# What is the Highest Spacetime Curvature Near Us?

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In principle, we could unravel exotic physics by irritating spacetime and inducing curvature in it.

What is the highest spacetime curvature accessible to us in our cosmic neighborhood? To answer this question, we should first define a measure of that curvature scale.

In his General Theory of Relativity, Albert Einstein wrote equations that relate the curvature of spacetime to the matter-energy density. As the physicist John Wheeler noted: “Matter tells spacetime how to curve, and spacetime tells matter how to move.” Gravity is not a force but rather spacetime curvature that affects the motion of matter. A marble with a proper speed will move on a circle on the flexible surface of a trampoline, which is curved by a bowling ball at its center. When the bowling ball is removed, the rubber surface would turn flat and the marble would move away in a straight line, just the way that Earth would fly out of the Solar system if the Sun dispersed.

Einstein was inspired by the equivalence principle, whereby all test objects follow the same universal motion under gravity, irrespective of their material composition. Galileo allegedly tested this principle from the top of the Leaning Tower of Pisa in Italy, and so did the astronaut David Scott during the Apollo 15 mission in 1971 by dropping a hammer and a feather in vacuum and verifying they both reached the surface of the Moon at the same time. In 2022, the MICROSCOPE satellite confirmed that two masses of titanium and platinum aboard a satellite orbiting Earth fall exactly in the same way to a precision of one part in quadrillion (10 to the power of -15).

The curvature in the surface of a beach ball is defined by its radius. The larger the radius is, the flatter is the surface from the vantage point of an ant walking on it. The local curvature scale of spacetime, R, is given by the speed of light divided by the square-root of Newton’s constant times the local mass density, R~[c/sqrt(G*rho)]. The denser the matter is, the shorter R is.

At the nuclear density of a proton, the curvature scale R is about 45 kilometers, comparable to the size of a large city. It is a few times larger than the size of a neutron star with the same mass density. Nuclear density is nearly a quadrillion times larger than the density of water, because the size of an atom is ~100,000 times bigger than the size of a proton and mass density scales inversely with size cubed.

Coincidentally, the average density of water is close to the average density of the Sun or of Jupiter. At this matter density, the curvature scale is about 8 times the Earth-Sun separation. The spacetime curvature scale R produced by the denser rock of planet Earth is half that value.

The highest spacetime curvature near astrophysical objects is just outside the event horizon of black holes. The lowest-mass black holes form by the collapse of massive stars, and carry a few times the mass of the Sun. Their curvature scale is about 10 kilometers, a factor of 6 smaller than that induced by a proton or a neutron star. The value of R for bigger black holes scales up in proportion to the black hole mass and reaches a tenth of the Earth-Sun separation for Sgr A*, the 4-million solar mass black hole at the center of the Milky-Way. The most massive black holes in the Universe with about 40 billion solar masses induce very weak curvature on a scale that extends out to 30 times the distance of Neptune from the Sun. An astronaut falling into their event horizon would barely sense their tidal field.

All in all, it is remarkable that protons curve spacetime almost as much as the smallest astrophysical black holes in our cosmic neighborhood.

Of course, colliders like CERN’s Large Hadron Collider, can smash particles at ultra-high energy and reach higher spacetimes curvatures for a short period of time. The mass-energy density can obtain yet higher values in collisions of the highest energy cosmic rays, with up to a trillion times the energy equivalent of the proton mass.

If new physics is associated with corrections at high spacetime curvature, it is likely to emerge near the Planck scale, which is 39 orders of magnitude smaller than 10 kilometers, the value of R outside the horizon of the smallest known black holes. The fact that this scale is well beyond our reach also in colliders explains why it is difficult to test theories which attempt to unify quantum mechanics and gravity, like string theory.

Alas, we have a lot to learn. This morning, I had received an email sent by Dr. Alexander Ross from Yale who asked: “What is a life not dedicated to learning?”, to which I replied: “A life not dedicated to learning is a life not fulfilled.”

Our current knowledge of new physics when the spacetime curvature reaches the Planck scale is a tiny island in a vast ocean of ignorance. The difficulty of accessing this knowledge through experimentation is one more reason to search for a smarter student in our class of intelligent civilizations within the classroom of the Milky-Way galaxy.