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Further reading

Every paper and book the explainer pages point to, gathered by strand. Each page lists its own at the end of its deepest fold, with a line on why each one matters.

Energy

Feeding a black hole

  1. Shakura and Sunyaev (1973), Black holes in binary systems. Observational appearance, Astronomy and Astrophysics 24, 337The thin disk, its temperature profile, and why stellar black holes fed hard are X-ray sources.
  2. Pringle (1981), Accretion discs in astrophysics, Annual Review of Astronomy and AstrophysicsThe standard review of how discs work.
  3. Balbus and Hawley (1991), A powerful local shear instability in weakly magnetized disks. I. Linear analysis, The Astrophysical JournalA weak magnetic field makes a rotating disk unstable, growing at about the rate the disk turns.
  4. Balbus and Hawley (1998), Instability, turbulence, and enhanced transport in accretion disks, Reviews of Modern PhysicsThe magnetic instability and the turbulence it drives, as the disk's viscosity.
  5. Narayan and Yi (1994), Advection-dominated accretion: a self-similar solution, The Astrophysical JournalFlows where most of the heat stays in the gas instead of being radiated.
  6. Narayan and Yi (1995), Advection-dominated accretion: underfed black holes and neutron stars, The Astrophysical JournalHot, optically thin flows exist only below a critical feed; below it, a thin disk and a hot flow are both possible solutions.
  7. Xie and Yuan (2012), The radiative efficiency of hot accretion flows, Monthly Notices of the Royal Astronomical SocietyA fit of the share of energy a hot flow radiates against its feed, from the faintest flows up to the thin-disk regime.
  8. Yuan and Narayan (2014), Hot accretion flows around black holes, Annual Review of Astronomy and AstrophysicsThe critical feeds in units of the Eddington feed; hot flows radiate less than thin disks, and less as the feed falls; Sgr A* is the prototype.
  9. Abramowicz, Czerny, Lasota and Szuszkiewicz (1988), Slim accretion disks, The Astrophysical JournalDisks fed past the ceiling, cooled by heat carried inward with the gas.
  10. Page and Thorne (1974), Disk-accretion onto a black hole. Time-averaged structure of accretion disk, The Astrophysical JournalThe fully relativistic flux from each ring of a thin disk.
  11. Gammie (1996), Layered accretion in T Tauri disks, The Astrophysical JournalMagnetic turbulence only where the gas is ionized: near the star and in a surface layer, with a quiet layer between.
  12. Landsberg and Tonge (1980), Thermodynamic energy conversion efficiencies, Journal of Applied PhysicsHow much work light can give, including the limit used above for blackbody radiation.

A gram is a fortune

  1. Einstein (1905), Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig? (Does the inertia of a body depend upon its energy content?), Annalen der PhysikA body that gives off energy loses mass: the first statement of mass-energy.
  2. Eddington (1920), The internal constitution of the stars, NatureSuggests that stars shine by turning hydrogen into helium, since a helium atom weighs less than four hydrogen atoms.
  3. Bethe (1939), Energy production in stars, Physical ReviewWorks out the reaction chains that make helium from hydrogen in stars.
  4. Huang, Wang, Kondev, Audi and Naimi (2021), The AME 2020 atomic mass evaluation (I), Chinese Physics CHow the atomic masses behind the mass defects are evaluated.
  5. Wang, Huang, Kondev, Audi and Naimi (2021), The AME 2020 atomic mass evaluation (II), Chinese Physics CThe table of atomic masses used on this page.
  6. Bardeen, Press and Teukolsky (1972), Rotating black holes: locally nonrotating frames, energy extraction, and scalar synchrotron radiation, The Astrophysical JournalThe orbits around a spinning hole, including the innermost stable orbit and its energy.
  7. Thorne (1974), Disk-accretion onto a black hole. II. Evolution of the hole, The Astrophysical JournalThe disc light a hole swallows stops its spin just short of the maximum.
  8. Gammie, Shapiro and McKinney (2004), Black hole spin evolution, The Astrophysical JournalMagnetised accretion simulations in which the spin settles below Thorne's limit.
  9. Narayan and Yi (1994), Advection-dominated accretion: a self-similar solution, The Astrophysical Journal LettersWhen gas cannot radiate fast enough, most of the energy is stored and carried into the hole.
  10. Blandford and Znajek (1977), Electromagnetic extraction of energy from Kerr black holes, Monthly Notices of the Royal Astronomical SocietyMagnetic fields can draw energy out of a spinning hole.
  11. Tchekhovskoy, Narayan and McKinney (2011), Efficient generation of jets from magnetically arrested accretion on a rapidly spinning black hole, Monthly Notices of the Royal Astronomical Society: LettersSimulated jets that carry out more energy than the gas brings in.

Mining a spinning hole

  1. Penrose (1969, reprinted 2002), Gravitational collapse: the role of general relativity, General Relativity and GravitationWhere the idea of taking energy from a hole's ergosphere first appeared.
  2. Christodoulou (1970), Reversible and irreversible transformations in black-hole physics, Physical Review LettersDefines the irreducible mass and shows what can be taken out reversibly.
  3. Hawking (1971), Gravitational radiation from colliding black holes, Physical Review LettersThe area theorem: the total horizon area never decreases.
  4. Wald (1974), Energy limits on the Penrose process, The Astrophysical JournalA break-up gains energy only if the pieces separate at more than about half the speed of light.
  5. Blandford and Znajek (1977), Electromagnetic extraction of energy from Kerr black holes, Monthly Notices of the Royal Astronomical SocietyMagnetic fields threading a spinning hole can pull energy out of its rotation.
  6. Tchekhovskoy, Narayan and McKinney (2011), Efficient generation of jets from magnetically arrested accretion on a rapidly spinning black hole, Monthly Notices of the Royal Astronomical SocietyThe jet power formula, and a simulated hole giving out more energy than it is fed.
  7. Lasota, Gourgoulhon, Abramowicz, Tchekhovskoy and Narayan (2014), Extracting black-hole rotational energy: the generalized Penrose process, Physical Review DThe Blandford-Znajek mechanism is a Penrose process, absorbing negative energy and angular momentum.
  8. Cromb, Gibson, Toninelli, Padgett, Wright and Faccio (2020), Amplification of waves from a rotating body, Nature PhysicsZel'dovich's amplification of waves by a rotating absorber, seen with sound.
  9. Brito, Cardoso and Pani (2015), Superradiance, Lecture Notes in PhysicsReview of superradiance and the Penrose process, with its efficiency limits, and the black-hole bomb.
  10. Isi, Farr, Giesler, Scheel and Teukolsky (2021), Testing the black-hole area law with GW150914, Physical Review LettersData from the first detected merger agree with the area theorem.

Heat

Recycling heat

  1. Carnot (1824), Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance, Bachelier, ParisThe waterfall picture: work comes from heat falling from hot to cold, and the most a perfect engine can get depends only on the two temperatures.
  2. Curzon and Ahlborn (1975), Efficiency of a Carnot engine at maximum power output, American Journal of PhysicsHeat flowing through finite gaps limits an engine's power; at the maximum, its efficiency is one minus the square root of the temperature ratio, and existing engines come close to that.
  3. Esposito, Kawai, Lindenberg and Van den Broeck (2010), Efficiency at maximum power of low-dissipation Carnot engines, Physical Review LettersBounds on the efficiency at maximum power for engines with low dissipation; the Curzon-Ahlborn value appears when the dissipation is symmetric.
  4. Shiraishi, Saito and Tasaki (2016), Universal trade-off relation between power and efficiency for heat engines, Physical Review LettersNonvanishing heat current always dissipates, so a heat engine with nonvanishing power never attains the Carnot efficiency.
  5. Pietzonka and Seifert (2018), Universal trade-off between power, efficiency, and constancy in steady-state heat engines, Physical Review LettersOf large power, near-Carnot efficiency and small fluctuations, a steady-state heat engine can have only two.
  6. Bejan (2002), Fundamentals of exergy analysis, entropy generation minimization, and the generation of flow architecture, International Journal of Energy ResearchExergy accounting, and the proportionality between exergy destruction and entropy generation.

Entropy is missing information

  1. Sharp and Matschinsky (2015), Translation of Ludwig Boltzmann's paper "On the relationship between the second fundamental theorem of the mechanical theory of heat and probability calculations regarding the conditions for thermal equilibrium", EntropyBoltzmann's 1877 paper in English, where entropy first became a count of arrangements.
  2. Planck (1914), The theory of heat radiation, translated by Morton Masius, P. Blakiston's SonPlanck's statement of S = k log W and his constant k, in the sections on entropy and probability.
  3. Maruyama, Nori and Vedral (2009), Colloquium: The physics of Maxwell's demon and information, Reviews of Modern PhysicsThe demon's history from its birth in 1867, and what it taught physics about information.
  4. Shannon (1948), A mathematical theory of communication, Bell System Technical JournalWhere the formula for the average information of a source, H = −Σ p log p, comes from.
  5. Szilard (1929), Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen (on the decrease of entropy by intelligent beings), Zeitschrift für PhysikThe one-molecule engine, and the first link between information and entropy.
  6. Brillouin (1951), Maxwell's demon cannot operate: information and entropy I, Journal of Applied PhysicsThe older answer to the demon: seeing the molecules costs entropy.
  7. Landauer (1961), Irreversibility and heat generation in the computing process, IBM Journal of Research and DevelopmentLogically irreversible steps, such as erasing, must release heat.
  8. Bennett (1982), The thermodynamics of computation: a review, International Journal of Theoretical PhysicsMeasurement can be reversible; the demon pays when it erases its memory.
  9. Jaynes (1957), Information theory and statistical mechanics, Physical ReviewStatistical mechanics recast as inference: pick the distribution of maximum entropy.
  10. Jarzynski (1997), Nonequilibrium equality for free energy differences, Physical Review LettersAn exact relation between the work done in repeated runs and the free-energy change.
  11. Crooks (1999), Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences, Physical Review EThe forward and reverse work distributions differ by a fixed exponential factor.
  12. Sagawa and Ueda (2008), Second law of thermodynamics with discrete quantum feedback control, Physical Review LettersThe maximum work under feedback: the free-energy change plus the mutual information gained.
  13. Sagawa and Ueda (2010), Generalized Jarzynski equality under nonequilibrium feedback control, Physical Review LettersJarzynski's equality with a demon: the information gained by measuring enters the exponent.
  14. Toyabe, Sagawa, Ueda, Muneyuki and Sano (2010), Experimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equality, Nature PhysicsA particle steered uphill by feedback gains more free energy than the work done on it, matching the generalized equality.
  15. Parrondo, Horowitz and Sagawa (2015), Thermodynamics of information, Nature PhysicsA review of how information, measurement and feedback fit into thermodynamics.

The black hole is not your freezer

  1. Hawking (1975), Particle creation by black holes, Communications in Mathematical PhysicsBlack holes radiate with a temperature set by their mass, which is why Vesper is colder than the sky.
  2. Darwin (1959), The gravity field of a particle, Proceedings of the Royal Society AOrbits of particles and light around a point mass, and the result that no ray from infinity escapes unless it is aimed wider than 3√3 GM/c², the capture radius used here.
  3. Synge (1966), The escape of photons from gravitationally intense stars, Monthly Notices of the Royal Astronomical SocietyPhotons leaving the surface of a very compact star escape only within a slender cone, which closes as the star nears its horizon; by reversibility, the same cone limits light coming in.
  4. Sanchez (1978), Absorption and emission spectra of a Schwarzschild black hole, Physical Review DThe total absorption cross-section behaves like a diffraction pattern, approaching the geometric-optics value 27πr_s²/4 in an oscillating way at high frequency.
  5. Page (1976), Particle emission rates from a black hole: massless particles from an uncharged, nonrotating hole, Physical Review DEmission rates of neutrinos, photons and gravitons from an uncharged, non-rotating hole, built on how strongly the hole absorbs each kind of wave.
  6. Winston (1970), Light collection within the framework of geometrical optics, Journal of the Optical Society of AmericaDerives the limit on light collection from phase space, a sine inequality, and designs nonimaging collectors that reach it.
  7. Winston, Miñano and Benítez (2005), Nonimaging Optics, ElsevierThe standard textbook on collecting light, including the étendue limit and the designs that reach it.
  8. Lloyd (2000), Ultimate physical limits to computation, NatureThe speed and memory limits of a 1 kg, one-litre computer, taken down to the smallest volume general relativity allows.

Cooling is the economy

  1. Dyson (1960), Search for artificial stellar sources of infrared radiation, ScienceThe original suggestion: a civilization using a star's light must re-radiate it as infrared.
  2. Carrigan (2009), IRAS-based whole-sky upper limit on Dyson spheres, The Astrophysical JournalBlackbody fits to the 1980s IRAS whole-sky catalogue, and upper limits on Dyson spheres.
  3. Wright, Mullan, Sigurdsson and Povich (2014), The Ĝ infrared search for extraterrestrial civilizations with large energy supplies. I. Background and justification, The Astrophysical JournalThe case for searching for the waste heat of civilizations with large energy supplies.
  4. Wright, Griffith, Sigurdsson, Povich and Mullan (2014), The Ĝ infrared search for extraterrestrial civilizations with large energy supplies. II. Framework, strategy, and first result, The Astrophysical JournalThe method for turning mid-infrared photometry into limits on energy use, and a first null result.
  5. Griffith, Wright, Maldonado, Povich, Sigurdsson and Mullan (2015), The Ĝ infrared search for extraterrestrial civilizations with large energy supplies. III. The reddest extended sources in WISE, The Astrophysical Journal Supplement SeriesA catalogue of the reddest galaxies in WISE, none showing the signature of nearly total starlight reprocessing.
  6. Suazo, Zackrisson, Mahto, Lundell, Nettelblad, Korn, Wright and Majumdar (2024), Project Hephaistos II. Dyson sphere candidates from Gaia DR3, 2MASS, and WISE, Monthly Notices of the Royal Astronomical SocietySeven M-dwarf candidates from a screen of millions of objects.
  7. Ren, Garrett and Siemion (2024), Background contamination of the Project Hephaistos Dyson spheres candidates, Research Notes of the AASThree candidates sit near radio sources; dust-obscured background galaxies could explain all seven.
  8. Wang, Ge, Chen, Feng and Yu (2021), Optimal heat exchanger area distribution and low-temperature heat sink temperature for power optimization of an endoreversible space Carnot cycle, EntropyWith the total heat-exchanger area fixed, a space power cycle has an optimal low-temperature heat sink temperature for maximum power.
  9. Wright et al. (2010), The Wide-field Infrared Survey Explorer (WISE): mission description and initial on-orbit performance, The Astronomical JournalThe survey behind the infrared searches, with its longest band at 22 micrometres.

Limits of computing

The cost of forgetting

  1. Landauer (1961), Irreversibility and heat generation in the computing process, IBM Journal of Research and DevelopmentThe original argument: logically irreversible steps must release heat.
  2. Szilard (1929), Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen (on the decrease of entropy by intelligent beings), Zeitschrift für PhysikThe one-molecule engine, and the first link between information and entropy.
  3. Bennett (1973), Logical reversibility of computation, IBM Journal of Research and DevelopmentAny computation can be done reversibly, and so without the erasure cost.
  4. Bennett (1982), The thermodynamics of computation: a review, International Journal of Theoretical PhysicsMaxwell's demon pays when it wipes its memory, not when it measures.
  5. Bérut et al. (2012), Experimental verification of Landauer's principle linking information and thermodynamics, NatureA trapped bead erased slowly releases heat approaching k T ln 2, and never less on average.
  6. Jun, Gavrilov and Bechhoefer (2014), High-precision test of Landauer's principle in a feedback trap, Physical Review LettersThe same test at high precision, with single runs dipping below the floor as theory predicts.
  7. Hong, Lambson, Dhuey and Bokor (2016), Experimental test of Landauer's principle in single-bit operations on nanomagnetic memory bits, Science AdvancesMagnetic memory bits erase at an energy consistent with the limit.
  8. Proesmans, Ehrich and Bechhoefer (2020), Finite-time Landauer principle, Physical Review LettersThe least extra cost of erasing in a finite time; in the slow limit it falls as one over the time.
  9. Ho, Erdil and Besiroglu (2023), Limits to the energy efficiency of CMOS microprocessors, arXiv preprintWhere the chips of the 2020s spend their energy, and how far they sit above the physical floors.
  10. Reeb and Wolf (2014), An improved Landauer principle with finite-size corrections, New Journal of PhysicsThe exact bound for small, quantum surroundings.
  11. Vaccaro and Barnett (2011), Information erasure without an energy cost, Proceedings of the Royal Society AErasure paid for in angular momentum instead of heat.
  12. Norton (2011), Waiting for Landauer, Studies in History and Philosophy of Modern PhysicsThe case that the usual derivations are circular.

Big minds are slow minds

  1. Sandberg (1999), The Physics of Information Processing Superobjects: Daily Life Among the Jupiter Brains, Journal of Evolution and Technology 5Limits on density, speed, size, heat and communication for planet-sized minds; section 7.1 measures delay as clock cycles spent waiting, and argues for layered minds.
  2. Lloyd (2000), Ultimate physical limits to computation, Nature 406Size limits parallelization: the cross-the-computer time against the bit-flip time, and why only a black hole is fully serial.
  3. Amdahl (1967), Validity of the single processor approach to achieving large scale computing capabilities, AFIPS Spring Joint Computer ConferenceThe serial part of a job caps the gain from more processors.
  4. Lamport (1978), Time, clocks, and the ordering of events in a distributed system, Communications of the ACM 21Order in a distributed system comes from messages, not from a shared clock.
  5. Fischer, Lynch and Paterson (1985), Impossibility of distributed consensus with one faulty process, Journal of the ACM 32In an asynchronous system, no deterministic protocol can guarantee agreement if one process may fail.
  6. Fischer and Lynch (1982), A lower bound for the time to assure interactive consistency, Information Processing Letters 14Tolerating f faulty parts needs at least f + 1 rounds of messages.

How much can a region remember?

  1. Bekenstein (1973), Black holes and entropy, Physical Review DThe proposal that a black hole's entropy is proportional to its horizon area.
  2. Bardeen, Carter and Hawking (1973), The four laws of black hole mechanics, Communications in Mathematical PhysicsBlack holes obey laws that mirror thermodynamics, including a first law linking mass and horizon area.
  3. Hawking (1975), Particle creation by black holes, Communications in Mathematical PhysicsBlack holes radiate at a temperature, which fixes the coefficient of the area law.
  4. Bekenstein (1981), Universal upper bound on the entropy-to-energy ratio for bounded systems, Physical Review DThe bound on information in a region, derived with a box lowered into a black hole.
  5. Unruh and Wald (1982), Acceleration radiation and the generalized second law of thermodynamics, Physical Review DArgues that buoyancy in the thermal atmosphere changes the lowered-box argument.
  6. 't Hooft (1993), Dimensional reduction in quantum gravity, arXiv preprintProposes that a world with gravity could be described by variables on a two-dimensional surface.
  7. Susskind (1995), The world as a hologram, Journal of Mathematical PhysicsDevelops 't Hooft's proposal as the holographic principle, with one degree of freedom per Planck area.
  8. Strominger and Vafa (1996), Microscopic origin of the Bekenstein-Hawking entropy, Physics Letters BCounts the microstates of certain extremal string-theory black holes and recovers A/4.
  9. Maldacena (1998), The large N limit of superconformal field theories and supergravity, Advances in Theoretical and Mathematical PhysicsConjectures that string theory in anti-de Sitter space is dual to a field theory on its boundary.
  10. Bousso (1999), A covariant entropy conjecture, Journal of High Energy PhysicsThe light-sheet form of the entropy bound, conjectured for all spacetimes that obey Einstein's equations.
  11. Bousso (2002), The holographic principle, Reviews of Modern PhysicsThe review: the covariant bound, its tests, and the limit of about 10^69 bits per square metre.
  12. Bousso (2003), Light-sheets and Bekenstein's entropy bound, Physical Review LettersDerives a version of Bekenstein's bound for weakly gravitating systems from the covariant bound.
  13. Casini (2008), Relative entropy and the Bekenstein bound, Classical and Quantum GravityRecasts the bound in terms of relative entropy, so that it holds automatically in flat-space field theory.
  14. Bekenstein (2005), How does the entropy/information bound work?, Foundations of PhysicsWhy purported counterexamples to the bound fail.
  15. Bartol et al. (2015), Nanoconnectomic upper bound on the variability of synaptic plasticity, eLifeAt least 26 distinguishable synaptic strengths, or 4.7 bits per synapse, in rat hippocampus.

What a quantum computer is, and is not

  1. Deutsch and Jozsa (1992), Rapid solution of problems by quantum computation, Proceedings of the Royal Society AA problem a quantum computer solves with certainty exponentially faster than any deterministic classical method.
  2. Cleve, Ekert, Macchiavello and Mosca (1998), Quantum algorithms revisited, Proceedings of the Royal Society ADeutsch-Jozsa with a single query, and quantum algorithms read as interference.
  3. Grover (1996), A fast quantum mechanical algorithm for database search, Proceedings of the 28th ACM Symposium on Theory of ComputingUnstructured search in about the square root of the list length.
  4. Shor (1997), Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM Journal on ComputingFactoring in polynomial time on a quantum computer.
  5. Bennett, Bernstein, Brassard and Vazirani (1997), Strengths and weaknesses of quantum computing, SIAM Journal on ComputingRelative to a random oracle, NP needs about 2^(n/2) quantum time, which Grover's algorithm meets.
  6. Holevo (1973), Bounds for the quantity of information transmitted by a quantum communication channel, Problems of Information Transmission 9(3), 177 to 183The most that can be learned about a quantum message by measuring it.
  7. Fowler, Mariantoni, Martinis and Cleland (2012), Surface codes: towards practical large-scale quantum computation, Physical Review AThe surface code and its tolerance of error rates as high as about one percent.
  8. Acharya et al. (2024), Quantum error correction below the surface code threshold, NatureLogical error falls by a factor of 2.14 per step in code size, down to 0.143% per cycle at distance 7.
  9. Krantz, Kjaergaard, Yan, Orlando, Gustavsson and Oliver (2019), A quantum engineer's guide to superconducting qubits, Applied Physics ReviewsTypical qubits near 5 GHz, run at about 20 mK, and the noise that limits them.
  10. Preskill (2018), Quantum computing in the NISQ era and beyond, QuantumNoisy 50 to 100 qubit machines: what they might do, and why noise caps the circuit size.
  11. Arute et al. (2019), Quantum supremacy using a programmable superconducting processor, NatureThe 53-qubit sampling experiment: about 200 seconds against an estimated 10,000 years.
  12. Pan, Chen and Zhang (2022), Solving the sampling problem of the Sycamore quantum circuits, Physical Review LettersA tensor-network method that reproduces the samples in about 15 hours on 512 GPUs.
  13. Pednault, Gunnels, Nannicini, Horesh and Wisnieff (2019), Leveraging secondary storage to simulate deep 54-qubit Sycamore circuits, arXiv preprintIBM's answer: an analysis that the Summit supercomputer could simulate such circuits in days.
  14. Gidney and Ekerå (2021), How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits, QuantumA resource estimate for running Shor's algorithm.
  15. Gidney (2025), How to factor 2048 bit RSA integers with less than a million noisy qubits, arXiv preprintThe same task in under a week with under a million noisy qubits, at the same 0.1% gate error rate.

Computing without forgetting

  1. Bennett (1973), Logical reversibility of computation, IBM Journal of Research and DevelopmentAny computation can be made reversible by keeping a history, copying out the answer and running backwards.
  2. Bennett (1989), Time/space trade-offs for reversible computation, SIAM Journal on ComputingA pebbling argument: time O(T^(1+ε)) and space O(S log T), or linear time and space O(S T^ε).
  3. Fredkin and Toffoli (1982), Conservative logic, International Journal of Theoretical PhysicsComputing under reversibility and conservation laws, the Fredkin gate and the billiard-ball model.
  4. Bennett (1982), The thermodynamics of computation: a review, International Journal of Theoretical PhysicsBallistic and Brownian computers; a Brownian computer's dissipation goes to zero only as its speed does.
  5. Athas, Svensson, Koller, Tzartzanis and Chou (1994), Low-power digital systems based on adiabatic-switching principles, IEEE Transactions on VLSI SystemsSwitching transistor logic adiabatically so that the charge on the circuit's capacitances is recycled.
  6. Lange, McKenzie and Tapp (2000), Reversible space equals deterministic space, Journal of Computer and System SciencesA reversible machine can simulate an ordinary one in the same space.
  7. Takeuchi, Yamanashi and Yoshikawa (2013), Measurement of 10 zJ energy dissipation of adiabatic quantum-flux-parametron logic using a superconducting resonator, Applied Physics LettersThe bit energy of an adiabatic superconducting gate, measured with a high-Q resonator.
  8. Takeuchi, Yamanashi and Yoshikawa (2017), Reversibility and energy dissipation in adiabatic superconductor logic, Scientific ReportsIn simulation, a reversible superconducting gate dissipates less than the Landauer bound at low frequency; irreversible ones cannot.
  9. Tolpygo, Golden and Semenov (2026; arXiv preprint 2512.22791, 2025), Demonstration of superconductor shift registers with energy dissipation below Landauer's thermodynamic limit, IEEE Transactions on Applied SuperconductivityA circular shift register of Josephson vortices with energy per bit-shift below k T ln 2, because no information is destroyed.
  10. Frank and Edwards (2025), Industry perspective: Limits of energy efficiency for conventional CMOS and the need for adiabatic reversible computing, APL Electronic DevicesWhy conventional CMOS efficiency is nearly capped, and the case for adiabatic reversible CMOS.

How fast can anything think?

  1. Margolus and Levitin (1998), The maximum speed of dynamical evolution, Physica DThe proof that a system of mean energy E needs at least πħ/2E to reach an orthogonal state, and twice that per step along a long chain of states.
  2. Mandelstam and Tamm (1945), The uncertainty relation between energy and time in non-relativistic quantum mechanics, J. Phys. USSR 9, 249; the DOI is the 1991 reprint in I. E. Tamm, Selected Papers (Springer), pp. 115 to 123The first speed limit, written in terms of the spread of the energy.
  3. Lloyd (2000), Ultimate physical limits to computation, NatureThe speed and memory limits of a one-kilogram "ultimate laptop".
  4. Levitin and Toffoli (2009), Fundamental limit on the rate of quantum dynamics: the unified bound is tight, Physical Review LettersThe two bounds combined, and proved tight: approached by a family of states, never beaten.
  5. Giovannetti, Lloyd and Maccone (2003), Quantum limits to dynamical evolution, Physical Review ASpeed limits for evolving by a given amount; entanglement is needed to reach the ultimate speed when the energy is spread over many parts.
  6. del Campo, Egusquiza, Plenio and Huelga (2013), Quantum speed limits in open system dynamics, Physical Review LettersA time-energy uncertainty relation for open systems with completely positive, trace-preserving dynamics.
  7. Taddei, Escher, Davidovich and de Matos Filho (2013), Quantum speed limit for physical processes, Physical Review LettersA speed limit for non-unitary processes, linked to quantum Fisher information.
  8. Deffner and Campbell (2017), Quantum speed limits: from Heisenberg's uncertainty principle to optimal quantum control, Journal of Physics AThe review of the whole subject, closed and open systems.
  9. Jordan (2017), Fast quantum computation at arbitrarily low energy, Physical Review AEnergy alone does not bound a computer's clock speed; other assumptions are needed.

The ultimate computer is a black hole

  1. Lloyd (2000), Ultimate physical limits to computation, NatureThe ultimate laptop, the black-hole limit of speed and memory, why it is serial, and the box on whether a hole can compute.
  2. Lloyd and Ng (2004), Black Hole Computers, Scientific AmericanThe same ideas for general readers: the hole as the most compressed computer, how its information might get out, and the universe as a computer.
  3. Hawking (1975), Particle creation by black holes, Communications in Mathematical PhysicsBlack holes radiate at a temperature, which fixes the lifetime.
  4. Page (1976), Particle emission rates from a black hole: massless particles from an uncharged, nonrotating hole, Physical Review DThe emission rates that set the lifetime prefactor.
  5. Hawking (1976), Breakdown of predictability in gravitational collapse, Physical Review DThe argument that evaporation destroys information.
  6. Hawking (2005), Information loss in black holes, Physical Review DHawking's later argument that the evolution is unitary and information is preserved.
  7. Page (1993), Information in black hole radiation, Physical Review LettersIf evaporation is unitary, information may come out so slowly at first, or so spread out, that small-correction calculations miss it.
  8. Page (2013), Time dependence of Hawking radiation entropy, Journal of Cosmology and Astroparticle PhysicsFor a hole emitting photons and gravitons, the radiation's entropy peaks a little past half the lifetime, when the hole has lost about two fifths of its entropy.
  9. Hayden and Preskill (2007), Black holes as mirrors: quantum information in random subsystems, Journal of High Energy PhysicsAfter the half-way point, information thrown in is revealed very fast; before it, it stays hidden.
  10. Sekino and Susskind (2008), Fast scramblers, Journal of High Energy PhysicsThe conjecture that black holes are the fastest scramblers, in a time logarithmic in the number of degrees of freedom.
  11. Penington (2020, preprint 2019), Entanglement wedge reconstruction and the information paradox, Journal of High Energy PhysicsDerives the Page curve and the Hayden-Preskill criterion from a phase transition at the Page time.
  12. Almheiri, Engelhardt, Marolf and Maxfield (2019), The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole, Journal of High Energy PhysicsA quantum extremal surface transition at the Page time in a two-dimensional model.
  13. Almheiri, Hartman, Maldacena, Shaghoulian and Tajdini (2021), The entropy of Hawking radiation, Reviews of Modern PhysicsThe review of how the entropy of Hawking radiation is calculated.

Black holes

A pair too wide to hear

  1. Peters (1964), Gravitational radiation and the motion of two point masses, Physical ReviewThe decay of an orbit under gravitational radiation, and the merger time for circular and eccentric orbits.
  2. Peters and Mathews (1963), Gravitational radiation from point masses in a Keplerian orbit, Physical ReviewThe power radiated by an orbiting pair.
  3. Hulse and Taylor (1975), Discovery of a pulsar in a binary system, Astrophysical Journal LettersThe pulsar whose orbit later gave the first evidence of gravitational radiation.
  4. Taylor and Weisberg (1982), A new test of general relativity: gravitational radiation and the binary pulsar PSR 1913+16, Astrophysical JournalThe orbital period decays as general relativity predicts.
  5. Weisberg and Huang (2016), Relativistic measurements from timing the binary pulsar PSR B1913+16, Astrophysical JournalDecades of timing: observed and predicted decay agree to a fraction of a percent.
  6. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration, 2016), Observation of gravitational waves from a binary black hole merger, Physical Review LettersGW150914: the first direct detection of gravitational waves.
  7. Abbott et al. (2023), GWTC-3: compact binary coalescences observed by LIGO and Virgo during the second part of the third observing run, Physical Review XThe catalogue of candidate mergers through 2020.
  8. LIGO Scientific Collaboration (Aasi et al., 2015), Advanced LIGO, Classical and Quantum GravityThe ground detectors and their band.
  9. Amaro-Seoane et al. (2017), Laser Interferometer Space Antenna, arXiv preprintThe space detector proposal and its millihertz band.
  10. Agazie et al. (NANOGrav, 2023), The NANOGrav 15 yr data set: evidence for a gravitational-wave background, Astrophysical Journal LettersPulsar timing finds a nanohertz background of waves.
  11. Sesana et al. (2021), Unveiling the gravitational universe at μ-Hz frequencies, Experimental AstronomyThe microhertz gap between pulsar timing and LISA, and a detector proposed for it.
  12. El-Badry et al. (2023), A Sun-like star orbiting a black hole, Monthly Notices of the Royal Astronomical SocietyGaia BH1: a dormant black hole found from a star's astrometric orbit.
  13. El-Badry et al. (2023), A red giant orbiting a black hole, Monthly Notices of the Royal Astronomical SocietyGaia BH2, on a wider orbit.
  14. Gaia Collaboration (Panuzzo et al., 2024), Discovery of a dormant 33 solar-mass black hole in pre-release Gaia astrometry, Astronomy and AstrophysicsGaia BH3, the most massive stellar black hole known in the Galaxy in 2024.

Black holes glow

  1. Hawking (1974), Black hole explosions?, NatureThe announcement: black holes should radiate like a body with a temperature, and small ones should end in an explosion.
  2. Hawking (1975), Particle creation by black holes, Communications in Mathematical PhysicsThe full calculation of the glow, from quantum fields in the geometry of a collapsing star.
  3. Bekenstein (1973), Black holes and entropy, Physical Review DThe argument that a hole's entropy is proportional to the area of its horizon.
  4. Unruh (1976), Notes on black-hole evaporation, Physical Review DThe evaporation worked out again, and the thermal bath felt by an accelerating observer.
  5. Page (1976), Particle emission rates from a black hole: massless particles from an uncharged, nonrotating hole, Physical Review DHow fast a hole sheds photons, neutrinos and gravitons, with the filtering by the hole's own gravity included, and which primordial holes are gone by now.
  6. Page (1976), Particle emission rates from a black hole. II. Massless particles from a rotating hole, Physical Review DA spinning hole sheds its angular momentum several times faster than its energy, so it is nearly spinless before most of its mass is gone.
  7. Carr, Kohri, Sendouda and Yokoyama (2010), New cosmological constraints on primordial black holes, Physical Review DThe lifetime with every particle a hot hole makes, and the mass that evaporates in the universe's age.
  8. Hawking (1976), Breakdown of predictability in gravitational collapse, Physical Review DThe case that complete evaporation destroys information.
  9. Page (1993), Information in black hole radiation, Physical Review LettersIf evaporation preserves information, the radiation's entropy must rise and then fall: the Page curve.
  10. Penington (2020, preprint 2019), Entanglement wedge reconstruction and the information paradox, Journal of High Energy PhysicsA way to compute the radiation's entropy that follows the Page curve, and how the information comes out.
  11. Almheiri, Engelhardt, Marolf and Maxfield (2019), The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole, Journal of High Energy PhysicsThe same turn in the entropy, found in a solvable model of an evaporating hole.
  12. Almheiri, Mahajan, Maldacena and Zhao (2020, preprint 2019), The Page curve of Hawking radiation from semiclassical geometry, Journal of High Energy PhysicsThe island rule: count a region inside the hole as part of the radiation, and its entropy follows the Page curve.
  13. Almheiri, Hartman, Maldacena, Shaghoulian and Tajdini (2021), The entropy of Hawking radiation, Reviews of Modern PhysicsA review of the island calculations and what they do and do not show.
  14. Unruh (1981), Experimental black-hole evaporation?, Physical Review LettersSound in a flowing fluid has a horizon of its own, and should radiate.
  15. Weinfurtner, Tedford, Penrice, Unruh and Lawrence (2011), Measurement of stimulated Hawking emission in an analogue system, Physical Review LettersWater waves meeting a horizon in a tank behave as the stimulated Hawking effect predicts.
  16. Steinhauer (2016), Observation of quantum Hawking radiation and its entanglement in an analogue black hole, Nature PhysicsSpontaneous emission from a sonic horizon in a Bose-Einstein condensate, with correlated partners on either side.
  17. Muñoz de Nova, Golubkov, Kolobov and Steinhauer (2019), Observation of thermal Hawking radiation and its temperature in an analogue black hole, NatureThe spectrum of the radiation from a condensate horizon is thermal, at the temperature predicted from the flow.

Why the hole looks like that

  1. Dyson, Eddington and Davidson (1920), A determination of the deflection of light by the sun's gravitational field, from observations made at the total eclipse of May 29, 1919, Philosophical Transactions of the Royal Society AThe 1919 eclipse measurement of starlight bending at the Sun's edge.
  2. Einstein (1936), Lens-like action of a star by the deviation of light in the gravitational field, ScienceThe ring and double image made by a star acting as a lens.
  3. Darwin (1959), The gravity field of a particle, Proceedings of the Royal Society ANo ray from infinity escapes unless it is aimed wider than 3√3 GM/c², and every star seen past the mass shows a series of faint ghosts just outside that radius.
  4. Synge (1966), The escape of photons from gravitationally intense stars, Monthly Notices of the Royal Astronomical SocietyWhich photons escape a Schwarzschild star, and how large the dark region looks from each radius.
  5. Luminet (1979), Image of a spherical black hole with thin accretion disk, Astronomy and Astrophysics 75, 228The first computed image of a black hole with a glowing disc, with its lopsided brightness from the disc's motion.
  6. Luminet (2019), An illustrated history of black hole imaging: personal recollections (1972-2002), arXiv preprintHis own account of the 1979 image and of the thirty years of computed pictures that followed.
  7. James, von Tunzelmann, Franklin and Thorne (2015), Gravitational lensing by spinning black holes in astrophysics, and in the movie Interstellar, Classical and Quantum GravityHow the film's renderer traced ray bundles round a spinning hole, and what lensing looks like with the camera close in.
  8. Gralla, Holz and Wald (2019), Black hole shadows, photon rings, and lensing rings, Physical Review DThe size of the dark patch is set by the glowing gas, and the photon ring adds only a little light.
  9. Johnson et al. (2020), Universal interferometric signatures of a black hole's photon ring, Science AdvancesThe subrings leave a signature on long baselines that could measure mass and spin.
  10. Event Horizon Telescope Collaboration (2019), First M87 Event Horizon Telescope results. I. The shadow of the supermassive black hole, The Astrophysical Journal LettersThe M87 ring, 42 ± 3 microarcseconds across, around a dark depression.
  11. Event Horizon Telescope Collaboration (2019), First M87 Event Horizon Telescope results. VI. The shadow and mass of the central black hole, The Astrophysical Journal LettersThe ring size gives the angular gravitational radius and, with the distance, the mass.
  12. Event Horizon Telescope Collaboration (2022), First Sagittarius A Event Horizon Telescope results. I. The shadow of the supermassive black hole in the center of the Milky Way*, The Astrophysical Journal LettersA ring 51.8 ± 2.3 microarcseconds across, consistent with a black hole of about 4 million Suns.
  13. El-Badry et al. (2023), A Sun-like star orbiting a black hole, Monthly Notices of the Royal Astronomical SocietyGaia BH1: a dark companion of 9.62 Suns found from the star's astrometric orbit.
  14. El-Badry et al. (2023), A red giant orbiting a black hole, Monthly Notices of the Royal Astronomical SocietyGaia BH2: a dark companion of 8.9 Suns on a 1,277-day orbit.
  15. Gaia Collaboration (2024), Discovery of a dormant 33 solar-mass black hole in pre-release Gaia astrometry, Astronomy and AstrophysicsGaia BH3, 32.70 Suns at 590 parsecs.
  16. Sahu et al. (2022), An isolated stellar-mass black hole detected through astrometric microlensing, The Astrophysical JournalHubble sees a background star shift as a dark lens of 7.1 Suns passes in front.
  17. Lam et al. (2022), An isolated mass-gap black hole or neutron star detected with astrometric microlensing, The Astrophysical Journal LettersThe same event, analysed independently, gives 1.6 to 4.4 Suns.
  18. Mróz, Udalski and Gould (2022), Systematic errors as a source of mass discrepancy in black hole microlensing event OGLE-2011-BLG-0462, The Astrophysical Journal LettersSystematic errors explain the disagreement; the lens is a black hole of 7.88 Suns.
  19. Lam and Lu (2023), A reanalysis of the isolated black hole candidate OGLE-2011-BLG-0462/MOA-2011-BLG-191, The Astrophysical JournalNew Hubble data and a bias from a nearby star: the lens is a black hole of about 6 Suns.

Where you can live around a hole and a star

  1. Holman and Wiegert (1999), Long-term stability of planets in binary systems, The Astronomical Journal 117, 621The simulations and the fitted formulas for the close and far limits, the islands of instability beyond the far limit, and the L4/L5 mass condition.
  2. Mudryk and Wu (2006), Resonance overlap is responsible for ejecting planets in binary systems, The Astrophysical Journal 639, 423Why the edge around one star is where it is, why it is jagged, and why the Hill test works for a circular pair.
  3. Domingos, Winter and Yokoyama (2006), Stable satellites around extrasolar giant planets, Monthly Notices of the Royal Astronomical Society 373, 1227How far out a moon can orbit a planet and survive: about half the Hill radius going forward, about the whole Hill radius going backward.
  4. Murray and Dermott (1999), Solar System Dynamics, Cambridge University PressThe restricted three-body problem, the Jacobi constant and the Lagrange points, in a standard textbook.
  5. Doyle et al. (2011), Kepler-16: a transiting circumbinary planet, Science 333, 1602A Saturn-sized planet seen crossing both stars of a pair, and the masses and orbits of all three bodies.
  6. Welsh et al. (2012), Transiting circumbinary planets Kepler-34 b and Kepler-35 b, Nature 481, 475Two more such planets, and the estimate that at least about 148 percent of close pairs host coplanar giant planets.

Ships die of tides, not of curvature

  1. Eggleton (1983), Approximations to the radii of Roche lobes, The Astrophysical Journal 268, 368A simple formula, good to about one percent, for the size of a star's Roche lobe at any mass ratio.
  2. Pirani (1956; republished 2009), On the physical significance of the Riemann tensor, General Relativity and GravitationThe Riemann tensor read as the relative acceleration of neighbouring free particles, which is the tidal field.
  3. Marck (1983), Solution to the equations of parallel transport in Kerr geometry; tidal tensor, Proceedings of the Royal Society AA frame carried along any free-fall path around a spinning hole, in closed form, and the tidal tensor written in it.
  4. Hills (1975), Possible power source of Seyfert galaxies and QSOs, NatureStars broken up inside a massive black hole's Roche limit as the engine of quasars and Seyfert nuclei; the swallowed-whole boundary bears his name.
  5. Rees (1988), Tidal disruption of stars by black holes of 10⁶ to 10⁸ solar masses in nearby galaxies, NaturePart of a shredded star's debris escapes and the rest is swallowed, making a bright flare.
  6. Kesden (2012), Tidal-disruption rate of stars by spinning supermassive black holes, Physical Review DRelativistic tides at closest approach, and how spin raises the heaviest hole that can disrupt a Sun-like star.
  7. Gezari (2021), Tidal Disruption Events, Annual Review of Astronomy and AstrophysicsA census of 56 observed events and what they show.
  8. Event Horizon Telescope Collaboration (2019), First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole, The Astrophysical Journal LettersThe image of M87*'s shadow, and the mass this page uses for it.

Clocks near the horizon

  1. Pound and Rebka (1960), Apparent weight of photons, Physical Review LettersGamma rays sent up and down a tower at Harvard shift in frequency as the equivalence principle predicts.
  2. Pound and Snider (1965), Effect of gravity on gamma radiation, Physical ReviewThe same experiment, redone ten times more precisely.
  3. Hafele and Keating (1972), Around-the-world atomic clocks: observed relativistic time gains, ScienceCaesium clocks flown round the world eastward and westward lost and gained time as relativity predicts.
  4. Chou, Hume, Rosenband and Wineland (2010), Optical clocks and relativity, ScienceAluminium-ion optical clocks see time dilation from a height change of less than a metre and from speeds of a few metres a second.
  5. Ashby (2003), Relativity in the Global Positioning System, Living Reviews in RelativityHow the satellite clocks are set for gravity and for speed, and why the system would not work without it.
  6. Kerr (1963), Gravitational field of a spinning mass as an example of algebraically special metrics, Physical Review LettersThe spacetime of a spinning black hole.
  7. Bardeen, Press and Teukolsky (1972), Rotating black holes: locally nonrotating frames, energy extraction, and scalar synchrotron radiation, The Astrophysical JournalThe zero-angular-momentum observers of a spinning hole.
  8. Event Horizon Telescope Collaboration (2019), First M87 Event Horizon Telescope results. VI. The shadow and mass of the central black hole, The Astrophysical Journal LettersThe size of the shadow of the hole in M87, and the mass it gives.

What is a black hole?

  1. Michell (1784), On the means of discovering the distance, magnitude, &c. of the fixed stars, in consequence of the diminution of the velocity of their light, Philosophical Transactions of the Royal Society of LondonThe first argument that a star could be massive and compact enough to keep its light in, and found only by its pull on a companion.
  2. Schwarzschild (1916), Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, pages 189 to 196; linked is the English translation by S. Antoci and A. Loinger (1999), On the gravitational field of a mass point according to Einstein's theory, arXivThe exact solution for empty space around a non-spinning mass, found within weeks of Einstein's field equations.
  3. Oppenheimer and Snyder (1939), On continued gravitational contraction, Physical ReviewA massive star with no fuel left collapses without limit and closes itself off from outside observers.
  4. Finkelstein (1958), Past-future asymmetry of the gravitational field of a point particle, Physical ReviewCoordinates in which the horizon is a one-way membrane and not a singular wall.
  5. Penrose (1965), Gravitational collapse and space-time singularities, Physical Review LettersOnce light is trapped, a singularity must form, with no need for perfect symmetry.
  6. Kerr (1963), Gravitational field of a spinning mass as an example of algebraically special metrics, Physical Review LettersThe exact solution for a spinning black hole.
  7. Event Horizon Telescope Collaboration (2019), First M87 Event Horizon Telescope results. I. The shadow of the supermassive black hole, The Astrophysical Journal LettersThe first image of a horizon-scale shadow, the size general relativity predicts for a hole of 6.5 billion Suns.
  8. Isi, Giesler, Farr, Scheel and Teukolsky (2019), Testing the no-hair theorem with GW150914, Physical Review LettersThe ringdown read as a main tone and an overtone, both fixed by one mass and one spin.
  9. Abbott et al. (2016), Observation of gravitational waves from a binary black hole merger, Physical Review LettersGW150914: two holes of about 36 and 29 Suns merging into one of 62.
  10. Abbott et al. (2023), Population of merging compact binaries inferred using gravitational waves through GWTC-3, Physical Review XRates and masses of merging black holes, with chirp-mass peaks near 8 and 28 Suns, and no verdict on the lower mass gap.
  11. Özel, Psaltis, Narayan and McClintock (2010), The black hole mass distribution in the Galaxy, The Astrophysical JournalSixteen holes in X-ray binaries fit a narrow distribution near 7.8 Suns, with none between 2 and 5.
  12. Abac et al. (2024), Observation of gravitational waves from the coalescence of a 2.5 to 4.5 solar-mass compact object and a neutron star, The Astrophysical Journal LettersGW230529: a merger with an object, most probably a black hole, inside the supposed mass gap.

The voyage

Pushed by light

  1. Maxwell (1873), A Treatise on Electricity and Magnetism, volume 2 (Cambridge reissue, 2010)The electromagnetic theory of light, with the prediction (article 792) that light presses on a surface it falls on.
  2. Lebedew (1901), Untersuchungen über die Druckkräfte des Lichtes (investigations of the pressure forces of light), Annalen der PhysikThe first laboratory measurement of the pressure of light.
  3. Tsuda, Mori, Funase et al. (2011), Flight status of IKAROS deep space solar sail demonstrator, Acta AstronauticaThe first solar sail flown in deep space, unfolded on the way to Venus.
  4. Forward (1984), Roundtrip interstellar travel using laser-pushed lightsails, Journal of Spacecraft and RocketsThe proposal for a laser-pushed sail to the stars and back.
  5. Lubin (2016), A Roadmap to Interstellar Flight, Journal of the British Interplanetary Society 69, 40A programme to send wafer-scale craft to a large fraction of light speed on directed energy.
  6. Parkin (2018), The Breakthrough Starshot system model, Acta AstronauticaThe cost-optimal Starshot design, and what its beam and sail would have to be.
  7. Kulkarni, Lubin and Zhang (2018), Relativistic spacecraft propelled by directed energy, The Astronomical JournalA full relativistic solution of the sail's motion, and the diffraction limit on its speed.
  8. Manchester and Loeb (2017), Stability of a light sail riding on a laser beam, The Astrophysical Journal LettersConical sails are unstable on a beam; a spherical shell is passively stable.
  9. Atwater et al. (2018), Materials challenges for the Starshot lightsail, Nature MaterialsThe criteria a Starshot sail film must meet, and the materials research they call for.

Bits travel, atoms do not

  1. Shannon (1948), A Mathematical Theory of Communication, Bell System Technical JournalThe founding paper on information and channel capacity.
  2. Kogelnik and Li (1966), Laser Beams and Resonators, Applied OpticsHow a laser beam is described and how it spreads.
  3. Gordon (1962), Quantum Effects in Communications Systems, Proceedings of the IREAn early quantum-mechanical treatment of the limits of optical communication.
  4. Giovannetti, Guha, Lloyd, Maccone, Shapiro and Yuen (2004), Classical Capacity of the Lossy Bosonic Channel: The Exact Solution, Physical Review LettersThe exact capacity of a beam that loses light, reached by coherent states, with far-field free-space links as the example.
  5. Giovannetti, García-Patrón, Cerf and Holevo (2014), Ultimate classical communication rates of quantum optical channels, Nature PhotonicsThe vacuum gives the least output entropy, which settles the capacity of Gaussian channels with noise.
  6. Wootters and Zurek (1982), A single quantum cannot be cloned, NatureAn unknown quantum state cannot be copied.
  7. Parfit (1986; first published 1984), Reasons and Persons, Oxford University PressThe teleporter and branch-line cases: duplication and what matters in personal identity.
  8. NASA, Deep Space Optical Communications (DSOC), mission pageThe flight transceiver, the Palomar receiver and the goals of the demonstration.
  9. NASA (December 2023), NASA's Tech Demo Streams First Video from Deep Space via LaserThe 267 megabit-per-second video downlink from 31 million kilometres.

Stopping from 0.1c

  1. Gros (2017), Universal scaling relation for magnetic sails: momentum braking in the limit of dilute interstellar media, Journal of Physics CommunicationsProtons traced through a current loop's field; fast interstellar braking needs of the order of a thousand tonnes of sail.
  2. Zubrin and Andrews (1991), Magnetic sails and interplanetary travel, Journal of Spacecraft and RocketsThe magnetic sail in the solar wind, and the fluid model of its drag.
  3. Zubrin and Martin (2000), The Magnetic Sail, final report to the NASA Institute for Advanced ConceptsThe fluid model applied to braking from a tenth of light speed.
  4. Frisch, Redfield and Slavin (2011), The Interstellar Medium Surrounding the Sun, Annual Review of Astronomy and AstrophysicsThe warm, partly ionized clouds around the Sun, their densities and temperatures.
  5. Heller and Hippke (2017), Deceleration of high-velocity interstellar photon sails into bound orbits at α Centauri, The Astrophysical Journal LettersStarlight and gravity assists from three stars park a graphene-thin sail arriving at thousands of kilometres per second.
  6. Forward (1984), Roundtrip interstellar travel using laser-pushed lightsails, Journal of Spacecraft and RocketsA staged sail that brakes on the launch beam, reflected from its own outer ring.

Ideas

Inward, not outward

  1. Kardashev (1964), Transmission of information by extraterrestrial civilizations, Soviet Astronomy 8, 217The paper that proposed ranking civilizations by their energy use, as Types I, II and III.
  2. Sagan (1973), The Cosmic Connection: An Extraterrestrial Perspective, DoubledayWhere Sagan proposed fractional types, with Type 1.0 as a fixed power and the Earth of his day at about Type 0.7.
  3. Barrow (1998), Impossibility: The Limits of Science and the Science of Limits, Oxford University PressThe ladder of civilizations ranked by the smallest scale they can manipulate, Type I-minus to Type Ω-minus.
  4. Ćirković (2015), Kardashev's classification at 50+: a fine vehicle with room for improvement, Serbian Astronomical Journal 191A review of the scale's first fifty years, its modifications and its difficulties.
  5. Dyson (1960), Search for artificial stellar sources of infrared radiation, Science 131The proposal to look for the waste heat of civilizations that use a star's output.
  6. Wright, Mullan, Sigurdsson and Povich (2014), The Ĝ infrared search for extraterrestrial civilizations with large energy supplies. I. Background and justification, The Astrophysical Journal 792The case for looking for large energy supplies in mid-infrared surveys.
  7. Griffith et al. (2015), The Ĝ infrared search for extraterrestrial civilizations with large energy supplies. III. The reddest extended sources in WISE, The Astrophysical Journal Supplement Series 217A WISE infrared search of a hundred thousand galaxies; none shows a civilization reprocessing nearly all of its starlight.
  8. Sandberg, Armstrong and Ćirković (2017), That is not dead which can eternal lie: the aestivation hypothesis for resolving Fermi's paradoxWaiting for a colder universe could multiply the computation available, which would explain a quiet sky.
  9. Smart (2012), The transcension hypothesis: sufficiently advanced civilizations invariably leave our universe, and implications for METI and SETI, Acta Astronautica 78The argument that advanced civilizations move toward ever smaller scales instead of spreading outward.
  10. Dennard, Gaensslen, Yu, Rideout, Bassous and LeBlanc (1974), Design of ion-implanted MOSFET's with very small physical dimensions, IEEE Journal of Solid-State Circuits 9The scaling rules under which switching energy fell with the cube of size.

Thinking in a cooling universe

  1. Dyson (1979), Time without end: physics and biology in an open universe, Reviews of Modern PhysicsThe scaling hypothesis and hibernation: in an ever-cooling open universe, unlimited subjective time on a finite energy supply.
  2. Adams and Laughlin (1997), A dying universe: the long-term fate and evolution of astrophysical objects, Reviews of Modern PhysicsThe timeline of the far future, from the last stars to the black-hole era.
  3. Krauss and Starkman (2000), Life, the Universe, and Nothing: life and death in an ever-expanding universe, The Astrophysical JournalWith a cosmological constant, a civilization's total conscious lifetime and recoverable information are finite.
  4. Gibbons and Hawking (1977), Cosmological event horizons, thermodynamics, and particle creation, Physical Review DA cosmological horizon has a temperature, which becomes the floor under the sky.
  5. DESI Collaboration (2025), DESI DR2 results II: measurements of baryon acoustic oscillations and cosmological constraints, Physical Review D 112, 083515Combined with other data, a preference short of discovery for dark energy that weakens with time, which would reopen the question of the final fate.

Copy, merge, choose your clock

  1. Sandberg and Bostrom (2008), Whole Brain Emulation: A Roadmap, Technical Report 2008-3, Future of Humanity Institute, OxfordThe levels of emulation, storage and compute by level (Tables 8 and 9), and a survey of published estimates of the brain's capacity (Table 10); it leaves identity and ethics aside.
  2. Bartol, Bromer, Kinney, Chirillo, Bourne, Harris and Sejnowski (2015), Nanoconnectomic upper bound on the variability of synaptic plasticity, eLife 4At least 26 distinguishable synaptic strengths in rat hippocampus, about 4.7 bits per synapse.
  3. Landauer (1986), How much do people remember? Some estimates of the quantity of learned information in long-term memory, Cognitive Science 10Estimates of the functional information in human long-term memory ranged around a billion bits.
  4. Deutsch (1985), Quantum theory, the Church-Turing principle and the universal quantum computer, Proceedings of the Royal Society A 400States the Church-Turing hypothesis as a physical principle: every finitely realizable physical system can be perfectly simulated by a universal computing machine.
  5. Bennett (1973), Logical reversibility of computation, IBM Journal of Research and Development 17Any computation can be done reversibly, so the logic never forces an erasure.
  6. Parfit (1986; first published 1984), Reasons and Persons, Oxford University PressThe teleporter and branch-line cases: duplication and what matters in personal identity.

Doubling

  1. von Neumann (1966, edited and completed by Burks), Theory of Self-Reproducing Automata, University of Illinois PressThe 1949 Illinois lectures and the 29-state cellular automaton: a constructor, a tape and a copier make a machine that builds a copy of itself.
  2. Freitas and Merkle (2004), Kinematic Self-Replicating Machines, Landes BioscienceA survey of physical replicators, with chapters on replication time against mass and on closure.
  3. Freitas and Gilbreath (eds.) (1982), Advanced Automation for Space Missions, NASA Conference Publication 2255The 1980 summer study of machine intelligence and robotics for space missions, including a self-replicating lunar factory.
  4. Zykov, Mytilinaios, Adams and Lipson (2005), Self-reproducing machines, NatureModular robot cubes that reproduce by assembling a copy from a set of cubes.
  5. Armstrong and Sandberg (2013), Eternity in six hours: Intergalactic spreading of intelligent life and sharpening the Fermi paradox, Acta AstronauticaIntergalactic colonization looks feasible, which sharpens the Fermi paradox.
  6. Hart (1975), An explanation for the absence of extraterrestrials on Earth, Quarterly Journal of the Royal Astronomical Society 16, 128Colonists at a tenth of light speed would reach most of the Galaxy in about a million years, so their absence suggests there are none.
  7. Pearl and Reed (1920), On the rate of growth of the population of the United States since 1790 and its mathematical representation, Proceedings of the National Academy of SciencesThe logistic curve fitted to a growing population.
  8. Tipler (1980), Extraterrestrial intelligent beings do not exist, Quarterly Journal of the Royal Astronomical Society 21, 267Self-replicating probes would explore the Galaxy in under 300 million years, so their absence means we are probably alone.

Why they might be asleep

  1. Sandberg, Armstrong and Ćirković (2017), That is not dead which can eternal lie: the aestivation hypothesis for resolving Fermi's paradox, arXiv 1705.03394, in the Journal of the British Interplanetary Society 69, 406The hypothesis: expand, then wait for the cold far future; the six assumptions, the leaks of waiting tallied, radiator and deadline limits, and what observation could show.
  2. Bennett, Hanson and Riedel (2019), Comment on 'The aestivation hypothesis for resolving Fermi's paradox', Foundations of Physics 49, 820The rebuttal: erasure can be paid from bounded out-of-equilibrium reservoirs at one bit of negentropy per bit, at any time, so waiting helps only once all reachable matter is controlled and spent.
  3. Dyson (1979), Time without end: physics and biology in an open universe, Reviews of Modern Physics 51, 447The older argument for slowing down and hibernating as the universe cools, and the radiator limit that forces the sleep.
  4. Krauss and Starkman (2000), Life, the Universe, and Nothing: life and death in an ever-expanding universe, The Astrophysical Journal 531, 22With a cosmological constant, a civilization's total conscious lifetime and recoverable information are finite.

Other pages

The scale ladder

  1. Barrow (1998), Impossibility: The Limits of Science and the Science of Limits, Oxford University PressThe ladder of civilizations ranked by the smallest scale they can manipulate, from BI to BΩ.
  2. Mohr, Newell, Taylor and Tiesinga (2025), CODATA recommended values of the fundamental physical constants: 2022, Reviews of Modern Physics 97The values of ħ, G, c and the constants the Planck length is built from.
  3. Navas et al. (2024), Review of Particle Physics, Physical Review D 110The reference tables for quarks and leptons, including the limits on any substructure.
  4. Landauer (1961), Irreversibility and heat generation in the computing process, IBM Journal of Research and DevelopmentThe original argument that erasing a bit releases heat.
  5. Ho, Erdil and Besiroglu (2023), Limits to the energy efficiency of CMOS microprocessors, arXiv preprintWhere the gates of the 2020s spend their energy, and how far they sit above the floor.
  6. Takeuchi, Yamanashi and Yoshikawa (2013), Measurement of 10 zJ energy dissipation of adiabatic quantum-flux-parametron logic using a superconducting resonator, Applied Physics LettersThe bit energy of an adiabatic superconducting gate, measured at about ten zeptojoules.
  7. Margolus and Levitin (1998), The maximum speed of dynamical evolution, Physica DThe proof that the steps a system can take each second are limited by its energy.
  8. Susskind (1995), The world as a hologram, Journal of Mathematical PhysicsThe holographic principle: one degree of freedom per Planck area of horizon.