 ## GRAVITATION

GRAVITATION IS CURVATURE OF SPACE-TIME
So how does one measure the curvature of space-time? Do not waste your time trying to visualize four dimensions. Do not try this at home. It’s not doable by humans, but it is possible to do so mathematically, using ideas I can show you in two-dimensions. THE PYTHAGOREAN THEOREM says that on the surface of a plane, a right triangle always has the interesting property that the sum of the squares of the sides equals the square of the hypotenuse.
Stretching or mapping a plane triangle onto the surface of a sphere (as in the nearby diagram), is a simplified resemblance in two dimensions of a “curved space” in four dimensions. The departure from “Pythagorean-icity” is a measure of the curvature of two-dimensional space. This can be done with equations, but sort of visualizable. with the help of the diagram. The curvature of the surface of the Earth is perpendicular to the surface of the Earth. Similarly, the “curvature of three-dimensional space” occurs so-to-speak “perpendicularly”… that is to say, in the “direction” of time. (See what I mean by trying to visualize four dimensions? We imagine it only by analogy with two dimensions on the Earth’s surface.)

NOW WE’RE READY to look at how space curvature and gravity works. You don’t have to read music to enjoy and hear music, and you don’t have to know math to understand the meaning of these two equations. SO THE FIRST EQUATION tells what gravity DOES: it’s a FORCE between masses: Newton described it as “action at a distance”. The second equation tells us what gravity IS, as Einstein understood it. A simple way to understand this equation: “Mass-energy tells space-time how to curve, and space-time tells mass how to move”. Of course, it’s a bit more complicated because gravity is energy, which is another form of mass, which creates more gravity.

CHEKOV advised his brother on the difference between “show” and “tell”, saying: “Don’t tell me the moon is shining; show me the glints of silver on the shards of broken glass.”  So Newton’s Law of Gravity TELLS us what gravity DOES. Einstein’s Field Equations of General Relativity SHOWS us what gravity IS… The curvature of space!

THESE ILLUSTRATIONS show us how gravitational waves were detected with the Laser Interferometer GravitationalWave Observatory (LIGO). A laser beam is split into two long equal perpendicular arms. The beams re-combine. If they are in phase they reinforce each other. If they’re out of phase, they cancel. The sensitivity is so extraordinary that if the wave’s induced space curvature distorts one of the two-mile arms by less than a thousandth the diameter of a proton, it’s detected.

AS EINSTEIN EXPLAINED elsewhere on this website, he said his “greatest blunder” was adding the “cosmological constant” Lambda, circled in red, to his field equations. In 1915, the known Universe was thought to be in a steady-state… neither expanding nor shrinking. But Newton’s gravitational force was purely attractive. Pure attraction, Einstein thought, would not give us a Universe in steady state or equilibrium. Everything should be squeezing together. So he added the Lambda term as a possible force of repulsion to counteract attraction. After Hubble’s discovery that the Universe is expanding, and recent results that show accelerating expansion, some cosmologists suggest that Lambda might be connected to pervasive dark energy, which is repulsive.

THE DIAGRAM THAT LOOKS LIKE TWO DONUTS or bagels is, in reality, a computer simulation of a binary pair of black holes that merge and emit gravitational waves, predicted by Einstein a century ago, and detected by heroic observational efforts spanning over 40 years by 1000 scientists at a cost of a billion dollars. Kip Thorne, Rai Weiss, & Barry Barish received the 2017 Nobel for this monumental work. When the black holes merge, they send a unique detectible signal that could come ONLY from this event. THIS VERY IMPRESSIVE and exciting research work has led to the detection and discovery of scores of black-hole and neutron star mergers. In a mere six-month period during 2019, gravitational-wave observatofies (like LIGO and VIRGO) have observed ripples in the fabric of space-time released by accelerating matter (like binary black-hole collisions) at the rate of one per week. Because of these observations, astronomers have gained a new way of seeing the cosmos.*. *www.nature.com/articles/d41586-020-03047-0.

## GRAVITATION

GRAVITATION IS CURVATURE OF SPACE-TIME
So how does one measure the curvature of space-time? Do not waste your time trying to visualize four dimensions. Do not try this at home. It’s not doable by humans, but it is possible to do so mathematically, using ideas I can show you in two dimensions.

THE PYTHAGOREAN THEOREM says that on the surface of a plane, a right triangle always has the interesting property that the sum of the squares of the sides equals the square of the hypotenuse. Stretching or mapping a plane triangle onto the surface of a sphere (as in the nearby diagram), is a simplified resemblance in two dimensions of a “curved space” in four dimensions. The departure from “Pythagorean-icity” is a measure of the curvature of two-dimensional space. This can be done with equations, but sort of visualizable. with the help of the diagram. The curvature of the surface of the Earth is perpendicular to the surface of the Earth. Similarly, the “curvature of three-dimensional space” occurs so-to-speak “perpendicularly”… that is to say, in the “direction” of time. (See what I mean by trying to visualize four dimensions? We imagine it only by analogy with two dimensions on the Earth’s surface.) NOW WE’RE READY to look at how space curvature and gravity works. You don’t have to read music to enjoy and hear music, and you don’t have to know math to understand the meaning of these two equations. SO THE FIRST EQUATION tells what gravity DOES: it’s a FORCE between masses: Newton described it as “action at a distance”. The second equation tells us what gravity IS, as Einstein understood it. A simple way to understand this equation: “Mass-energy tells space-time how to curve, and space-time tells mass how to move”. Of course, it’s a bit more complicated because gravity is energy, which is another form of mass, which creates more gravity.

CHEKOV advised his brother on the difference between “show” and “tell”, saying: “Don’t tell me the moon is shining; show me the glints of silver on the shards of broken glass.”  So Newton’s Law of Gravity TELLS us what gravity DOES. Einstein’s Field Equations of General Relativity SHOWS us what gravity IS… The curvature of space!

THESE ILLUSTRATIONS show us how gravitational waves were detected with the Laser Interferometer GravitationalWave Observatory (LIGO). A laser beam is split into two long equal perpendicular arms. The beams re-combine. If they are in phase they reinforce each other. If they’re out of phase, they cancel. The sensitivity is so extraordinary that if the wave’s induced space curvature distorts one of the two-mile arms by less than a thousandth the diameter of a proton, it’s detected.

AS EINSTEIN EXPLAINED elsewhere on this website, he said his “greatest blunder” was adding the “cosmological constant” Lambda, circled in red, to his field equations. In 1915, the known Universe was thought to be in a steady-state… neither expanding nor shrinking. But Newton’s gravitational force was purely attractive. Pure attraction, Einstein thought, would not give us a Universe in steady-state or equilibrium. Everything should be squeezing together. So he added the Lambda term as a possible force of repulsion to counteract attraction. After Hubble’s discovery that the Universe is expanding, and recent results that show accelerating expansion, some cosmologists suggest that Lambda might be connected to pervasive dark energy, which is repulsive.

THE DIAGRAM THAT LOOKS LIKE TWO DONUTS or bagels is, in reality, a computer simulation of a binary pair of black holes that merge and emit gravitational waves, predicted by Einstein a century ago, and detected by heroic observational efforts spanning over 40 years by 1000 scientists at a cost of a billion dollars. Kip Thorne, Rai Weiss, & Barry Barish received the 2017 Nobel for this monumental work. When the black holes merge, they send a unique detectible signal that could come ONLY from this event.

THIS VERY IMPRESSIVE and exciting research work has led to the detection and discovery of scores of black-hole and neutron star mergers. In a mere six-month period during 2019, gravitational-wave observatofies (like LIGO and VIRGO) have observed ripples in the fabric of space-time released by accelerating matter (like binary black-hole collisions) at the rate of one per week. Because of these observations, astronomers have gained a new way of seeing the cosmos.*. *www.nature.com/articles/d41586-020-03047-0. 