Life is dependent on the income of energy with low entropy and the disposal of energy with high entropy. On Earth, the low-entropy energy is provided by solar radiation and the high-entropy energy is disposed of as infrared radiation emitted into cold space. Here, we turn the situation around and imagine the cosmic background radiation as the low-entropy source of energy for a planet orbiting a black hole into which the high-entropy energy is expelled. We estimate the power that can be produced by thermodynamic processes on such a planet, with a particular interest in planets orbiting a fast rotating Kerr black hole as in the science fiction movie Interstellar. We also briefly discuss a reverse Dyson sphere absorbing cosmic background radiation from the outside and dumping waste energy to a black hole inside.
References
The energy input/output of Earth is the solar constant ∼1.36 kW/m2 multiplied by the disk area of Earth πR2 ≈ 1.28 × 1014 m2, i.e., 1.74 × 1017 J each second; if this is divided by the temperature ∼300 K we obtain the estimate of the entropy production.
The celestial sphere is an imaginary sphere of arbitrarily large radius concentric with the observer. It is a practical tool in spherical astronomy for projecting objects on the sky. In Fig. 3, we show the Mollweide projection of the celestial sphere with the light (dark) area representing its hot (cold) part, respectively. The Mollweide projection generally used for global maps of the world or night sky is an equal-area, pseudocylindrical map projection preserving the proportions of areas, which is important for our considerations. It represents the central meridian as a straight line segment with other meridians longer and bowed outward away from the central one. This projection maps the sphere into a proportional 2:1 ellipse (the length of the central meridian is a half of the equator length).
The rotation parameter a (spin) is defined as a specific intrinsic angular momentum of the Kerr black hole; see, e.g., Hartle, Ref. 19, Sec. 15.2. For a = 0 the Kerr solution becomes the Schwarzschild non-rotating spherically symmetric spacetime [see Eq. (B1)]. The maximum possible value a = 1 corresponds to an extreme Kerr black hole having a rotation velocity of the event horizon equal to the speed of light. The case a > 1 gives a hypothetical naked singularity.
As shown in Ref. 33, the apparent angular size of a Schwarzschild black hole for a distant stationary observer is , where r is the distance of the black hole center in units of GM/c2. For r ≫ 1 the angular size is the same as that of a sphere of radius seen from the same distance.
The photosphere of a Kerr black hole is a region of unstable spherical or circular photon orbits wrapping the black hole. The photosphere reaches its maximum extent in the equatorial plane between corotating and counter-rotating circular photon orbits, while it becomes infinitesimally thin on the polar axis (see Ref. 32).
