ΕΑΝ Η ΑΙΝΣΤΑΝΙΑ ΒΑΡΥΤΗΤΑ ΗΤΑΝ ΕNAΣ ΚΑΛΥΤΕΡΟΣ ΤΡΟΠΟΣ ΓΙΑ ΝΑ ΠΕΡΙΓΡΆΨΟΥΜΕ ΤΗΝ ΒΑΡΥΤΗΤΑ ΓΙΑΤΙ ΟΙ ΜΑΘΗΤΕΣ ΣΤΟ ΣΧΟΛΕΙΟ ΔΙΔΆΣΚΟΝΤΑΙ ΤΗΝ ΒΑΡΥΤΗΤΑ ΤΟΥ ΝΕΥΤΩΝΑ;

ΕΑΝ Η ΑΙΝΣΤΑΝΙΑ ΒΑΡΥΤΗΤΑ ΗΤΑΝ ΕΝΑΣ ΚΑΛΥΤΕΡΟΣ ΤΡΟΠΟΣ ΓΙΑ ΝΑ ΠΕΡΙΓΡΆΨΟΥΜΕ ΤΗΝ ΒΑΡΥΤΗΤΑ ΓΙΑΤΙ ΟΙ ΜΑΘΗΤΕΣ ΣΤΟ ΣΧΟΛΕΙΟ ΔΙΔΆΣΚΟΝΤΑΙ ΤΗΝ ΒΑΡΥΤΗΤΑ ΤΟΥ ΝΕΥΤΩΝΑ; 
TOY VICTOR V. TOTH (PHYSISIST)

There is a very simple reason why (initially anyway) students are taught Newtonian gravity instead of Einstein gravity.

The most elegant way to formulate Newtonian gravity is through Poisson's equation. Let me show you what it looks like, in three dimensions:

∇2ϕ=4πGρ,

where ϕ

is the gravitational field, ρ is the mass density, G is Newton's constant of gravitation, and ∇

is the gradient operator. To solve this equation in three dimensions, you need to know vector calculus (and its prerequisites, such as basic algebra and calculus) but not much else.

The equation that defines Einstein's gravity is, in turn, as follows:

Rμν−12gμνR=8πGc4Tμν,

where Rμν=∂αΓαμν−∂νΓαμα+ΓαμνΓβαβ−ΓαμβΓβαν

is the Ricci tensor, R is its trace, Γαμν=12gαβ(∂μgβν+∂νgβμ−∂βgμν) are the Christoffel-symbols associated with the metric gμν, ∂μ is the partial derivative with respect to the μ-th coordinate, c is the speed of light, and Tμν

is the stress-energy-momentum tensor of matter that incorporates values describing the mass-energy density, momentum, pressure and stress that characterize the matter continuum. To make sense of this equation, you need to know Riemannian geometry; you need to be able to do differential geometry in four-dimensional curved spacetime; you need to be familiar with tensor algebra and calculus.

And what do you gain from all this? Under most normal circumstances, a tiny, tiny correction. For something like the Earth, the parameter of interest is the dimensionless value U=GM/c2r

, where M is the Earth's mass and r is the distance from its center; at the surface of the Earth, U∼7×10−10. The first post-Newtonian correction due to relativity is comparable in magnitude to the square of this value, which is about 5×10−19

. A tiny, tiny, tiny number.

So all that excess mathematical machinery brings you is a tiny correction. Unless you actually have the ability to measure that tiny correction (e.g., you are navigating GPS satellites or a precision gravitational experiment like GRACE, or calculating the per-century tiny precession anomaly of Mercury) this machinery is unneeded. Instead, you can happily continue using Poisson's equation, which for small, compact masses yields the nice solution ϕ=−GM/r

, which you can use to exquisite precision without having to worry about relativity.

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Update (responding to question details that weren't there when I began writing the answer above): Why is Einstein gravity not taught intuitively/abstractly? Because the idea that two bodies interact with each other through a field vs. the idea that the interaction is a result of distortions of spacetime geometry are really equivalent. Every force can be represented using geometry; but if the force is not universal, e.g., electromagnetism, then different geometries are needed to describe the motion of charged vs. uncharged objects, for instance. Gravity is universal: the equivalence principle states that all objects respond to gravity the same way regardless of their composition. Therefore, the geometry "sensed" by objects is the only geometry in town. But these are just pretty words; when you try to teach gravity, these words don't mean much unless you can translate them into usable mathematics. And that you cannot do without the full apparatus of Riemannian geometry, tensor calculus and the like. Whereas if you treat gravity as a force field, that is an intuitive picture that translates into much more accessible mathematics, and the results remain very accurate except in rather exceptional cases.

ANAΔΗΜΟΣΙΕΥΣΗ ΑΠΟ ΤΟ QUORA 9/6/2017