Last month, a new Nature paper reported data indicating that the Earth’s inner core is less solid than previously thought. This could have implications for a futuristic plan to reach the opposite side of the Earth through a tunnel. If we were to drill a tunnel through the center of Earth, how much time would it take a U.S. resident to travel to the neighborhood of Australia in the Indian Ocean “down under”?
The travel time can be estimated by taking the ratio between Earth’s radius, R, and the gravitational free-fall speed, v. Since the kinetic energy per passenger mass, (1/2)*v², is induced by the gravitational potential energy, GM/R, the travel time is of the order of (R/v) ~ [R/(GM/R)^{1/2}]~(G*D)^{-1/2}, where G is Newton’s constant and D~(M/R³) is the mean mass-density of Earth. Substituting 5.5 grams per cubic centimeter for D, this approximate time-density relation estimates the travel time to be of order 28 minutes. The exact calculation based on the actual density profile of Earth gives a travel time of 38 minutes through the Earth’s diameter. Satellites in low Earth orbits take a few times longer to go around the Earth, because the circumference of their orbit is pi=3.14 times the Earth’s diameter whereas their orbital speed is constant.
Remarkably, the Sun’s mean density is about 4 times smaller than that of the Earth, suggesting a travel time that is twice (square root of 4) longer than for Earth. Even though the Sun’s diameter is a hundred times bigger than that of Earth, a free-falling object would cross it within ~80 minutes. The travel time is dictated by the mean density, irrespective of how large the object is.
As passengers in freely-falling train cabins recede away from the center of Earth in the second half of their journey, they may feel like galaxies in an expanding Universe. This might inspire them to use the same time-density relation for estimating the mean density of the Universe based on the time that elapsed since the Big Bang. To obtain this estimate, they may use the fact that the oldest stars in the Milky-Way galaxy are about 14 billion years old, providing a good approximation to the age of the Universe. Adopting the approximate time-density relation gives an average cosmic mass density of 8×10^{-29} grams per cubic centimeter. The actual value for all forms of matter and energy is nearly an order of magnitude smaller because our simplified time-density relation is missing the factor of (8*pi/3) within the cosmic context.
The mass of the Milky-Way galaxy equals a trillion Solar masses. The mean density of the Universe implies that the distance to the closest galaxy with a similar mass would be about 4 million light years. The actual distance of the nearest massive galaxy, Andromeda, is 2.5 million light years. Andromeda is closer than expected because it is falling towards the Milky-Way. The two galaxies will merge within a few billion years.
The orbital time of the Sun around the center of the Milky-Way is about 200 million years. The time-density relation suggests that the mean density interior to the Sun’s orbit is 4×10^{-25} grams per cubic centimeter. Based on this density value and the mass of Sun, the nearest Sun-like star should be at a distance of order 10 light-years. In reality, the nearest Sun-like stars are Alpha-Centauri A & B at a distance of 4.34 light-years.
The highest density of a self-gravitating object is that of a black hole, when the free-fall speed approaches the speed of light, v~c. Equating the kinetic and potential energy gives the black hole radius, R~(2*G*M/c²), which is exactly the horizon radius derived by Karl Schwarzschild in 1916 as a solution to Albert Einstein’s equations of General Relativity. The dynamical time here is the Schwarzschild radius divided by the speed of light, equal to 10 microseconds for a black hole with the mass of the Sun. The corresponding density in the time-density relation sets the minimum value for the formation of a black hole out of a self-gravitating system. This threshold density scales inversely with the square of the black hole mass. It is higher than the mass density of an atomic nucleus for a black hole below 3 solar masses. This sets the lower limit on the mass of astrophysical black holes, since no star can exceed the nuclear density reached by a neutron star. Indeed, LIGO-Virgo-KAGRA did not detect gravitational waves from lower-mass black holes.
But based on the time-density relation, the Universe must have had higher densities at shorter times after the Big Bang. This is obvious since cosmic expansion rarefies matter and radiation over time. In particular, when the age of the Universe was shorter than 10 microseconds, the light crossing time of the Schwarzschild radius for a Solar-mass, the mean cosmic density was comparable to that of a solar-mass black hole. Since the Schwarzschild radius is proportional to mass, and the dynamical time is the light-crossing time of that radius, the maximum black hole mass that can be made at any time after the Big Bang is proportional that time. In other words, a black hole with a tenth of the mass of the Sun could have been made one microsecond after the Big Bang, and so on.
To make primordial black holes, the local density of matter and radiation in some rare regions must have been larger than average so as to induce collapse rather than cosmic expansion. Whether such conditions were induced by an early phase transition or unknown physics is unknown. If dark matter is made of primordial black holes, astrophysical constraints restrict these black holes to have asteroid masses in the range between 10^{17–10^{22} grams. Primordial black hole in this mass range can kill a person by passing surgically through the human body. It would traverse the Sun in 80 minutes.
The approximate relationship between dynamical time and average density applies to all systems that are marginally bound by gravity. The relation does not apply to objects bound by electromagnetic interactions like atoms, or by the strong interaction like atomic nuclei.
Physics works. Some philosophers argue that physics does not really explain physical reality; it forecasts phenomena but does not get to the nature of things. Perhaps this is good enough because our interactions with the world are mediated through phenomena. By making sense of the physical reality, physics serves as a tool that allows us to shape the world.
In the same fashion, some philosophers argue that artificial intelligence (AI) is not truly intelligent and will never possess the human qualities of consciousness and free will. Nevertheless, once our mental interactions with AI will be indistinguishable from our mental interactions with humans, this philosophical distinction will be irrelevant. At that time, AI will shape our mental world.
ABOUT THE AUTHOR
Avi Loeb is the head of the Galileo Project, founding director of Harvard University’s — Black Hole Initiative, director of the Institute for Theory and Computation at the Harvard-Smithsonian Center for Astrophysics, and the former chair of the astronomy department at Harvard University (2011–2020). He is a former member of the President’s Council of Advisors on Science and Technology and a former chair of the Board on Physics and Astronomy of the National Academies. He is the bestselling author of “Extraterrestrial: The First Sign of Intelligent Life Beyond Earth” and a co-author of the textbook “Life in the Cosmos”, both published in 2021. The paperback edition of his new book, titled “Interstellar”, was published in August 2024.
