Astronomy & Celestial Mechanics
the sky, computed
Cosmology

Two horizons that differ only in who is inside

A black hole's horizon and the cosmological one share an entropy formula exactly and differ in temperature by precisely a factor of two. The factor of two is the whole of the difference, and what it encodes is which side of the surface the observer stands on.

Assumes Horizons, Black hole spin and Dark energy.

16 min read 7 figures Scales that defeat intuitionMeasuring the unreachable

The previous rung established when a cosmological event horizon exists: precisely when the expansion accelerates, because that is precisely when one integral converges. It said nothing about what such a horizon is, beyond a surface past which signals do not go.

There is more to say, and the more is thermodynamic. A horizon has a temperature and an entropy, and the formulas for both are the same formulas that describe a black hole. The sameness is not an analogy that holds approximately in a limit. One of the two quantities is identical between the two cases and the other differs by a factor of two, and both of those facts have reasons attached.

Two horizons, one formula, and a factor of two. Horizon temperature against horizon radius, on logarithmic axes, for the two kinds of horizon this collection has. The upper line is a de Sitter horizon at T = ħc/2πk_BR and the lower is a Schwarzschild horizon at T = ħc/4πk_BR — the same expression with the same constants, differing by exactly two, and both falling as one over the radius so that a bigger horizon is a colder one. The cosmological horizon today has a radius of 4,451 megaparsecs and a temperature of 2.65e-30 K, which is thirty orders of magnitude below the microwave background and will never be measured by anything. A solar-mass black hole sits at 6.2e-8 K, and a black hole as cold as the sky would weigh 2.32e+22 solar masses — of the order of the mass inside the observable universe, which is not a coincidence, since both numbers are c³/GH up to factors of order one. The factor of two between the two lines is the one place the analogy is not exact, and it is not a convention: it comes from the periodicity of the Euclidean time coordinate, which is 8πGM/c³ for a black hole and 2π/H for de Sitter space. The entropies, by contrast, agree exactly.
Fig. 1 Horizon temperature against horizon radius, on logarithmic axes. The upper line is a de Sitter horizon at T = ħc/2πk_BR and the lower is a Schwarzschild horizon at T = ħc/4πk_BR — the same expression with the same constants, differing by exactly two, and both falling as one over the radius. The cosmological horizon today is at 2.65 × 10⁻³⁰ K, thirty orders of magnitude below the microwave background. A black hole as cold as the sky would weigh 2.3 × 10²² solar masses, which is of order the mass inside the observable universe.

Where a temperature comes from at all

Neither temperature is a temperature of anything material. Both come from the same argument, and the argument is worth stating because it explains the factor of two rather than leaving it as a coincidence.

Take the metric, continue the time coordinate to imaginary values, and require that the resulting Euclidean geometry be smooth at the horizon rather than having a conical singularity there. That requirement fixes the period of the imaginary time coordinate, and a quantum field theory periodic in imaginary time with period β is a thermal field theory at temperature ħ/k_Bβ. The temperature is a smoothness condition, and nothing about it refers to any particle.

For Schwarzschild the period comes out at 8πGM/c³. For de Sitter it comes out at 2π/H. Convert both to the radius of the horizon in question — R_s = 2GM/c² and R_H = c/H — and the two temperatures are ħc/4πk_BR_s and ħc/2πk_BR_H. Same expression, factor of two.

The factor traces to a geometric difference and not to a physical one. In the Schwarzschild case the surface gravity at the horizon is c⁴/4GM = c²/2R_s; in de Sitter it is c²/R_H. Both temperatures are ħ times the surface gravity over 2πck_B, which is the general result — so the two agree about the relationship between temperature and surface gravity exactly, and differ only because the geometry gives a different surface gravity at a given radius.

There is a second route to the same numbers that is worth having because it makes the temperature feel less like a formal trick. A horizon is where an accelerating observer’s causal access ends, and an observer with proper acceleration a in flat space already sees a thermal bath at ħa/2πck_B — the Unruh temperature, which is the same expression again with the surface gravity replaced by the acceleration. The three results are one result: a horizon of any kind, seen by the observer for whom it is a horizon, is warm at ħκ/2πck_B where κ is whatever plays the role of surface gravity. The equivalence principle is doing the work, and that is why an argument about black holes transfers to a cosmology without any new physics.

The entropies do not differ

Where the analogy is exact rather than nearly exact is the harder of the two quantities.

One entropy formula for two kinds of horizon. Entropy against horizon radius, in units of Boltzmann's constant, on logarithmic axes. There is one curve because there is one formula: S = k_B A/4ℓ_P², a quarter of the horizon area in Planck units, and it applies unchanged to a black hole's horizon and to the cosmological one. A solar-mass black hole has 1.0e+77 k_B, which already exceeds the thermodynamic entropy of the star it was made from by some twenty orders of magnitude. The cosmological horizon at 4,451 Mpc has 2.27e+122 — larger than everything else in the observable universe put together by about a hundred orders of magnitude, and larger than the sum of all the black holes in it by around twenty. The slope is two: entropy goes as the area rather than as the volume, which is the statement the holographic principle is named for and which is as true of the sky as of a black hole. What the shared formula does not settle is what the entropy counts. For a black hole the horizon hides an interior and the entropy is plausibly a count of what is inside; for a cosmological horizon the hidden region is the outside, it is different for every observer, and no observer's horizon is preferred. The formula is the same and the object it describes may not be.
Fig. 2 Entropy against horizon radius, in units of Boltzmann’s constant. There is one curve because there is one formula: S = k_B A/4ℓ_P², a quarter of the horizon area in Planck units, unchanged between the two cases. A solar-mass black hole has 1.0 × 10⁷⁷ k_B, which exceeds the thermodynamic entropy of the star it was made from by some twenty orders of magnitude. The cosmological horizon has 2.3 × 10¹²², larger than everything else in the observable universe put together by about a hundred orders of magnitude.

That the two entropies agree exactly while the two temperatures differ by two is a genuinely odd fact, and it is not an accident of the derivations. The first law relates them: dE = T dS, and the energies differ in a way that compensates. What the agreement means is that the entropy is a property of the horizon surface — a quarter of its area in Planck units — while the temperature is a property of the observer’s relationship to it.

The comparison of the two panels is worth making explicit, because it is the sort of thing that is easy to state and hard to feel. A solar-mass black hole is 3 kilometres across and carries 10⁷⁷ units of entropy. The cosmological horizon is 10²³ times larger in radius and carries 10⁴⁵ times more entropy — which is 10²³ squared, because the entropy is an area. The area law is not a small correction to a volume law; it is a completely different scaling, and it is the reason the sky dominates.

The number 2.3 × 10¹²² deserves to be read slowly. It is the largest entropy in the observable universe by a very wide margin. All the black holes together contribute of order 10¹⁰⁴; all the photons in the microwave background about 10⁸⁸; all the baryons and their thermal degrees of freedom rather less. The sky’s own horizon carries eighteen orders of magnitude more than the largest of those and a hundred more than the rest.

It is also the number that the second law of thermodynamics has to be stated against if it is to be stated cosmologically at all. The usual puzzle about why the early universe had low entropy is normally posed in terms of matter, where the answer is that a nearly uniform gas is a high-entropy state and the low entropy was gravitational. Posed in terms of horizons the arithmetic changes: the horizon at recombination was small, so its entropy was small, and the total has been climbing ever since simply because the horizon has been growing. Whether that constitutes an explanation or a restatement is disputed, and the dispute is about whether a horizon entropy is a real entropy.

Which side the observer is on

The difference the factor of two encodes is one of orientation, and it is worth stating carefully because the two situations are not mirror images.

A black hole’s horizon is a surface that an observer surrounds. It is at a definite place, every observer agrees where it is, and what it hides is an interior — a bounded region. An observer can in principle go and look, at the cost of not coming back.

A cosmological horizon surrounds the observer. It is not at a definite place: every observer has their own, centred on themselves, and two observers a billion light years apart have overlapping but different horizons — the same observer-dependence that makes the dipole in the microwave background a statement about motion rather than about the sky. What it hides is the exterior — an unbounded region, most of the universe — and no observer can go and look, because the horizon recedes from any attempt to approach it in the sense that a signal sent toward it never crosses.

That difference has a consequence for what the entropy could be counting. For a black hole, the natural reading is that the horizon area counts the microstates of what fell in, and that reading has been made precise in string theory for certain extremal holes. For a cosmological horizon there is nothing analogous: the hidden region is different for each observer, no observer’s is preferred, and a count of states behind it would have to be observer-dependent. The formula is the same and what it counts may not be.

Beyond redshift 1.87, what a galaxy does today will never be seen. For a galaxy at each redshift, the cosmic time of the last event on it that will ever be visible from here — not the last that has arrived, but the last that ever arrives, integrated to infinite future time. The horizontal line is the present, 13.8 billion years. A galaxy below redshift 1.87 has its curve above that line: its entire future will be seen from here, arriving ever more slowly and ever more redshifted, so it never quite disappears. A galaxy above redshift 1.87 has its curve below the line, and that is the whole content of an event horizon: what such a galaxy is doing today will never be seen, ever, by anybody here. Only a finite slice of its history is coming, and when the last of that light arrives the object stops changing. The redshift at which the curve crosses is 1.87, the comoving distance there is 16.7 billion light years against a particle horizon of 46.1, and the ratio of the volumes says that 95 per cent of the galaxies now observable are already beyond reach. Superluminal recession is not what does this. Everything past about redshift 1.5 has always been receding faster than light and is seen perfectly well, because the Hubble sphere grows to meet the photon; what closes the horizon is that with a cosmological constant the comoving Hubble radius stops growing and begins to shrink, so a photon that has not already been overtaken never will be.
Fig. 3 For a galaxy at each redshift, the last event on it that will ever be visible from here. The crossing at redshift 1.87 is the event horizon expressed as a property of other objects rather than as a distance, and it is the concrete form of the observer-dependence: the crossing is at 1.87 as measured from here, and an observer in one of those galaxies sees this one crossing their own horizon at their own redshift 1.87. Neither is right about the other being beyond reach; both are.
The same history, on the clock that straightens light. Conformal time upward against comoving distance sideways, for the Planck 2018 cosmology. Conformal time is ∫dt/a, which is exactly the comoving distance light covers, so on these axes every photon moves at forty-five degrees — at every epoch, whatever the expansion is doing. That single property turns every curved thing in the ordinary space-time diagram into a straight one. The universe began at η = 0 and is now at η = 46.1 Gly of conformal time; it will ever accumulate only 62.8, because the integral ∫dt/a converges once Λ dominates, and that finite ceiling is the whole reason an event horizon exists. The particle horizon is the 45° line from the origin and the event horizon is the 45° line back from the ceiling, so the two horizons that were curves are now the two edges of one light cone drawn twice. The shaded wedges are the past light cones of two points on the last scattering surface, at conformal time 0.914 Gly and comoving distance 45.2 Gly from here. They do not overlap. Two points on that surface separated by more than 2η_rec were never in causal contact, which subtends 2.31° on the sky, and the microwave sky therefore contains about 9,805 patches that have no common past and the same temperature to one part in a hundred thousand. That is the horizon problem, and in these coordinates it is a statement about whether two triangles intersect.
Fig. 4 The same geometry in conformal coordinates, where the event horizon is the 45° line running down and outward from the end of conformal time. What that line makes obvious is the observer-dependence: the line is drawn from this worldline, and a diagram drawn from a galaxy ten billion light years away would have its own 45° line from its own position, cutting the diagram differently. The particle horizon shares that property and the Hubble sphere does not, which is one more reason the three should not be run together.

The one horizon in this collection that is warm enough to matter

Nothing above will ever be observed, and it is worth setting beside a horizon temperature that has been observed, because the contrast says what the obstacle is.

The microwave background is a horizon of a sort — the surface of last scattering is where the universe becomes opaque looking backwards — and it is at 2.7255 K, measured to four decimal places and the most perfect blackbody ever recorded. It is not a horizon in the causal sense: it is an opacity boundary, it recedes as the universe ages, and light from beyond it exists and simply has not escaped. But it is the surface that bounds what can be seen, and it has a temperature that instruments read.

The de Sitter horizon is thirty orders of magnitude colder, and the reason is a ratio of two lengths. The last scattering surface’s temperature is set by atomic physics — hydrogen recombines at about 3,000 K and the redshift does the rest — while the de Sitter temperature is set by the size of the horizon itself, and the horizon is 10²⁷ times a hydrogen atom’s Bohr radius many times over. A temperature that goes as the inverse of a cosmological length is a temperature that cannot be large.

The scale of the numbers, and one coincidence that is not one

The cosmological horizon’s temperature is 2.65 × 10⁻³⁰ K. Nothing will measure it. The energy of a typical quantum of that radiation is about 10⁻³³ electronvolts, and its wavelength is the size of the horizon, so detecting it would require an apparatus larger than the region it is trying to measure.

Every black hole evaporates, and the sky decides when it may start. Evaporation time against mass, on logarithmic axes, from the standard result that the lifetime goes as the cube of the mass. The slope is three and the range is absurd: an object of 8.7e-20 solar masses — 173 million tonnes, roughly a cubic kilometre of rock — takes the present age of the universe, and a solar-mass hole takes 2.1e+58 billion years. A hole only evaporates if it is losing more than it gains, and what it gains is set by the temperature of the bath it sits in. Today that bath is the microwave background at 2.7255 K, so every hole below 2.26e-8 solar masses is hotter than its surroundings and evaporating while everything above it is absorbing and growing — which is every astrophysical black hole there is. The background cools as the universe expands, so the threshold rises with time and eventually passes every hole. What it cannot pass is the floor: in a universe with a cosmological constant the temperature does not fall below the de Sitter value of 2.65e-30 K, and a hole as cold as that would weigh 2.32e+22 solar masses, which is more than the observable universe contains. So the floor never protects anything real, every black hole eventually evaporates, and the last of them goes after about 2.1e+91 billion years.
Fig. 5 Evaporation time against black-hole mass, on logarithmic axes, with the slope of three that the mass-cubed law requires. An object of 8.7 × 10⁻²⁰ solar masses — 173 million tonnes, roughly a cubic kilometre of rock — evaporates in the present age of the universe, and a solar-mass hole takes 2 × 10⁵⁸ billion years. A hole only evaporates when it is losing more than it gains, and today the bath is the microwave background at 2.7 K, so every hole above 2.3 × 10⁻⁸ solar masses is absorbing rather than radiating — which is every astrophysical black hole there is.

The coincidence flagged in the first figure — that a black hole as cold as the sky would weigh about what the observable universe contains — is worth resolving rather than admiring. Both quantities are c³/GH up to factors of order unity. The mass inside the Hubble radius is the critical density times the volume, which is (3H²/8πG)(4π/3)(c/H)³ = c³/2GH. The mass of a black hole whose temperature equals the de Sitter temperature is, from setting ħc/4πk_BR_s equal to ħc/2πk_BR_H, a hole of radius 2R_H, which has mass c²R_H/G = c³/GH. So the two agree to a factor of two by construction, and the coincidence is that the critical density is what it is — which is to say, no coincidence at all.

What is inside the sky’s horizon

One number in the first figure is worth converting into something a reader can hold, because the abstraction hides it.

The cosmological horizon is 4,450 megaparsecs in radius — 14.5 billion light years, which is the Hubble distance rather than the event horizon, since the temperature is set by H rather than by the integral of the previous rung. Inside it are of order 10⁸⁰ baryons, 10⁸⁸ photons, perhaps 10²³ stars and 10¹² galaxies. All of that, thermodynamically, is a rounding error against the surface that bounds it — including the supermassive black holes at the centres of those galaxies, which are individually the largest entropy reservoirs anything material has and collectively eighteen orders of magnitude short.

That is the content of the holographic principle in its original and least speculative form: the maximum entropy of a region is not proportional to its volume, as it would be for any ordinary system, but to the area of its boundary. A box of gas twice as wide holds eight times the entropy; a horizon twice as wide holds four times. Somewhere between the two the scaling has to change, and the crossover is where the matter in the box would collapse into a black hole. For a region the size of the sky the difference between the two scalings is 40 orders of magnitude, and the area law wins by all of it.

The thermodynamics that follows

If a horizon has a temperature and an entropy then it has a thermodynamics, and the entries in it are unfamiliar.

Adding mass to a region of de Sitter space decreases the entropy, because it shrinks the horizon: a Schwarzschild–de Sitter spacetime has both a black-hole horizon and a cosmological one, and the sum of their areas is maximised when the black hole has zero mass. So empty de Sitter space is the maximum-entropy state, and putting anything into it — a star, a galaxy, an observer — lowers the total. That is the opposite of the behaviour of a self-gravitating gas, which raises its entropy by clumping, and the two coexist because they are statements about different things: the gas raises the entropy of its own degrees of freedom while lowering that of the horizon it sits inside, and the sums do not obviously cancel. That is the reverse of the usual intuition about entropy and matter, and it follows directly from the area law.

It also produces the one genuinely disturbing consequence in this area, which is worth stating because it is a live problem rather than a curiosity. A thermal system with finite entropy explores its states, and a system with 10¹²² of them, given infinite time, produces every fluctuation compatible with its energy — including fluctuations that look like observers. Those observers would be vastly more numerous than the ones produced by ordinary cosmological evolution, because ordinary evolution happens once and fluctuations happen forever. That the reasoning leads somewhere absurd is generally taken as evidence that one of its premises is wrong, and the candidates are that de Sitter space is not eternal, that the entropy does not count states in this sense, or that the counting argument itself is invalid. No consensus exists on which.

Whether an event horizon exists at all, against one number. The comoving event horizon today — the distance a signal sent now will ever cover — against the equation of state of the dark energy, for a flat universe with the measured matter density. The curve runs away at w = −1/3 and does not exist above it: that is where the expansion stops accelerating, and in a universe that does not accelerate the integral ∫da/a²E diverges and every galaxy is eventually reachable, however far away. Below −1/3 the horizon is finite and shrinks as w falls, because a more negative equation of state makes the dark energy density grow with time rather than stay constant. At the cosmological constant's w = −1 the horizon is 16.7 billion light years against a particle horizon of 46.1, so 4.7 per cent of the volume now observable is still reachable. The band is the measured −1.03 ± 0.03. What the figure is for is the asymmetry in what the measurement still allows: two sigma toward zero puts the horizon at 17.5 Gly and two sigma the other way at 14.7, and the shape of the curve means that the closer the true value sits to −1/3 the more violently the answer moves. The reachable fraction is not a robust number in the way the particle horizon is.
Fig. 6 The comoving event horizon against the equation of state, from the previous rung. It belongs here too, because every number in this essay scales with the horizon radius: the temperature as its inverse and the entropy as its square. A universe with w = −0.6 has a horizon 3.9 times larger than the one drawn, so a temperature 3.9 times lower and an entropy fifteen times larger — and one with w above −1/3 has no horizon, no temperature and no entropy at all.

What the analogy does not reach

Three things are true of a black hole’s horizon and not of the cosmological one, and each marks where the correspondence ends.

A black hole has other quantities. It has a mass, an angular momentum and a charge, and the horizon area depends on all three; the innermost stable orbit, the ergosphere and the accretion efficiency all move with the spin. A cosmological horizon has a radius and nothing else, because there is nothing outside it against which to define a rotation and no matter inside it that the horizon belongs to.

A black hole’s temperature rises as it radiates. Losing mass shrinks it, and a smaller horizon is hotter, so evaporation runs away and ends in a burst. The cosmological horizon’s temperature is fixed by Λ, which does not change, so it radiates forever at a constant temperature into a space that is not getting emptier. There is no runaway and no endpoint.

And a black hole’s horizon is a one-way membrane for matter as well as for signals. Something can fall in. Nothing crosses a cosmological horizon in the corresponding sense — a galaxy that passes beyond it has not gone anywhere, and from its own point of view nothing happened at all. The surface is defined by a failure of communication, not by a change of location.

What the picture cannot show

Every number here is computed for exact de Sitter space, which the universe is not. It is a universe with matter in it, expanding toward de Sitter, and the horizon temperature of an asymptotically de Sitter spacetime is a well-defined quantity only in the asymptotic limit.

That matters for the entropy in particular. The 2.3 × 10¹²² is computed from today’s Hubble radius as though the geometry were already the final one; the true asymptotic horizon is set by Λ alone rather than by the present expansion rate, and since matter still contributes about thirty per cent of the density today, the two differ by a factor of about 1.2 in radius and 1.4 in entropy. That is small on a scale where the exponent is 122 and it is not nothing, and the figure computes the first quantity while the sentence about maximum entropy is about the second.

There is a larger caveat about the whole framework. Hawking’s derivation for a black hole is a calculation in quantum field theory on a fixed background, and its extension to a cosmological horizon is the same calculation in a different spacetime. Both are semiclassical: they treat the geometry as classical and the fields as quantum, and they are known to fail when the horizon becomes comparable to the Planck length. Nothing here is near that regime, so the derivations are as trustworthy as they are anywhere. What is genuinely unsettled is not the temperature but its interpretation — whether the entropy counts anything, and if so what.

From six gravitational radii to one. The radius of the innermost stable circular orbit against the dimensionless spin a = Jc/GM², in units of GM/c², for orbits prograde and retrograde with the hole's rotation. Both curves are the Bardeen–Press–Teukolsky expression and are checked at the three places it has exact values: 6 at zero spin, and 1 and 9 at the extremal limit. The separation is the observable consequence of frame dragging — space near the hole is itself circulating, so an orbit going the same way can stay closer before it becomes unstable, and one going the other way cannot come as close as a non-rotating hole allows. The prograde branch is required to fall and the retrograde branch to rise at every step drawn, which is a claim about the direction of the effect rather than about its size. The marked spin of 0.998 is not the extremal value but the equilibrium a hole fed by a thin disc actually reaches, because photons emitted by the disc are preferentially captured on retrograde orbits and spin the hole down again. Nothing here depends on what the hole is made of: two numbers fix the whole geometry, and this figure is the first of them holding still while the second moves.
Fig. 7 And the black hole from the other side, for contrast: spin, and the four things it changes. A rotating hole has two horizons, an ergosphere, an innermost stable orbit that moves with the spin, and an efficiency that reaches 42 per cent — none of which has a cosmological counterpart, because a cosmological horizon has no angular momentum and nothing outside it to define one against. The analogy between the two kinds of horizon is exact in thermodynamics and stops immediately outside it.

Where this ladder goes next

Five rungs have taken the horizons of this cosmology from an arithmetic surprise about the size of the observable universe to a thermodynamics of the surface that bounds it. What is left on the anchor divides into two.

One direction is the fair-sample question — and it is sharpened rather than softened by everything above, since an entropy of 10¹²² attached to the boundary of one observer’s region is a strange thing to have if the region is not special: whether the contents of one observer’s horizon are representative of anything larger, which is a question about how far homogeneity extends and about what could be said if it does not. The other is the interior of the diagram rather than its edges — the fact that a signal’s reach shrinks while a photon’s arrival does not, so the set of galaxies from which light is still arriving grows while the set that can still be signalled shrinks, and the two sets are diverging.

And below both: the habit this ladder has been about. A horizon is a property of an integral, not of a speed. Nothing in five rungs has been about anything travelling faster than light. Every one of them has been about whether a particular integral converges, and the answers changed when a quarter of a magnitude turned up in a supernova survey.

Where this goes next

Whether an event horizon exists at all, against one number. The comoving event horizon today — the distance a signal sent now will ever cover — against the equation of state of the dark energy, for a flat universe with the measured matter density. The curve runs away at w = −1/3 and does not exist above it: that is where the expansion stops accelerating, and in a universe that does not accelerate the integral ∫da/a²E diverges and every galaxy is eventually reachable, however far away. Below −1/3 the horizon is finite and shrinks as w falls, because a more negative equation of state makes the dark energy density grow with time rather than stay constant. At the cosmological constant's w = −1 the horizon is 16.7 billion light years against a particle horizon of 46.1, so 4.7 per cent of the volume now observable is still reachable. The band is the measured −1.03 ± 0.03. What the figure is for is the asymmetry in what the measurement still allows: two sigma toward zero puts the horizon at 17.5 Gly and two sigma the other way at 14.7, and the shape of the curve means that the closer the true value sits to −1/3 the more violently the answer moves. The reachable fraction is not a robust number in the way the particle horizon is. Whether there is a horizon at all An event horizon exists precisely when the expansion accelerates, and its size is not a smooth function of how much. The integral that defines it runs away as the equation of state approaches minus a third, so two sigma either way on a measured number are two very different futures. Beyond redshift 1.87, what a galaxy does today will never be seen. For a galaxy at each redshift, the cosmic time of the last event on it that will ever be visible from here — not the last that has arrived, but the last that ever arrives, integrated to infinite future time. The horizontal line is the present, 13.8 billion years. A galaxy below redshift 1.87 has its curve above that line: its entire future will be seen from here, arriving ever more slowly and ever more redshifted, so it never quite disappears. A galaxy above redshift 1.87 has its curve below the line, and that is the whole content of an event horizon: what such a galaxy is doing today will never be seen, ever, by anybody here. Only a finite slice of its history is coming, and when the last of that light arrives the object stops changing. The redshift at which the curve crosses is 1.87, the comoving distance there is 16.7 billion light years against a particle horizon of 46.1, and the ratio of the volumes says that 95 per cent of the galaxies now observable are already beyond reach. Superluminal recession is not what does this. Everything past about redshift 1.5 has always been receding faster than light and is seen perfectly well, because the Hubble sphere grows to meet the photon; what closes the horizon is that with a cosmological constant the comoving Hubble radius stops growing and begins to shrink, so a photon that has not already been overtaken never will be. The galaxies that are already out of reach With a cosmological constant the comoving distance a photon can ever cover converges, so there is a redshift beyond which light leaving today never arrives. It is 1.87, and about ninety-five per cent of the galaxies now visible are past it — which superluminal recession has nothing to do with. From six gravitational radii to one. The radius of the innermost stable circular orbit against the dimensionless spin a = Jc/GM², in units of GM/c², for orbits prograde and retrograde with the hole's rotation. Both curves are the Bardeen–Press–Teukolsky expression and are checked at the three places it has exact values: 6 at zero spin, and 1 and 9 at the extremal limit. The separation is the observable consequence of frame dragging — space near the hole is itself circulating, so an orbit going the same way can stay closer before it becomes unstable, and one going the other way cannot come as close as a non-rotating hole allows. The prograde branch is required to fall and the retrograde branch to rise at every step drawn, which is a claim about the direction of the effect rather than about its size. The marked spin of 0.998 is not the extremal value but the equilibrium a hole fed by a thin disc actually reaches, because photons emitted by the disc are preferentially captured on retrograde orbits and spin the hole down again. Nothing here depends on what the hole is made of: two numbers fix the whole geometry, and this figure is the first of them holding still while the second moves. The second number a black hole has A black hole in equilibrium is described by its mass and its spin, and nothing else. The mass decides how strongly it pulls. The spin decides how much light a kilogram of infalling matter can emit before it disappears — and between the two extremes that figure changes by a factor of seven.

About the same objects

Not linked from either essay — found by the objects both name.

The objects this essay names

Each one links to every other essay that touches it.

Black hole entropyCosmological constantDe sitter spaceEvent horizonHawking radiationHolographic principleHorizon temperatureObservable universePlanck lengthSchwarzschild radiusSurface gravityThermodynamics