Re: [Phys-L] Two questions on General Relativity

[John Denker <jsd@av8n.com>]

On 3/26/22 12:42 AM, Antti Savinainen via Phys-l wrote:

> Is it really so that the Equivalence Principle, if correctly applied,
> fails to yield the correct value in the experiments carried out by
> Eddington and others?

True. Here's another mention of the idea. Not really
an explanation but at least a mention:

https://en.wikipedia.org/wiki/Shapiro_time_delay

> Has the Equivalence Principle only pedagogical value anymore, that
> is, is it virtually useless for research purposes?

That seems like an overstatement. The equivalence principle is
valid locally, to first order. In particular, there exists:
 ++ a local frame corresponding to weightlessness in Spain.
 ++ a local frame corresponding to weightlessness in New Zealand.
 −− no global frame that works for both.

First-order approximations are very heavily used for research
purposes. Not "useless" at all. Here's an analogy: Electrostatics
is not the state-of-the-art theory. It's useful for a few things,
even though it is not consistent with special relativity. If you
start with electrostatics and try to make it consistent with SR,
you wind up re-inventing magnetism, which might come as a bit of
a surprise. Eventually you might say that the electric field E
and the magnetic field B never really existed; instead there is
nothing but the electromagnetic field F. But even so, there remain
situations where it makes sense to measure the electrostatic piece
of F and ignore the rest.

By the same token, if you start with the equivalence principle
and try to patch it up to make it consistent with SR, you wind
up reinventing general relativity. This is a lot of work, and I
don't recommend it. It's easier to start with the full general
relativity formalism, and derive the equivalence principle as a
corollary.

==============

Things that correctly describe the local situation can be rather
misleading about the global situation. For example, consider
geodesics on a cone.

  https://www.av8n.com/physics/geodesics.htm#sec-cone

The cone is locally flat (zero intrinsic curvature) everywhere
except at the apex. Euclidean geometry everywhere, locally.
Yet you can easily have a geodesic that crosses another geodesic
twice (without going anywhere near the apex). Very non-Euclidean
globally.

It's amusing and instructive to make such cones out of paper
and just stare at them for a while. Keep one printout flat, i.e
not made into a cone, to convince yourself and your students
that the alleged geodesics really are geodesics, i.e. perfectly
straight lines.

You can probably surprise the local geometry teacher with this.

===========

Curvature has a /direction/. For example, a saddle curves upward
in one direction and downward in the perpendicular direction.

As for gravitation, have you ever wondered in *which direction*
the local spacetime is curved????? Spoiler:

  https://www.av8n.com/cgi-bin/spoiler?msg=Sbe%20na%20beqvanel%20tenivgngvbany%20svryq%2C%20gur%20snoevp%20bs%0D%0Afcnprgvzr%20vf%20pheirq%20zbfgyl%20va%20gur%20%2Agvzr%2A%20qverpgvba.%0D%0A

This makes it kinda hard to visualize.