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

EVIDENCE:
- Somewhere in its center the energies made available by the nuclear force, a thousand times more intense than the electromagnetic force, are somehow being tapped.
- This meant that the Sun was actually gaining energy from some source other than its own contraction, that it was radiating away this energy as light and heat, and that it could have continued to radiate for the entire history of the Earth without getting any cooler.
- The Sun is not, therefore, a ball of ordinary fire at all.
- The atomic nucleus was discovered, and it came to be clear that there is energy packed inside the nucleus in amounts far greater than exists among the electrons, from which the more common forms of energy are derived.

Somewhere in its center the energies made available by the nuclear force, a thousand times more intense than the electromagnetic force, are somehow being tapped.

This meant that the Sun was actually gaining energy from some source other than its own contraction, that it was radiating away this energy as light and heat, and that it could have continued to radiate for the entire history of the Earth without getting any cooler.

The Sun is not, therefore, a ball of ordinary fire at all.

The atomic nucleus was discovered, and it came to be clear that there is energy packed inside the nucleus in amounts far greater than exists among the electrons, from which the more common forms of energy are derived.

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:
- Overview of consecutive fusion processes in massive stars Duration of the main phases of fusion for a 20 M star[211] ☉ Fuel Temperature Density Burn duration material (million kelvins) (kg/cm3) (τ in years) H 37 0.0045 8.1 million He 188 0.97 1.2 million C 870 170 976 Ne 1,570 3,100 0.6 O 1,980 5,550 1.25 S/Si 3,340 33,400 0.0315 (~11.5 days) See also List of proper names of stars Outline of astronomy Sidereal time Star clocks

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.

Overview of consecutive fusion processes in massive stars Duration of the main phases of fusion for a 20 M star[211] ☉ Fuel Temperature Density Burn duration material (million kelvins) (kg/cm3) (τ in years) H 37 0.0045 8.1 million He 188 0.97 1.2 million C 870 170 976 Ne 1,570 3,100 0.6 O 1,980 5,550 1.25 S/Si 3,340 33,400 0.0315 (~11.5 days) See also List of proper names of stars Outline of astronomy Sidereal time Star clocks

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- The small dots indicate specific (past and future) events
- Einstein (The Meaning of Relativity): "Two events taking place at the points A and B of a system K are simultaneous if they appear at the same instant when observed from the Relativity of simultaneity: Event B is middle point, M, of the interval AB.
- According to general relativity, time also runs slower in stronger gravitational fields; this is gravitational time dilation.[99] The effect of the dilation becomes more noticeable in a mass-dense object.
- As a consequence of how gravitational fields warp spacetime, the subject will experience gravitational time dilation.

The small dots indicate specific (past and future) events

Einstein (The Meaning of Relativity): "Two events taking place at the points A and B of a system K are simultaneous if they appear at the same instant when observed from the Relativity of simultaneity: Event B is middle point, M, of the interval AB.

According to general relativity, time also runs slower in stronger gravitational fields; this is gravitational time dilation.[99] The effect of the dilation becomes more noticeable in a mass-dense object.

As a consequence of how gravitational fields warp spacetime, the subject will experience gravitational time dilation.

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:
- Overview of consecutive fusion processes in massive stars Duration of the main phases of fusion for a 20 M star[211] ☉ Fuel Temperature Density Burn duration material (million kelvins) (kg/cm3) (τ in years) H 37 0.0045 8.1 million He 188 0.97 1.2 million C 870 170 976 Ne 1,570 3,100 0.6 O 1,980 5,550 1.25 S/Si 3,340 33,400 0.0315 (~11.5 days) See also List of proper names of stars Outline of astronomy Sidereal time Star clocks

Overview of consecutive fusion processes in massive stars Duration of the main phases of fusion for a 20 M star[211] ☉ Fuel Temperature Density Burn duration material (million kelvins) (kg/cm3) (τ in years) H 37 0.0045 8.1 million He 188 0.97 1.2 million C 870 170 976 Ne 1,570 3,100 0.6 O 1,980 5,550 1.25 S/Si 3,340 33,400 0.0315 (~11.5 days) See also List of proper names of stars Outline of astronomy Sidereal time Star clocks

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- In tabletop lab experiments ultrahighintensity lasers have generated >> gigagauss magnetic fields, ≥ 1016 Volt/cm electric field strengths, >> terabar light pressures and >> 1022 m/sec2 subatomic particle accelerations.
- The power intensity of these lasers has reached the point to where they actually probe QED vacuum physics and general relativistic physics, and they have even modified the vacuum itself.
- However, new ultrahighintensity lasers became available in the 1990s that have achieved extreme physical conditions in the lab that are comparable to the extreme astrophysical conditions expected to be found in stellar cores and on black hole event horizons (Perry, 1996; Mourou et al., 1998; Perry, 2000).
- The Mead and Nachamkin (1996) device is actually designed to extract electrical energy from the higher frequency/higher energy density ZPE modes.

In tabletop lab experiments ultrahighintensity lasers have generated >> gigagauss magnetic fields, ≥ 1016 Volt/cm electric field strengths, >> terabar light pressures and >> 1022 m/sec2 subatomic particle accelerations.

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

However, new ultrahighintensity lasers became available in the 1990s that have achieved extreme physical conditions in the lab that are comparable to the extreme astrophysical conditions expected to be found in stellar cores and on black hole event horizons (Perry, 1996; Mourou et al., 1998; Perry, 2000).

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

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:
- Time of flight techniques One of the last and most accurate time of flight measurements, Michelson, Pease and Pearson's 1930–1935 experiment used a rotating mirror and a one-mile (1.6 km) long vacuum chamber which the light beam traversed 10 times.
- From this difference in angle, the known speed of rotation and the distance to the distant mirror the speed of light may be calculated.[110] Foucault used this apparatus to measure the speed of light in air versus water, based on a suggestion by François Arago.[111] Today, using oscilloscopes with time resolutions of less than one nanosecond, the speed of light can be directly measured by timing the delay of a light pulse from a laser or an LED reflected from a mirror.
- Toothed the distance between the wheel and the mirror, the number of 2.
- Beam- wheelteeth on the wheel, and the rate of rotation, the speed of light splitting semi- breaker of the can be calculated.[109] transparent light beam mirror 4.

Time of flight techniques One of the last and most accurate time of flight measurements, Michelson, Pease and Pearson's 1930–1935 experiment used a rotating mirror and a one-mile (1.6 km) long vacuum chamber which the light beam traversed 10 times.

From this difference in angle, the known speed of rotation and the distance to the distant mirror the speed of light may be calculated.[110] Foucault used this apparatus to measure the speed of light in air versus water, based on a suggestion by François Arago.[111] Today, using oscilloscopes with time resolutions of less than one nanosecond, the speed of light can be directly measured by timing the delay of a light pulse from a laser or an LED reflected from a mirror.

Toothed the distance between the wheel and the mirror, the number of 2.

Beam- wheelteeth on the wheel, and the rate of rotation, the speed of light splitting semi- breaker of the can be calculated.[109] transparent light beam mirror 4.

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- Noether's theorem provides a test for theoretical models of the phenomenon: If the theory has a continuous symmetry, then Noether's theorem guarantees that the theory has a conserved quantity, and for the theory to be correct, this conservation must be observable in experiments.[8]
- a fundamental tool of modern theoretical physics, because of the insight it gives into conservation laws, and also as a practical calculation tool.
- Her theorem allows researchers to determine the conserved quantities from the observed symmetries of a physical system.
- For illustration, suppose that a new physical phenomenon is discovered.

Noether's theorem provides a test for theoretical models of the phenomenon: If the theory has a continuous symmetry, then Noether's theorem guarantees that the theory has a conserved quantity, and for the theory to be correct, this conservation must be observable in experiments.[8]

a fundamental tool of modern theoretical physics, because of the insight it gives into conservation laws, and also as a practical calculation tool.

Her theorem allows researchers to determine the conserved quantities from the observed symmetries of a physical system.

For illustration, suppose that a new physical phenomenon is discovered.

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:
- Weinberg 2002, Equation 5.4.7 7.
- Weinberg 2002, Equations (5.4.17) 8.
- The division by two is just a normalization.
- Weinberg 2002, Equation 5.4.6 6.

The division by two is just a normalization.

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- The Higgs field gives masses to fundamental fermions by flipping their d-bit between up and down.
- A particle with properties consistent with being the Higgs boson, withamassof125GeV,wasdiscoveredin2012bytheCMSandATLAScollaborationsattheLargeHadron Collider (Chatrchyan et al., 2012; Aad et al., 2012).
- The Higgs field must be a Lorentz scalar to allow it to have a non-zero expectationvalueinthevacuum.IfinsteadtheHiggsfieldwereforexampleaLorentzspinororvector,then its presence would define a preferred direction and rest frame, contradicting the observed Lorentz symmetry of the laws of physics.
- The remaining 1 degree of freedom of the Higgs field becomes a massive particle, the Higgs scalar boson.

The Higgs field gives masses to fundamental fermions by flipping their d-bit between up and down.

A particle with properties consistent with being the Higgs boson, withamassof125GeV,wasdiscoveredin2012bytheCMSandATLAScollaborationsattheLargeHadron Collider (Chatrchyan et al., 2012; Aad et al., 2012).

The Higgs field must be a Lorentz scalar to allow it to have a non-zero expectationvalueinthevacuum.IfinsteadtheHiggsfieldwereforexampleaLorentzspinororvector,then its presence would define a preferred direction and rest frame, contradicting the observed Lorentz symmetry of the laws of physics.

The remaining 1 degree of freedom of the Higgs field becomes a massive particle, the Higgs scalar boson.

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:
- "Proof of the Umbral Moonshine Conjecture" (htt ps://doi.org/10.1186%2Fs40687-015-0044-7).
- doi:10.1007/BF01232032 (https://doi.org/10.1007%2FBF01232032).
- CiteSeerX 10.1.1.165.2714 (https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.165.2714).
- arXiv:0706.3359 (https://arxiv.org/ab s/0706.3359) [hep-th (https://arxiv.org/archive/hep-th)].

"Proof of the Umbral Moonshine Conjecture" (htt ps://doi.org/10.1186%2Fs40687-015-0044-7).

arXiv:0706.3359 (https://arxiv.org/ab s/0706.3359) [hep-th (https://arxiv.org/archive/hep-th)].

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:
- A simple empirical fix is to use the bulk photon velocity v ≡ 3Θ in place of the 1 γ 1 baryon velocity to compute S .
- The left and right panels are the same, butwithwavenumberk plottedlinearlyontheleft,logarithmicallyontheright.Thetransferfunctionsplottedhere includemonopole,dipole,quadrupole,andearlyISWcontributions,butexcludethelateISWcontribution.Thedipole contributiontotherapidrecombinationapproximationiscomputedfromthephotonvelocity3Θ1 (thetight-coupling limit),notthebaryonvelocityv b.ThecosmologicalmodelisthestandardflatΛCDMmodeldescribedin§32.3.
- As a result, the rapid approximation tends to overestimate the true dipole contribution if the baryon bulk velocity v is used for the b dipole source term S .
- 34.2 Harmonics of the CMB photon distribution 911 10−1 10−1 20 20 10−2 10−2 200 200 500 500 10−3 10−3 T l (η 0 ,k) 2000 T l (η 0 ,k) 2000 1000 1000 10−4 10−4 4000 4000 10−5 10−5 5000 5000 10−6 10−6 10−7 10−7 0 1000 2000 3000 4000 5000 6000 10 102 103 104 wavenumberkη 0 wavenumberkη 0 Figure34.6 CMBtransferfunctionscomputedfromaBoltzmanntreatment(solidlines)includingphotonandneutrino multipoles up to lmax =16, compared to their values in the rapid recombination approximation (dotted lines) with thesourcefunctionstakenfromthehydrodynamicapproximation,foraselectionofharmonicsl,asmarked.Alllines are the envelopes of the underlying rapidly oscillating transfer functions.

A simple empirical fix is to use the bulk photon velocity v ≡ 3Θ in place of the 1 γ 1 baryon velocity to compute S .

The left and right panels are the same, butwithwavenumberk plottedlinearlyontheleft,logarithmicallyontheright.Thetransferfunctionsplottedhere includemonopole,dipole,quadrupole,andearlyISWcontributions,butexcludethelateISWcontribution.Thedipole contributiontotherapidrecombinationapproximationiscomputedfromthephotonvelocity3Θ1 (thetight-coupling limit),notthebaryonvelocityv b.ThecosmologicalmodelisthestandardflatΛCDMmodeldescribedin§32.3.

As a result, the rapid approximation tends to overestimate the true dipole contribution if the baryon bulk velocity v is used for the b dipole source term S .

34.2 Harmonics of the CMB photon distribution 911 10−1 10−1 20 20 10−2 10−2 200 200 500 500 10−3 10−3 T l (η 0 ,k) 2000 T l (η 0 ,k) 2000 1000 1000 10−4 10−4 4000 4000 10−5 10−5 5000 5000 10−6 10−6 10−7 10−7 0 1000 2000 3000 4000 5000 6000 10 102 103 104 wavenumberkη 0 wavenumberkη 0 Figure34.6 CMBtransferfunctionscomputedfromaBoltzmanntreatment(solidlines)includingphotonandneutrino multipoles up to lmax =16, compared to their values in the rapid recombination approximation (dotted lines) with thesourcefunctionstakenfromthehydrodynamicapproximation,foraselectionofharmonicsl,asmarked.Alllines are the envelopes of the underlying rapidly oscillating transfer functions.

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:
- gravitational repulsion or antigravity) is needed to keep the throat open and stable against collapse.
- The negative surface tension (= positive outward pressure, a.k.a.
- Even in a vacuum, where the average energy density is zero, the energy density fluctuates.
- Equations (2.10a-c) imply that (for ∂Ω convex) we are dealing with negative surface energy density and negative surface tensions.

gravitational repulsion or antigravity) is needed to keep the throat open and stable against collapse.

The negative surface tension (= positive outward pressure, a.k.a.

Even in a vacuum, where the average energy density is zero, the energy density fluctuates.

Equations (2.10a-c) imply that (for ∂Ω convex) we are dealing with negative surface energy density and negative surface tensions.

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- Archived from the original (https://www.nasa.gov/centers/armstrong/features/teachers-learn-the-dif ference-between-mass-and-weight-even-in-space) on 20 March 2023.
- Gibbs, Yvonne (31 March 2017).
- S2CID 125953814 (https://api.semanticscholar.org/CorpusID:12595381 4).
- Coelho, Ricardo Lopes (January 2010).

Archived from the original (https://www.nasa.gov/centers/armstrong/features/teachers-learn-the-dif ference-between-mass-and-weight-even-in-space) on 20 March 2023.

S2CID 125953814 (https://api.semanticscholar.org/CorpusID:12595381 4).

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- Popper on the website of Karl Popper Archives at the University of Klagenfurt (see also External links).
- ISBN 0415285895, 0415285909) The Self and Its Brain: An Argument for Interactionism (with Sir John C.
- The manipulative function could take precedence over the cognitive function [...] How could Popper take it for granted that free political discourse is aimed at understanding reality?
- Eccles), 1977, ISBN 0415058988 In Search of a Better World, 1984, ISBN 0415135486 Die Zukunft ist offen (The Future is Open) (with Konrad Lorenz), 1985 (in German), ISBN

Popper on the website of Karl Popper Archives at the University of Klagenfurt (see also External links).

ISBN 0415285895, 0415285909) The Self and Its Brain: An Argument for Interactionism (with Sir John C.

The manipulative function could take precedence over the cognitive function [...] How could Popper take it for granted that free political discourse is aimed at understanding reality?

Eccles), 1977, ISBN 0415058988 In Search of a Better World, 1984, ISBN 0415135486 Die Zukunft ist offen (The Future is Open) (with Konrad Lorenz), 1985 (in German), ISBN

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:
- Toobtainamoreappropriaterepresentationoftheblackhole,weshouldtakeasliceof(1)withθandφboth constant(incontrasttothefunnelpicturebasedonaslicewitht andθ constant)andthenembedtheslicein (2+1)-dimensionalMinkowskispacetimeM2,1.Theresultingpicturecontainstwoflanges.8 Appendix4:Wormholesandsuch Confusiogrumbles,“Didwenotcheat?IcanseethatthemetricintheKruskal-Szekerescoordinates (cid:12) (cid:14) ds2=− 4r S e−r/rS dV2−dU2 +r2d(cid:24)2 (13) r [reproducedhereforconvenience]isfreeofthecoordinatesingularityatr=r ,butthetransformation(8)and S (9)from(t,r)to(V,U)isnotsmoothaswecrossoverr=r (thatis,dV/dr,dU/draresingularatr=r ).” S S Indeed,Confusioisright,butthat’sjustthelawofcalculus:ifwetransformfromsingulartononsingular coordinates,thetransformationnecessarilymustbesingular.Thinkaboutitthisway.Imagineacivilizationfarfar
- S Appendix3:Misleadingtoshowtheblackholeasafunnelor asarubbersheet You’veprobablyseenapictureofablackholedepictedasakindoffunnel,oralternativelyasarubbersheet depressedbyaheavyroundmass.Farawayfromthefunnel,orthedepressionintherubbersheet,thesurface issupposedtobeflat.Iwillpointedlynotshowthispicture(youcandrawityourselfbasedonthemathematical descriptiongivenbelow),butitanditsvariantshaveappearedincountlessmagazines,newspapers,popular books,andevenonthecoverofatextbook.Inmanysciencemuseums,visitorsareinvitedtotossasmallball ontothesurfaceofanactualfunnelshapedconstruction.Ifyoutosstheballwithsufficientspeedinanangular direction,itwillorbitaroundthecentralfunnel,slowlyspiralingintothedark“bottomless”pitinthecenter.And ofcourse,ifyoutosstheballintheradialdirection,itwillfallrightin,“suckedinbytheirresistibleforce”of theblackhole,oftenthoughtofasa“sourceofevil”inthevisitor’smind.Youknowofcoursethatthisdisplay depictsthesunequallywell.
- BlackHoles Good.Aswe’venoticed,Confusioisgettinglessconfusedbytheday.Indeed,mostpeoplethinkofthehorizon asamathematical2-spheresurroundingtheblackhole.Youaddtimeandit’s2+1dimensionalforsure.
- Thismuseumdisplayentertainsthevisitorsandeducatesthemtosomeextent,butD.Marolfhaspointedout thatitismisleadingatbest.Forsure,ithasseriouslyconfusedsomestudents.

Toobtainamoreappropriaterepresentationoftheblackhole,weshouldtakeasliceof(1)withθandφboth constant(incontrasttothefunnelpicturebasedonaslicewitht andθ constant)andthenembedtheslicein (2+1)-dimensionalMinkowskispacetimeM2,1.Theresultingpicturecontainstwoflanges.8 Appendix4:Wormholesandsuch Confusiogrumbles,“Didwenotcheat?IcanseethatthemetricintheKruskal-Szekerescoordinates (cid:12) (cid:14) ds2=− 4r S e−r/rS dV2−dU2 +r2d(cid:24)2 (13) r [reproducedhereforconvenience]isfreeofthecoordinatesingularityatr=r ,butthetransformation(8)and S (9)from(t,r)to(V,U)isnotsmoothaswecrossoverr=r (thatis,dV/dr,dU/draresingularatr=r ).” S S Indeed,Confusioisright,butthat’sjustthelawofcalculus:ifwetransformfromsingulartononsingular coordinates,thetransformationnecessarilymustbesingular.Thinkaboutitthisway.Imagineacivilizationfarfar

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:
- Matching the exact time component of the two metrics, we can determine the value of the (cid:15) mass correction: (cid:32) (cid:33) (cid:18) r (cid:19)2 2(cid:15) 1− − dt2 = r2(1+ρ2)dτ2 ⇒ r r e e e 2(cid:15) ⇒ − dt2 = r2dτ2 r e e 2(cid:15) dt2 − dt2 = r2 ⇒ r e l2 e r3 q2G2 1 q2G2 ⇒ (cid:15) = − e = − 2 256r6 2256r3 e e The energy will then be: 2(cid:15) 2 1 q2G2 q2G E = = − = − , G G2256r3 256r3 e e which is exactly what was obtained using the first method.
- This means that the calculation above is valid only when d << 16r e 3 .
- Then the total length L of the fermion’s trajectory will be: L = πl+df(ν) and the energy has a more complicated expression depending on the particular an43
- TRAVERSABLE WORMHOLE WITH FERMIONS Section 5.

Matching the exact time component of the two metrics, we can determine the value of the (cid:15) mass correction: (cid:32) (cid:33) (cid:18) r (cid:19)2 2(cid:15) 1− − dt2 = r2(1+ρ2)dτ2 ⇒ r r e e e 2(cid:15) ⇒ − dt2 = r2dτ2 r e e 2(cid:15) dt2 − dt2 = r2 ⇒ r e l2 e r3 q2G2 1 q2G2 ⇒ (cid:15) = − e = − 2 256r6 2256r3 e e The energy will then be: 2(cid:15) 2 1 q2G2 q2G E = = − = − , G G2256r3 256r3 e e which is exactly what was obtained using the first method.

This means that the calculation above is valid only when d << 16r e 3 .

Then the total length L of the fermion’s trajectory will be: L = πl+df(ν) and the energy has a more complicated expression depending on the particular an43

TRAVERSABLE WORMHOLE WITH FERMIONS Section 5.

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:
- For reference, the commutation relations of so(3,1) are (M1) with the spacetime metric η = diag(−1, 1, 1, 1).
- Language and terminology is used as in Representation theory of the Lorentz group.
- The Dirac equation can be derived from this Lagrangian by using the Euler–Lagrange equation.
- Derivation of a Dirac spinor representation This outline describes one type of Dirac spinors as elements of a particular representation space of the (1/2 , 0) ⊕ (0, 1/2 ) representation of the Lorentz group.

For reference, the commutation relations of so(3,1) are (M1) with the spacetime metric η = diag(−1, 1, 1, 1).

Language and terminology is used as in Representation theory of the Lorentz group.

The Dirac equation can be derived from this Lagrangian by using the Euler–Lagrange equation.

Derivation of a Dirac spinor representation This outline describes one type of Dirac spinors as elements of a particular representation space of the (1/2 , 0) ⊕ (0, 1/2 ) representation of the Lorentz group.

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:
- Consideraphotonmovingalongthex-axis.Fromds2= 1 (−dt2+dx2+dw2), wehavedx=dt, x=t, w2 andw=w∗(withw∗arbitrary).NotethatantideSitterspacetimeistranslationinvariantintandxbutnotin w.Referringto(13),wetranslatex=tintoX=(2t,2t,w ∗ 2−1,w ∗ 2+1)/(2w∗),sothata=(0,0,w ∗ 2−1,w ∗ 2+ 1)/(2w∗)andb=(1,1,0,0).Notethatthenormalizationofbcanbeabsorbedintothedefinitionofζ.Wesee thatindeed,a2=−1,b2=0,anda.b=0.
- Appendix4:ContinuingfromdeSittertoantideSitterspacetimes Fromthedefiningequations(1)and(2),weseethatformallywecangofromdeSittertoantideSitterspacetime byanalyticallycontinuingL→iLandXd→iXd.(Bynow,XdisperhapsbetterknowntousasW.)Alternatively, insteadofthinkingabouttheembeddingspacetimes,wecanconsideraspecificsetofcoordinatesxμandsolve Einstein’sfieldequation.Evidently,asolutionofR μν =−(d L − 2 1)g μν,whichafterallisabunchofcoupledpartial(or ordinary)differentialequations,becomesasolutionofR μν =+(d L − 1)g μνwhenweformallysetL→iL.However, thisproceduremayormaynotresultinaspacetime(seebelow),andfurthercontinuationsinthecoordinates xμwillingeneralbeneeded.
- ∗YoumighthavenoticedthatIavoidusingtheterm“embeddingspacetime”inthischapter,sinceIwouldn’t knowwhattocallaspacewithtwotimecoordinates.
- Appendix5:Geodesicsintheembeddingspace As in appendix 3 in chapter IX.10, we can discuss geodesics as visualized with the embedding coordinates XMsatisfyingX2=η MN XMXN=−1.Gothroughthesamestepsasinthatappendix,introducingaLagrange multiplierandsoforthbutkeepinginmindthatinthepresentcase, η=(−++...+−).Inparticular, in theembeddingspace∗aphotonzipsmerrilyalongastraightlineinthesensethatitfollowsX ̈ =0.Asinthe precedingchapter,wecanverifythatthegeodesicsintheembeddingspacedodescribegeodesicsinwhatever coordinatesareusedtomapantideSitterspacetime.

Consideraphotonmovingalongthex-axis.Fromds2= 1 (−dt2+dx2+dw2), wehavedx=dt, x=t, w2 andw=w∗(withw∗arbitrary).NotethatantideSitterspacetimeistranslationinvariantintandxbutnotin w.Referringto(13),wetranslatex=tintoX=(2t,2t,w ∗ 2−1,w ∗ 2+1)/(2w∗),sothata=(0,0,w ∗ 2−1,w ∗ 2+ 1)/(2w∗)andb=(1,1,0,0).Notethatthenormalizationofbcanbeabsorbedintothedefinitionofζ.Wesee thatindeed,a2=−1,b2=0,anda.b=0.

Appendix4:ContinuingfromdeSittertoantideSitterspacetimes Fromthedefiningequations(1)and(2),weseethatformallywecangofromdeSittertoantideSitterspacetime byanalyticallycontinuingL→iLandXd→iXd.(Bynow,XdisperhapsbetterknowntousasW.)Alternatively, insteadofthinkingabouttheembeddingspacetimes,wecanconsideraspecificsetofcoordinatesxμandsolve Einstein’sfieldequation.Evidently,asolutionofR μν =−(d L − 2 1)g μν,whichafterallisabunchofcoupledpartial(or ordinary)differentialequations,becomesasolutionofR μν =+(d L − 1)g μνwhenweformallysetL→iL.However, thisproceduremayormaynotresultinaspacetime(seebelow),andfurthercontinuationsinthecoordinates xμwillingeneralbeneeded.

Appendix5:Geodesicsintheembeddingspace As in appendix 3 in chapter IX.10, we can discuss geodesics as visualized with the embedding coordinates XMsatisfyingX2=η MN XMXN=−1.Gothroughthesamestepsasinthatappendix,introducingaLagrange multiplierandsoforthbutkeepinginmindthatinthepresentcase, η=(−++...+−).Inparticular, in theembeddingspace∗aphotonzipsmerrilyalongastraightlineinthesensethatitfollowsX ̈ =0.Asinthe precedingchapter,wecanverifythatthegeodesicsintheembeddingspacedodescribegeodesicsinwhatever coordinatesareusedtomapantideSitterspacetime.

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 had no idea what the application was.” “You understand the concept behind Schrödinger’s cat?” “Sure.” “And how observation determines reality?” “Yes.” “This other version of me was trying to put a human being into superposition.
- They ran a tox screen that returned traces of a mysterious psychoactive compound.
- “So the stuff you said at Daniela’s—that was all true?” Leaning in close, I whisper, “Every word.
- “How is that possible?” “I just need you to listen.

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

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

“So the stuff you said at Daniela’s—that was all true?” Leaning in close, I whisper, “Every word.

“How is that possible?” “I just need you to listen.

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:
- Under the cash and the credit cards, there’s an Illinois driver’s license with a headshot that’s me but that I’ve never seen, an ID for a gym I’ve never been to, and a health insurance card from a carrier I’ve never used.
- I finish counting my funds—$332 and three expired credit cards.
- Some of the bar patrons have noticed the cab stopped in the intersection, and presumably needing rides, they are drifting over, shouting for me to hold the car.
- “The rest when we get there.” He snatches the money and accelerates through the intersection, past the drunks.

Under the cash and the credit cards, there’s an Illinois driver’s license with a headshot that’s me but that I’ve never seen, an ID for a gym I’ve never been to, and a health insurance card from a carrier I’ve never used.

I finish counting my funds—$332 and three expired credit cards.

Some of the bar patrons have noticed the cab stopped in the intersection, and presumably needing rides, they are drifting over, shouting for me to hold the car.

“The rest when we get there.” He snatches the money and accelerates through the intersection, past the drunks.

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:
- Because photon 3 is part of an entangled pair, observation at 1 2 d and d is equally probable.
- The chance of classical teleportation outside the region of temporal overlap of 1 2 photon 1 and 2 results in 25% for d f f (+45o) coincidence Fig.(11).
- The EPR pair in this scheme Fig.(12) was not based on polarisation, but kvector entanglement.
- Only in the region of temporal overlap quantum teleportation is possible, while outside this region photon 1 and 2 travel independently towards f and f .

Because photon 3 is part of an entangled pair, observation at 1 2 d and d is equally probable.

The chance of classical teleportation outside the region of temporal overlap of 1 2 photon 1 and 2 results in 25% for d f f (+45o) coincidence Fig.(11).

The EPR pair in this scheme Fig.(12) was not based on polarisation, but kvector entanglement.

Only in the region of temporal overlap quantum teleportation is possible, while outside this region photon 1 and 2 travel independently towards f and f .

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:
- JohnL.Petersen,ChairmanoftheTAIboard,deniedFitts’ allegationsandJoeFirmage,anotherBoardmember,didnotrecallthemeetingdescribedinthe minutes.
- ● https://medium.com/@throwmasterjohn/jacques-vallees-details-interactions-with-joe-fir mage-in-the-late-1990s-a87ecc3c20ca Note:In2017,CatherineAustinFittspublishedapaperwithaMichiganStateUniversity economistthatfound$21trillioninunauthorizedspendinginthedepartmentsofDefenseand HousingandUrbanDevelopmentbetween1998and2015.
- Auniversitypressreleasestatesthe DODIGdisabledkeydocumentsshowingunsupportedspendingafterthepaperwaspublished, andsoonafter,theDODannounceditsfirsteverdepartment-wideindependentauditinits
- TheLindberghFoundationwasfoundedbyNeilArmstrongand Gen.JamesDoolittle.

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:
- that the monitoring instruments could scarcely detect it) – it was discovered that there was a definite correlation between the change in strength (i.e., radical frequency shifts were observed) of the monitored radio signal and the teleportation of the test specimen, such that the weak or absent signal indicated that the specimen was “nonexistent” (or in an altered physical state) during teleportation (note: the monitored signal amplitude and frequency of the micro-transmitter specimen were stable before and after teleportation); (cid:190) before and after “passing through the container wall/barrier”, the test specimen and the container’s wall/barrier are both complete solid objects; (cid:190) the gifted psychics were never allowed to see (they were blindfolded in many experiments) or touch each of the test specimens or the sealed containers before and after experiments were conducted, and only the experimenters touched the specimens and containers (using both blind and double-blind protocols); (cid:190) the experimental results were all repeatable Approved for public release; distribution unlimited.

that the monitoring instruments could scarcely detect it) – it was discovered that there was a definite correlation between the change in strength (i.e., radical frequency shifts were observed) of the monitored radio signal and the teleportation of the test specimen, such that the weak or absent signal indicated that the specimen was “nonexistent” (or in an altered physical state) during teleportation (note: the monitored signal amplitude and frequency of the micro-transmitter specimen were stable before and after teleportation); (cid:190) before and after “passing through the container wall/barrier”, the test specimen and the container’s wall/barrier are both complete solid objects; (cid:190) the gifted psychics were never allowed to see (they were blindfolded in many experiments) or touch each of the test specimens or the sealed containers before and after experiments were conducted, and only the experimenters touched the specimens and containers (using both blind and double-blind protocols); (cid:190) the experimental results were all repeatable Approved for public release; distribution unlimited.

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 you use the coordinates xμ, somebody else could perfectly well use coordinates x (cid:5)μ, withthetwosetsofcoordinatesrelatedbytheDfunctionsofDvariablesx (cid:5)μ(x)andtheir inversesxμ(x (cid:5) ).
- To economize on ds, you should curve away from the edge at y=0 into the region of larger y and hence smaller ds for a given stretch of (dx,dy).Atthispoint,wecanonlydiscussthiscurvedpathqualitatively.InchapterII.2, wewilllearnhowtodeterminethiscurve.Inthissci-fimovie,youseeyourfavoriteperson inthedistance.Youruntohimorher, butyousuredon’twantthispersontothinkyou are an idiot, which you would be if you ran along a “straight” line expressed in (x,y) coordinates.YoutrytobelikeFeynmanthelifeguardintheprologue, soyoufollowthe (cid:8) curvethatminimizes ds.
- Generalcoordinatetransformation In elementary discussions of change of coordinates, brute force substitution, as in (1), suffices,butasyouwillsee,itpaystoformalizethestepsinvolved,sothattheformalism appliestomoregeneralsituations.Ihavealreadyadvertisedthatmanyconceptsinvolved inchangingcoordinatesarealsorelevantfordiscussingcurvedspaces.
- Againwecometothepervasivethemeoftheoreticalphysicsalreadyalludedtoinchapter I.3:transformations.Givenasetofphysicallaws, weask:Whatarethetransformations thatleavetheselawsunchanged?Howdotherelevantphysicalquantitiestransform?What combinationsofthesequantitiesareleftinvariant?Hereweareconcernedwithcoordinate transformations.

If you use the coordinates xμ, somebody else could perfectly well use coordinates x (cid:5)μ, withthetwosetsofcoordinatesrelatedbytheDfunctionsofDvariablesx (cid:5)μ(x)andtheir inversesxμ(x (cid:5) ).

To economize on ds, you should curve away from the edge at y=0 into the region of larger y and hence smaller ds for a given stretch of (dx,dy).Atthispoint,wecanonlydiscussthiscurvedpathqualitatively.InchapterII.2, wewilllearnhowtodeterminethiscurve.Inthissci-fimovie,youseeyourfavoriteperson inthedistance.Youruntohimorher, butyousuredon’twantthispersontothinkyou are an idiot, which you would be if you ran along a “straight” line expressed in (x,y) coordinates.YoutrytobelikeFeynmanthelifeguardintheprologue, soyoufollowthe (cid:8) curvethatminimizes ds.

Generalcoordinatetransformation In elementary discussions of change of coordinates, brute force substitution, as in (1), suffices,butasyouwillsee,itpaystoformalizethestepsinvolved,sothattheformalism appliestomoregeneralsituations.Ihavealreadyadvertisedthatmanyconceptsinvolved inchangingcoordinatesarealsorelevantfordiscussingcurvedspaces.

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:
- Mercuryandheartpalpitations Imagine my joy at .
- AplanetnamedVulcanwassimilarlypredictedtoorbitbetweenthesunandMercury,but itwasneverfound.
- (cid:5)(cid:5) ThenEinsteinproposedhiscurvedspacetime,andoutpopsthe43 percentury.Amazing!
- the result that the equations give the perihelion motion of Mercury correctly.

Mercuryandheartpalpitations Imagine my joy at .

the result that the equations give the perihelion motion of Mercury correctly.

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- In coordinates X can be expanded at each point p in
- an assignment of a cotangent vector to each point p, by definition such that for each vector field X.
- operators of the first order, where p is an event.
- The exterior derivative df of a function f is a covector field, i.e.

In coordinates X can be expanded at each point p in

an assignment of a cotangent vector to each point p, by definition such that for each vector field X.

operators of the first order, where p is an event.

The exterior derivative df of a function f is a covector field, i.e.

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:
- Thus, In the Eulerian description, the physical properties are expressed as
- the current configuration is taken as the reference configuration.
- This approach is conveniently applied in the study of fluid flow where the kinematic property of greatest interest is the rate at which change is taking place rather than the shape of the body of fluid at a reference time.[14] Mathematically, the motion of a continuum using the Eulerian description is expressed by the mapping function which provides a tracing of the particle which now occupies the position in the current configuration to its original position in the initial configuration .
- All physical quantities characterizing the continuum are described this way.

Thus, In the Eulerian description, the physical properties are expressed as

the current configuration is taken as the reference configuration.

This approach is conveniently applied in the study of fluid flow where the kinematic property of greatest interest is the rate at which change is taking place rather than the shape of the body of fluid at a reference time.[14] Mathematically, the motion of a continuum using the Eulerian description is expressed by the mapping function which provides a tracing of the particle which now occupies the position in the current configuration to its original position in the initial configuration .

All physical quantities characterizing the continuum are described this way.

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:
- Nielsen, Michael A.; Chuang, Isaac L.
- S2CID 55579565 (https://api.semanticscholar.org/CorpusID:55579565).
- doi:10.1364/JOSAB.32.000A50 (https://doi.org/10.1364%2FJOSAB.32.0 00A50).
- arXiv:0806.0270 (https://arxiv.org/abs/0806.0270).

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:
- Similarly, the2-dimensionalballB2isthedisk,withthecircleS1asitsboundary.) Thus, the spatial sections of AdSd are bounded by Sd−2, which may be thought of as Euclidean space Ed−2 with spatial infinity identified as a single point.
- Space covers only the northern hemisphere of Sd−1, with a boundary at the equator.
- Canyoufigureitoutnow?) Hence the slang expression “tin can” for anti de Sitter spacetime and the explosive developmentoftheAdS/CFTcorrespondence.Inparticular,inthispicture,wearetoregard thegoodoldM3,1spacetimeweliveandplayinastheboundaryofa(4+1)-dimensional AdS5spacetime.Wouldn’tthatbequitealaugh,ifallthistimewewereactuallylivingon theboundaryofAdS5withoutknowingit?Sortoflikethetimewhenwerealizedthatwe werenotlivingonE2,butactuallyontheboundaryofB3.
- We finally conclude that the anti de Sitter spacetimeAdSdisboundedbyMd−2,1.(Inappendix1,Igiveyouaslightlymoreintuitive demonstrationofthisstatement.Inappendix3,youwillseethatAdS2has2boundaries.

Similarly, the2-dimensionalballB2isthedisk,withthecircleS1asitsboundary.) Thus, the spatial sections of AdSd are bounded by Sd−2, which may be thought of as Euclidean space Ed−2 with spatial infinity identified as a single point.

Space covers only the northern hemisphere of Sd−1, with a boundary at the equator.

Canyoufigureitoutnow?) Hence the slang expression “tin can” for anti de Sitter spacetime and the explosive developmentoftheAdS/CFTcorrespondence.Inparticular,inthispicture,wearetoregard thegoodoldM3,1spacetimeweliveandplayinastheboundaryofa(4+1)-dimensional AdS5spacetime.Wouldn’tthatbequitealaugh,ifallthistimewewereactuallylivingon theboundaryofAdS5withoutknowingit?Sortoflikethetimewhenwerealizedthatwe werenotlivingonE2,butactuallyontheboundaryofB3.

We finally conclude that the anti de Sitter spacetimeAdSdisboundedbyMd−2,1.(Inappendix1,Igiveyouaslightlymoreintuitive demonstrationofthisstatement.Inappendix3,youwillseethatAdS2has2boundaries.

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:
- Since the density of the dust is increasing with distance from this axis, the solution is rather artificial, but as one of the simplest known solutions in general relativity, it stands as a pedagogically important example.
- That is, we demand that the world lines of the fluid particles form a timelike congruence having nonzero vorticity but vanishing expansion and shear.
- (In fact, since dust particles feel no forces, this will turn out to be a timelike geodesic congruence, but this needs not be assumed in advance.) A simple ansatz corresponding to this demand is expressed by the following frame field, which contains two undetermined functions of : To prevent misunderstanding, we should emphasize that taking the dual coframe gives the metric tensor in terms of the same two undetermined functions: Multiplying out gives We compute the Einstein tensor with respect to this frame, in terms of the two undetermined functions, and demand that the result have the form appropriate for a perfect fluid solution with the timelike unit vector everywhere tangent to the world line of a fluid particle.
- Derivation One way of obtaining this solution is to look for a cylindrically symmetric perfect fluid solution in which the fluid exhibits rigid rotation.

Since the density of the dust is increasing with distance from this axis, the solution is rather artificial, but as one of the simplest known solutions in general relativity, it stands as a pedagogically important example.

That is, we demand that the world lines of the fluid particles form a timelike congruence having nonzero vorticity but vanishing expansion and shear.

(In fact, since dust particles feel no forces, this will turn out to be a timelike geodesic congruence, but this needs not be assumed in advance.) A simple ansatz corresponding to this demand is expressed by the following frame field, which contains two undetermined functions of : To prevent misunderstanding, we should emphasize that taking the dual coframe gives the metric tensor in terms of the same two undetermined functions: Multiplying out gives We compute the Einstein tensor with respect to this frame, in terms of the two undetermined functions, and demand that the result have the form appropriate for a perfect fluid solution with the timelike unit vector everywhere tangent to the world line of a fluid particle.

Derivation One way of obtaining this solution is to look for a cylindrically symmetric perfect fluid solution in which the fluid exhibits rigid rotation.

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:
- were used for interferometry measurements of the speed of light.[120] Interferometric determination of wavelength becomes less precise with wavelength and the experiments were thus limited in precision by the long wavelength (~4 mm [0.16 in]) of the radiowaves.
- By adjusting the path length while observing the interference pattern and carefully measuring the change in path length, the wavelength of the light λ can be determined.
- Since then, scientists have provided increasingly accurate measurements.
- The speed of light is then calculated using the equation c = λf.

were used for interferometry measurements of the speed of light.[120] Interferometric determination of wavelength becomes less precise with wavelength and the experiments were thus limited in precision by the long wavelength (~4 mm [0.16 in]) of the radiowaves.

By adjusting the path length while observing the interference pattern and carefully measuring the change in path length, the wavelength of the light λ can be determined.

Since then, scientists have provided increasingly accurate measurements.

The speed of light is then calculated using the equation c = λf.

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:
- object 2 is a rocket, object 1 the Earth), we simply write r instead of r and m instead of m and define the gravitational 12 2 Gravity field surrounding Earth from a field g(r) as: so that we can write: macroscopic perspective This formulation is dependent on the objects causing the field.
- Gravitational fields are also conservative; that is, the work done by gravity from one position to another is path-independent.
- It is actually equal to the gravitational acceleration at that point.
- that would be applied on an object in any given point in space, per unit mass.

Newton's law of universal gravitation - Wikipedia.pdf

object 2 is a rocket, object 1 the Earth), we simply write r instead of r and m instead of m and define the gravitational 12 2 Gravity field surrounding Earth from a field g(r) as: so that we can write: macroscopic perspective This formulation is dependent on the objects causing the field.

Gravitational fields are also conservative; that is, the work done by gravity from one position to another is path-independent.

It is actually equal to the gravitational acceleration at that point.

that would be applied on an object in any given point in space, per unit mass.

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- Force 4-vector: This is the derivative of the momentum 4-vector with respect to As expected, the final components of the above 4-vectors are all standard 3-vectors corresponding to spatial 3-momentum, 3-force etc.[89]: 36–59 4-vectors and physical law The first postulate of special relativity declares the equivalency of all inertial frames.
- Energy–momentum 4-vector: As indicated before, there are varying treatments for the energy–momentum 4-vector so that one may also see it expressed as or .
- This means that physical laws may take the form of equations connecting scalars, which are always frame independent.
- Newtonian momenta fail to behave properly under Lorentzian transformation, and Einstein preferred to change the definition of momentum to one involving 4-vectors rather than give up on conservation of momentum.

Force 4-vector: This is the derivative of the momentum 4-vector with respect to As expected, the final components of the above 4-vectors are all standard 3-vectors corresponding to spatial 3-momentum, 3-force etc.[89]: 36–59 4-vectors and physical law The first postulate of special relativity declares the equivalency of all inertial frames.

Energy–momentum 4-vector: As indicated before, there are varying treatments for the energy–momentum 4-vector so that one may also see it expressed as or .

This means that physical laws may take the form of equations connecting scalars, which are always frame independent.

Newtonian momenta fail to behave properly under Lorentzian transformation, and Einstein preferred to change the definition of momentum to one involving 4-vectors rather than give up on conservation of momentum.

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:
- ytisrevinU aveneG euqitpO PAG Photon pairs source λ λ p s,i laser nonlinear filtre birefringent crystal (cid:17)Parametric fluorescence (cid:17)Energy and momentum conservation r r r ω = ω + ω = + k k k p s i p s i (cid:17)Phase matching determines the wavelengths and propagation directions of the down-converted photons Energy conservation: ⇒ each photon from the pair has an uncertain frequency, but the sum of the two frequencies is precisely that of the pump laser ⇒ each photon from the pair has an uncertain age, but the age’s difference is precisely zero ⇒ similar to the original EPR state

ytisrevinU aveneG euqitpO PAG Photon pairs source λ λ p s,i laser nonlinear filtre birefringent crystal (cid:17)Parametric fluorescence (cid:17)Energy and momentum conservation r r r ω = ω + ω = + k k k p s i p s i (cid:17)Phase matching determines the wavelengths and propagation directions of the down-converted photons Energy conservation: ⇒ each photon from the pair has an uncertain frequency, but the sum of the two frequencies is precisely that of the pump laser ⇒ each photon from the pair has an uncertain age, but the age’s difference is precisely zero ⇒ similar to the original EPR state

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:
- Jan Faye (November 18, 2015), "Backward Causation" (https://plato.stanford.edu/entries/causatio n-backwards/), Stanford Encyclopedia of Philosophy, retrieved May 25, 2019 2.
- "Time Travel Paradoxes" (https://web.archive.org/web/19980118212457/ht tp://www.friesian.com/paradox.htm).
- (2001), "The time travel paradox", Phys.
- 1103%2FPhysRevD.65.064013), S2CID 18460829 (https://api.semanticscholar.org/CorpusID:184 60829) 13.

Jan Faye (November 18, 2015), "Backward Causation" (https://plato.stanford.edu/entries/causatio n-backwards/), Stanford Encyclopedia of Philosophy, retrieved May 25, 2019 2.

"Time Travel Paradoxes" (https://web.archive.org/web/19980118212457/ht tp://www.friesian.com/paradox.htm).

(2001), "The time travel paradox", Phys.

1103%2FPhysRevD.65.064013), S2CID 18460829 (https://api.semanticscholar.org/CorpusID:184 60829) 13.

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- This quest may reveal insights into the nature of singularities, the birth and death of universes, and the connection between gravity and quantum mechanics.
- 2.Question How does John Archibald Wheeler view the importance of studying black holes?
- Answer:Understanding what is inside a black hole is akin to seeking knowledge about the fundamental laws of the universe and the unification of general relativity and quantum mechanics.
- Wormholes and Time Machines| Q&A 1.Question What is the significance of understanding what is inside a black hole?

This quest may reveal insights into the nature of singularities, the birth and death of universes, and the connection between gravity and quantum mechanics.

2.Question How does John Archibald Wheeler view the importance of studying black holes?

Answer:Understanding what is inside a black hole is akin to seeking knowledge about the fundamental laws of the universe and the unification of general relativity and quantum mechanics.

Wormholes and Time Machines| Q&A 1.Question What is the significance of understanding what is inside a black hole?

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:
- Ψ Ψ   α     β        (23) It is not necessary that the three particles are of the same kind.
- To teleport the state of a spin-1⁄2 particle, the use of an entangled pair of particles, shared between Alice and Bob, is unavoidable.
- These states Eq.(24) form a basis that represent entangled states (25).
- It was of course trivial that Alice could send the particle to Bob, but assume that this took too long or that this classical channel was not good enough for the state to fully survive.

Ψ Ψ   α     β        (23) It is not necessary that the three particles are of the same kind.

To teleport the state of a spin-1⁄2 particle, the use of an entangled pair of particles, shared between Alice and Bob, is unavoidable.

These states Eq.(24) form a basis that represent entangled states (25).

It was of course trivial that Alice could send the particle to Bob, but assume that this took too long or that this classical channel was not good enough for the state to fully survive.

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:
- Seismic exploration One early commercial application of information theory was in the field of seismic oil exploration.
- The measure of sufficient randomness in extractors is minentropy, a value related to Shannon entropy through Rényi entropy; Rényi entropy is also used in evaluating randomness in cryptographic systems.
- They are, almost universally, unsuited to cryptographic use as they do not evade the deterministic nature of modern computer equipment and software.
- Pseudorandom number generation Pseudorandom number generators are widely available in computer language libraries and application programs.

Seismic exploration One early commercial application of information theory was in the field of seismic oil exploration.

The measure of sufficient randomness in extractors is minentropy, a value related to Shannon entropy through Rényi entropy; Rényi entropy is also used in evaluating randomness in cryptographic systems.

They are, almost universally, unsuited to cryptographic use as they do not evade the deterministic nature of modern computer equipment and software.

Pseudorandom number generation Pseudorandom number generators are widely available in computer language libraries and application programs.

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- The traveling twin's frame is not inertial during periods when she is accelerating.
- This result appears puzzling because each twin observes the other twin as moving, and so at first glance, it would appear that each should find the other to have aged less.
- 2-11 presents the simple case of a twin going straight out along the x axis and immediately turning back.
- The proper time measured along the traveling twin's world line from O to C, plus the proper time measured from C to B, is less than the stay-at-home twin's proper time measured from O to A to B.

The traveling twin's frame is not inertial during periods when she is accelerating.

This result appears puzzling because each twin observes the other twin as moving, and so at first glance, it would appear that each should find the other to have aged less.

2-11 presents the simple case of a twin going straight out along the x axis and immediately turning back.

The proper time measured along the traveling twin's world line from O to C, plus the proper time measured from C to B, is less than the stay-at-home twin's proper time measured from O to A to B.

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 particles are studied in particle physics.
- Composition Particles can also be classified according to composition.
- In the latter case, those particles are called "observationally stable".
- In general, a particle decays from a high-energy state to a lower-energy state by emitting some form of radiation, such as the emission of photons.

These particles are studied in particle physics.

Composition Particles can also be classified according to composition.

In the latter case, those particles are called "observationally stable".

In general, a particle decays from a high-energy state to a lower-energy state by emitting some form of radiation, such as the emission of photons.

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:
- Lines and segments In a Euclidean space, a line is a Euclidean subspace of dimension one.
- A more symmetric representation of the line passing through P and Q is where O is an arbitrary point (not necessary on the line).
- Since a vector space of dimension one is spanned by any nonzero vector, a line is a set of the form where P and Q are two distinct points of the Euclidean space as a part of the line.
- Linear subspaces are Euclidean subspaces and a Euclidean subspace is a linear subspace if and only if it contains the zero vector.

Lines and segments In a Euclidean space, a line is a Euclidean subspace of dimension one.

A more symmetric representation of the line passing through P and Q is where O is an arbitrary point (not necessary on the line).

Since a vector space of dimension one is spanned by any nonzero vector, a line is a set of the form where P and Q are two distinct points of the Euclidean space as a part of the line.

Linear subspaces are Euclidean subspaces and a Euclidean subspace is a linear subspace if and only if it contains the zero vector.

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- It is just that we humans move through our world at speeds that are so much smaller than the speed of light that we are not aware of relativistic effects.Thecorrectnessofspecialrelativityisconfirmedeverydayinparticleacceleratorsthatsmashparticles together at highly relativistic speeds.
- Everydayexperiencegivestheimpressionthattimeisabsolute,andthatspaceisentirelydistinctfromtime, as Galileo and Newton postulated.
- General relativity postulates that the local structure of spacetime is that of special relativity.
- If you are interested in the history, I recommend starting with the popular account by Thorne (1994).

It is just that we humans move through our world at speeds that are so much smaller than the speed of light that we are not aware of relativistic effects.Thecorrectnessofspecialrelativityisconfirmedeverydayinparticleacceleratorsthatsmashparticles together at highly relativistic speeds.

Everydayexperiencegivestheimpressionthattimeisabsolute,andthatspaceisentirelydistinctfromtime, as Galileo and Newton postulated.

General relativity postulates that the local structure of spacetime is that of special relativity.

If you are interested in the history, I recommend starting with the popular account by Thorne (1994).

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- that permitted them to view a video display of themselves relayed by a camera placed behind them.
- Participants reported the eerie sensation that their video double was also being touched, thus sensing they were at a location outside their physical bodies.
- This setup created the weird illusion that their bodies, viewed from the rear, actually were standing in front of them.
- In other words, participants could literally “see” their bodies at a second location, separate from their physical selves.

that permitted them to view a video display of themselves relayed by a camera placed behind them.

Participants reported the eerie sensation that their video double was also being touched, thus sensing they were at a location outside their physical bodies.

This setup created the weird illusion that their bodies, viewed from the rear, actually were standing in front of them.

In other words, participants could literally “see” their bodies at a second location, separate from their physical selves.

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:
- (cid:3) The trouble is that while the 3-vector V lives in the tangent plane at x, it doesn’t necessarily live in the tangent plane at y: it will in general have a component sticking (cid:3) out of that tangent plane.
- ThatisthebeastwewanttosubtractfromW (cid:3) (x+δx).
- TothemoreknowledgeablebeingslivingintheambientE3,itisobvious.Justparallel (cid:3) transportV intheambientEuclideanspace.Anychildcoulddoit,theyyell.
- andwedoitwithoutgivingitasecondthought.Butifwearelivingonacurvedsurface,it (cid:3) isnotsosimpletoparalleltransport.GivenavectorV atthepointx,howdoweparallel transportittosomeotherpointy?

(cid:3) The trouble is that while the 3-vector V lives in the tangent plane at x, it doesn’t necessarily live in the tangent plane at y: it will in general have a component sticking (cid:3) out of that tangent plane.

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:
- Nonetheless, we do attempt to introduce some selected topics of quantum informationtheory,whichthenserve astheconceptualfootingforourdetailed descriptionsofthemostrecentquantuminformationexperiments.Inthissense,on the one hand, we are biased concerning the chosen topics.
- On the other hand, as our goal is to advertise a rather new concept for the realization of quantum information protocols, namely, the combination of notionsand techniques from twocomplementaryapproaches,ourpresentationofthebasicsofquantuminformationshouldalsoprovideanewperspectiveonquantuminformation.Thetwo complementaryapproachesarethetwomostcommonlyusedencodingsofquantuminformation:theonebasedupondiscretetwo-levelsystems(so-calledqubits), certainly by far the most popular and well-known approach, in analogyto classiQuantumTeleportationandEntanglement.AkiraFurusawa,PetervanLoock Copyright©2011WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim ISBN:978-3-527-40930-3
- Regarding the more abstract, mathematicalfoundationsofquantuminformation,thereare variousexcellentsources alreadyexisting[4–8].
- Further, thesequential distributionofmanycopiesofentangledstatesinaquantumrepeaterallowforextendingquantumcommunicationtolargedistances,evenwhenthephysicalquantumchannelisimperfectsuchasalossy,opticalfiber[2,3].

Nonetheless, we do attempt to introduce some selected topics of quantum informationtheory,whichthenserve astheconceptualfootingforourdetailed descriptionsofthemostrecentquantuminformationexperiments.Inthissense,on the one hand, we are biased concerning the chosen topics.

On the other hand, as our goal is to advertise a rather new concept for the realization of quantum information protocols, namely, the combination of notionsand techniques from twocomplementaryapproaches,ourpresentationofthebasicsofquantuminformationshouldalsoprovideanewperspectiveonquantuminformation.Thetwo complementaryapproachesarethetwomostcommonlyusedencodingsofquantuminformation:theonebasedupondiscretetwo-levelsystems(so-calledqubits), certainly by far the most popular and well-known approach, in analogyto classiQuantumTeleportationandEntanglement.AkiraFurusawa,PetervanLoock Copyright©2011WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim ISBN:978-3-527-40930-3

Regarding the more abstract, mathematicalfoundationsofquantuminformation,thereare variousexcellentsources alreadyexisting[4–8].

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- Forces of the Quantum Vacuum: An introduction to Casimir physics.
- "On the attraction between two perfectly conducting plates" (http://www.d wc.knaw.nl/DL/publications/PU00018547.pdf) (PDF).
- arXiv:quantph/0203002 (https://arxiv.org/abs/quant-ph/0203002).
- "The Influence of Retardation on the London-van der Waals Forces".

Forces of the Quantum Vacuum: An introduction to Casimir physics.

"On the attraction between two perfectly conducting plates" (http://www.d wc.knaw.nl/DL/publications/PU00018547.pdf) (PDF).

"The Influence of Retardation on the London-van der Waals Forces".

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:
- Then, when the origin of M is displaced by the translation vector d relative to the origin of F and rotated by the angle φ relative to the x-axis of F, the new coordinates in F of points in M are given by:
- A smooth curve from one position to another in this configuration space is a continuous set of displacements, called the motion of M relative to F.
- The motion of a body consists of a continuous set of rotations and translations.
- The set of rigid transformations in an n-dimensional space is called the special Euclidean group on Rn, and denoted SE(n).

Then, when the origin of M is displaced by the translation vector d relative to the origin of F and rotated by the angle φ relative to the x-axis of F, the new coordinates in F of points in M are given by:

A smooth curve from one position to another in this configuration space is a continuous set of displacements, called the motion of M relative to F.

The motion of a body consists of a continuous set of rotations and translations.

The set of rigid transformations in an n-dimensional space is called the special Euclidean group on Rn, and denoted SE(n).

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 should cause the image quality of very distant objects observed through telescopes to degrade.
- This would violate Lorentz invariance.
- But observations of radiation from nearby quasars by Floyd Stecker of NASA's Goddard Space Flight Center failed to find evidence of violation of Lorentz invariance.[13] A foamy spacetime also sets limits on the accuracy with which distances can be measured because photons should diffuse randomly through a spacetime foam, similar to light diffusing by passing through fog.
- doi:10.1103/PhysRev.97.511 (https://doi.org/10.1103%2FPhysRev.97.511).

This should cause the image quality of very distant objects observed through telescopes to degrade.

But observations of radiation from nearby quasars by Floyd Stecker of NASA's Goddard Space Flight Center failed to find evidence of violation of Lorentz invariance.[13] A foamy spacetime also sets limits on the accuracy with which distances can be measured because photons should diffuse randomly through a spacetime foam, similar to light diffusing by passing through fog.

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:
- 64 BIBLIOGRAPHY [80] Pre-Amiga Commodore Computer History ‘The Amazing Commodore PET 2001 : The First ComputertobeAnnouncedThatwasMoreThanaCircuitBoard’ Webarticle,http://www.commodore.ca/gallery/magazines/popular science review oct 77/popu larscienceoct 77.htm [81] Berry,M.V.,Geim,A.K.()‘Offlyingfrogsandlevitrons’ EuropeanJournalofPhysics18pp.
- 307-313 [82] JohnAtanasoff: Themanwhoinventedthecomputer‘TheProcess’ Webpage,http://www.johnatanasoff.com/the process.php [83] Cray : the Supercomputer Company ‘Oak Ridge National Laboratory’s Cray XT5 ‘Jaguar’ Supercomputer’ Webarticle,http://www.cray.com/Products/XT/ORNLJaguar.aspx [84] Wesley,D.,Barczak,G.()‘The$2millionPlayStation3Supercomputer’ Webarticle,http://performancetrap.org/2010/01/28/ps3-supercomputer/ [85] Sabuncu, M., Andersen, U.L., Leuchs, G.
- () ‘Experimental Demonstration of Continuous VariableCloningwithPhase-ConjugateInputs PhysicalReviewLetters98(17)pp.

64 BIBLIOGRAPHY [80] Pre-Amiga Commodore Computer History ‘The Amazing Commodore PET 2001 : The First ComputertobeAnnouncedThatwasMoreThanaCircuitBoard’ Webarticle,http://www.commodore.ca/gallery/magazines/popular science review oct 77/popu larscienceoct 77.htm [81] Berry,M.V.,Geim,A.K.()‘Offlyingfrogsandlevitrons’ EuropeanJournalofPhysics18pp.

307-313 [82] JohnAtanasoff: Themanwhoinventedthecomputer‘TheProcess’ Webpage,http://www.johnatanasoff.com/the process.php [83] Cray : the Supercomputer Company ‘Oak Ridge National Laboratory’s Cray XT5 ‘Jaguar’ Supercomputer’ Webarticle,http://www.cray.com/Products/XT/ORNLJaguar.aspx [84] Wesley,D.,Barczak,G.()‘The$2millionPlayStation3Supercomputer’ Webarticle,http://performancetrap.org/2010/01/28/ps3-supercomputer/ [85] Sabuncu, M., Andersen, U.L., Leuchs, G.

() ‘Experimental Demonstration of Continuous VariableCloningwithPhase-ConjugateInputs PhysicalReviewLetters98(17)pp.

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 ê and ê are the polar unit vectors.
- Differentiating with respect to time gives the velocity r θ dr with radial component and an additional component rω due to the rotation.
- Two have already been discussed above, in the cases that either the radial components or the angular components are zero, and the non-zero component of motion describes uniform acceleration.
- dt Special cases of motion described by these equations are summarized qualitatively in the table below.

where ê and ê are the polar unit vectors.

Differentiating with respect to time gives the velocity r θ dr with radial component and an additional component rω due to the rotation.

Two have already been discussed above, in the cases that either the radial components or the angular components are zero, and the non-zero component of motion describes uniform acceleration.

dt Special cases of motion described by these equations are summarized qualitatively in the table below.

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 1948, Dutch physicists Hendrik B.
- Note that this method of computing vacuum energy is mathematically equivalent to having a quantum harmonic oscillator at each point and, therefore, suffers the same renormalization problems.
- This loss is cumulative and could result in the black hole's disappearance over time.
- Vacuum energy can also be thought of in terms of virtual particles (also known as vacuum fluctuations) which are created and destroyed out of the vacuum.

Note that this method of computing vacuum energy is mathematically equivalent to having a quantum harmonic oscillator at each point and, therefore, suffers the same renormalization problems.

This loss is cumulative and could result in the black hole's disappearance over time.

Vacuum energy can also be thought of in terms of virtual particles (also known as vacuum fluctuations) which are created and destroyed out of the vacuum.

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:
- On the other hand, as our goal is to advertise a rather new concept for the realization of quantum information protocols, namely, the combination of notionsand techniques from twocomplementaryapproaches,ourpresentationofthebasicsofquantuminformationshouldalsoprovideanewperspectiveonquantuminformation.Thetwo complementaryapproachesarethetwomostcommonlyusedencodingsofquantuminformation:theonebasedupondiscretetwo-levelsystems(so-calledqubits), certainly by far the most popular and well-known approach, in analogyto classiQuantumTeleportationandEntanglement.AkiraFurusawa,PetervanLoock Copyright©2011WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim ISBN:978-3-527-40930-3
- In thisintroductorychapter, we shall give abrief, certainlyincomplete, and in somesensebiasedoverviewofquantuminformation.Itwillbeincomplete,asthe focusofthisbookisonopticalquantuminformationprotocols,andtheirexperimental realizations,includingmanyexperiment-oriented details otherwise missing in textbooks on quantum information.
- Further, thesequential distributionofmanycopiesofentangledstatesinaquantumrepeaterallowforextendingquantumcommunicationtolargedistances,evenwhenthephysicalquantumchannelisimperfectsuchasalossy,opticalfiber[2,3].
- Apart from its conceptual importance, however, quantum information may alsoleadtoreal-worldapplicationsforcommunication(quantumcommunication) andcomputation(quantumcomputation)byexploitingquantumpropertiessuch as the superposition principle and entanglement.

On the other hand, as our goal is to advertise a rather new concept for the realization of quantum information protocols, namely, the combination of notionsand techniques from twocomplementaryapproaches,ourpresentationofthebasicsofquantuminformationshouldalsoprovideanewperspectiveonquantuminformation.Thetwo complementaryapproachesarethetwomostcommonlyusedencodingsofquantuminformation:theonebasedupondiscretetwo-levelsystems(so-calledqubits), certainly by far the most popular and well-known approach, in analogyto classiQuantumTeleportationandEntanglement.AkiraFurusawa,PetervanLoock Copyright©2011WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim ISBN:978-3-527-40930-3

In thisintroductorychapter, we shall give abrief, certainlyincomplete, and in somesensebiasedoverviewofquantuminformation.Itwillbeincomplete,asthe focusofthisbookisonopticalquantuminformationprotocols,andtheirexperimental realizations,includingmanyexperiment-oriented details otherwise missing in textbooks on quantum information.

Apart from its conceptual importance, however, quantum information may alsoleadtoreal-worldapplicationsforcommunication(quantumcommunication) andcomputation(quantumcomputation)byexploitingquantumpropertiessuch as the superposition principle and entanglement.

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- Answer:In 1975, Chandrasekhar immersed himself in developing a comprehensive mathematical description of
- Answer:The golden age of black hole research brought a radical shift in the understanding of black holes from being seen as simple holes in space, where nothing can escape, to dynamic objects that can spin, pulsate, create gravitational waves, and interact with their environment in complex ways.
- Physicists realized that all properties of a black hole could be predicted with just three parameters: mass, spin, and electric charge, fundamentally altering the theoretical landscape of general relativity.
- 2.Question How did Subrahmanyan Chandrasekhar contribute to black hole research at the University of Chicago in 1975?

Answer:In 1975, Chandrasekhar immersed himself in developing a comprehensive mathematical description of

Answer:The golden age of black hole research brought a radical shift in the understanding of black holes from being seen as simple holes in space, where nothing can escape, to dynamic objects that can spin, pulsate, create gravitational waves, and interact with their environment in complex ways.

Physicists realized that all properties of a black hole could be predicted with just three parameters: mass, spin, and electric charge, fundamentally altering the theoretical landscape of general relativity.

2.Question How did Subrahmanyan Chandrasekhar contribute to black hole research at the University of Chicago in 1975?

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:
- Finkelstein time t is defined by F t +r =t+r* , (8.19) F
- The more or less diagonal lines (black) are outgoing and infalling null geodesics.
- The outgoing and infalling null geodesics appear not to cross the horizon, but thisisanartefactoftheReissner-Nordströmcoordinatesystem.
- The geometry has two horizons (pink), an outer horizon, and an inner horizon at r− =0.25r+.

Finkelstein time t is defined by F t +r =t+r* , (8.19) F

The more or less diagonal lines (black) are outgoing and infalling null geodesics.

The outgoing and infalling null geodesics appear not to cross the horizon, but thisisanartefactoftheReissner-Nordströmcoordinatesystem.

The geometry has two horizons (pink), an outer horizon, and an inner horizon at r− =0.25r+.

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- The exterior derivative of the Lorentz connection 1-form is, equation (15.70), the 2-form ( ) ( ) ∂Γ ∂Γ ∂Γ ∂Γ dΓ= λ − κ d2xκλ = mnλ − mnκ γγm∧γγnd2xκλ , (16.206) ∂xκ ∂xλ ∂xκ ∂xλ implicitly summed over distinct antisymmetric indices mn and κλ.
- (16.208) Equation (16.208) is Cartan’s second equation of structure.
- Equations (16.206) and (16.207) imply that the Riemann 2-form R is related to the Lorentz connection 1-form Γ by R≡dΓ+ 1[Γ,Γ] .
- The torsion vector 2-form S is defined by (the minus sign ensures that Cartan’s equation (16.212) takes conventional form, given the definition (11.48) of the components Sm) κλ S ≡−S γγmd2xκλ , (16.209) mκλ implicitly summed over distinct antisymmetric indices κλ.

The exterior derivative of the Lorentz connection 1-form is, equation (15.70), the 2-form ( ) ( ) ∂Γ ∂Γ ∂Γ ∂Γ dΓ= λ − κ d2xκλ = mnλ − mnκ γγm∧γγnd2xκλ , (16.206) ∂xκ ∂xλ ∂xκ ∂xλ implicitly summed over distinct antisymmetric indices mn and κλ.

(16.208) Equation (16.208) is Cartan’s second equation of structure.

Equations (16.206) and (16.207) imply that the Riemann 2-form R is related to the Lorentz connection 1-form Γ by R≡dΓ+ 1[Γ,Γ] .

The torsion vector 2-form S is defined by (the minus sign ensures that Cartan’s equation (16.212) takes conventional form, given the definition (11.48) of the components Sm) κλ S ≡−S γγmd2xκλ , (16.209) mκλ implicitly summed over distinct antisymmetric indices κλ.

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:
- How could a planet know ∗ instantlyanychangeinthepositionofitsstar?InthePrincipia,heleft thisconundrum“to theconsiderationofthereader.”Buthedidfret,andina1693lettertohisfriendRichard Bentley,heopined: Thatgravityshouldbeinnate,inherentandessentialtomattersothatonebodymayactupon anotheratadistancethroughavacuumwithoutthemediationofanythingelsebyandthrough whichtheiractionorforcemaybeconveyedfromonetoanotheristomesogreatanabsurdity thatIbelievenomanwhohasinphilosophicalmattersanycompetentfacultyofthinkingcan everfallintoit.1 Tellme,whenyoufirstlearnedabouttheinversesquarelaw,didyounotfinditbizarre?
- In chapters II.1 and II.3, I reminded you that in Newtonian gravity, the gravitational potential(cid:26)isdeterminedintermsofthemassdistributionρ by ∇2(cid:26)(x(cid:3),t)=4πGρ(x(cid:3),t) (1) Lookatthisequation:anychangeinthemassdistributionwillbeinstantaneouslycommunicatedtotheNewtonianpotential.Thegravitationalpotential(cid:26)isslavishlyyokedto thematterdistribution.
- WouldNewtonhavedescribedyouaslackingin“facultyofthinking”?
- ∗Thereisperhapsalessonheresomewherefortheyoungtheoreticalphysicistsreadingthisbook.Newton wascontenttopostulatetheinversesquarelawandthenexploreitsconsequences.Heleftitsdynamicalorigin2to otherslikeDescartes,whosetheoryofvorticessweepingtheplanetsalongwassweptintothedustbinofhistory.

In chapters II.1 and II.3, I reminded you that in Newtonian gravity, the gravitational potential(cid:26)isdeterminedintermsofthemassdistributionρ by ∇2(cid:26)(x(cid:3),t)=4πGρ(x(cid:3),t) (1) Lookatthisequation:anychangeinthemassdistributionwillbeinstantaneouslycommunicatedtotheNewtonianpotential.Thegravitationalpotential(cid:26)isslavishlyyokedto thematterdistribution.

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:
- “On the collapse of charged scalar fields”.
- doi: 10.1103/PhysRevLett.83.5423.
- doi: 10.1103/PhysRevD.68.044013.
- “Strongly hyperbolic second order Einstein’s evolution equations”.

“On the collapse of charged scalar fields”.

“Strongly hyperbolic second order Einstein’s evolution equations”.

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 double-slit experiment for a classical particle, a wave, and a quantum particle demonstrating wave-particle duality In the double-slit experiment, as originally performed by Thomas Young in 1803,[32] and then Augustin Fresnel a decade The diffraction pattern produced when light is shone through later,[32] a beam of light is directed through one slit (top) and the interference pattern produced by two slits two narrow, closely spaced slits, producing (bottom).
- The same behavior can be demonstrated in water waves: the double-slit experiment was seen as a demonstration of the wave nature of light.
- The double slit pattern is more dramatic.
- Quantum radiation, quantum fields In 1923 Compton demonstrated that the Planck-Einstein energy quanta from light also had momentum; three years later the "energy quanta" got a new name "photon".[25] Despite its role in almost all stages of the quantum revolution, no explicit model for light quanta existed until 1927 when Paul Dirac began work on a quantum theory of radiation[26] that became quantum electrodynamics.

Introduction to quantum mechanics - Wikipedia.pdf

The double-slit experiment for a classical particle, a wave, and a quantum particle demonstrating wave-particle duality In the double-slit experiment, as originally performed by Thomas Young in 1803,[32] and then Augustin Fresnel a decade The diffraction pattern produced when light is shone through later,[32] a beam of light is directed through one slit (top) and the interference pattern produced by two slits two narrow, closely spaced slits, producing (bottom).

The same behavior can be demonstrated in water waves: the double-slit experiment was seen as a demonstration of the wave nature of light.

The double slit pattern is more dramatic.

Quantum radiation, quantum fields In 1923 Compton demonstrated that the Planck-Einstein energy quanta from light also had momentum; three years later the "energy quanta" got a new name "photon".[25] Despite its role in almost all stages of the quantum revolution, no explicit model for light quanta existed until 1927 when Paul Dirac began work on a quantum theory of radiation[26] that became quantum electrodynamics.

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- One team of investigators surveyed adults between the ages of 21 and 40 or over age 60 about their happiness and the happiness of the average person at their current age, age 30, and at age 70.
- Even in the oldest group, people aged 75 to 85 years, 26% reported still being sexually active.
- As overall health declined, so did sexual activity.
- Contrary to the myth of older people as lacking in sexual desire, a national survey (Laumann, Das, & Waite, in press) of about 3,000 people indicated that more than three quarters of men aged 75 to 85 and half of their women count erparts reported still being interested in sex.

One team of investigators surveyed adults between the ages of 21 and 40 or over age 60 about their happiness and the happiness of the average person at their current age, age 30, and at age 70.

Even in the oldest group, people aged 75 to 85 years, 26% reported still being sexually active.

As overall health declined, so did sexual activity.

Contrary to the myth of older people as lacking in sexual desire, a national survey (Laumann, Das, & Waite, in press) of about 3,000 people indicated that more than three quarters of men aged 75 to 85 and half of their women count erparts reported still being interested in sex.

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- In Dunedin he met the Professor of Physiology John Carew Eccles and formed a lifelong friendship with him.
- Popper was president of the Aristotelian Society from 1958 to 1959.
- He resided in Penn, Buckinghamshire.[28]
- It was here that he wrote his influential work The Open Society and Its Enemies.

In Dunedin he met the Professor of Physiology John Carew Eccles and formed a lifelong friendship with him.

Popper was president of the Aristotelian Society from 1958 to 1959.

He resided in Penn, Buckinghamshire.[28]

It was here that he wrote his influential work The Open Society and Its Enemies.

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 is special abouttheGullstrand-Painlevé-Cartesiantetradisthatthetetrad-frameconnections,computedbytheusual formula (11.54), are given by the coordinate gradient of the radial velocity (the following equation is valid component-by-component despite the non-matching up-down placement of indices) ∂βa Γ0 =Γa =∂ βa =δβ (a,b=1,2,3) .
- This proper distance δξn = en δxν = δn(δxν −βνδt ) = δxn −βnδτ equals the distance ν ν ff δxn moved relative to the background Gullstrand-Painlevé-Cartesian coordinates, minus the distance βnδτ moved by the river.
- The geodesic equation (19.27) says that the change δpk in the tetrad 4-vector pk in the time δτ is δpk =−Γk δξnpm .
- Consider a fishy swimming in the Gullstrand-Painlevé river, with some arbitrary tetrad-frame 4-velocity um,andconsideratetrad-frame4-vectorpk attachedtothefishy.Ifthefishyisinfree-fall,thenthegeodesic equation of motion for pk is as usual dpk +Γk unpm =0 .

What is special abouttheGullstrand-Painlevé-Cartesiantetradisthatthetetrad-frameconnections,computedbytheusual formula (11.54), are given by the coordinate gradient of the radial velocity (the following equation is valid component-by-component despite the non-matching up-down placement of indices) ∂βa Γ0 =Γa =∂ βa =δβ (a,b=1,2,3) .

This proper distance δξn = en δxν = δn(δxν −βνδt ) = δxn −βnδτ equals the distance ν ν ff δxn moved relative to the background Gullstrand-Painlevé-Cartesian coordinates, minus the distance βnδτ moved by the river.

The geodesic equation (19.27) says that the change δpk in the tetrad 4-vector pk in the time δτ is δpk =−Γk δξnpm .

Consider a fishy swimming in the Gullstrand-Painlevé river, with some arbitrary tetrad-frame 4-velocity um,andconsideratetrad-frame4-vectorpk attachedtothefishy.Ifthefishyisinfree-fall,thenthegeodesic equation of motion for pk is as usual dpk +Γk unpm =0 .

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:
- 3 Showthat(18)and(20)followfrom(29).
- sin2θ 2 Showthatfordiagonalmetrics,atmostoneofthethreetermsontherighthandsideof(18)definingthe Christoffelsymbolactuallysurvives.
- (cid:8) (cid:6) 6 I (cid:8) nsteadofvaryingtheintegral dλ g μν (X(λ))d d X λ μd d X λ ν in(13),wecould,asanalternative,varytheintegral dλg μν (X(λ))d d X λ μd d X λ ν .Showthatthisleadstothesamegeodesicequationsasinthetext.Althoughthis approachisarithmeticallysimpler,itis,asexplainedinthetext,conceptuallyopaque,sincethisintegralis notparametrizationinvariantanddoesnotrepresentageometricquantity,suchasthelengthofthecurve.
- Atinybitofsubtletygetsglossedoverhere.SeeS.Weinberg,GravitationandCosmology.

(cid:8) (cid:6) 6 I (cid:8) nsteadofvaryingtheintegral dλ g μν (X(λ))d d X λ μd d X λ ν in(13),wecould,asanalternative,varytheintegral dλg μν (X(λ))d d X λ μd d X λ ν .Showthatthisleadstothesamegeodesicequationsasinthetext.Althoughthis approachisarithmeticallysimpler,itis,asexplainedinthetext,conceptuallyopaque,sincethisintegralis notparametrizationinvariantanddoesnotrepresentageometricquantity,suchasthelengthofthecurve.

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:
- formulation of M-theory and a tool for investigating the properties of M-theory in a relatively simple setting.[48] The development of the matrix model formulation of M-theory has led physicists to consider various connections between string theory and a branch of mathematics called noncommutative geometry.
- This subject is a generalization of ordinary geometry in which mathematicians define new geometric notions using tools from noncommutative algebra.[49] In a paper from 1998, Alain Connes, Michael R.
- Douglas, and Albert Schwarz showed that some aspects of matrix models and M-theory are described by a noncommutative quantum field theory, a special kind of physical theory in which spacetime is described mathematically using noncommutative geometry.[50] This established a link between matrix models and M-theory on the one hand, and noncommutative geometry on the other hand.
- It quickly led to the discovery of other important links between noncommutative geometry and various physical theories.[51][52]

formulation of M-theory and a tool for investigating the properties of M-theory in a relatively simple setting.[48] The development of the matrix model formulation of M-theory has led physicists to consider various connections between string theory and a branch of mathematics called noncommutative geometry.

This subject is a generalization of ordinary geometry in which mathematicians define new geometric notions using tools from noncommutative algebra.[49] In a paper from 1998, Alain Connes, Michael R.

Douglas, and Albert Schwarz showed that some aspects of matrix models and M-theory are described by a noncommutative quantum field theory, a special kind of physical theory in which spacetime is described mathematically using noncommutative geometry.[50] This established a link between matrix models and M-theory on the one hand, and noncommutative geometry on the other hand.

It quickly led to the discovery of other important links between noncommutative geometry and various physical theories.[51][52]

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

List tests or derivations implied by the evidence.


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

EVIDENCE:
- I’m now standing less than ten feet away, my heart thumping, wondering if she’ll spot me, and then— Eye contact.
- Hers go wide and her mouth opens, and I can’t tell if she’s horrified or delighted or just surprised to see my face.
- She pushes through the crowd, throws her arms around my neck, and pulls me in tight with, “Oh my God, I can’t believe you came.
- I’d heard you left the country for a while or were missing or something.” I’m not sure how to respond to that, so I just say, “Well, here I am.” Daniela hasn’t worn perfume in years, but she’s wearing it tonight, and she smells like Daniela without me, like Daniela before our separate scents merged into us.

I’m now standing less than ten feet away, my heart thumping, wondering if she’ll spot me, and then— Eye contact.

Hers go wide and her mouth opens, and I can’t tell if she’s horrified or delighted or just surprised to see my face.

She pushes through the crowd, throws her arms around my neck, and pulls me in tight with, “Oh my God, I can’t believe you came.

I’d heard you left the country for a while or were missing or something.” I’m not sure how to respond to that, so I just say, “Well, here I am.” Daniela hasn’t worn perfume in years, but she’s wearing it tonight, and she smells like Daniela without me, like Daniela before our separate scents merged into us.

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:
- St Albans, Hertfordshire.[34][35] There, the family was considered highly intelligent and somewhat eccentric;[34][36] meals were often spent with each person silently reading a book.[34] They lived a frugal existence in a large, cluttered, and poorly maintained house and travelled in a converted London taxicab.[37][38] During one of Hawking's father's frequent absences working in Africa,[39] the rest of the family spent four months in Mallorca visiting his mother's friend Beryl and her husband, the poet Robert Graves.[40]

St Albans, Hertfordshire.[34][35] There, the family was considered highly intelligent and somewhat eccentric;[34][36] meals were often spent with each person silently reading a book.[34] They lived a frugal existence in a large, cluttered, and poorly maintained house and travelled in a converted London taxicab.[37][38] During one of Hawking's father's frequent absences working in Africa,[39] the rest of the family spent four months in Mallorca visiting his mother's friend Beryl and her husband, the poet Robert Graves.[40]

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

List tests or derivations implied by the evidence.