Physics Shorts: Why We Still Use Newtonian Gravity

GR is notoriously difficult mathematically (because the geometry of spacetime alters the geometry of spacetime as mass moving through the geometry of spacetime and… are you confused yet?)

GR is very slippery because it’s dynamic. Newtonian gravity is really, really simple because it very much isn’t, with one of the most rudimentary quadratic equations you’re likely to find in physics. If you understand the order of operations, the calculations are trivial.

And when it boils down to it, at the sort of mundane speeds and distances we deal with in most circumstances, the difference that shows Newton’s error is way off down the decimal expansion. The errors are so tiny that you need really enormous distances for them to show up at the speeds we can observe. I have two favourite examples to highlight the discrepancy at mundane velocities.

The first was in fact known about in and/or shortly after Newton’s time, namely what Newtonian gravity predicts for the precession of Mercury’s perihelion, namely 5557 seconds of arc per century. A second of arc is 1/3600th of a degree. The actual value is 5600 seconds of arc. Newton is out by 43 seconds in a century.

The other is the Cassini-Huygens probe, which is spectacular because, after executing four gravitational slingshot manoeuvres (two around Venus, one around Earth and one around Jupiter), and a journey through space of something in the order of 15×1011 kilometres, inserted itself into Saturn's orbit within twenty metres of its intended target. That's a fairly spectacular demonstration of just how piddling the errors are at those speeds and distances.

It’s a bit like asking why we do this

When we have a sextant and trigonometry.