I'm no expert on the mathematical details, but from what I've read, my
understanding is that while the "general covariance" (also called
'diffeomorphism invariance') of general relativity was originally
interpreted by Einstein to be a new physical principle of general
relativity akin to Lorentz-invariance, later work showed that in fact
it's a consequence of the mathematical form used to express general
relativity in terms of tensor equations with a metric tensor that
gives you the spacetime geometry (the metric can be used to calculate
coordinate-invariant 'geometric' facts like proper time on timelike
curves and proper distance on spacelike curves), in fact *any* theory
including Newtonian physics can be expressed in a similar generally
covariant way. See the paper at
http://philsci-archive.pitt.edu/11504/1/BIforarchive.pdf which
discusses this (and on pages 4-6 it talks about the fact that Einstein
came to acknowledge this point), there's also some discussion in the
physics forums thread at
https://www.physicsforums.com/threads/spacetime-symmetries-vs-diffeomorphism-invariance.739006/
In that thread someone also talks about how general relativity may be
"simpler" when expressed in generally covariant form than other
theories like Newtonian physics, something the physicist Julian
Barbour also talks about on p. 2 of the paper at
https://arxiv.org/pdf/gr-qc/0309089.pdf where he notes that "within
two years Einstein had been forced by a critique of Kretschmann [7] to
acknowledge that any physical theory must, if it is to have any
content, be expressible in generally covariant form. He argued [8]
that the principle nevertheless had great heuristic value. One should
seek only those theories that are simple when expressed in generally
covariant form. However, Einstein gave no definition of simplicity."
Barbour then goes on to give his own suggestion about a way to refine
the "simplicity" intuition into something more rigorous. I think this
is related to his "shape dynamics" research program at
https://en.wikipedia.org/wiki/Shape_dynamics which aims to reformulate
general relativity in a way that's more friendly to quantizing gravity
(and is also more 'Machian'), but the discussion at
https://physics.stackexchange.com/questions/662370/what-is-shape-dynamics
says shape dynamics can always be made locally equivalent to general
relativity but is not necessarily globally equivalent (though the
example they give is the interior region of a black hole, so I dunno
if there are any situations where you'd get different predictions
about aspects of general relativity that have actually been
empirically tested).
Also, one more basic point here is that the tensor notation that's
used to express the Einstein field equations in their most general
form just includes a single symbol each to represent the metric tensor
and the stress-energy tensor, but to actually do calculations using a
particular coordinate system you need to know the components of the
tensor in the coordinate system you're using, similar to linear
algebra equations where you may use a single symbol to represent a
matrix or vector but you need their components to do
coordinate-dependent calculations. And the form of the components does
change depending on the choice of coordinate system, for example if
you are dealing with the flat spacetime Minkowski metric of SR, the
components of the metric tensor will have the same form in any
inertial coordinate system, but a different form in non-inertial
coordinate systems.
On Sat, Nov 19, 2022 at 3:03 AM Alan Grayson <agrayson2...@gmail.com>
wrote:
I see. TY. Let's assume AE was aware of what you write after he
developed SR. That is, assume he knew that the laws of physics are
NOT invariant when one or more frames are non-inertial. What
prompted him to developed a theory, GR, based on tensors where the
laws of physics*are *invariant under coordinate transformations?
Based on your comments, it would seem that a theory which*is*
invariant for non-inertial frames, namely GR, is impossible.AG
On Friday, November 18, 2022 at 11:03:17 PM UTC-7 jessem wrote:
If both frames are inertial, any relativistic laws will have
the same form under the coordinate transformation. If either
is non-inertial, then there is no reason to expect the laws to
have the same form. What allows us to say that SR still
applies is that all predictions about physical,
coordinate-independent facts, like the proper time elapsed on
a given worldline between two events on that worldline, will
be exactly the same as the predictions made by inertial
observers.
As an analogy, we could use some non-Cartesian coordinate
system on a Euclidean plane, like one with curved coordinate
axes, but shapes on the plane would still be said to obey the
rules of Euclidean geometry even if some of the formulas that
work in Cartesian coordinate systems don't work in this
non-Cartesian coordinate system. For example, in a Cartesian
coordinate system, if you have a straight line segment whose
ends are at coordinates (x1, y1) and (x2, y2), then the length
of the line segment will be given by (x2 - x1)^2 + (y2 -
y1)^2, but this formula won't work if you describe the
selfsame line segment in a non-Cartesian coordinate system.
On Sat, Nov 19, 2022 at 12:11 AM Alan Grayson
<agrays...@gmail.com> wrote:
Are you claiming that we can have two coordinate systems,
one or both non-inertial, and a transformation from one to
another such that the laws of physics will have the same
form under this transformation? Is so, what allows us to
say "SR can be applied to non-inertial frames"? What has
SR to do with this result? AG
On Friday, November 18, 2022 at 4:26:20 PM UTC-7 jessem wrote:
If you have two coordinate systems A and B, and you
know the coordinates of some physical event in system
A, then the coordinate transformation can give you the
coordinates of the same event in system B--the
transformation is just a mapping from one system to
the other. But from what I understand you can use the
same coordinate transformation not just for individual
events, but for dynamical laws expressed in system A
(as differential equations, perhaps), giving equations
for the same physical laws expressed in system B. So
if someone has the coordinates of some initial
configuration of matter in system A, along with the
dynamical laws governing that matter in system A, then
we can use the transformation to get the equivalent
initial configuration in system B, and the equivalent
dynamical laws in B. And then if someone uses the
initial conditions and laws in A to predict later
events, and someone else uses the initial conditions
and laws in B to predict later events, all their
predictions will correctly map to one another using
the same coordinate transformation.
To say some laws of physics are "Lorentz invariant"
means if you write down the laws in one inertial frame
A and then apply the Lorentz transformation to see the
equivalent laws in a different inertial frame B, the
equations will have exactly the same form in both
frames. If you transform the dynamical laws into a
non-inertial frame C, the equations won't generally
have the same form, but they are still the same
relativistic laws and will yield predictions about
later events that map correctly back to what the
inertial observer predicts, for example if the
inertial observer predicts two physical clocks will
cross paths when one shows a proper time of 15 seconds
and the other shows a proper time of 40 seconds, then
the non-inertial observer will get the same prediction.
On Fri, Nov 18, 2022 at 5:16 AM Alan Grayson
<agrays...@gmail.com> wrote:
Thanks for that! You seem to know the subject
well. What exactly does it *mean* to say SR can be
used for non-inertial frames? Or, do you deny the
claim that SR *can* be used for non-inertial
frames? AG
On Thursday, November 17, 2022 at 8:25:41 AM UTC-7
jessem wrote:
The Lorentz transformation is specifically
meant for transforming between inertial
frames, it can't generally be used to
transform between non-inertial frames (you
could construct a pair of non-inertial
coordinate systems that were related by the
Lorentz transformation if you wanted, just
like you could construct a pair of
non-inertial frames related by the Galilei
transformation or whatever transformation you
wish--as I said to John Clark, there's no
'canonical' way to construct a non-inertial
coordinate system in relativity, you can
define one basically however you like).
However, all the physical consequences of the
postulate that the laws of physics are
Lorentz-invariant can be specified in terms of
different equations that would apply in a
non-inertial frame, and these equations can be
derived by using whatever coordinate
transformation was used to define the
non-inertial frame's coordinates relative to
an inertial coordinate system.
On Thu, Nov 17, 2022 at 8:07 AM Alan Grayson
<agrays...@gmail.com> wrote:
Jessem: I was wondering if the LT can be
used to determine how the laws of physics
change between two accelerating frames,
accelerating at the same rate but moving
in opposite directions. AG
On Wednesday, November 16, 2022 at 7:58:31
AM UTC-7 jessem wrote:
That doesn't address my specific
question about whether you define the
"predictions of SR" in terms of the
same specific equations that work in
inertial frames, like the time
dilation equation delta-tau = delta-t
* sqrt(1 - v^2/c^2), or whether you
would define the predictions of SR in
terms of new equations for the
non-inertial frame (which could be
obtained by applying a coordinate
transformation that maps the
coordinates of an inertial frame to
those of the non-inertial frame).
On Wed, Nov 16, 2022 at 5:18 AM Alan
Grayson <agrays...@gmail.com> wrote:
And when I used the word "true", I
just meant that no observations
exist which contradict the
predictions of SR. AG
On Tuesday, November 15, 2022 at
11:09:25 PM UTC-7 Alan Grayson wrote:
I just mean, if both frames
are accelerating at the same
rate, will the v in the LT, be
the instantaneous relative
velocity? AG
On Tuesday, November 15, 2022
at 11:05:42 PM UTC-7 Alan
Grayson wrote:
Specifically, will the
time dilation of a clock
in an accelerating frame,
be the same as a clock as
measured for a clock in a
the observer's
accelerating frame, where
v in the LT is the
instantaneous velocity of
the clock in the
observer's frame at every
time t in the observer's
frame?
On Tuesday, November 15,
2022 at 10:54:06 PM UTC-7
Alan Grayson wrote:
By "valid", I mean
"true". IOW, is SR
limited to
non-accelerating
frames? If the frames
are accelerating, will
the LT still hold for
relating the laws of
physics between those
frames? AG
On Tuesday, November
15, 2022 at 9:58:27 PM
UTC-7 jessem wrote:
It depends what
you mean by
"valid". Certainly
all the physical
laws of relativity
such as time
dilation can be
expressed in a
non-inertial
coordinate system,
like Rindler
coordinates. But
the equations
expressing these
laws will not be
the same in
non-inertial
coordinate
systems, for
example you can no
longer assume that
a clock moving at
constant
coordinate
velocity for a
coordinate time
interval of
delta-t will
elapse a proper
time of delta-tau
= delta-t * sqrt(1
- v^2/c^2).
On Tue, Nov 15,
2022 at 9:50 PM
Alan Grayson
<agrays...@gmail.com>
wrote:
Wormholes have
nothing to do
with my
question.
Please answer
the question
defining this
thread. TY.
On Tuesday,
November 15,
2022 at
1:00:50 PM
UTC-7
meeke...@gmail.com
wrote:
A stable
wormhole
requires
threading
by
negative
energy
density.
Since no
such
negative
energy
field is
know and
it's
existence
would
imperil
the
stability
of matter,
its
existence
seems
highly
unlikely.
Brent
On
11/15/2022
11:17 AM,
spudboy100
via
Everything
List wrote:
Me:
Forget
acronyms,
or even
Einstein's
gravitic
Reference
Frame
dragging
(His
movie
reel
analogy),
Instead
ask
yourselves
are these
physicists
correct
in
proposing
that some
black
holes are
wormholes?
Objects
We
Thought
Were
Black
Holes May
Actually
Be
Wormholes,
Scientists
Say
(futurism.com)
<https://futurism.com/objects-black-holes-wormholes>
For this
science
fiction
boy, I
say
interesting
and
maybe,
hopeful?
Let the
hard
science
Bohr
flavor of
quantum
mechanics
and
relativity
yield for
in
objection,
how this
is
fictional,
improbable,
and
crapola?
For
reference
frames, I
know
Einstein
locked
this in
with
time,
which he
discussed
with
Michele
Besso
(remember
the
letter to
Beso's
family?)
but
otherwise,
how
valuable
to
astronomers
and
physicists
is ref
frame
dragging
and all
that?
Does it
predict
do you think?
The
validity
of a
science
is its
ability
to
predict-Vanevar
Bush.
-----Original
Message-----
From:
Alan
Grayson
<agrays...@gmail.com>
To:
Everything
List
<everyth...@googlegroups.com>
Sent:
Tue, Nov
15, 2022
1:30 pm
Subject:
Re: Is
Special
Relativity
valid for
accelerating
frames of
reference?
TY.
RA.
On
Tuesday,
November
15, 2022
at
6:19:02
AM UTC-7
johnk...@gmail.com
wrote:
On
Tue,
Nov
15,
2022
at
6:38
AM
Alan
Grayson
<agrays...@gmail.com>
wrote:
/>
IHA
= ?/
I
Hate
Acronyms.
John
K
Clark
See
what's on
my
new
list
at
Extropolis
<https://groups.google.com/g/extropolis>
8gfk
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