# Losing Weight by Relocating

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In Albert Einstein’s General Theory of Relativity, the Earth’s gravitational acceleration toward the floor we stand on, g, is indistinguishable from what we would feel if the floor was pushed against our feet by a rocket. And if the rocket happened to accelerate away from our feet and towards Earth from space, then the Earth’s gravity would be negated. In other words, one can reduce the strength of gravity by an opposite acceleration. This might sound hypothetical, but this phenomenon has measurable consequences in our daily life.

For example, the centrifugal acceleration from the Earth’s rotation pushes us outwards and counteracts Earth’s gravitational pull. The value of g at the equator is 9.780 meters per second squared, smaller than the value of g at the poles — which is 9.832 meters per second squared. The centrifugal force at the equator accounts for about two-thirds of the difference in g between the equator and the poles. The rest of the difference is due to the oblateness of Earth. The change in the net gravitational acceleration g reduces the weight of the human body by 0.53% near the equator compared to the poles. Not a bad approach for losing half a pound just by changing your residency from Greenland to Kenya.

There should also be a fractional change in weight between daytime — when we are closer to the Sun, and nighttime — when we are farther away from the Sun by roughly the Earth’s diameter. This much smaller effect amounts to a fractional change of fifty billionths or an equivalent weight change by a few milligrams. The Moon’s weight differential on the two sides of Earth is roughly twice that of the Sun.

But circumstances were more extreme at earlier times in Earth’s history. After the Moon creation impact, the rotation period of the Earth was about four hours, making the centrifugal acceleration 36 times larger than it is today. A weight measurement at the equator would have been smaller by nearly 20% compared to the poles, about as much weight loss as my body encountered as a result of an elaborate low-carb diet.

At a fixed rotation period, the centrifugal acceleration increases with distance from the center of Earth, and at the high altitude of 35,786 kilometers it balances gravity. This gives rise to geosynchronous orbits for satellites at that altitude with an orbital period of a day. For a stationary observer on Earth, these satellites return to the same position on the sky after a day.

But one can trade gravity for rotation even better. In a spaceship, artificial gravity can be created through rotation. With one rotation per minute, the outer boundary of a circular spacecraft with a radius of 893 meters would mimic a gravitational acceleration of g. At 4 rotations per minute, the spacecraft radius has to be 56 meters, or equivalently a diameter comparable to a football field. This was roughly the size of the interstellar object `Oumuamua, but this object was tumbling every 8 hours with a centrifugal acceleration that is 4 million times smaller than g. In a football-field-scale spacecraft, a human’s head and feet would sense a few percent difference in g, which is insignificant. Of course, the design could be even larger to make the differential force for its residents smaller. For example, it could include two small parts connected by a straight long cable and rotating around the midpoint.

In Einstein’s famous thought experiment, residing in a free-falling elevator would feel like there is no gravity at all because the elevator’s floor would not push against the feet of the travelers. This is true as long as differential acceleration is negligible. Freely falling through the horizon of a very massive black hole would pose no health risks. But low-mass black holes would generate huge differential gravity across the human body and could tear it apart. When I visited my daughter’s class in elementary school a decade ago, one of the kids asked: “What will happen to my body if I fall into a black hole?” As I was going through the details, the teacher interrupted and said: “Lets stop here, because the kids will have nightmares if we continue.”

Assuming that my readers are grown-up, let me complete my answer here. The difference between the pull on the feet and head of an astronaut would be negligible near the horizon of supermassive black holes with masses larger than millions of Suns, as found at the centers of galaxies. Their event horizons are on the scale of planetary orbits in our solar system. However, the horizon of stellar-mass black holes, formed as a result of the collapse of the cores of massive stars, is the size of a city. This generates a differential gravity of millions of g across the height of an astronaut that would spaghettify a free-falling body into a stream of debris. The telltale signatures of the tidal disruption of stars by black holes are observed routinely.

The simplest way to mimic Earth’s gravity is by a rocket that accelerates at g. Doing it for a year would bring the astronauts onboard to the speed of light. Continuing the acceleration for half a century in the spacecraft’s frame would make them travel up to a sizable fraction of the horizon of the Universe. Time dilation makes the travel time equivalent to billions of years in the frame of the solar system. If the spacecraft would return after ten billion years, it would find a cold white dwarf as the relic of the old Sun.

All good things come to an end, except for the laws of physics. There is no need for a judiciary system in the Universe because the laws of physics cannot be broken. All we need to do as humans, is to imagine the best we can do with these laws. Others in our cosmic neighborhood might have had even better imagination.

As I suggested to my brilliant hosts, Jim Braude and Margery Eagan, in today’s interview on Boston’s NPR, we must check if there are any objects sent by neighbors to our backyard. Their rotating spacecraft might be more imaginative than ours. The awe it would inspire might convince us to stop wasting resources on wars over the limited territory of the rock we were born on. With suitable transportation, we might realize that there is a wealth of attractive real estate among the stars, offering any value of g that we fancy for our desired body weight.