Gravity II

 

Little Green Apples

God didn't make little green apples
And it don't rain in Indianapolis in the summertime
And there's no such thing as Doctor Seuss
Or Disneyland, and Mother Goose, no nursery rhyme.

What does a falling apple tell us about gravity?

Space-time tells mass-energy how to move. Mass-energy tells space-time how to curve. If space‑time did not curve, then we would live in a Euclidean world. In most applications, the curvature is so small that we tend to approximate spacetime as flat, i.e. our approximate Frame of Reference is Euclidean. But the curvature is there nonetheless, which makes the absolute Frame of Reference non-Euclidean.

If space-time is curved, then the question is that curvature positive, i.e. spherical, or negative, i.e. hyperbolic. If the curvature is positive and the radius of the sphere is exceptionally large compared to a typical distance, then space-time can be treated as virtually flat. If the curvature is negative, its radius is NOT a factor.

Mass-energy should move along the shortest path in space-time. That is what Newton’s First Law says: “An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force.”  The shortest path in in the same direction in spacetime is the hypotenuse between two points, i.e. a geodesic. If space‑time is curved hyperbolically, and the curve of space-time determines the geodesic over which mass-energy will move, then gravity is an apparent force, like Centrifugal Force or the Coriolis Force. If there are two objects, then the geodesic between those two objects is the distance in space-time between those two objects. If spacetime is curved, then the object with less mass‑energy will move towards the object with more mass‑energy. Thus what we interpret as gravity, is because we interpret that movement as Euclidean and it is really an apparent force if spacetime has a hyperbolic curvature. If two objects have exactly the same mass-energy, if both objects are not moving, then they should  NOT move towards each other.[1]

Consider Newton’s apocryphal apple falling from a tree. How might that be interpreted in hyperbolic, non-Euclidean space-time? In Euclidean space-time, the apple moves from the tree to the surface of the Earth. It would keep moving towards the center of the Earth, but the Pauli Exclusion principle, that two objects can not occupy the same space at the same time, says it can not pass through the surface of the Earth. In non-Euclidean space-time, the apple and the Earth are both moving,  e.g. the Earth is moving around the Sun, etc. It is just that both are moving at the same speed and in the same direction. Thus to an observer in our Euclidean frame of reference on the Earth, it only appears that the apple is not moving before it falls. Since the apple is much smaller than the earth, it should fall towards the earth. The path that it follows is the geodesic in space‑time. What appears to be an attraction between the two objects is merely the smaller object moving according to the curvature of space-time.

The difference between a Euclidean geodesic and a non-Euclidean, hyperbolic, geodesic is that the hyperbolic, non-Euclidean, geodesic will follow an exponential formula. In my field of Travel Demand Forecasting, trips are distributed according to the “Gravity” Model. It was called this because the impedance between the production of a trip and the attraction of a trip seemed to follow Newton’s Law of Gravity. A. G. Wilson later showed that this is because the trips actually randomly followed an  exponential function that looked like the gravity function. I was working at the Boston Metropolitan Planning Organization when I presented this ( I am amazed now that I was not thrown out for being particularly nerdy) and my favorite comment was by a manager who said that he had a tough time believing that people made trips like apples, but he had no problem believing that people made trips randomly. It appears that even Newton’s apples don’t behave like apples, but instead follow a similar exponential function. If gravity is an apparent force, then trying to find a Quantum Theory of Gravity may make no more sense than trying to find a Quantum Theory of Centrifugal Force. Gravity as a force may not fit into a Unified Field Theory because Gravity is not a real force, it is only an a apparent force.

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[1] Assuming those objects both have the same charge, etc. If they have opposite electrical charges, then the electromagnetic force will attract them, which IS an application of a force.