Life under a black sun

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by  Tomáš Opatrný

Life under a black sun
Toma´ˇs Opatrny´ and Luka´ˇs Richterek∗
Faculty of Science, Palack´y University, 17. Listopadu 12, 77146 Olomouc, Czech Republic
Pavel Bakala†
Institute of Physics, Faculty of Philosophy and Science,
Silesian University in Opava, Bezruˇcovo n´am. 13, CZ-74601 Opava, Czech Republic
(Dated: January 13, 2016)
Life isdependenton theincomeof energy with low entropyandthedisposal ofenergy with high
entropy. On Earth, the low-entropy energy is provided by solar radiation and the high-entropy
energy is disposed as infrared radiation emitted into the cold space. Here we turn the situation
around and assume cosmic background radiation as the low-entropy source of energy for a planet
orbiting a black hole into which the high-entropy energy is disposed. We estimate the power that
canbeproducedbythermodynamicprocessesonsuchaplanet,withaparticularinterestinplanets
orbiting a fast rotating Kerr black hole as in the science fiction movie Interstellar. We also briefly
discussareverseDysonsphereabsorbingcosmicbackgroundradiationfromtheoutsideanddumping
waste energy to a black hole inside.
6
1
0
I. INTRODUCTION The considerations presented here can also be rel-
2
evant for the discussion of the early stages of the
n Universe: recently Loeb8 suggested that a habitable
a Life onEarthis possible thanks to the hotSun and
epoch occurred when the Universe was about 15 mil-
J the cold sky. Their temperature difference makes it
lion years old and the background radiation had the
2 possible to drive processes far from thermodynamic
temperature of 273–300K, allowing for rocky planets
1 equilibrium by increasing the entropy elsewhere in
] the Universe. Absorbing photons from the Sun at wsuigthgelsitqiuonidwwasatceorvecrheedmbiysttrhyeoNnattuhreeirmsaugrfaazcinese.9mTehnis-
c 6000K and emitting about 20 times more pho-
∼ tioning also some criticism pointing out that the cold
q tons at 300K to the cold sky makes the entropy
∼ sky is thermodynamically as important for life as the
- balance sufficient to sustain complex processes in
r hotSun. Althoughthereisnodoubtthatsomesource
g which entropy locally drops. As explained by Ervin
of negative entropy is necessary for life, the question
[ Schr¨odinger in his book “What is life?”,1 organisms
remainswhatcouldbethesource. Theblack-holesun
feed on negative entropy. The hot Sun and cold skies
1 is an option to be discussed here.
v provide the Earth with a great deal of negative en-
7 tropy: in this way, the Earth produces 5 1014 Thepaperisorganizedasfollows: InSec.IIgeneral
9 J/K of entropy each second.2 ∼ × formulasforconvertingtheincomingbackgroundradi-
8 ationenergyintousefulworkarediscussed,inSec.III
Here we playwith the idea ofa worldupside down:
2 twospecialregimesforthesolidangleoftheincoming
the sun is cold and skies are “hot”. Let us imag-
0 radiation—largeand smallare studied, in Sec. IV the
ine a planet orbiting a black hole in a universe filled
. results for anisotropic radiation due to fast orbiting
1
with background radiation. The inhabitants accept
0 closetotheblackholearepresented,inSec.VaDyson
low-entropy energy from the sky and dispose waste
6 sphere encapsulating a black hole is considered, and
heat to the black hole. The protagonists of the re-
1 in Sec. VI the conclusions are provided. Details of
v: coernbtitimngovaiesIunpteerrmseallsasrivwehbolawckanhtotleoGcoalrognainzetuaapmlaignhett the numerical computations of the temperature map
i of the sky for an observer orbiting a black hole are
X find these results vital (for physical details of their
given in Appendix A.
trip and destination we recommend the book “The
r
a Science of Interstellar”4 by Kip Thorne, the scientific
consultant and an executive producer of the film).
II. RADIATION HEAT EXCHANGE
Apart from the pedagogical value of simple ther-
modynamic exercises, these speculations might be
A. Sky projection and etendue conservation
relevant in the distant future when stars exhaust
their nuclear fuel and die, and black holes may be-
comedominantconstituentsoftheentropyproduction Letusassumethatheatexchangeofthe planetcan
processes.5,6 This kind of energetics might be useful occur radiativelyby surfaceS. The celestialsphere is
until the expansion of the Universe cools the cosmic divided into two parts: one is “hot” at temperature
background radiation below the temperature of the T and the other cold, for simplicity at temperature
1
black hole Hawking radiation. After that, the black ofabsolutezero. Thezero-temperatureblackholeisa
hole becomes a net radiator and thus the nearby in- goodapproximationifHawkingradiation10,11 isnegli-
habitantsmightagainliveundera“hot”sunandcold gible. Letthe solidanglesspannedbythesetwoparts
sky. Recently, the mechanism of extracting mechani- be Ω andΩ ,with Ω +Ω =4π. To allowfor the
H C H C
cal work of radiating black holes has been discussed;7 most efficient thermal energy exchange of the planet
however,thefocushereisonthenotsodistantfuture. with the sky, we assume the whole planet to be cov-
2
propagates through non-absorbing non-scattering en-
(a)
vironment (such as through our idealized optical sys-
tems), the etendue is conserved. This is analogous to
the phase-space conservation in conservative systems
according to the Liouville theorem. For our scheme
the consequence of etendue conservation is that
S Ω
H H
= . (1)
S Ω
C C
The surfaces S and S then serve as the hot and
H C
cold terminals of heat engines.
Without the idealization leading to Eq. (1), one
T T T
H C H wouldhavetotakeintoaccountthefactthatthe cold
terminalalsointeractswithobjects athighertemper-
(b) aturewhichwoulddecreasetheefficiencyofworkpro-
duction. Letusnotethatinthephotovoltaicindustry
oneofthe goalsisthe constructionoflightconcentra-
torsapproachingtheetenduelimit(see,e.g.,Ref. 13),
soweassumethattheadvancedcivilizationinhabiting
the planet under study has reached this goal.
B. Temperature optimization for given SH
and SC
Let the surfaces of the heat exchangers S and S
H C
be at temperatures T and T and let them serve
H C
as heat exchangers for a heat engine. The task is to
find such T and T that the power of the heat en-
H C
FIG. 1. (a) Projection system for interaction of a Lam-
gine is maximized. The procedure is analogousto the
bertianradiatorwiththeblackhole. Thearraysrepresent
poweroptimizationofirreversibleenginesasfirststud-
radiationfromthecoldsurfaceattemperatureTc directed ied by Novikov14 and later independently by Curzon
to the black hole. The remaining surface at temperature
and Ahlborn15 (for more general considerations see,
TH interactsbyradiation(notshown)withthehotsky. (b)
e.g., Ref. 16). Our case is different in the tempera-
Schemeofthethermodynamicsystem: theplanetcovered
with the light concentration systems of the above picture ture dependence of the thermal energy exchange rate
accepts highenergy photons(longarrows) from thespace ( T4), and also in having fixed the ratio of the ex-
∝
and sendslow energy photons (short arrows) tothe black changer surfaces S /S .
H C
hole. ThermalenergyreceivedbyS fromthehotpartof
H
the sky during a time interval ∆t is Q = σS (T4
1 H 1 −
T4)∆t and the waste energy sent to the cold part of
H
eredwithlightconcentrationsystemsthatprojectthe the sky is Q = σS T4∆t, where σ is the Stefan-
2 C C
celestial sphere into its images. We assume the nu- Boltzmann constant, σ 5.68 10−8Wm−2K−4.
≈ ×
mericalapertureofthesedevicesapproaching1which Their difference can be converted into work W =
means that each point of the image is uniformly illu- Q Q provided that Q /Q = T /T . A relation
1 2 1 2 H C
−
minated from the 2π solid angle by rays coming from between T and T follows from this assumption,
H C
the object (see Fig. 1). This means that each part of
the image can interact as a Lambertian radiatorwith S T4 1/3
thecorrespondingpartofthesky. Inthisway,allrays TC = SH T14 −1 TH, (2)
coming from the hot part of the sky are projected to (cid:20) C (cid:18) H (cid:19)(cid:21)
onepartofthesurfaceoftheplanetandraysfromthe
and the average power P = W/∆t can be expressed
cold partto another. Letus denote the areasof these
as
surfaces S and S with S +S =S.
H C H C
Ageneraltheoremofrayopticssaysthataquantity P =σS (3)
H
called etendue for propagating light cannot decrease
S 1/3 T4 4/3
(see, e.g., Ref. 12). For a part of a beam propa- T4 T4 H 1 1 T4 .
gating in directions within a solid angle δΩ across a ×" 1 − H −(cid:18)SC(cid:19) (cid:18)TH4 − (cid:19) H#
surface element δA, the element of etendue is defined
as δ =n2cosθδΩδA, where n is the refraction index AssumingthatS , S ,andT arefixed,onecanfind
H C 1
E
of the medium and θ is the angle between the direc- such T that the power of the engine is maximized.
H
tionofpropagationandthe normaltoδA. Ifthe light Setting thederivativeequaltozero,dP/dT =0,one
H
3
1.0 (see Fig. 2), and the efficiency η of the process ap-
1.0
proach 1 as
T / T
0.8 0.8 H 1 η 1 (3q)1/4. (12)
≈ −
0.6
Rather than to a “standard” black hole, this limit
T / T
0.6 C 1
η 0.4 corresponds to a distant star illuminating a planet in
otherwiseempty universe(the relevancetoa lesscon-
0.4 0.2 ventional black hole situation will be discussed later
in Sec. IVB). The results can be applied to estimate
0
0.2 0 0.2 0.4 q 0.6 0.8 1.0 Earth’s energy income from our Sun. In this case
q 5.4 10−6 andT =5778K,whichleadsto work-
1
≈ ×
ingtemperaturesT =0.385T 2,200KandT =
H 1 C
0 0.0239 T 138 K, yielding the≈efficiency 92 %, and
0 0.2 0.4 q 0.6 0.8 1.0 maximum1 ≈available power P 1.603 1017 W.
≈ ×
FIG. 2. Efficiency η in dependence on the hot area frac- B. Large heating area
tion q for optimized power according to Eq. (9). Inset:
temperatures of the hot and cold surfaces in dependence
In the opposite limit S S, on solving Eq. (6)
on q. H →
for 1 q 1 we find
− ≪
33
finds that u 1 (1 q) (13)
≈ − 26 −
T =u1/4T , (4)
H 1 which leads to
SH 1/3 (1 u)1/3 33
TC =(cid:18)SC(cid:19) u−1/12 T1, (5) TH ≈(cid:20)1− 28(1−q)(cid:21)T1, (14)
where u solves the equation 3 34
T 1 (1 q) T , (15)
27 C ≈ 4 − 28 − 1
u4 18u2 8u 1=0 (6) (cid:20) (cid:21)
q − − − 33
η (1 q). (16)
in the interval 0<u<1, and ≈ 28 −
S Ω Using these expressions in (8), we find the power of
H H
q = (7) maximum work generation
≡ S 4π
is the “hot” fraction of the sky. The resulting power 33
P σS T4. (17)
is then max ≈ 28 C 1
P =ηqσST4, (8) As an illustration, let us assume a planet of the
max 1
Earth’ssizeorbitingablackholewhoseangularsizeas
where qσST4 is the total power of the incoming radi-
1 seenfromthe planetequalsto the angularsize ofSun
ation, and
asseenfromEarth. Thisgivesq 1 5.4 10−6and
1/3 4/3 η 5.7 10−7. Thus, S 2,7≈60k−m2, s×o that the
q 1 u C
η 1 u u − (9) pla≈netw×ouldradiateitswast≈ethermalenergyfroman
≡ − − 1 q u
(cid:18) − (cid:19) (cid:18) (cid:19) area comparable to Rhode Island. If the background
istheefficiencywithwhichtheincomingradiationcan radiationisatroomtemperatureT =300Kasinthe
1
be converted into useful work. 15 million years old universe assumed in Ref. 8, then
Even though Eq. (6) as a quartic equation has an T 225K,i.e. about 48◦C. FromEq.(17) we find
C
≈ −
explicitsolution,wesolveitnumericallyandtheresult that useful work could then be obtained with power
is usedtofindthe dependence oftheworkingtemper- P 130GW. This is two orders of magnitude be-
max
≈
atures TH, TC and of the efficiency η on the hot-sky lowthepresentworldenergyconsumption,andsixor-
fraction q, as shown in Fig. 2. dersofmagnitudebelowthe powerpresentlysupplied
to Earth by Sun. The situation goes much worse for
today’s two orders of magnitude colder background
III. SPECIAL OPERATING REGIMES radiation at T = 2.725K, resulting in eight orders
1
smaller available useful power P 910W.
max
≈
A. Small heating area
Assuming S S , or q 1, Eq. (6) yields u IV. CIRCULAR MOTION CLOSE TO THE
H C
3−3/4q1/4 and the≪temperatur≪es approachzero as ≈ BLACK HOLE
T 3−3/16q1/16T , (10)
H ≈ 1 Since η increases with decreasing q, to get as much
T 31/16q5/16T (11) poweraspossible,weshouldplacetheorbitascloseto
C 1
≈
4
(a) θ [ ’ ’ ]
30
1.25 1.5
1.0
1.75
20
2.0
0.75
150,000 50,000 10
200,000 100,000
250,000
−6 −5 −4 −3 −2 −1 0 1 ∆φ[ ’ ’ ]
−10
(b) 2.0 3.0 4.0
−20
5.0
1.0 6.0
−30
0.75
FIG.4. Detailoftheshadowboundaryofablackholefor
thecaseofFig.3(c)nearθ=0andφ=150◦,where∆φ
φ 150◦. The contour lines of constant temperature ar≡e
−
shown, the number indicating relative temperature shift
with respect to the temperature of background radiation
measured by a distant observer. The light spot on the
(c) upperleft shows theangularsize ofNeptuneas seenfrom
the Earth, for comparison (the shape is deformed due to
1.0 4 different scales of θ and ∆φ).
2
0.75
A. Schwarzschild black hole
8
16 We first consider motion along a timelike circu-
lar geodesic around a static, spherically symmetric
black hole of a mass M and Schwarzschild radius
R = 2GM/c2, where G = 6.67 10−11Nm2kg−2
FIG. 3. Mollweide projection of the sky of an observer isSthe gravitational constant and c×= 3 108m/s is
orbiting a black hole. The dark area is the shadow of ×
the speed of light. Stable bound circular orbits of ra-
the black hole, the full lines are contour lines of constant
temperature,wherethenumberindicatesrelativetemper- dius r exist for r 3RS = 6GM/c2 (see, e.g., Refs.
≥
atureshiftwithrespecttothetemperatureofbackground 17 and 18). In order to obtain the maximum power,
radiation measured by a distant observer. The symbols the innermost stable circular orbit at r = 6GM/c2
and indicate the direction to which and from which hasbeenchosen. Theresultingpictureoftheskyisin
⊙ ⊗
the observer is moving, respectively. The dotted lines Fig. 3(a). The shadow of the black hole covers12.2%
are parallels of latitude θ = 0, 30◦, 60◦ and meridi- of the sky, i.e., q = 0.878. If the temperature shifts
ans of longitude φ = 0, 60◦, 1±20◦. (±a) A nonrotating
black hole, orbit at r =±6GM±/c2, (b) a rotating black were disregarded, Eqs. (6), (8), and (9) would yield
η 1.39%andP 0.012σST4. ForanEarth-size
hole with rotation parameter a=1 1.3 10−14 and or- ≈ max ≈ 1
bit radius r = 2.2GM/c2, (c) a rot−ating×black hole with planetandbackgroundradiationatT1 =2.725K,this
a=1 1.3 10−14 and r=1.0000379GM/c2. would be Pmax 19MW.
≈
− × However, fast motion around the black hole makes
parts of the sky warmer which brings some advan-
tage. As seen in Fig. 3(a), in the direction of motion
the black hole as possible. However, things get com- the observer sees the temperature of the sky to be
plicatedthere: theobservermovesfastandrelativistic morethantwice aslargecomparedtowhatwouldsee
effects become important. The absorbed radiation is an observerat rest, far from the black hole (the max-
Doppler-shifted, as well as blue-shifted by falling to imum blue shift being 2.12). Thus, the power avail-
the black hole vicinity. The sky then ceases to be able from this part of the sky can be more than 24
isothermal: the radiation comes hotter from the di- timeslargerthanwithouttheblueshift. However,the
rectionwherethe observermovesandcolder fromthe hottest part of the sky is relatively small andthe rest
rear. To get the precise picture of the sky, advanced ofthe sky is colder. To find the totalavailablepower,
numerical computations are necessary which are ex- wehavedividedthe hotskyinto30segmentsofequal
plained in Appendix A; the results are in Figs. 3 and temperature-spanintervals. Eachsegmentthenserves
4. Having the temperature map of the sky, one can asaheaterofaseparateheatengine. Sincethepower
use segments of the sky with different temperature as of each engine also depends on the cold area used for
heaters andapply anotheroptimization to allocate to dumpingitswasteenergy,onehastofindtheoptimum
them various parts of the cooling area so as to get allocationofthecoldareatotheindividualenginesto
maximum power. obtain a maximum total power. The result of the nu-
5
(a) B. Fast rotating Kerr black hole
0.1
0.2
S/S
tot
0.08 0.15 P/P How can we come closer to the black hole so as to
tot
increasethecoldskyproportion? Wewereinspiredby
0.1
themovieInterstellarwherethecharacterscomeclose
0.06
0.05 to a fast-rotating giant black hole named Gargantua.
Rotating black holes havestable circular orbits closer
0
0.04 1 1.2 1.4 1.6 1.8 2 2.2 than6GM/c2. ForGargantua,therotationparameter
a was extraordinarilylarge, a=1 1.3 10−14, thus
− ×
allowingthecharacterstoenjoyextraordinarilystrong
0.02
relativistic effects close to the black hole.3,4 We have
computedthe resultsfortwospecialcasesofthe orbit
0 radius, r = 2.2GM/c2 (Fig. 3(b) and 5(b)) and r =
0.5 1 1.5 2
1.0000379GM/c2 (Fig. 3(c) and 4). The latter case
T / T
0 corresponds to the orbit of Miller’s planet where the
(b)
characters of Interstellar spend three hours while 21
0.25
years pass on their base station which is sufficiently
S/S distant from the reach of Gargantua’s gravitational
tot
0.2 0.08 time shift.
P/P
0.06 tot
We can see that the shadow of the black hole be-
0.15
0.04 comes deformed and covers a large part of the ob-
server’s sky, including the direction of motion of the
0.02
0.1 planet. Nevertheless,the planet does not fall into the
0
holeasitisdraggedbyitsrotatinggravitationalfield.
0 1 2 3 4 5 6 7
0.05 Theblueshiftbecomesmuchstronger,althoughinrel-
ativelysmallstripsoftheskyjustabovetheshadowof
the black hole in the direction of the planet motion.
0
0 1 2 3 4 5 6 7 In the case of r = 2.2GM/c2 the black hole covers
T / T 26 % of the sky and the maximum blue shift is 6.90.
0
Uponusingthesameoptimizationprocedureasinthe
FIG.5. Emptybars: fractionsofobserver’sskyofdifferent preceding subsection, one finds the maximum attain-
temperatures, where the “hot” sky was divided into 30 able power to be Pmax ≈4.2σST14. For an Earth-size
segments of equal temperature span. Full bars: fractions planetandbackgroundradiationatT1 =2.725Kthis
of observer’s sky used as cold reservoirs for heat engines would be P 6.7 GW, i.e., enough for a small
max
≈
of uppertemperatureT. Allbarssum upto1,theempty country.
bars sum up to the fraction of the radiating sky and the
full bars sum up to the fraction covered by the shadow The case of Miller’s planet with r =
of black hole. Inset: fractions of the power of the heat 1.0000379GM/c2 leads to extreme blue shifts,
engines with uppertemperature T, the bars sum up to 1.
reaching up to 275,000. The black hole covers 40 %
(a) Schwarzschild black hole and orbit at r =6GM/c2 as
of the sky and most of the radiation energy comes
in Fig. 3(a), (b) Kerrblack hole with a=1 1.3 10−14
and orbit at r=2.2GM/c2 as in Fig. 3(b). − × from a very narrow strip of a few arcseconds (see
Fig. 4): our numerical results show that 99% of the
energy comes from a strip of longitude span 2.3′′ and
latitude span 5.8′′. This size is comparable, e.g., to
the angular size of the planet Neptune as seen from
mericaloptimizationis showninFig.5(a). The white Earth,with angulardiameter 2.2′′. Thus, the good
≈
barsaretheareasofthehotsegmentswhicharegiven news is that Miller’s planet enjoys small heating
as the input data. The blackbarsarethe areasofthe area regime (see Sec. IIIA), meaning that most of
correspondingcoldsegments obtainedas the result of the incoming radiation energy can be converted into
the optimizationprocedure. As canbe seen,the opti- useful work. The badnews for the visiting astronauts
mumallocationassumesabsorbingradiationfromthe is that it is too much energy: the incoming flux
14 hottest segments of blueshifts above 1.4. Colder density (power per unit area perpendicular to the
partsofthe“hot”skyarenotworthusingashotreser- incoming radiation) is Φ 420 kW/m2, i.e., about
≈
voirs. TheresultingpoweristhenP 0.126σST4, 300 times bigger than the solar constant. This value
max ≈ 1
i.e., one order of magnitude larger than the above can be used to find the equilibrium temperature
estimate based on disregarding the frequency shifts. of a planet radiating its energy as a black body,
For anEarth-sizeplanet andbackgroundradiationat T = 4 Φ/(4σ) 890◦C. Thus, the tidal waves
≈
T = 2.725 K, this would be P 200 MW, i.e., observed on the planet might be, e.g., of melted
1 max
≈ p
still rather low for a comfortable life of more than a aluminum. Moreover,the astronautswould be grilled
few small towns. by extreme-UV radiation.
6
presentworldenergyconsumption(ignoring,however,
the solar power harvested by the ecosystems). As
the lastexample we considerthe earlyUniverseback-
ground radiation with T 300K and a black hole
1
≈
with the apparent radius equal to the radius of Sun,
i.e., (√27/2)R 6.96 108m. This would lead to
S
≈ ×
P 2.9 1020W, which is three orders of magni-
max
≈ ×
tude above the present solar energy income of Earth.
VI. CONCLUSION
FIG. 6. Scheme of the Dyson sphere and of its black-
hole version. In the original version19 the shell captures
radiation emitted by the star inside and radiates waste Fornonrotatingblackholesandthepresenttemper-
heat to the space. In the black-hole version, the shell ab- ature of the cosmic microwave background, the avail-
sorbsbackgroundradiationcomingfromoutsideandemits able power appears to be rather small for the living
waste heat to theblack hole inside. standards of our civilization. One might speculate
of the distant future when hydrogen as the nuclear
fuel for stars is exhausted and black holes together
V. BLACK HOLE DYSON SPHERE with backgroundradiationbecome one ofthe few rel-
evant sources of negative entropy. Nevertheless, with
Rather than assuming a planet, one could imagine the accelerated expansion of the Universe the back-
a spherical shell enclosing the black hole. In 1960, ground radiation becomes colder so that even less
Freeman Dyson speculated about possible signatures power would be available. One might also speculate
of intelligent extraterrestrial life that would build a of hypothetical Earth-like planets orbiting primordial
structure around a star to capture all of its power.19 black holes in the early stages of the Universe filled
The waste thermal energy would be emitted as in- with room-temperature background radiation. The
frared radiation detectable by our observatories. We availablepowerbudgetbecomesmuchmoregenerous,
can turn this idea upside down: the inhabitants of however,onecouldhardlyexpectthatorganismswith
the shell collect energy from the background radia- necessaryradiation-focusingequipmentwouldstanda
tionandsendthe wastethermalenergyto the central chance to evolve.
black hole (see Fig. 6). ThesituationisdifferentforfastrotatingKerrblack
To explore the properties of such a scheme, we can holes and planets on close orbits: gravitational and
use results of Sec. IIB, however, we assume that the Dopplershiftschangethetemperaturemapofthesky
total area is not fixed, but variable — determined by to allow for harvesting much more power. Although
the radius of the Dyson sphere R . The sphere col- the highly inspiring idea of the movie Interstellar to
D
lectsthermalenergyfromtheouterareaS =4πR2 . exploreplanetsorbitingKerrblackholesisappealing,
H D
With respect to the cold area, the situation is now theconditionsonMiller’splanetofthefilmprovetobe
simpler than in Sec. IIA since no light concentra- rather harsh. This could be expected: since the time
tors are necessary. The waste energy is radiatedfrom dilatation on the planet is about sixty thousand, the
thewholeinnersurfaceandtheemittedphotoneither astronautswouldreceivesignalsfromthedistantout-
hits the black hole or is absorbed by another part of side arriving about sixty thousand times faster than
the inner surface. Thus, the waste energyis absorbed emitted. Suchafrequencyshiftmustapplyalsotothe
by the shadow of the black hole which for a distant cosmicbackground,turningitmuchhotter. Neverthe-
observer looks as a sphere of radius (√27/2)R (see less,witha suitably chosenorbitslightlyfarther from
S
Ref. 20). Therefore, S =108πG2M2/c4. Gargantua,onecanhopetofindtheskyconditionsof
C
the planet much closer to terrestrial.
The available power increases with increasing the
Dyson sphere radius, but up to a limit given by the
fixed area of the heat sink. To calculate the limiting
power, we assume R R and apply the results Appendix A: Construction of the sky temperature
D S
of Subsection IIIB with≫S /S 1. In this case map for an observer orbiting Kerr black hole
C H
≪
the limiting power follows from Eq. (17) as P
max
0.1055Pref, where Pref = σSCT14 is the power of th≈e Here we describe the computation of the temper-
background radiation that would be absorbed by the ature map of the observer’s local sky. The relevant
black hole if there were no Dyson sphere. quantity is the relativistic frequency-ratio factor g of
As an illustration, we first assume a black hole raybundlescomingfromdistantuniverseandforming
with the mass of Sun M = 2 1030 kg and tem- multiple relativistic images on the observer’s celestial
×
perature of the background radiation T = 2.725K: sphere.21 The frequency ratio between the locally ob-
1
the limiting power is P 250W. A supermas- served and emitted frequencies of such ray bundles
max
≈
sive black hole of the size of 4 106 solar masses g = ν /ν = phti/p also corresponds to the ratio
obs ∞ t
∼ ×
(e.g., the one in the center of our Galaxy) could give of the locally measured and emitted energies (time-
us P 4 1015W, i.e., about 200 times our components of photon four-momentum). Here and
max
∼ ×
7
hereafter, the angle brackets in the index denote the One can obtain the coordinate covariant components
local frame of the observer on the Keplerian orbit. of the four-momentum and related constants of mo-
The source intensity divided by the third power of tion (A4) by transforming the local components of
the frequency is conserved as a Lorentz invariant.22 p , using appropriate frame tetrads of one-form by
hµi
Moreover,Planck’slawdescribestheblackbodyradi- the relation
ation as having a spectral intensity in frequency pro-
portional to ν3/(ehν/kT 1), where T is the source pµ =ωµhαiphαi. (A6)
−
temperature, ν the frequency, h the Planck constant
Then the frequency-ratio factor g(λ,q) can be ex-
andkistheBoltzmannconstant. Therefore,theblack
pressed as a function of constants of motion corre-
bodyspectrumofthebackgroundradiationwillbelo-
sponding to the projection position on the observer’s
cally observed as a black body spectrum with a tem-
sky as
peraturemultipliedbythefactorg relatedtoapartic-
ular ray bundle. Naturally, the bolometric intensity
1
amplification of raybundles coming from distant uni- g(λ,q)= . (A7)
−p (λ,q)
verse is given by the fourth power of g. t
Usingthe( +++)signatureandgeometricalunits
The local tetrad of one-forms related to the ob-
−
(c=G=M =1), the line element of the Kerr space-
serveronthecorotatingKeplerianorbitaroundaKerr
timeinBoyer-Lindquistcoordinatesparameterizedby
black hole can be obtained by the Lorentz boost of
specific angular momentum (spin) a reads
ZAMO (zero angular momentum observers, locally
non-rotatingobservers)24,26tetrad,whichintheequa-
2r 4ra Σ
ds2 = 1 dt2 sin2θdtdϕ+ dr2+ torial plane (θ =π/2) takes the form27
− − Σ − Σ ∆
(cid:18) (cid:19)
2ra2sin2θ ∆Σ
+Σdθ2+ r2+a2+ sin2θdϕ2, (A1) ω(t) = ,0,0,0 ,
Σ A
(cid:18) (cid:19) (r )
whereΣ r2+a2cos2θand∆ r2 2r+a2. Compo- ω(r) = 0, Σ/∆,0,0 ,
≡ ≡ −
nents of the four-momentum of a photon in the Kerr
(A8)
spacetime are given by ω(θ) =n0,0p,√Σ,0 , o
pr =r˙ =srΣ−1 Rλ,q(r), (A2) ω(ϕ) =n 2ar ,0o,0, A ,
pθ =θ˙ =s Σ−1qΘ (θ), (−√AΣ rΣ)
θ λ,q
pφ =φ˙ =Σ−1∆q−1 2ar+λ Σ2 2r cosec2θ , whereA (r2+a2)2 a2∆. Thevelocityβ ofcorotat-
− ingKeple≡rianobserve−rswithrespecttothe equatorial
pt =t˙=Σ−1∆−1 (cid:2)Σ2−2ar(cid:0)λ , (cid:1) (cid:3) ZAMO frame can be written as27
where the dotted q(cid:0)uantities d(cid:1)enote differentiation
r2+a2 2a√r
with respect to some affine parameter, and the sign β = − . (A9)
pair s ,s describes the orientation of radial and lat- √∆(r3/2+a)
r θ
itudinal evolution, respectively.23–25 Radial and lati-
The tetrad (A8) straightforwardly transformed into
tudinal effective potentials read as
the frame of corotating Keplerian observer reads
R (r)= r2+a2 aλ 2 ∆ q+(λ a)2 ,
λ,q
− − − ωhti =γ ω(t) βω(ϕ),0,0, βω(ϕ) ,
Θ (θ)=(cid:0)q+a2cos2θ (cid:1)λ2cot2hθ. i(A3) t − t − ϕ
λ,q
− ωhri =ω(nr), ωhθi =ω(θ), o (A10)
Here, λ and q are constants of motion related to the
covariant components of the photon four-momentum ωhϕi =γ ω(ϕ) βω(t),0,0,ω(ϕ) ,
t − t ϕ
(A6) by the relations
n o
pφ where γ (1 β2)−1/2.
λ= , ≡ −
−p For such an observer located in the close vicinity
t
2 (A4) of the event horizon, the factor g varies depending
p
q = θ +(λtan(π/2 θ))2 a2cos2θ. on angular coordinates in the local sky, as a result of
p − −
(cid:18) t(cid:19) interplay between extreme gravitational lensing and
In the local reference frame related to an arbitrary optical effects of special relativity caused by the or-
observer, the azimuthal and latitudinal components bital motion of the Keplerian frame. In principle,
p , p of the photon four-momentum fully deter- applying the relations discussed above, it is possible
hϕi hθi
mine a projection of the corresponding ray onto the to construct the so-called critical loci curve, which
local sky. Assuming the photon energy is normalized forms a boundary between a projection of a distant
toone,theremainingcomponentsofp canbewrit- universe and a shadow of the black hole in the lo-
hµi
ten as follows: cal sky.28,29 To avoid significant analytical difficulties
associated with such an approach, we used our rela-
phti =−1, phri = 1−p2hθi−p2hϕi. (A5) tivistic ray-tracing code LSDplus, which performs a
q
8
time-reverse direct numerical integrationof the equa- tion(seeFigs3,4). Thecodealsocalculatesrelativis-
tions (A2).30,31 The LSDplus code allows to draw a tic frequency-ratio factor g by (A7) for ray bundles
boundaryoftheshadowoftheblackholeontheMoll- coming from distant universe.
weideprojectionofthelocalskywitharbitraryresolu-
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