Re: Experiments on the validity of Relativity

Il 16/05/2024 09:24, Tom Roberts ha scritto:

[...] two massive bodies (which fall gravitationally and freely
towards each other) *accelerate* both in the reference of one and
both in that of the other.

 
Sure. This does not refute GR, because of your confusion between
coordinate acceleration and proper acceleration.

In my animation https://www.geogebra.org/m/ttgky8xu there are two small
planets (A and B) that accelerate gravitationally and simultaneously
towards the common center of mass P.

Is this acceleration what you call "coordinate acceleration"?

And what is your "proper acceleration"?

Is it, perhaps, the acceleration of planet A with respect to itself and
of planet B with respect to itself?

Or is it something else?

Luigi Fortunati


[[Mod. note --

For your specific questions, a careful reading of
  https://en.wikipedia.org/wiki/Proper_acceleration
together with its references, would likely be useful.



In your animation, while A and B are initially stationary (before the
animation starts):
* in the coordinate system of your animation, A's center of mass
  has zero coordinate acceleration
* in the coordinate system of your animation, B's center of mass
  has zero coordinate acceleration
* A's center of mass has nonzero proper acceleration
* B's center of mass has nonzero proper acceleration

In your animation, while A and B are free-falling towards each other:
* in the coordinate system of your animation, A's center of mass
  has a (time-dependent) nonzero coordinate acceleration
* in the coordinate system of your animation, B's center of mass
  has a (time-dependent) nonzero coordinate acceleration
* A's center of mass has zero proper acceleration
* B's center of mass has zero proper acceleration

Notice that I've referred explicitly to A's and B's *center of mass*:
both coordinate and proper acceleration are properties of a *point*
or an *observer* (more precisely, in GR, a "worldline").  Applying
these concepts to an extended body may be ok if the body has negligable
self-gravity.  But, it's tricky try to apply these concepts to extended
bodies with non-negligible self-gravity, precisely because the answer
to the question "what's the proper acceleration of this point on the
body?" differs from one point to another.



But more generally, as Tom Roberts has noted in this thread (and many
people have said in past threads in this newsgroup), to learn GR you
really need a more extensive treatment than newsgroup discussions.  That
is, you really need a good textbook or textbooks, and/or a university
course or courses.  For self-study, my suggestion would be to get copies
of several textbooks and work through them in parallel -- often seeing
multiple pedagogical treatments can help you better understand what's
going on.

A few book suggestions (in roughly increasing order of mathematical
difficulty):

  Robert Geroch
  "General Relativity from A to B"
  University of Chicago Press, 1981

  James B Hartle
  "Gravity: An Introduction to Einstein's General Relatiity"
  Cambridge University Press, 2021

  Ian R Kenyon
  "General Relativity"
  Oxford University Press, 1990
  (or you could go for the 2nd edition of this,
   which includes some cosmology as well as GR)

  Bernard F Schutz
  "A First Course in General Relativity", 3rd edition
  Cambridge University Press, 2022

  Sean M Carroll,
  "Spacetime and Geometry: An Introduction to General Relativity"
  Addison-Wesley, 2004

Of these, the book by Geroch is notable for being almost completely
non-mathematical and for being very cheap.  The other books all introduce
a certain amount of tensor calculus, because that's what's needed to
cleanly present GR.
-- jt]]