not reproducible and are thus widely believed to be due to experimental errors.[98] Special relativity is mathematically self-consistent, and it is an organic part of all modern physical theories, most notably quantum field theory, string theory, and general relativity

On the other hand, the existence of antiparticles leads to the conclusion that relativistic quantum mechanics is not enough for a more accurate and complete theory of particle interactions.

This equation not only described the intrinsic angular momentum of the electrons called spin, it also led to the prediction of the antiparticle of the electron (the positron),[p 19][p 20] and fine structure could only be fully explained with special relativity.

But at macroscopic scales and in the absence of strong gravitational fields, special relativity is experimentally tested to extremely high degree of accuracy (10−20)[97] and thus accepted by the physics community.

Experimental results that appear to contradict it are not reproducible and are thus widely believed to be due to experimental errors.[98] Special relativity is mathematically self-consistent, and it is an organic part of all modern physical theories, most notably quantum field theory, string theory, and general relativity

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:
- Wheeler inspired his students with visionary ideas, Zel’dovich led a focused research team, and Sciama created a nurturing environment for students, encouraging their intellectual growth.
- The Role of Chandrasekhar Chandrasekhar returned to studying black holes after years of absence while contributing significantly to various fields within astrophysics.
- Install Bookey App to Unlock Full Text and Research indicated that black holes lack distinguishing Audio features (or "hair"), retaining only mass, charge, and spin.
- His presence was welcomed among younger researchers during this exciting period of discovery.

Wheeler inspired his students with visionary ideas, Zel’dovich led a focused research team, and Sciama created a nurturing environment for students, encouraging their intellectual growth.

The Role of Chandrasekhar Chandrasekhar returned to studying black holes after years of absence while contributing significantly to various fields within astrophysics.

Install Bookey App to Unlock Full Text and Research indicated that black holes lack distinguishing Audio features (or "hair"), retaining only mass, charge, and spin.

His presence was welcomed among younger researchers during this exciting period of discovery.

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:
- This computation accurately reproduced the energy levels of the Bohr model.
- The post-measurement wave function generally cannot be known prior to the measurement, but the probabilities for the different possibilities can be calculated using the Born rule.[27][52][note 4] Other, more recent interpretations of quantum mechanics, such as relational quantum mechanics and QBism, also give the Schrödinger equation a status of this sort.[55][56] Schrödinger himself suggested in 1952 that the different terms of a superposition evolving under the Schrödinger equation are "not alternatives but all really happen simultaneously".
- computed the hydrogen spectral series by treating a hydrogen atom's electron as a wave , moving in a potential well , created by the proton.
- The meaning of the Schrödinger equation and how the mathematical entities in it relate to physical reality depends upon the interpretation of quantum mechanics that one adopts.

This computation accurately reproduced the energy levels of the Bohr model.

The post-measurement wave function generally cannot be known prior to the measurement, but the probabilities for the different possibilities can be calculated using the Born rule.[27][52][note 4] Other, more recent interpretations of quantum mechanics, such as relational quantum mechanics and QBism, also give the Schrödinger equation a status of this sort.[55][56] Schrödinger himself suggested in 1952 that the different terms of a superposition evolving under the Schrödinger equation are "not alternatives but all really happen simultaneously".

computed the hydrogen spectral series by treating a hydrogen atom's electron as a wave , moving in a potential well , created by the proton.

The meaning of the Schrödinger equation and how the mathematical entities in it relate to physical reality depends upon the interpretation of quantum mechanics that one adopts.

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:
- Retrieved June 11, 2008 from http://www.sciam.com/article.cfm?id=rupertsresonance&colID=13 Shimamura, A.
- Coping with an undesirable life event: A study of early reactions to physical disability.
- Chicago: The University of Chicago Press.
- Silver, E., Cirincione, C., & Steadman, H.

Retrieved June 11, 2008 from http://www.sciam.com/article.cfm?id=rupertsresonance&colID=13 Shimamura, A.

Coping with an undesirable life event: A study of early reactions to physical disability.

Chicago: The University of Chicago Press.

Silver, E., Cirincione, C., & Steadman, H.

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- This is a weaker energy condition than ANEC.
- Anachronalgeodesic is one which doesn’t contain any points connected by a timelike path.
- Constraints paragraph above, this possibility was excluded; thus short traversable wormholes cannot exist.
- Graham and Olum provide arguments in their 2007 paper about the validity of this new energy condition, concluding the paper by stating that this condition is sufficient to rule out CTCs (thus ruling out short traversable wormholes) and wormholes connecting different asymptotically flat regions [38].

This is a weaker energy condition than ANEC.

Anachronalgeodesic is one which doesn’t contain any points connected by a timelike path.

Constraints paragraph above, this possibility was excluded; thus short traversable wormholes cannot exist.

Graham and Olum provide arguments in their 2007 paper about the validity of this new energy condition, concluding the paper by stating that this condition is sufficient to rule out CTCs (thus ruling out short traversable wormholes) and wormholes connecting different asymptotically flat regions [38].

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:
- In relativitythistranslates,foramassiveparticle,intothelongestpropertimebetweentwopoints.Theproper √ √ time along any path is dτ = −ds2 = −g dxμdxν.
- Since λ can be chosen arbitrarily, it is common to choose it in some convenient fashion.
- 98 Action principle for point particles is that the integrand must be identically zero.
- (4.6) μ ∂(dxμ/dλ) 4.3 Lagrangian for a test particle According to the principle of equivalence, a test particle in a gravitating system moves along a geodesic, a straight line relative to local free-falling frames.

In relativitythistranslates,foramassiveparticle,intothelongestpropertimebetweentwopoints.Theproper √ √ time along any path is dτ = −ds2 = −g dxμdxν.

Since λ can be chosen arbitrarily, it is common to choose it in some convenient fashion.

98 Action principle for point particles is that the integrand must be identically zero.

(4.6) μ ∂(dxμ/dλ) 4.3 Lagrangian for a test particle According to the principle of equivalence, a test particle in a gravitating system moves along a geodesic, a straight line relative to local free-falling frames.

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:
- For example, uncertainty relations in which one of the observables is an angle has little physical meaning for fluctuations larger than one period.[22][75][76][77] Other examples include highly bimodal distributions, or unimodal distributions with divergent variance.
- It is also possible to derive an uncertainty relation that, as the Ozawa's one, combines both the statistical and systematic error components, but keeps a form very close to the Heisenberg original inequality.
- A solution that overcomes these issues is an uncertainty based on entropic uncertainty instead of the product of variances.
- The two simultaneous measurements on A and B are necessarily[73] unsharp or weak.

For example, uncertainty relations in which one of the observables is an angle has little physical meaning for fluctuations larger than one period.[22][75][76][77] Other examples include highly bimodal distributions, or unimodal distributions with divergent variance.

It is also possible to derive an uncertainty relation that, as the Ozawa's one, combines both the statistical and systematic error components, but keeps a form very close to the Heisenberg original inequality.

A solution that overcomes these issues is an uncertainty based on entropic uncertainty instead of the product of variances.

The two simultaneous measurements on A and B are necessarily[73] unsharp or weak.

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:
- 5.2.1 Traversable wormholes via a double trace deformation In 2017 Gao, Jefferis and Wall published a paper discussing a novel construction of a traversable black hole [32].
- They begin the set-up with an eternal BTZ black hole (Ban ̃ados, Teitelboim and Zanelli) - a black hole solution for (2 + 1)-dimensional topological gravity with a negative cosmological constant [41], which contains a non-traversable Einstein-Rosen bridge.
- The interaction is kept active only for a short period of time.
- In order to maintain it stable for long periods of time, multiple methods are examined, including rotation of the wormhole’s mouths around each other and placing the mouths in AdS spacetimes.

5.2.1 Traversable wormholes via a double trace deformation In 2017 Gao, Jefferis and Wall published a paper discussing a novel construction of a traversable black hole [32].

They begin the set-up with an eternal BTZ black hole (Ban ̃ados, Teitelboim and Zanelli) - a black hole solution for (2 + 1)-dimensional topological gravity with a negative cosmological constant [41], which contains a non-traversable Einstein-Rosen bridge.

The interaction is kept active only for a short period of time.

In order to maintain it stable for long periods of time, multiple methods are examined, including rotation of the wormhole’s mouths around each other and placing the mouths in AdS spacetimes.

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:
- murder was legally responsible for the act.
- The first trick to understanding the insanity defense is to recognize that sanity and insanity are legal, not psychiatric, terms.
- Despite their informal use in everyday language (“That man on the street is talking to himself; he must be insane”), these terms don’t refer to whether a person is psychotic—that is, out of touch with reality—but rather to whether that person was legally responsible at the time of the crime.
- Yet | Chapter 10 Disorder in the Court 217

murder was legally responsible for the act.

The first trick to understanding the insanity defense is to recognize that sanity and insanity are legal, not psychiatric, terms.

Despite their informal use in everyday language (“That man on the street is talking to himself; he must be insane”), these terms don’t refer to whether a person is psychotic—that is, out of touch with reality—but rather to whether that person was legally responsible at the time of the crime.

Yet | Chapter 10 Disorder in the Court 217

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:
- where T is the temperature of the vacuum and k is the Boltzmann constant.
- And a general rule became apparent from their analysis that is applicable to modified vacua for massive and massless quantum field theories, for low energy: c* > c (vacuum refraction index < 1) when the modified vacuum has a lower energy density c* < c (vacuum refraction index > 1) when the modified vacuum has a higher energy density c* = c (vacuum refraction index = 1) when the vacuum is free (or un-modified) with ρ = 0 0 vac The first two rules explain the sign of the change
- (1995) discovered that the low energy modification of the speed of light is proportional to the ratio of the modified vacuum energy density (as compared to the standard vacuum energy density, ρ = 0) over m 4, vac e with a universal numerical coefficient and the corresponding coupling constants.
- c 135 m 4 0 0 0   0 e (cid:137) For light (photons) propagating in an anisotropic vacuum given by an external constant uniform electric field E, the polarization-averaged modified speed of light is: c∗  22 E2  = 1− α2  <1 ((cid:61)=c =ε = μ =1) (2.19).

where T is the temperature of the vacuum and k is the Boltzmann constant.

And a general rule became apparent from their analysis that is applicable to modified vacua for massive and massless quantum field theories, for low energy: c* > c (vacuum refraction index < 1) when the modified vacuum has a lower energy density c* < c (vacuum refraction index > 1) when the modified vacuum has a higher energy density c* = c (vacuum refraction index = 1) when the vacuum is free (or un-modified) with ρ = 0 0 vac The first two rules explain the sign of the change

(1995) discovered that the low energy modification of the speed of light is proportional to the ratio of the modified vacuum energy density (as compared to the standard vacuum energy density, ρ = 0) over m 4, vac e with a universal numerical coefficient and the corresponding coupling constants.

c 135 m 4 0 0 0   0 e (cid:137) For light (photons) propagating in an anisotropic vacuum given by an external constant uniform electric field E, the polarization-averaged modified speed of light is: c∗  22 E2  = 1− α2  <1 ((cid:61)=c =ε = μ =1) (2.19).

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:
- S2CID 118518558 (https://ap i.semanticscholar.org/CorpusID:118518558).
- S2CID 7779840 (ht tps://api.semanticscholar.org/CorpusID:7779840).
- Proceedings of the Conference Foundations of Probability and Physics - 2: Växjö (Soland), Sweden, June 2-7, 2002.
- ISSN 1367-2630 (https://search.worldcat.org/issn/1367-2630).

S2CID 118518558 (https://ap i.semanticscholar.org/CorpusID:118518558).

S2CID 7779840 (ht tps://api.semanticscholar.org/CorpusID:7779840).

Proceedings of the Conference Foundations of Probability and Physics - 2: Växjö (Soland), Sweden, June 2-7, 2002.

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:
- Accelerating observers It is often, incorrectly, asserted that special relativity cannot handle accelerating particles or accelerating reference frames.
- The accelerating observer (origin).
- Bends in the world line represent particle acceleration.
- It is only in the presence of gravitation that general relativity is required.[29] An accelerating particle's 4-vector acceleration is the derivative with respect to proper time of its 4-velocity.

Accelerating observers It is often, incorrectly, asserted that special relativity cannot handle accelerating particles or accelerating reference frames.

Bends in the world line represent particle acceleration.

It is only in the presence of gravitation that general relativity is required.[29] An accelerating particle's 4-vector acceleration is the derivative with respect to proper time of its 4-velocity.

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:
- Ancient and medieval thinkers, however, considered the celestial orbs to be thick spheres of rarefied matter nested one within the other, each one in complete contact with the sphere above it and the sphere below.[2] When scholars applied Ptolemy's epicycles, they presumed that each planetary sphere was exactly thick enough to accommodate them.[2] By combining this nested sphere model with astronomical observations, scholars calculated what became generally accepted values at the time for the distances to the Sun: about 4 million miles (6.4 million kilometres), to the other planets, and to the edge of the universe: about 73 million miles (117 million kilometres).[3] The nested sphere model's distances to the Sun and planets differ significantly from modern measurements of the distances,[4] and the size of the universe is now known to be inconceivably large and continuously expanding.[5] Albert Van Helden has suggested that from about 1250 until the 17th century, virtually all educated Europeans were familiar with the Ptolemaic model of "nesting spheres and the cosmic dimensions derived from it".[6] Even following the adoption of Copernicus's heliocentric model of the universe, new versions of the celestial sphere model were introduced, with the planetary spheres following this sequence from the Sun at the centre: Mercury, Venus, Earth-Moon, Mars, Jupiter and Saturn.
- Celestial spheres The celestial spheres, or celestial orbs, were the fundamental entities of the cosmological models developed by Plato, Eudoxus, Aristotle, Ptolemy, Copernicus, and others.
- Mainstream belief in the theory of celestial spheres did not survive the Scientific Revolution.
- Since it was believed that the fixed stars were unchanging in their positions relative to one another, it was argued that they must be on the surface of a single starry sphere.[1] Context In modern thought, the orbits of the planets are Geocentric celestial spheres; Peter Apian's viewed as the paths of those planets through Cosmographia (Antwerp, 1539) mostly empty space.

Ancient and medieval thinkers, however, considered the celestial orbs to be thick spheres of rarefied matter nested one within the other, each one in complete contact with the sphere above it and the sphere below.[2] When scholars applied Ptolemy's epicycles, they presumed that each planetary sphere was exactly thick enough to accommodate them.[2] By combining this nested sphere model with astronomical observations, scholars calculated what became generally accepted values at the time for the distances to the Sun: about 4 million miles (6.4 million kilometres), to the other planets, and to the edge of the universe: about 73 million miles (117 million kilometres).[3] The nested sphere model's distances to the Sun and planets differ significantly from modern measurements of the distances,[4] and the size of the universe is now known to be inconceivably large and continuously expanding.[5] Albert Van Helden has suggested that from about 1250 until the 17th century, virtually all educated Europeans were familiar with the Ptolemaic model of "nesting spheres and the cosmic dimensions derived from it".[6] Even following the adoption of Copernicus's heliocentric model of the universe, new versions of the celestial sphere model were introduced, with the planetary spheres following this sequence from the Sun at the centre: Mercury, Venus, Earth-Moon, Mars, Jupiter and Saturn.

Celestial spheres The celestial spheres, or celestial orbs, were the fundamental entities of the cosmological models developed by Plato, Eudoxus, Aristotle, Ptolemy, Copernicus, and others.

Mainstream belief in the theory of celestial spheres did not survive the Scientific Revolution.

Since it was believed that the fixed stars were unchanging in their positions relative to one another, it was argued that they must be on the surface of a single starry sphere.[1] Context In modern thought, the orbits of the planets are Geocentric celestial spheres; Peter Apian's viewed as the paths of those planets through Cosmographia (Antwerp, 1539) mostly empty space.

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:
- Beyond the quite beyond the circumference of the Earth, and return to the escape velocity, it will leave the Earth mountain from which it was projected."[21] without returning (E).
- The mass of the Earth is approximately three-millionths of the mass of the Sun.
- It appeared in Newton's 1728 book A Treatise of the System of the World.
- Universal gravitational mass In contrast to earlier theories (e.g.

Beyond the quite beyond the circumference of the Earth, and return to the escape velocity, it will leave the Earth mountain from which it was projected."[21] without returning (E).

The mass of the Earth is approximately three-millionths of the mass of the Sun.

It appeared in Newton's 1728 book A Treatise of the System of the World.

Universal gravitational mass In contrast to earlier theories (e.g.

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:
- To me, the logical fallacy in this argument by pure thought is that it may take an infinite amount of energy to make something go at infinitespeed.Indeed, thatiswhathappensinaNewtonianuniverse, whichislogically consistentifyoudon’taskdisallowedquestionssuchashowtheuniversalclockwasset up.(Or,whosetitup?) Thisbriefdigressionshowswhyitmaybewisetofocusonphysics.
- But another observermovingataspeeduinadirectionoppositetothisthingwould,accordingtothe Galileantransformation,seeitmovingataspeedc+u,butthiswouldcontradictcbeing aspeedlimit.Ergo,thecommonsenseGalileantransformationhastobemodified.
- I am inclined to believehisstatementthathedidn’t.Theindependenceofcontheobserverfollowsfrom Maxwell’s theory, while begging the question of what medium electromagnetic waves propagatein.Instead,Idiscusstheexistenceofaspeedlimit.
- This is not how the Lorentz transformation was discovered in our civilization, but it could have happened this way elsewhere.

To me, the logical fallacy in this argument by pure thought is that it may take an infinite amount of energy to make something go at infinitespeed.Indeed, thatiswhathappensinaNewtonianuniverse, whichislogically consistentifyoudon’taskdisallowedquestionssuchashowtheuniversalclockwasset up.(Or,whosetitup?) Thisbriefdigressionshowswhyitmaybewisetofocusonphysics.

I am inclined to believehisstatementthathedidn’t.Theindependenceofcontheobserverfollowsfrom Maxwell’s theory, while begging the question of what medium electromagnetic waves propagatein.Instead,Idiscusstheexistenceofaspeedlimit.

This is not how the Lorentz transformation was discovered in our civilization, but it could have happened this way elsewhere.

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- I’m going to make sure the back door is locked, then I’ll join you.” They head down the hallway.
- How many versions of me tracked us here?
- The only other way into the house is through the French doors that lead from the screened-in porch into the living room.
- What do we need to bring with us?

I’m going to make sure the back door is locked, then I’ll join you.” They head down the hallway.

How many versions of me tracked us here?

The only other way into the house is through the French doors that lead from the screened-in porch into the living room.

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- The gauge-fixing fixes the two subspaces, but allows tetrad transformations within each subspace.
- Thus the components R^ of the restricted Riemann curvature klmn azkl do not vanish even though the components R^ do vanish.
- (17.193) mnl mnl The vanishing of the mixed components γ of the tetrad metric implies that the generalized extrinsic az curvature is antisymmetric in its first two indices, K =−K .
- Under this restricted group of tetrad transformations, the tetrad connections Γ with first two indices az from opposite subspaces form a tensor, the generalized extrinsic curvature azm K , azm K ≡Γ =γγ ·∂ γγ .

The gauge-fixing fixes the two subspaces, but allows tetrad transformations within each subspace.

Thus the components R^ of the restricted Riemann curvature klmn azkl do not vanish even though the components R^ do vanish.

(17.193) mnl mnl The vanishing of the mixed components γ of the tetrad metric implies that the generalized extrinsic az curvature is antisymmetric in its first two indices, K =−K .

Under this restricted group of tetrad transformations, the tetrad connections Γ with first two indices az from opposite subspaces form a tensor, the generalized extrinsic curvature azm K , azm K ≡Γ =γγ ·∂ γγ .

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:
- subsystem of 2-by-2 dimension (single qubit density matrix).
- Whereas the trace is a scalarvalued function on operators, the partial trace is an operatorvalued function.
- Now for indices k, i in the range 1, ..., m, consider the sum This gives a matrix b .
- Partial trace In linear algebra and functional analysis, the partial trace is a generalization of the trace.

subsystem of 2-by-2 dimension (single qubit density matrix).

Whereas the trace is a scalarvalued function on operators, the partial trace is an operatorvalued function.

Now for indices k, i in the range 1, ..., m, consider the sum This gives a matrix b .

Partial trace In linear algebra and functional analysis, the partial trace is a generalization of the trace.

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:
- doi:10.1119/1.5089208 (https://doi.org/10.1119%2F1.5089208).
- Bibcode:1999FoPh...29..631Z (https://ui.adsabs.harvard.edu/abs/1999FoPh...29..631Z).
- "'A Practical Tool,' But Puzzling, Too" (https://www.nytime s.com/2000/12/12/science/essay-a-practical-tool-but-puzzling-too.html).
- Stanford Encyclopedia of Philosophy.

Copenhagen interpretation - Wikipedia.pdf

"'A Practical Tool,' But Puzzling, Too" (https://www.nytime s.com/2000/12/12/science/essay-a-practical-tool-but-puzzling-too.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:
- As of 2018, other than in nuclear Heat of fusion 8.5 kJ/mol medicine, radium has no commercial applications.
- Its Van der Waals 283 pm physical and chemical properties most closely radius resemble its lighter congener, barium.[5] Pure radium is a volatile, lustrous silvery-white Spectral lines of radium metal, even though its lighter congeners calcium, Other properties strontium, and barium have a slight yellow tint.[5] Natural occurrence from decay Radium's lustrous surface rapidly becomes black upon exposure to air, likely due to the formation of Crystal structure body-centered cubic (bcc) radium nitride (Ra N ).[6] Its melting point is either (cF4) 3 2 700 °C (1,292 °F) or 960 °C (1,760 °F)[a] and its Lattice constant a = 514.8 pm boiling point is 1,737 °C (3,159 °F); however, this is (near r.t.)[2] not well established.[7] Both of these values are slightly lower than those of barium, confirming Thermal 18.6 W/(m⋅K) periodic trends down the group 2 elements.[8] Like conductivity barium and the alkali metals, radium crystallizes in Electrical resistivity 1 μΩ⋅m (at 20 °C) the body-centered cubic structure at standard temperature and pressure: the radium–radium bond Magnetic ordering nonmagnetic distance is 514.8 picometers.[9] Radium has a density CAS Number 7440-14-4 of 5.5 g/cm3, higher than that of barium, and the two History elements have similar crystal structures (bcc at Naming after Latin radius (ray), for standard temperature and pressure).[10][11] emitting in the form of rays as it decays Discovery Pierre and Marie Curie
- Formerly, from the 1910s to the 1970s, it was used as Heat of vaporization 113 kJ/mol a radioactive source for radioluminescent devices Vapor pressure and also in radioactive quackery for its supposed P (Pa) 1 10 100 1 k 10 k 100 k curative power.
- Electronegativity Pauling scale: 0.9 Ionization energies 1st: 509.3 kJ/mol Bulk properties 2nd: 979.0 kJ/mol Radium is the heaviest known alkaline earth metal Covalent radius 221±2 pm and is the only radioactive member of its group.

As of 2018, other than in nuclear Heat of fusion 8.5 kJ/mol medicine, radium has no commercial applications.

Its Van der Waals 283 pm physical and chemical properties most closely radius resemble its lighter congener, barium.[5] Pure radium is a volatile, lustrous silvery-white Spectral lines of radium metal, even though its lighter congeners calcium, Other properties strontium, and barium have a slight yellow tint.[5] Natural occurrence from decay Radium's lustrous surface rapidly becomes black upon exposure to air, likely due to the formation of Crystal structure body-centered cubic (bcc) radium nitride (Ra N ).[6] Its melting point is either (cF4) 3 2 700 °C (1,292 °F) or 960 °C (1,760 °F)[a] and its Lattice constant a = 514.8 pm boiling point is 1,737 °C (3,159 °F); however, this is (near r.t.)[2] not well established.[7] Both of these values are slightly lower than those of barium, confirming Thermal 18.6 W/(m⋅K) periodic trends down the group 2 elements.[8] Like conductivity barium and the alkali metals, radium crystallizes in Electrical resistivity 1 μΩ⋅m (at 20 °C) the body-centered cubic structure at standard temperature and pressure: the radium–radium bond Magnetic ordering nonmagnetic distance is 514.8 picometers.[9] Radium has a density CAS Number 7440-14-4 of 5.5 g/cm3, higher than that of barium, and the two History elements have similar crystal structures (bcc at Naming after Latin radius (ray), for standard temperature and pressure).[10][11] emitting in the form of rays as it decays Discovery Pierre and Marie Curie

Formerly, from the 1910s to the 1970s, it was used as Heat of vaporization 113 kJ/mol a radioactive source for radioluminescent devices Vapor pressure and also in radioactive quackery for its supposed P (Pa) 1 10 100 1 k 10 k 100 k curative power.

Electronegativity Pauling scale: 0.9 Ionization energies 1st: 509.3 kJ/mol Bulk properties 2nd: 979.0 kJ/mol Radium is the heaviest known alkaline earth metal Covalent radius 221±2 pm and is the only radioactive member of its group.

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:
- Shot Noise Power The theory behind noise analysis of photon statistics (sometimes called the forward Kolmogorov equation) starts from the Masters equation from Shimoda et al.[15] where corresponds to the emission cross section and upper population number product , and the is the absorption cross section .
- The dynamic only considers neighboring modes and as the photons travel through a medium of excited and ground state atoms from position to .
- That is the product of transmission window efficiency to our photodetector, and quantum efficiency.
- is the spontaneous emission factor that typically corresponds relative strength of spontaneous emission to stimulated emission.

Shot Noise Power The theory behind noise analysis of photon statistics (sometimes called the forward Kolmogorov equation) starts from the Masters equation from Shimoda et al.[15] where corresponds to the emission cross section and upper population number product , and the is the absorption cross section .

The dynamic only considers neighboring modes and as the photons travel through a medium of excited and ground state atoms from position to .

That is the product of transmission window efficiency to our photodetector, and quantum efficiency.

is the spontaneous emission factor that typically corresponds relative strength of spontaneous emission to stimulated emission.

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:
- (Pedantic note: alert readers will have noticed that we ignored the fact that neither of our frame fields is well defined on the axis.
- Indeed, as we would expect, the required acceleration diverges as these timelike circles approach the null circles lying in the critical cylinder .
- A genuine paradox Let us draw the light cones for some typical events in the van Stockum dust, to see how their appearance (in our comoving cylindrical chart) depends on the radial coordinate: As the figure shows, at , the cones become tangent to the coordinate plane , and we obtain a closed null curve (the red circle).
- These closed timelike curves are not timelike geodesics, so these paradoxical observers must accelerate to experience these effects.

(Pedantic note: alert readers will have noticed that we ignored the fact that neither of our frame fields is well defined on the axis.

Indeed, as we would expect, the required acceleration diverges as these timelike circles approach the null circles lying in the critical cylinder .

A genuine paradox Let us draw the light cones for some typical events in the van Stockum dust, to see how their appearance (in our comoving cylindrical chart) depends on the radial coordinate: As the figure shows, at , the cones become tangent to the coordinate plane , and we obtain a closed null curve (the red circle).

These closed timelike curves are not timelike geodesics, so these paradoxical observers must accelerate to experience these effects.

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:
- Altogether there are 6 + 9 = 15 distinct non-vanishing a00 [ab]0 connections.
- Table 17.1 lists the distinct possibilities for the 3 eigenvalues n , and gives the corresponding traditional e Bianchi type.
- The extrinsic curvature K is by definition ab K ≡Γ = 1γ ̇ , (17.99) ab a0b 2 ab with trace √ dln γ K ≡Ka = 1γabγ ̇ = , (17.100) a 2 ab dt where γ ≡ |γ | is the determinant
- The spatialconnectionsΓ alsovaryintimebecause,whereascc withoneindexraisedisconstantintime,the abc ab coefficients c ≡ γ cd with all indices lowered depend on time through the time-dependent metric γ .

Altogether there are 6 + 9 = 15 distinct non-vanishing a00 [ab]0 connections.

Table 17.1 lists the distinct possibilities for the 3 eigenvalues n , and gives the corresponding traditional e Bianchi type.

The extrinsic curvature K is by definition ab K ≡Γ = 1γ ̇ , (17.99) ab a0b 2 ab with trace √ dln γ K ≡Ka = 1γabγ ̇ = , (17.100) a 2 ab dt where γ ≡ |γ | is the determinant

The spatialconnectionsΓ alsovaryintimebecause,whereascc withoneindexraisedisconstantintime,the abc ab coefficients c ≡ γ cd with all indices lowered depend on time through the time-dependent metric γ .

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:
- Geometry, topology, and physics.
- Think for example of the case where is a torus , and is constructed by winding a straight line round at an irrational slope.
- Topology A technical complication is that as a subspace of may carry a topology that is coarser than that on ; this may happen in cases where is injective.
- Hall 2015 Corollary 3.50 Retrieved from "https://en.wikipedia.org/w/index.php?title=One-parameter_group&oldid=1327589361"

Think for example of the case where is a torus , and is constructed by winding a straight line round at an irrational slope.

Topology A technical complication is that as a subspace of may carry a topology that is coarser than that on ; this may happen in cases where is injective.

Hall 2015 Corollary 3.50 Retrieved from "https://en.wikipedia.org/w/index.php?title=One-parameter_group&oldid=1327589361"

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:
- Analysis of individual photons confirm that information cannot travel faster than the speed of light.[41][42] This is experimentally established in many tests of relativistic energy and momentum.[43] More generally, it is impossible for signals or energy to travel faster than c.
- One argument for this is known as causality.
- The γ factor approaches infinity as v approaches c, and it would take an infinite amount of energy to accelerate an object with mass to the speed of light.[40]: 13.3 The speed of light is the upper limit for the speeds of objects with positive rest mass.
- If the spatial distance between two events A and B is greater than the time interval between them multiplied by c then there are frames of reference in which A precedes B, others in which B precedes A, and others in which they are simultaneous.

Analysis of individual photons confirm that information cannot travel faster than the speed of light.[41][42] This is experimentally established in many tests of relativistic energy and momentum.[43] More generally, it is impossible for signals or energy to travel faster than c.

One argument for this is known as causality.

The γ factor approaches infinity as v approaches c, and it would take an infinite amount of energy to accelerate an object with mass to the speed of light.[40]: 13.3 The speed of light is the upper limit for the speeds of objects with positive rest mass.

If the spatial distance between two events A and B is greater than the time interval between them multiplied by c then there are frames of reference in which A precedes B, others in which B precedes A, and others in which they are simultaneous.

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:
- These same photons can still be before they detected as radio waves today.
- Therefore, the cosmic microwave background is a glowing fog of a picture of the universe at the end of this epoch including the tiny fluctuations color similar to generated during inflation (see 9-year WMAP image), and the spread of objects such this and a temperature of as galaxies in the universe is an indication of the scale and size of the universe as it 4000 K, at the developed over time.[42] time.
- The entire baryon plasma—known as baryon acoustic oscillations—became embedded in the universe would distribution of matter as it condensed, giving rise to a very slight preference in have appeared as a brilliantly distribution of large-scale objects.
- The photons released by these newly formed hydrogen atoms initially had a temperature/energy of around ~ 4000 K.

Chronology of the universe - Wikipedia.pdf

These same photons can still be before they detected as radio waves today.

Therefore, the cosmic microwave background is a glowing fog of a picture of the universe at the end of this epoch including the tiny fluctuations color similar to generated during inflation (see 9-year WMAP image), and the spread of objects such this and a temperature of as galaxies in the universe is an indication of the scale and size of the universe as it 4000 K, at the developed over time.[42] time.

The entire baryon plasma—known as baryon acoustic oscillations—became embedded in the universe would distribution of matter as it condensed, giving rise to a very slight preference in have appeared as a brilliantly distribution of large-scale objects.

The photons released by these newly formed hydrogen atoms initially had a temperature/energy of around ~ 4000 K.

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:
- 16.8 Variation of the gravitational action 431 tensors, Exercise 11.2.
- Under a local Lorentz transformation generated by the infinitesimal antisymmetric tensor ε , a tetrad vector a varies as nm n a →a′ =a +δa =a +ε ma .
- mnλ m n Its variation under an infinitesimal Lorentz transformation generated by the antisymmetric tensor ε is nm ( ∂γγ ) ∂(ε pγγ ) ∂γγ ∂ε δΓ =δ γγ · n =γγ · n p +ε pγγ · n = nm +ε pΓ +ε pΓ mnλ m ∂xλ m ∂xλ m p ∂xλ ∂xλ n mpλ m pnλ =D ε a coordinate and tetrad tensor .
- 16.8 Variation of the gravitational action The gravitational action S with the Lagrangian (16.91) is g 1 ∫ ( ∂Γ ) S = emnκλ mnλ +Γp Γ ed4x0123 .

16.8 Variation of the gravitational action 431 tensors, Exercise 11.2.

Under a local Lorentz transformation generated by the infinitesimal antisymmetric tensor ε , a tetrad vector a varies as nm n a →a′ =a +δa =a +ε ma .

mnλ m n Its variation under an infinitesimal Lorentz transformation generated by the antisymmetric tensor ε is nm ( ∂γγ ) ∂(ε pγγ ) ∂γγ ∂ε δΓ =δ γγ · n =γγ · n p +ε pγγ · n = nm +ε pΓ +ε pΓ mnλ m ∂xλ m ∂xλ m p ∂xλ ∂xλ n mpλ m pnλ =D ε a coordinate and tetrad tensor .

16.8 Variation of the gravitational action The gravitational action S with the Lagrangian (16.91) is g 1 ∫ ( ∂Γ ) S = emnκλ mnλ +Γp Γ ed4x0123 .

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:
- (33.34) T(η) In most of this book Θ refers to photons, but in this section the temperature fluctuation Θ refers to any species, bosonic or fermionic, massless or massive (neutrinos have small masses, §42.4.15).
- For a relativistic species, the unperturbed occupation number in thermodynamic equilibrium is 0 1 f = , (33.35) ep/T ∓1 where the sign is − for bosons (photons) and + for fermions (neutrinos).
- 33.4 Boltzmann equation for the temperature fluctuation 885 33.4 Boltzmann equation for the temperature fluctuation The full hierarchy of Boltzmann equations is needed only to describe relativistic species (photons and neutrinos); non-relativistic species (non-baryonic dark matter and baryons) are described adequately by the equations of conservation of energy and momentum.
- Cosmological fluctuations in relativistic species are commonly characterized in terms of a temperature fluctuation Θ, δT(η,x,p) Θ(η,x,p)≡ .

(33.34) T(η) In most of this book Θ refers to photons, but in this section the temperature fluctuation Θ refers to any species, bosonic or fermionic, massless or massive (neutrinos have small masses, §42.4.15).

For a relativistic species, the unperturbed occupation number in thermodynamic equilibrium is 0 1 f = , (33.35) ep/T ∓1 where the sign is − for bosons (photons) and + for fermions (neutrinos).

33.4 Boltzmann equation for the temperature fluctuation 885 33.4 Boltzmann equation for the temperature fluctuation The full hierarchy of Boltzmann equations is needed only to describe relativistic species (photons and neutrinos); non-relativistic species (non-baryonic dark matter and baryons) are described adequately by the equations of conservation of energy and momentum.

Cosmological fluctuations in relativistic species are commonly characterized in terms of a temperature fluctuation Θ, δT(η,x,p) Θ(η,x,p)≡ .

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:
- and momentum p variables of a particle, 1927.
- Two alternative frameworks for quantum physics offer different explanations for the uncertainty principle.
- It states that there is a limit to the precision with which certain pairs of physical properties, such as position and momentum, can be simultaneously known.
- The energy–time relationship is widely used to relate quantum state lifetime to measured energy widths but its formal derivation is fraught with confusing issues about the nature of time.

and momentum p variables of a particle, 1927.

Two alternative frameworks for quantum physics offer different explanations for the uncertainty principle.

It states that there is a limit to the precision with which certain pairs of physical properties, such as position and momentum, can be simultaneously known.

The energy–time relationship is widely used to relate quantum state lifetime to measured energy widths but its formal derivation is fraught with confusing issues about the nature of time.

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- (27.4) ∂x a ∂xa Sometimes it will also be convenient to use ∇ to denote the 4-dimensional spacetime derivative m {∂ } ∇ ≡ ,∇ .
- (27.1) μν μν InthisChapterthetetradγγ istakentobeorthonormal,andalignedwiththeunperturbedcoordinateaxes m e0 , so that the unperturbed inverse vierbein is the unit matrix μ e0 μ =δμ .
- Perturbations in a flat space background General relativistic perturbation theory is simplest in the case that the unperturbed background space is Minkowski space.
- First, decompose perturbations into scalar, vector, and tensor parts.

(27.4) ∂x a ∂xa Sometimes it will also be convenient to use ∇ to denote the 4-dimensional spacetime derivative m {∂ } ∇ ≡ ,∇ .

(27.1) μν μν InthisChapterthetetradγγ istakentobeorthonormal,andalignedwiththeunperturbedcoordinateaxes m e0 , so that the unperturbed inverse vierbein is the unit matrix μ e0 μ =δμ .

Perturbations in a flat space background General relativistic perturbation theory is simplest in the case that the unperturbed background space is Minkowski space.

First, decompose perturbations into scalar, vector, and tensor parts.

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:
- Index | 831 388;weakfieldactionwithout,572;andWeyl’s of,developmentof,253;unificationwithgravity, ansatztoSchwarzschildsolution,374 674–676 Einstein-HilbertLagrangian,determinationof electromagnetismanalogy,Einstein’ssearchforthe Einstein’sgravity,581 metric,404 Einsteinpotential,comparedtoNewtonianpotential, electrons:collisionswithphotons,222f;degenerate, planetaryorbits,371 455;delayedrecombinationinearlyuniverse, Einstein-Rosenbridge,433; andcoordinate 516–517 singularities,91–92,92f electrostatics,mathematicaltreatmentleadingto Einsteintensor,388;asresultofvariationof Maxwell,582 Einstein-Hilbertactionwithrespecttometric, electroweakinteraction,527,765;inMversusRplot, 394 14f Einsteinianmechanics,andcubeofphysics,13f elementaryparticles,massesof,16n elasticstring:asexampleofvariationalcalculus,113; elementaryphysics,definitionofmass,213 hangingunderforceofgravity,114f elementaryscalarfields,759n electriccharge:roleforphoton,383.Seealsocharge embedding:ofcurvedspacesinhigherdimensional electricdipolemoment,ofatom,andaction,715 flatspaces,85–86;ofsurface,determinationof electricfield,245;relativisticunification,247 curvature,90 electrodynamics:initialvalueproblemin,404.See embeddingspace,geodesicsin,645 alsoMaxwellianelectromagnetism Emerson,RalphWaldo,dictumof,235n electromagneticaction,244,250–251;inexpanding emptyspacetime:Einstein’sfieldequationfor, universe,333;local,246.SeealsoMaxwellaction 347–348;gravityin,362 electromagneticcouplingconstant,767 energy:dark(seedarkenergy);elastic,ofhanging electromagneticcurrent:conservationof,225;asflat string,113–114;exactmeaningof,383;extraction spaceanalogoustoenergymomentumtensor, fromKerrblackholes,470;ofmembrane, 379;relativisticformof,226 rotationalinvariance,118;notconserved,27; electromagneticduality,255,483 searchforminimizationfunction,114;spatial electromagneticfield:ascollectionofinfinite densityof,228.Seealsogravitationalenergy numberofharmonicoscillators,382;couplingto energyconditions,557 chargedparticles,250;derivationofequationof energy conservation, 26, 153; historical motion,385;determinationof,338,342n;energy considerations,387n;aroundrotatingblack densityof,255;energymomentumtensorin,381; holes,459;instaticisotropicspacetime,310 inKałuza-Kleintheory,691;andLorentzvector energydensity:constant,fillinguniverse,356; potential,244;MaxwellactioninMinkowskian electromagneticfield,255;inflatspacetime, spacetime,381;andMaxwell’sequations,252; 382;ratiotocriticaldensity,505;replacingmass motionincurvedspacetime,301;andmystery density,inNewtoniangravitationalpotential, oflight,162;atparticleposition,246;treatedas 379n;asrotationalscalar,226–227;andscale superpositionofmodes,746 factorofuniverse,496f;ofuniverse,359,504; electromagneticfieldtensor,244;dual,255;gauge vacuum,749.Seealsodarkenergydensity invariance,249;relativisticcurlofa4–vector, energydistribution,governingcurvatureof 252 spacetime,390 electromagneticgaugetransformations,similarityto energyfunctional:boundaryconditions,116;ofa coordinatetransformations,564 membrane,118;minimizationforNewtonian electromagneticinteraction, comparedto gravity,119 gravitationalinteraction,768 energylevelsplitting,inverseof,atcosmological electromagneticpotential,infifthdimension,677 timescale,768 electromagneticwaves:crosssectionforscatteringon energymomentum,roleforgraviton,383 atomormolecule,715;momentumof,derivation energymomentumconservation,227;andBianchi ofEinstein’sformula,232 identity,393;derivationbyusingcosmological electromagnetism:4-dimensional,720–721;in action,387e;andgeneralinvarianceofmatter curvedspacetime,325–326; describedby action,383–384;ingravitationalfield,386 differentialforms,598;finitesizedobjectsin, energymomentumpseudotensor,386 714–715;fixedgauges,564;inflatspacetime,382; energymomentumtensor:assumptionsabout, gaugeinvariantderivativein,342,353n;Maxwell’s 557;calledstressenergytensor,228;covariant lawsof,andGalileantransformation,20; conservationof, 384; curvedspacetime restrictionsimposedonbyLorentzsymmetry,339; generalizationof,380;inelectromagneticfield, roleofsignsin,382;similaritiestogravitational 381;ofelectromagneticfield,tracelessness,381; waves,568;fromspecialrelativity,244–246;theory underLorentztransformation,226–227;“new

Index | 831 388;weakfieldactionwithout,572;andWeyl’s of,developmentof,253;unificationwithgravity, ansatztoSchwarzschildsolution,374 674–676 Einstein-HilbertLagrangian,determinationof electromagnetismanalogy,Einstein’ssearchforthe Einstein’sgravity,581 metric,404 Einsteinpotential,comparedtoNewtonianpotential, electrons:collisionswithphotons,222f;degenerate, planetaryorbits,371 455;delayedrecombinationinearlyuniverse, Einstein-Rosenbridge,433; andcoordinate 516–517 singularities,91–92,92f electrostatics,mathematicaltreatmentleadingto Einsteintensor,388;asresultofvariationof Maxwell,582 Einstein-Hilbertactionwithrespecttometric, electroweakinteraction,527,765;inMversusRplot, 394 14f Einsteinianmechanics,andcubeofphysics,13f elementaryparticles,massesof,16n elasticstring:asexampleofvariationalcalculus,113; elementaryphysics,definitionofmass,213 hangingunderforceofgravity,114f elementaryscalarfields,759n electriccharge:roleforphoton,383.Seealsocharge embedding:ofcurvedspacesinhigherdimensional electricdipolemoment,ofatom,andaction,715 flatspaces,85–86;ofsurface,determinationof electricfield,245;relativisticunification,247 curvature,90 electrodynamics:initialvalueproblemin,404.See embeddingspace,geodesicsin,645 alsoMaxwellianelectromagnetism Emerson,RalphWaldo,dictumof,235n electromagneticaction,244,250–251;inexpanding emptyspacetime:Einstein’sfieldequationfor, universe,333;local,246.SeealsoMaxwellaction 347–348;gravityin,362 electromagneticcouplingconstant,767 energy:dark(seedarkenergy);elastic,ofhanging electromagneticcurrent:conservationof,225;asflat string,113–114;exactmeaningof,383;extraction spaceanalogoustoenergymomentumtensor, fromKerrblackholes,470;ofmembrane, 379;relativisticformof,226 rotationalinvariance,118;notconserved,27; electromagneticduality,255,483 searchforminimizationfunction,114;spatial electromagneticfield:ascollectionofinfinite densityof,228.Seealsogravitationalenergy numberofharmonicoscillators,382;couplingto energyconditions,557 chargedparticles,250;derivationofequationof energy conservation, 26, 153; historical motion,385;determinationof,338,342n;energy considerations,387n;aroundrotatingblack densityof,255;energymomentumtensorin,381; holes,459;instaticisotropicspacetime,310 inKałuza-Kleintheory,691;andLorentzvector energydensity:constant,fillinguniverse,356; potential,244;MaxwellactioninMinkowskian electromagneticfield,255;inflatspacetime, spacetime,381;andMaxwell’sequations,252; 382;ratiotocriticaldensity,505;replacingmass motionincurvedspacetime,301;andmystery density,inNewtoniangravitationalpotential, oflight,162;atparticleposition,246;treatedas 379n;asrotationalscalar,226–227;andscale superpositionofmodes,746 factorofuniverse,496f;ofuniverse,359,504; electromagneticfieldtensor,244;dual,255;gauge vacuum,749.Seealsodarkenergydensity invariance,249;relativisticcurlofa4–vector, energydistribution,governingcurvatureof 252 spacetime,390 electromagneticgaugetransformations,similarityto energyfunctional:boundaryconditions,116;ofa coordinatetransformations,564 membrane,118;minimizationforNewtonian electromagneticinteraction, comparedto gravity,119 gravitationalinteraction,768 energylevelsplitting,inverseof,atcosmological electromagneticpotential,infifthdimension,677 timescale,768 electromagneticwaves:crosssectionforscatteringon energymomentum,roleforgraviton,383 atomormolecule,715;momentumof,derivation energymomentumconservation,227;andBianchi ofEinstein’sformula,232 identity,393;derivationbyusingcosmological electromagnetism:4-dimensional,720–721;in action,387e;andgeneralinvarianceofmatter curvedspacetime,325–326; describedby action,383–384;ingravitationalfield,386 differentialforms,598;finitesizedobjectsin, energymomentumpseudotensor,386 714–715;fixedgauges,564;inflatspacetime,382; energymomentumtensor:assumptionsabout, gaugeinvariantderivativein,342,353n;Maxwell’s 557;calledstressenergytensor,228;covariant lawsof,andGalileantransformation,20; conservationof, 384; curvedspacetime restrictionsimposedonbyLorentzsymmetry,339; generalizationof,380;inelectromagneticfield, roleofsignsin,382;similaritiestogravitational 381;ofelectromagneticfield,tracelessness,381; waves,568;fromspecialrelativity,244–246;theory underLorentztransformation,226–227;“new

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:
- Especially see equations (7.10), (7.11) and 29.
- "Distributional 1/https://pdfs.semanticscholar.org/dd72/c4b894cd582604e7c4b58 energy–momentum tensor of the Kerr–Newman
- Thibault Damour: Black Holes: Energetics and Thermodynamics 10.
- "Finally, results from Gravity Probe BAJ&pg=PA57).

Especially see equations (7.10), (7.11) and 29.

"Distributional 1/https://pdfs.semanticscholar.org/dd72/c4b894cd582604e7c4b58 energy–momentum tensor of the Kerr–Newman

Thibault Damour: Black Holes: Energetics and Thermodynamics 10.

"Finally, results from Gravity Probe BAJ&pg=PA57).

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:
- They are determined by Maxwell's equations, a set of differential equations which directly relate E and B to ρ and J.[18] Alternatively, one can describe the system in terms of its scalar and vector potentials V and A.
- A set of integral equations known as retarded potentials allow one to calculate V and A from ρ and J,[note 1] and from there the electric and magnetic fields are determined via the relations[19] The E fields and B fields due to electric charges (black/white) and magnetic poles (red/blue).[16][17] Top: E field due to an electric dipole moment d.
- Bottom right: B field due to understood as a collection of two vector fields in space.
- In motion (velocity v), an electric charge induces a B field while a Einstein's theory of gravity, called general relativity, is magnetic charge (not found in nature) another example of a field theory.

They are determined by Maxwell's equations, a set of differential equations which directly relate E and B to ρ and J.[18] Alternatively, one can describe the system in terms of its scalar and vector potentials V and A.

A set of integral equations known as retarded potentials allow one to calculate V and A from ρ and J,[note 1] and from there the electric and magnetic fields are determined via the relations[19] The E fields and B fields due to electric charges (black/white) and magnetic poles (red/blue).[16][17] Top: E field due to an electric dipole moment d.

Bottom right: B field due to understood as a collection of two vector fields in space.

In motion (velocity v), an electric charge induces a B field while a Einstein's theory of gravity, called general relativity, is magnetic charge (not found in nature) another example of a field theory.

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- Leighton says, “I hope you’ll be more comfortable here.
- — One of the sleeping quarters on sublevel one has been prepared for me, and it’s a welcome upgrade from the cell.
- A woman with long brown hair walks toward the box and opens the door.
- There will be people right outside this door to stop you, and then you’ll have to live in a straitjacket in that disgusting cell downstairs.

Leighton says, “I hope you’ll be more comfortable here.

— One of the sleeping quarters on sublevel one has been prepared for me, and it’s a welcome upgrade from the cell.

A woman with long brown hair walks toward the box and opens the door.

There will be people right outside this door to stop you, and then you’ll have to live in a straitjacket in that disgusting cell downstairs.

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:
- Notonlydotherichgetricher,buttherichalsocancountoninheritingfromthewimps.
- Firstacousticpeak After all this talk, let us do an actual calculation, as promised.
- Backinschool,welearnedthattheangularsizeofanobjectisgivenbyθ =λ/d,where λdenotesthelinearsizeoftheobject,thatis,thedistancebetweenitstwoends,anddthe distancefromustotheobjectalongthelineofsight.Thus,todetermineθ andhencel , 1 1 l(l + 1)C l 10 100 1,000 l Figure2 Schematicrepresentationoftheobservationaldata onangularcorrelationinthefluctuationsofthecosmic microwavebackground;seethetextfordetails.
- 1 1 1 Sincethesmallervaluesoflcorrespondtolargervaluesofθ,thevalueofθ tellsusabout themaximalangularsizeoftheprimevaldensityfluctuations.Letususethisobservation to estimate the position of the first acoustic peak.

Firstacousticpeak After all this talk, let us do an actual calculation, as promised.

Backinschool,welearnedthattheangularsizeofanobjectisgivenbyθ =λ/d,where λdenotesthelinearsizeoftheobject,thatis,thedistancebetweenitstwoends,anddthe distancefromustotheobjectalongthelineofsight.Thus,todetermineθ andhencel , 1 1 l(l + 1)C l 10 100 1,000 l Figure2 Schematicrepresentationoftheobservationaldata onangularcorrelationinthefluctuationsofthecosmic microwavebackground;seethetextfordetails.

1 1 1 Sincethesmallervaluesoflcorrespondtolargervaluesofθ,thevalueofθ tellsusabout themaximalangularsizeoftheprimevaldensityfluctuations.Letususethisobservation to estimate the position of the first acoustic peak.

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:
- What can they be μλνρ contractedwith,then?
- Inμλνρ μλση νρ deed, these correspond to E ≡F X ̇ν and B ≡F ̃ X ̇ν, respectively, in the electroμ μν μ μν magneticcase.Byanalogy,thefieldsE andB representthedecompositionofcurvature μν μν intoits“electric”and“magnetic”components,assuggestedbythenotation.Perhapsthe readerisnotsurprisedthatthesefieldsnowcarrytwoindicesinsteadofone.Wenowneed tosquarethemtoformscalarstoputintotheaction.
- Finitesizedobjectsandtidalforces Hence, to leading order in the size of the object, the action governing its motion in a gravitationalfieldisgivenby (cid:7) (cid:29) (cid:30) S = dτ −m+c E Eμν+c B Bμν+...
- Well, these two quantities both vanish, according to Einstein’s field equation, in the empty spacetime the black hole is moving through.

What can they be μλνρ contractedwith,then?

Inμλνρ μλση νρ deed, these correspond to E ≡F X ̇ν and B ≡F ̃ X ̇ν, respectively, in the electroμ μν μ μν magneticcase.Byanalogy,thefieldsE andB representthedecompositionofcurvature μν μν intoits“electric”and“magnetic”components,assuggestedbythenotation.Perhapsthe readerisnotsurprisedthatthesefieldsnowcarrytwoindicesinsteadofone.Wenowneed tosquarethemtoformscalarstoputintotheaction.

Finitesizedobjectsandtidalforces Hence, to leading order in the size of the object, the action governing its motion in a gravitationalfieldisgivenby (cid:7) (cid:29) (cid:30) S = dτ −m+c E Eμν+c B Bμν+...

Well, these two quantities both vanish, according to Einstein’s field equation, in the empty spacetime the black hole is moving through.

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:
- Notes and Records of the Royal Society.
- Isaac Newton: Adventurer in thought (https://archive.org/details/isaacne wtonadven0000hall/page/67).
- OCLC 53006741 (https://search.worldcat.org/oclc/53006741).
- Archived (https://web.archive.org/web/20141009051407/http://www.ne wtonproject.sussex.ac.uk/view/texts/normalized/NATP00004) from the original on 9 October 2014.

Isaac Newton: Adventurer in thought (https://archive.org/details/isaacne wtonadven0000hall/page/67).

Archived (https://web.archive.org/web/20141009051407/http://www.ne wtonproject.sussex.ac.uk/view/texts/normalized/NATP00004) from the original on 9 October 2014.

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:
- "Matter and Antimatter in the Universe".
- org/web/20060912123849/https://www.depauw.edu/sfs/interviews/williamson54interview.htm) from the original on 12 September 2006.
- doi:10.1525/sfs.18.2.230 (https://doi.org/10.1525%2Fsfs.18.2.230).
- Quantum Generations: A History of Physics in the Twentieth Century.

"Matter and Antimatter in the Universe".

org/web/20060912123849/https://www.depauw.edu/sfs/interviews/williamson54interview.htm) from the original on 12 September 2006.

Quantum Generations: A History of Physics in the Twentieth Century.

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:
- DEC : r3 p r • And the averaged energy conditions!
- • (Quantum inequalities, semi-classical energy conditions, etc) Pedro's Universes, 2018, Francisco Lobo
- 4.1.3 Exotic matter • The flaring-out condition entails r + p < 0 !!
- the violation of the NULL ENERGY r CONDITION (NEC) at the throat: (negative energy densities not essential) • In fact, it violates all of the pointwise energy conditions: 1.

DEC : r3 p r • And the averaged energy conditions!

• (Quantum inequalities, semi-classical energy conditions, etc) Pedro's Universes, 2018, Francisco Lobo

4.1.3 Exotic matter • The flaring-out condition entails r + p < 0 !!

the violation of the NULL ENERGY r CONDITION (NEC) at the throat: (negative energy densities not essential) • In fact, it violates all of the pointwise energy conditions: 1.

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- To determine the curvature tensor, we acted with the commutator [D ,D ]on some μ ν arbitraryvectorfieldS (x).Thisinvolvedtwopowersofderivativesandthemetrictensor ρ all over the place, and thus had the structure we were looking for.
- ToEinstein’sFieldEquationasQuicklyasPossible | 349 WearenowinapositiontoderivethecelebratedSchwarzschildmetricaroundamass distribution,studyblackholes,andevenplaywiththeuniverseundersomecircumstances.
- Andevenearlier,eversincechapterI.6,wehaveanticipated(morethanonceinfact)that thecurvatureisgivenbytwopowersofderivativesactingonthemetric.
- EversincechapterIV.3,wehavesuspectedthattheactionforthegravitationalfieldhas tocontaintwopowersofderivativesactingonthedynamicalvariable,namelythemetric.

To determine the curvature tensor, we acted with the commutator [D ,D ]on some μ ν arbitraryvectorfieldS (x).Thisinvolvedtwopowersofderivativesandthemetrictensor ρ all over the place, and thus had the structure we were looking for.

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:
- Miles up the shoreline, I can see the indistinct, nighttime glow of downtown, where the skyscrapers edge up against the water.
- Out of the bushes, I stumble into the middle of a road that parallels the lakeshore.
- That dog is still barking in the distance, but looking back toward the lake, I see no flashlights, hear no voices.
- On the edge of the infield, I collapse, sweat pouring off my body, every muscle in agony.

Miles up the shoreline, I can see the indistinct, nighttime glow of downtown, where the skyscrapers edge up against the water.

Out of the bushes, I stumble into the middle of a road that parallels the lakeshore.

That dog is still barking in the distance, but looking back toward the lake, I see no flashlights, hear no voices.

On the edge of the infield, I collapse, sweat pouring off my body, every muscle in agony.

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:
- Therigor-mindedreaderrealizesthatweactuallyneedtocheckthis.
- Forpictures,Toy/Universe,p.25.
- CananyspacebeembeddedinEN?Ifso,whatistheminimumvalueofNforagivenspace?Thesenontrivial questionswereansweredbyJohnNash,themathematicianportrayedinthefilmABeautifulMind:Ann.Math.
- Laterinlife,healsoappearsinQFTNut,olderbutnotwiser.

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:
- Prove the Jacobi identity (2.126).
- 2.22 Bianchi identities The Jacobi identity [D ,[D ,D ]]+[D ,[D ,D ]]+[D ,[D ,D ]]=0 (2.126) κ λ μ λ μ κ μ κ λ implies the Bianchi identities which, for vanishing torsion, are D R +D R +D R =0 .
- (2.127) κ λμνπ λ μκνπ μ κλνπ The torsion-free Bianchi identities can be written in shorthand D R =0 .
- There are 4 ways to pick [κλμ], and 6 ways to pick antisymmetric νπ, giving 4×6 = 24 Bianchi identities, but 4 of the identities, D R = 0, are implied by the cyclic symmetry (2.120), which is a consequence of vanishing torsion.

2.22 Bianchi identities The Jacobi identity [D ,[D ,D ]]+[D ,[D ,D ]]+[D ,[D ,D ]]=0 (2.126) κ λ μ λ μ κ μ κ λ implies the Bianchi identities which, for vanishing torsion, are D R +D R +D R =0 .

(2.127) κ λμνπ λ μκνπ μ κλνπ The torsion-free Bianchi identities can be written in shorthand D R =0 .

There are 4 ways to pick [κλμ], and 6 ways to pick antisymmetric νπ, giving 4×6 = 24 Bianchi identities, but 4 of the identities, D R = 0, are implied by the cyclic symmetry (2.120), which is a consequence of vanishing torsion.

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:
- Moreover, if T > 1, Tegmark maintains that protons and electrons would be unstable and could decay into particles having greater mass than themselves.
- By stating that simultaneity is a matter of convention, Poincaré meant that to talk about time at all, one must have synchronized clocks, and the synchronization of clocks must be established by a specified, operational procedure (convention).
- On the other hand, in view of creating black holes from an ideal monatomic gas under its self-gravity, Wei-Xiang Feng showed that (3 + 1)-dimensional spacetime is the marginal dimensionality.
- (This is not a problem if the particles have a sufficiently low temperature.)[71] Lastly, if N < 3, gravitation of any kind becomes problematic, and the universe would probably be too simple to contain observers.

Moreover, if T > 1, Tegmark maintains that protons and electrons would be unstable and could decay into particles having greater mass than themselves.

By stating that simultaneity is a matter of convention, Poincaré meant that to talk about time at all, one must have synchronized clocks, and the synchronization of clocks must be established by a specified, operational procedure (convention).

On the other hand, in view of creating black holes from an ideal monatomic gas under its self-gravity, Wei-Xiang Feng showed that (3 + 1)-dimensional spacetime is the marginal dimensionality.

(This is not a problem if the particles have a sufficiently low temperature.)[71] Lastly, if N < 3, gravitation of any kind becomes problematic, and the universe would probably be too simple to contain observers.

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- The Hubbard and Jaynes–Cummings–Hubbard (JCH) models support similar bound states.
- A nucleus is a bound state of protons and neutrons (nucleons).
- Then the evolution is bound with respect to if , where .[9] A quantum particle is in a bound state if at no point in time it is found "too far away" from any finite region .
- Define a one-parameter group of unitary operators , a density operator and an observable on .

The Hubbard and Jaynes–Cummings–Hubbard (JCH) models support similar bound states.

A nucleus is a bound state of protons and neutrons (nucleons).

Then the evolution is bound with respect to if , where .[9] A quantum particle is in a bound state if at no point in time it is found "too far away" from any finite region .

Define a one-parameter group of unitary operators , a density operator and an observable on .

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:
- (2005), The quantum theory of fields (3 vol.), vol.
- Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1973).
- The Feynman Lectures on Physics.
- 1, Cambridge University Press, ISBN 978-0-521-670531 Ohlsson, T.

(2005), The quantum theory of fields (3 vol.), vol.

Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1973).

1, Cambridge University Press, ISBN 978-0-521-670531 Ohlsson, T.

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- “What do you mean?” “In our life that you say we share together...am I happy?” “I thought you didn’t want to talk about it.” “I couldn’t sleep last night.
- The next time my eyes open, the room is filled with daylight and Daniela is standing over me, looking worried.
- “How does that feel?” she asks.
- She’s wearing makeup, and her clothes—jeans and a form-fitting T —age her down considerably from thirty-nine years.

“What do you mean?” “In our life that you say we share together...am I happy?” “I thought you didn’t want to talk about it.” “I couldn’t sleep last night.

The next time my eyes open, the room is filled with daylight and Daniela is standing over me, looking worried.

She’s wearing makeup, and her clothes—jeans and a form-fitting T —age her down considerably from thirty-nine years.

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- Bibcode:1990ISRv...15..364B (https://ui.adsabs.harvard.edu/abs/1990ISRv...15..364B).
- doi:10.3389/fpsyg.2019.03000 (https://d oi.org/10.3389%2Ffpsyg.2019.03000).
- The Fourth Dimension: A Guided Tour of the Higher Universe.
- Schläfli, Ludwig (2010) [1852, 1901].

The Fourth Dimension: A Guided Tour of the Higher Universe.

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:
- Also, in chapter V.3, in defining proper distance, we encountered the integral R≡ (cid:8) t 0 dt ,whichwecannowconverttoanintegraloverredshiftasusedbycosmologists: t a(t) e (cid:7) (cid:7) (cid:7) 1 da 1 da z dz(cid:5) R= = = (5) a(te) aa ̇ a(te) a2H H(z(cid:5)) Fillinguptheuniverse (cid:13) Theenergydensityρ= ρ mayconsistofseveralcomponents.Forexample,theindex j j jcantakeonseveralvalues:j =r,m,and(cid:27),indicatingradiation,matter,andcosmological constant,respectively.Forthepurposeofthischapter,baryonicmatter,luminousornot, islumpedinwithdarkmatterandcollectivelyreferredtoasmatter.Asalreadymentioned, we assume that the dark energy is a manifestation of the cosmological constant.
- Thus, H is the present value of the Hubble parameter.
- IntroductiontoOurUniverse ∗ givenintheprecedingchapter.DefinetheHubble parameter a ̇ R ̇ H ≡ = (2) a R NoticethatwesaidHubbleparameter,notHubbleconstant.Unlessa(t)isapureexponential(asinchapterVI.2),H willvarywithtime.Ingeneral,wewillindicatethepresentvalue of a variable by the subscript 0.
- Using(2),werewrite(1)as 8πG k H2= ρ− (3) 3 R2 Whattimeisitoverthere?

Also, in chapter V.3, in defining proper distance, we encountered the integral R≡ (cid:8) t 0 dt ,whichwecannowconverttoanintegraloverredshiftasusedbycosmologists: t a(t) e (cid:7) (cid:7) (cid:7) 1 da 1 da z dz(cid:5) R= = = (5) a(te) aa ̇ a(te) a2H H(z(cid:5)) Fillinguptheuniverse (cid:13) Theenergydensityρ= ρ mayconsistofseveralcomponents.Forexample,theindex j j jcantakeonseveralvalues:j =r,m,and(cid:27),indicatingradiation,matter,andcosmological constant,respectively.Forthepurposeofthischapter,baryonicmatter,luminousornot, islumpedinwithdarkmatterandcollectivelyreferredtoasmatter.Asalreadymentioned, we assume that the dark energy is a manifestation of the cosmological constant.

Thus, H is the present value of the Hubble parameter.

IntroductiontoOurUniverse ∗ givenintheprecedingchapter.DefinetheHubble parameter a ̇ R ̇ H ≡ = (2) a R NoticethatwesaidHubbleparameter,notHubbleconstant.Unlessa(t)isapureexponential(asinchapterVI.2),H willvarywithtime.Ingeneral,wewillindicatethepresentvalue of a variable by the subscript 0.

Using(2),werewrite(1)as 8πG k H2= ρ− (3) 3 R2 Whattimeisitoverthere?

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- The CDH also receives housekeeping data and science data from the other spacecraft subsystems and components, and packages the data for storage on a data recorder or transmission to the ground via the communications subsystem.
- A spacecraft needs an attitude control subsystem to detumble, be correctly oriented in space and respond to external torques and forces properly.
- In addition, some spacecraft payloads are explicitly for the purpose of ground–ground communication using receiver/retransmitter electronic technologies.
- GNC Guidance refers to the calculation of the commands (usually done by the CDH subsystem) needed to steer the spacecraft where it is desired to be.

The CDH also receives housekeeping data and science data from the other spacecraft subsystems and components, and packages the data for storage on a data recorder or transmission to the ground via the communications subsystem.

A spacecraft needs an attitude control subsystem to detumble, be correctly oriented in space and respond to external torques and forces properly.

In addition, some spacecraft payloads are explicitly for the purpose of ground–ground communication using receiver/retransmitter electronic technologies.

GNC Guidance refers to the calculation of the commands (usually done by the CDH subsystem) needed to steer the spacecraft where it is desired to be.

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:
- Furthermore, the only thing we have available to construct the
- If the spatial separation between particles and the mean time between collisions are much less than the length and time scales we are interested in, we have a fluid (a term used loosely here to include gases).
- (Reneging on our earlier a a a promise,weputbackthesubscriptN hereforemphasis.) Similarly, when we do the integral in (7), we encounter pν(τ )(dqμ/dτ )/(dq0/dτ ), a a a a a a whichwewriteaspμ(τ )pν(τ )/E (τ )(sinceE =m dq a 0 =p0 isjustanothernamefor a a a a a a a dτa a p0).Thus, a (cid:10) pμ(τ )pν(τ ) Tμν(x)= a a a a δ(3)(x(cid:3)−q(cid:3) (τ )) (13) a a E (τ ) a a a The denominator E may seem a bit off to you, but you will soon see that it plays a a necessaryrole.
- (See figure 3.) Since worldlines a cannot go backward, the sum in (11) reduces to just one term.

Furthermore, the only thing we have available to construct the

If the spatial separation between particles and the mean time between collisions are much less than the length and time scales we are interested in, we have a fluid (a term used loosely here to include gases).

(Reneging on our earlier a a a promise,weputbackthesubscriptN hereforemphasis.) Similarly, when we do the integral in (7), we encounter pν(τ )(dqμ/dτ )/(dq0/dτ ), a a a a a a whichwewriteaspμ(τ )pν(τ )/E (τ )(sinceE =m dq a 0 =p0 isjustanothernamefor a a a a a a a dτa a p0).Thus, a (cid:10) pμ(τ )pν(τ ) Tμν(x)= a a a a δ(3)(x(cid:3)−q(cid:3) (τ )) (13) a a E (τ ) a a a The denominator E may seem a bit off to you, but you will soon see that it plays a a necessaryrole.

(See figure 3.) Since worldlines a cannot go backward, the sum in (11) reduces to just one term.

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:
- Given any finite subgroup G thereof in which only the identity matrix possesses 1 as an eigenvalue, the natural group action of the orthogonal group on the n-sphere restricts to a group action of G, with the quotient n manifold S / G inheriting a geodesically complete Riemannian metric of constant curvature 1.
- Up to homothety, every spherical space form arises in this way; this largely reduces the study of spherical space forms to problems in group theory.
- For instance, this can be used to show directly that every

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

Up to homothety, every spherical space form arises in this way; this largely reduces the study of spherical space forms to problems in group theory.

For instance, this can be used to show directly that every

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- The multiplication rules for the vectors γγ given by equam tions (42.115) agree with the usual multiplication rules for Dirac γ-matrices: the vectors γγ anticommute, m and their scalar products form the Minkowski metric.
- 5 12 Thus the Dirac and SM algebras are subalgebras of the Spin(11,1) geometric algebra, such that all Dirac generators commute with all SM generators modified per (42.113).
- The Dirac pseudoscalar I coincides with the Spin(11,1) pseudoscalar J defined by equation (42.108), I ≡γγ γγ γγ γγ =J .
- (42.116) 0 1 2 3 Equivalently, the Dirac chiral operator γ ≡−iI coincides with the Spin(11,1) chiral operator κ ≡−iJ.

The multiplication rules for the vectors γγ given by equam tions (42.115) agree with the usual multiplication rules for Dirac γ-matrices: the vectors γγ anticommute, m and their scalar products form the Minkowski metric.

5 12 Thus the Dirac and SM algebras are subalgebras of the Spin(11,1) geometric algebra, such that all Dirac generators commute with all SM generators modified per (42.113).

The Dirac pseudoscalar I coincides with the Spin(11,1) pseudoscalar J defined by equation (42.108), I ≡γγ γγ γγ γγ =J .

(42.116) 0 1 2 3 Equivalently, the Dirac chiral operator γ ≡−iI coincides with the Spin(11,1) chiral operator κ ≡−iJ.

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

List tests or derivations implied by the evidence.