Flatness & Newtonian Gravity
John (Liberty) Bell (Change John to Liberty for email)
effect that Newtonian gravity equates to spatial curvature whereas GR
gravity equates to spacetime curvature.
The reason for my confusion is that Newton's inverse square law is
directly (and exactly) derivable from Newton's concept of lines of
force, only if space is Euclidean.
Could there, perhaps, be some confusion in this respect over the fact
that GR gives twice the bending of light grazing the Sun than
Newtonian gravity does?
Afaict, Newtonian curvature of light does not, therefore, mean that
space is curved, if it requires the assumption that space is flat, to
make the Newtonian prediction.
Can anyone clarify here?
Juan R. González-Álvarez
19 Apr 2009 23:32:12 +0000:
> I am a bit confused about a comment made here some time ago, to the
> effect that Newtonian gravity equates to spatial curvature
No, Newtonian gravity is an action-at-a-distance theory, where
gravity is a true force (as in electromagnetic interactions).
> whereas GR
> gravity equates to spacetime curvature.
Right
> The reason for my confusion is
> that Newton's inverse square law is directly (and exactly) derivable
> from Newton's concept of lines of force, only if space is Euclidean.
Right, Space in Newton theory is flat and static.
> Could there, perhaps, be some confusion in this respect over the
> fact that GR gives twice the bending of light grazing the Sun than
> Newtonian gravity does?
This is related to existence of an overall factor 2 (in General
relativity Schwarschild metric). Feynman field theory of gravity
and other non-geometrical approaches also explain the observation.
> Afaict, Newtonian curvature of light does not, therefore, mean that
> space is curved, if it requires the assumption that space is flat,
> to make the Newtonian prediction.
Right. I reproduced the field theory of gravity explanation of light
bending and compared with geometrical interpretation of GR in the next
link.
http://canonicalscience.blogspot.com/2007/08/relativistic-lagrangian-
and-limitations_20.html
If you want to donwload the images, the links are
http://4.bp.blogspot.com/_I-n4UWp0ZqM/RsmMD0c5CkI/AAAAAAAAAEc/
joqIPrW5O-k/s400/flatspacetime.gif
for light bending using a field theory of gravity over flat spacetime
and
http://3.bp.blogspot.com/_I-n4UWp0ZqM/RsmMDkc5CjI/AAAAAAAAAEU/
EdXg-C5_-Fo/s400/curvedspacetime.gif
for light bending using general relativity and curved spacetimes.
--
http://www.canonicalscience.org/
Usenet Guidelines:
http://www.canonicalscience.org/en/miscellaneouszone/guidelines.html
carlip...@physics.ucdavis.edu
> I am a bit confused about a comment made here some time ago, to the
> effect that Newtonian gravity equates to spatial curvature whereas GR
> gravity equates to spacetime curvature.
Almost the opposite is true. Newtonian gravity doesn't require
curvature at all, of course. But it can be written as a (rather
complicated) geometrical theory, as was first done by Cartan
in 1923. In this formulation, spacetime has a preferred time-
slicing, and the slices -- that is, the constant time hypersurfaces
-- are flat, while parallel transport in the time direction is
nontrivial. There's a nice description in chapter 12 of Misner,
Thorne, and Wheeler.
[...]
> Could there, perhaps, be some confusion in this respect over the
> fact that GR gives twice the bending of light grazing the Sun than
> Newtonian gravity does?
There's some connection, but again with space and time reversed;
the extra factor of two in the GR deflection traces back to spatial
curvature.
Steve Carlip
1000K
Newtonian garity is equivalent to TIME curvature in a flat Space :
ds2 = dx2 + dy2 + dz2 - (1- 2 GM/rc2) c2dt2
"John (Liberty) Bell (Change John to Liberty for email)"
<john...@accelerators.co.uk> a �crit dans le message de news:
70406247-d39f-4b2a...@z19g2000yqe.googlegroups.com...
Eric Gisse
> It is the reverse
> Newtonian garity is equivalent to TIME curvature in a flat Space :
>
> ds2 = dx2 + �dy2 + dz2 - (1- 2 GM/rc2) c2dt2
Lack of exponents aside, this is just the weak field approximation of
general relativity. Which is not the same thing as Newtonian
_gravitation_.
[...top posting is bad...]
of_1001...@hotmail.com
email)" <john.b...@accelerators.co.uk> wrote:
> I am a bit confused about a comment made here some time ago, to the
> effect that Newtonian gravity equates to spatial curvature whereas GR
> gravity equates to spacetime curvature.
> The reason for my confusion is that Newton's inverse square law is
> directly (and exactly) derivable from Newton's concept of lines of
> force, only if space is Euclidean.
Is that Faraday's concept of lines of force?
> Could there, perhaps, be some confusion in this respect over the fact
> that GR gives twice the bending of light grazing the Sun than
> Newtonian gravity does?
As Steve Carlip has said, it is essentially the other way around.
One way to see this is to note that in the Newtonian limit, there
is only a contribution from the time component of the metric tensor
to the equation of motion of a test particle - there is a direct
relation
between g_00 and the Newtonian potential V in this limit, with
g_00 ~ -(1 + 2 V) (taking c=1).
Hence, the Newtonian prediction for the deflection of light can
only take into account time-dilation effects (or, equivalently,
gravitational redshift effects), and is independent
of spatial curvature effects (corresponding to g_ij components
of the metric tensor).
It is interesting that in special relativity, one can write down
scalar gravity equations (similar to the 4-vector e-m
equations), and again only get a deflection of light equal to
that of the Newtonian prediction. This makes sense in
light of the above, since special relativity has time dilation
but not spatial curvature.
When one moves to GR, and looks at a (slightly generalised)
Schwarzschild space time, one has a metric of the form
g_00 = 1 + 2 aV, g_rr = 1/(1 + 2bV),
where V=-GM/r is the Newtonian potential (and a=b=1 in the
Schwarzschild case) - all isotropic metrics take this form
in a particular approximation. The first-order deflection is
proportional to a+b. Thus, time dilation contributes a
proportion a/(a+b), and spatial curvature a proportion b/(a+b).
In the Schwarzschild case, with a=b=1, this means each
effect contributes half the bending.
There is a good general discussion in the book Gravitation
and Cosmology by Steven Weinberg (sections 8.1 to 8.5).
The equations for the Newtonian and Schwarzschild cases
are also discussed at
http://www.mathpages.com/rr/s6-03/6-03.htm
harry
<of_1001...@hotmail.com> wrote in message
news:b6dbf5c7-5b2d-4c3b...@f41g2000pra.googlegroups.com...
But it does: special relativity has time dilation and length contraction.
However, in 1911 Einstein did not account for length contraction when
applying the equivalence principle, he only thought of time dilation. Adler,
Bazin and Schiffer (Introduction to General Relativity) pointed out that
including length contraction suffices to obtain the light bending of GRT
directly from SRT together with the equivalence principle.
> When one moves to GR, and looks at a (slightly generalised)
> Schwarzschild space time, one has a metric of the form
> g_00 = 1 + 2 aV, g_rr = 1/(1 + 2bV),
> where V=-GM/r is the Newtonian potential (and a=b=1 in the
> Schwarzschild case) - all isotropic metrics take this form
> in a particular approximation. The first-order deflection is
> proportional to a+b. Thus, time dilation contributes a
> proportion a/(a+b), and spatial curvature a proportion b/(a+b).
> In the Schwarzschild case, with a=b=1, this means each
> effect contributes half the bending.
>
> There is a good general discussion in the book Gravitation
> and Cosmology by Steven Weinberg (sections 8.1 to 8.5).
> The equations for the Newtonian and Schwarzschild cases
> are also discussed at
> http://www.mathpages.com/rr/s6-03/6-03.htm
Thanks,
Harald
John Bell (Change John to Liberty for email)
> On Apr 20, 9:32�am, "John (Liberty) Bell (Change John to Liberty for
> email)" <john.b...@accelerators.co.uk> wrote:
> > The reason for my confusion is that Newton's inverse square law is
> > directly (and exactly) derivable from Newton's concept of lines of
> > force, only if space is Euclidean.
>
> Is that Faraday's concept of lines of force?
On googling the below link, yes. Maxwell's interpretation of Newtonian
lines of force is a very different beast
Thanks for the heads-up.
Did Newton even use the concept of lines of force himself?
John Polasek
Actually it's g00 = -1 +2aV/c, and grr = 1/1-2bV/c (added 2 minus
signs where V = MG/r).
grr = -1/g00.
>> where V=-GM/r is the Newtonian potential (and a=b=1 in the
>> Schwarzschild case) - all isotropic metrics take this form
>> in a particular approximation. The first-order deflection is
>> proportional to a+b. Thus, time dilation contributes a
>> proportion a/(a+b), and spatial curvature a proportion b/(a+b).
>> In the Schwarzschild case, with a=b=1, this means each
>> effect contributes half the bending
You make it sound easy; so the algebra must be easy, probably just a
matter of working with g00 and grr properly. I can't quite get it.
>>
>> There is a good general discussion in the book Gravitation
>> and Cosmology by Steven Weinberg (sections 8.1 to 8.5).
>> The equations for the Newtonian and Schwarzschild cases
>> are also discussed at
>> http://www.mathpages.com/rr/s6-03/6-03.htm
Mathpages didn't quite get it either, seeing there's about 7 pages of
dense algebra before arriving at the double effect.
It should be possible to present an elegant 2 line proof, but I've
never seen one.
>>Thanks,
>Harald
John Polasek
John Bell (Change John to Liberty for email)
>
> email)" <john.b...@accelerators.co.uk> wrote:
> > I am a bit confused about a comment made here some time ago, to the
> > effect that Newtonian gravity equates to spatial curvature whereas GR
> > gravity equates to spacetime curvature.
> > The reason for my confusion is that Newton's inverse square law is
> > directly (and exactly) derivable from Newton's concept of lines of
> > force, only if space is Euclidean.
>
> Is that Faraday's concept of lines of force?
As introduced to me (as a schoolboy), I think the 2 are the same,
except that one applies to gravity, and the other to electromagnetism
(unknown in Newton's time).
> > Could there, perhaps, be some confusion in this respect over the fact
> > that GR gives twice the bending of light grazing the Sun than
> > Newtonian gravity does?
> As Steve Carlip has said, it is essentially the other way around.
Agreed. I have never known Steve Carlip to write something daft, and
you have apparently confirmed in a more simplistic/simple way (which
is the way I usually like it). This also confirms what I understood
intuitively, which is why I got confused by a prior poster who said
the opposite.
> There is a good general discussion in the book Gravitation
> and Cosmology by Steven Weinberg (sections 8.1 to 8.5).
> The equations for the Newtonian and Schwarzschild cases
> are also discussed athttp://www.mathpages.com/rr/s6-03/6-03.htm
Thanks
Rock Brentwood
> "John (Liberty) Bell (Change John to Liberty for email)" <john.b...@accel=
erators.co.uk> wrote:
> > I am a bit confused about a comment made here some time ago, to the
> > effect that Newtonian gravity equates to spatial curvature whereas GR
> > gravity equates to spacetime curvature.
>
> Almost the opposite is true. Newtonian gravity doesn't require
> curvature at all, of course. But it can be written as a (rather
> complicated) geometrical theory, as was first done by Cartan
> in 1923. In this formulation, spacetime has a preferred time-
> slicing, and the slices -- that is, the constant time hypersurfaces
> -- are flat, while parallel transport in the time direction is
> nontrivial.
The Cartan formulation is built atop an Riemann-Cartan manifold. The
connection is not trivial, so there is curvature (after all, at the
bottom line there is the Ricci tensor and Cartan's equation R_{mn} = T_
{mn}).
To answer these questions, it's best to understand the infrastructure
of geometry the way a mathematician sees it, because there's an
important point to be made from this standpoint.
Geometry is a multi-layered structure. At the bottommost layer is the
"bare manifold", which has all the formal apparatus associated with
diferential forms, exterior differentials, Lie derivative, integrals
and so on.
The objects seen in Riemann-Cartan geometry are structures defined at
this level.
Second, there is the level associated with the extra structure,
itself; in particular, the connection. Curvature is defined in terms
of a connection. At this level you have covariant derivatives and the
ability to define straight lines ("autoparallels"). What is NOT
present at this level is any notion of a metric.
What this also means -- in particular -- is that there is no notion of
spacetime signature at ths level either. That is, at this layer, or
substrate of geometry, you can't tell the difference between a
Relativistic and Non-Relativity (or even locally Euclidean) geometry.
They all look exactly the same.
But this is the level at which you speak of curvature.
Thus, the question of what curvature is (or is not) and whether (or
not) it is present actually has nothing to do with the question of
Relativity vs. Non-Relativity! We're still one layer below that level.
Finally, the next layer up, you have the metric (and its dual metric)
and all the extra structure it brings into the picture. With it, you
can define the causal structure (or signature) of the space (i.e., in
particular, you can distinguish between a relativistic theory and non-
relativistic theory).
The formulation of Newtonian gravity in Riemann-Cartan geometry is not
much different than (or more "complex" or "difficult") than the
formulation of general relativity. It's the very same ingredients
appearing in the very same way -- as long as one pays proper respect
to the fact that the metric and the dual metric are now independent
objects (each being degenerate) and that the orthogonal group is the 6-
parameter group -- of basically the same structure (with a minor
deformation) as the Lorentz group -- preserves both metrics. So the
connection is almost of the same form as in the relativistic theory.
I laid out the formulation in 3+1 language that provides a common
enveloping framework for Riemann-Cartan gravity in both Galilei and
Lorentzian signatures in part 3 of
Basic Topics of the Mathematics of General Relativity
http://federation.g3z.com/Physics/index.htm#Solutions
(It's in part 3: "quasi-Galilean" tetrads).
I don't include the dynamics of Cartan's theory however (that is, the
Cartan equation), just the geometric infrastructure that's
prerequisite to formulating any dynamics.
The question of providing a continuous bridging between GR and
Newtonian gravity is highly non-trivial, and it's in this sense that
"complication" actually arises; rather than in the formulation of
field laws.
It's still largely an open question how to continuously deform the
whole mathematical edifice associated with GR (that is, both the
geometry, the signature, AND (most importantly) the Lagrangian) into a
form that resides in a Galilean signature AND yet contains Newtonian
Gravity as a subset.
The best reference I've seen (so far) is the series:
Galilean Theories of Gravitation
R. De Pietri, L. Lusanna, M. Pauri
arXiv:gr-qc/9212002v2
Newtonian Gravity as a Gauge Theory. I: The Standard Theory
R. Di Pietri, L. Lusanna, M. Pauri
arXiv:gr-qc/9405046v1 22 May 94
Newtonian Gravity as a Gauge Theory. II: The Dynamical 3-Space
Theories
R. Di Pietri, L. Lusanna, M. Pauri
arXiv:gr-qc/9405047v1 22 May 94
(The first article from 1992, above, is the de facto part 0 of the
series).
An exercise I still haven't done is to pull the Utiyama Machine on the
infrastructure. That is: you start with the general framework of
Riemann-Cartan geometry, adapt it to a Galilean signature, write down
the MOST GENERAL Lagrangian invariant under the local Galilei group
and try to fit Newtonian gravity within it as a special case.
The second part of the exercise is to adopt Jacobson's approach:
assume the extra structure of a global velocity field. This is the
same thing that is required to get a consistent Galilean limit for
electromagnetism (and gauge theory) -- as Maxwell, himself, noted when
he introduced his absolute velocity vector G.
Here, the approach is more general. Write down the most general
Lagrangian that is locally invariant under the distinguished SO(3)
subgroup of Galilei. Include both G and its exterior derivative dG in
the theory (this is basically what Jacobson did in his Einstein-Aether
model). NOW try to provide a continuous link between the mathematical
structures associated with GR and Newtonian Gravity.
None of this is easy. A Riemann-Cartan geometry, when adapted to
either the Galilean or Lorentzian signature, has around 70 orthogonal
group invariants to play around with; and yet more when broadening the
scope to SO(3) invariants. The trick is to get the Lagrangian to
continuously deform, with respect to a "signature parameter" alpha = 0
-> alpha = (1/c)^2, while at the same time getting it so that the
dependencies on the SO(3) invariants meld into dependencies solely on
the Lorentz invariants, by the time you get to alpha = (1/c)^2.
The complication that was raised in the 3 references, above, by the
way, is that there are terms that not only go as O(alpha) that that
you need to retain, when contracting from the Lorentz to the Galilei
signature, but also terms that are O(alpha^2) -- that's because the
coupling coefficient is second-order in alpha.
(In fact, that's the root of all the complications).
Extending the Galilei group to its 11-parameter version handles the O
(alpha) terms (hence the extended titles of the papers, "Standard and
Generalized 'Newtonian' Gravities as Gauge Theories of the EXTENDED
Galilei Group".
To keep the O(alpha^2) terms in, the authors have to resort to twists
and contorsions that I still don't fully understand, and whose need I
still don't fully appreciate.
> There's some connection, but again with space and time reversed;
> the extra factor of two in the GR deflection traces back to spatial
> curvature.
With orbits, I think it's also true that the periapse shift can be
traced back to the non-closure of orbits in the special relativistic
form of the Kepler problem. Orbits don't closed in the relativized
theory.
of_1001...@hotmail.com
> <of_1001_nig...@hotmail.com> wrote in message
> > It is interesting that in special relativity, one can write down
> > scalar gravity equations (similar to the 4-vector e-m
> > equations), and again only get a deflection of light equal to
> > that of the Newtonian prediction. �This makes sense in
> > light of the above, since special relativity has time dilation
> > but not spatial curvature.
>
> But it does: special relativity has time dilation and length contraction.
> However, in 1911 Einstein did not account for length contraction when
> applying the equivalence principle, he only thought of time dilation. Adler,
> Bazin and Schiffer (Introduction to General Relativity) pointed out that
> including length contraction suffices to obtain the light bending of GRT
> directly from SRT together with the equivalence principle.
I'm reposting a quick reply here, as my first reply hasn't gone up in
a week.
The Adler et al argument sounds very interesting, although it
also sounds heuristic - i.e., not based on a theory as such.
I was thinking of a scalar field theory for gravity, within the
]context of special relativity, as discussed in the book
"Gravitation and Relativity" by Bowler (Pergamon, 1976). This
theory predicts half the bending of light given in GR.
Your point certainly raises the question, however, of where
the factor of 1/2 comes from - there may well be some sort
of effective length contraction going on, rather than time
dilation as I had assumed (although the latter is related
to conservation of energy in this context, and so the
more likely candidate I think). I will check out Bowler's
book again when I get some time.
harry
news:2fa64258-73a0-4dc2...@u39g2000pru.googlegroups.com...
> On Apr 30, 2:18 am, "harry" <harald.NOTTHISvanlin...@epfl.ch> wrote:
>> <of_1001_nig...@hotmail.com> wrote in message
>> > It is interesting that in special relativity, one can write down
>> > scalar gravity equations (similar to the 4-vector e-m
>> > equations), and again only get a deflection of light equal to
>> > that of the Newtonian prediction. This makes sense in
>> > light of the above, since special relativity has time dilation
>> > but not spatial curvature.
>>
>> But it does: special relativity has time dilation and length contraction.
>> However, in 1911 Einstein did not account for length contraction when
>> applying the equivalence principle, he only thought of time dilation.
>> Adler,
>> Bazin and Schiffer (Introduction to General Relativity) pointed out that
>> including length contraction suffices to obtain the light bending of GRT
>> directly from SRT together with the equivalence principle.
>
> I'm reposting a quick reply here, as my first reply hasn't gone up in
> a week.
>
> The Adler et al argument sounds very interesting, although it
> also sounds heuristic - i.e., not based on a theory as such.
It's based on his 1911 theoretical extension of SRT but without overlooking
length contraction (thus without the complex GR structure; the authors seem
to suggest that it doesn't work well for the perihelion). It has
qualitatively the same anisotropies as GR.
> I was thinking of a scalar field theory for gravity, within the
> ]context of special relativity, as discussed in the book
> "Gravitation and Relativity" by Bowler (Pergamon, 1976). This
> theory predicts half the bending of light given in GR.
In a parallel thread in sci.physics.foundations we discussed Einstein's
peculiar remark in his popular account that :
"It may be added that, according to the theory, half of this deflection
is produced by the Newtonian field of attraction of the sun, and the
other half by the geometrical modification ("curvature") of space caused
by the sun."
- http://www.bartleby.com/173/a3.html
I took suggestions from this thread (including from you) as a clue for what
Einstein likely meant with that remark; apparently he equated time dilation
with "Newtonian field of attraction"; and perhaps he used that equivalence
for years. IMHO, such mixing of explanations from different theories is
inconsistent and therefore counter-productive.
> Your point certainly raises the question, however, of where
> the factor of 1/2 comes from - there may well be some sort
> of effective length contraction going on, rather than time
> dilation as I had assumed (although the latter is related
> to conservation of energy in this context, and so the
> more likely candidate I think). I will check out Bowler's
> book again when I get some time.
I have been puzzled by the factor two for years and so I have been searching
the literature for this, although my knowledge of GRT is very limited.
Happily for an understanding of physical explanations it's not necessary to
master GRT's mathematical machinery; but in his 1916 paper Einstein doesn't
provide enough details and the general literature isn't very helpful either.
Now, in view of the straightforward, elaborate explanation by Adler et all
(which I do understand) as well as from purifying (back-translating to GRT)
Einstein's above remark, I think that it is safe to say that the right
answer must be as follows:
Whereas in 1911 he only accounted for gravitational time dilation, GRT takes
into account both time dilation (like a "Newtonian field of attraction") and
gravitational length contraction ("geometrical modification of space"); and
the calculation shows that this results in twice the bending.
Regards,
Harald
Juan R.
> <of_1001...@hotmail.com> wrote in message news:2fa64258-73a0-4dc2-
a14f-0a2...@u39g2000pru.googlegroups.com...
>> On Apr 30, 2:18 am, "harry" <harald.NOTTHISvanlin...@epfl.ch> wrote:
>>> <of_1001_nig...@hotmail.com> wrote in message
(...)
>> I was thinking of a scalar field theory for gravity, within the
>> ]context of special relativity, as discussed in the book "Gravitation
>> and Relativity" by Bowler (Pergamon, 1976). This theory predicts half
>> the bending of light given in GR.
>
> In a parallel thread in sci.physics.foundations we discussed Einstein's
> peculiar remark in his popular account that :
>
> "It may be added that, according to the theory, half of this deflection
> is produced by the Newtonian field of attraction of the sun, and the
> other half by the geometrical modification ("curvature") of space caused
> by the sun." - http://www.bartleby.com/173/a3.html
The overall 2phi/c^2 factor measures deviation from flatness.
It is *not* that a half that factor is Newtonian and the other half is a
correction due to deviation from flatness.
(...)
>> Your point certainly raises the question, however, of where the factor
>> of 1/2 comes from - there may well be some sort of effective length
>> contraction going on, rather than time dilation as I had assumed
>> (although the latter is related to conservation of energy in this
>> context, and so the more likely candidate I think). I will check out
>> Bowler's book again when I get some time.
>
> I have been puzzled by the factor two for years and so I have been
> searching the literature for this, although my knowledge of GRT is very
> limited. Happily for an understanding of physical explanations it's not
> necessary to master GRT's mathematical machinery; but in his 1916 paper
> Einstein doesn't provide enough details and the general literature isn't
> very helpful either.
(...)
In short, the reason which GR and other theories give the correct light
deflection, whereas Newtonian gravity and Einstein 1911 theory give only a
half of the deflection is due to that gravitational interaction is not
scalar but tensorial.
Regards
Rock Brentwood
> In short, the reason which GR and other theories give the correct
light
> deflection, whereas Newtonian gravity and Einstein 1911 theory give only a
> half of the deflection is due to that gravitational interaction is not
> scalar but tensorial.
This is a red herring and can clearly be seen as such, as follows:
Newton-Cartan gravity is both non-relativistic and tensorial, but the
discrepancy still exists. (Or to put it another way: if the
discrepancy had NOT remained, after generalizing Newtonian gravity to
a curved space-time theory, that would have led to a major coup back
in the 1920's and we'd all be talking about it, instead!)
Thus, whatever the cause of the discrepancy, the tensorial character
of gravity has nothing to do with it.
Juan R.
> On May 8, 4:32 pm, "Juan R." Gonz�lez-�lvarez
> > In short, the reason which GR and other theories give the correct
> light
>> deflection, whereas Newtonian gravity and Einstein 1911 theory give
>> only a half of the deflection is due to that gravitational interaction
>> is not scalar but tensorial.
>
> This is a red herring and can clearly be seen as such, as follows:
One central issue of my message (deleted in your response) was the
correction to Harry's incorrect claim (also corrected in spf).
Another other central issue of my message was my claim that the factor "2"
may be traced to the tensorial character of the interaction.
Below I give more details and correct your claims.
> Newton-Cartan gravity is both non-relativistic and tensorial, but the
> discrepancy still exists. (Or to put it another way: if the discrepancy
> had NOT remained, after generalizing Newtonian gravity to a curved
> space-time theory, that would have led to a major coup back in the
> 1920's and we'd all be talking about it, instead!)
>
> Thus, whatever the cause of the discrepancy, the tensorial character of
> gravity has nothing to do with it.
If you read my message above, you can see I was refering to the tensorial
character of the *interaction*, when wrote "[...] gravitational
interaction is not scalar but tensorial".
In Newton-Cartan theory one starts from geodesic equations and defines
connections like flat connections plus a correction term.
The correction is given (by definition) by a *spatial* metric, a
*temporal* metric and the gradient of a scalar potential phi (which is
also introduced in the theory).
The *spatial* and *temporal* metrics are degenerate and mutually
orthogonal with signatures (0 + + +) and (+ 0 0 0). This describes a
*static* and *universal* spacetime.
Thus gravitational 'interactions' [#] are really given by the *scalar*
potential phi.
Any theory of gravity I know using a *scalar* potential for interactions
has the problem that either (i) you choose a factor "Phi" or (ii) a factor
"2Phi" for the interaction.
(i)
Gives (a = -grad Phi) for massive particles but only half deflection for
light because lacks the famous factor "2".
(ii)
Includes the factor "2" and gives half deflection for light but not the
correct motion for massive particles (a = -grad 2Phi).
All scalar theories I know (including Einstein 1911) chose (i) and failed
to give light deflection. The famous factor "2" lacking in Newtonian
theory and in Einstein's 1911 theory.
Tensorial theories of interactions (field theory or GR [#]) give both
correct massive motion and correct light deflection.
I will not enter here in details (just study the theories) but I will
point that the motion of low velocity massive particles is given by a term
(2 T^00 - T^00) in the tensorial interaction. Thus the factor "2" is
cancelled giving the correct final (grad Phi) for low velocity massive
particles.
The motion for light signals is given by a term (2 T^00 - 0) in the
tensorial interaction. Now this gives the extra factor 2 for masless
particles and the correct light deflection.
Tensorial interactions have more 'freedom' and can give a term "Phi" for
massive particles and "2Phi" for masless at once. Scalar theories of
gravitational interaction may choose between (i) and (ii).
Regards.
[#] Recall GR is a metric theory.
harry
"Juan R. Gonz�lez-�lvarez" <juanR...@canonicalscience.com> wrote in
message news:pan.2009.05...@canonicalscience.com...
> Rock Brentwood wrote on Wed, 13 May 2009 07:17:34 +0000:
>
>> On May 8, 4:32� pm, "Juan R." Gonz�lez-�lvarez
>> > In short, the reason which GR and other theories give the correct
>> light
>>> deflection, whereas Newtonian gravity and Einstein 1911 theory give
>>> only a half of the deflection is due to that gravitational interaction
>>> is not scalar but tensorial.
>>
>> This is a red herring and can clearly be seen as such, as follows:
>
> One central issue of my message (deleted in your response) was the
> correction to Harry's incorrect claim (also corrected in spf).
Not only you apparently confuse me with Einstein but I already corrected his
imprecise formulation (in the part that you snipped!). And there was a kind
of red herring by you as your summary obscured my clarification:
Whereas in 1911 Einstein only accounted for gravitational time dilation, GRT
takes into account both time dilation and gravitational length contraction;
the calculation shows that this results in twice the bending.
[..]
> In Newton-Cartan theory one starts from geodesic equations and defines
> connections like flat connections plus a correction term.
>
> The correction is given (by definition) by a *spatial* metric, a
> *temporal* metric and the gradient of a scalar potential phi (which is
> also introduced in the theory).
>
> The *spatial* and *temporal* metrics are degenerate and mutually
> orthogonal with signatures (0 + + +) and (+ 0 0 0). This describes a
> *static* and *universal* spacetime.
>
> Thus gravitational 'interactions' [#] are really given by the *scalar*
> potential phi.
Thanks for the clarification about Newton-Cartan theory; I suspected that to
be the case but I wasn't sure.
> Any theory of gravity I know using a *scalar* potential for interactions
> has the problem that either (i) you choose a factor "Phi" or (ii) a factor
> "2Phi" for the interaction.
[..]
Regards,
Harald
Rock Brentwood
> In Newton-Cartan theory one starts from geodesic equations and defines
> connections like flat connections plus a correction term.
Newton-Cartan theory is gravitational theory on a Riemann-Cartan
background. The relevant objects (the connection and frame 1-forms,
the curvature and torsion 2-forms) are in place before one ever
considers a metric. Yes, there are equations for auto-parallels but,
at the level of these 4 objects, there is no distinction between
signature. One cannot tell the difference between a Galilean,
Lorentzian (or even Eucliidean or Platonic) signature.
> The correction is given (by definition) by a *spatial* metric, a
> *temporal* metric and the gradient of a scalar potential phi (which is
> also introduced in the theory).
The equation for gravity is just R_{mn} = k T_{mn} in Newton-Cartan
gravity. There is no scalar potential here.
There are no "spatial" and "temporal" metrics. It's just the covariant
and contravariant metric; the identification of them as "spatial" and
"temporal" is a red-herring.
For a degenreate signature, the two metrics are independent (as they
are, in effect, even for a non-degenerate signature -- this I
discussed in more detail a short while back in "Towards a General
Theory of Signature and Signature Change"). That's the general
phenomenon that's going on here. It has nothing per se to do with the
Galilean signature. All signatures have these two metrics (the
covariant and contravariant metric, that is).
> The *spatial* and *temporal* metrics are degenerate and mutually
> orthogonal with signatures (0 + + +) and (+ 0 0 0). This describes a
> *static* and *universal* spacetime.
None of this is necessarily the case; see the references cited a short
while back earlier in this thread.
> [#] Recall GR is a metric theory.
No. It's a theory of connections on a Riemann-Cartan geometry. The
metric has little or no bearing and plays no role, once the connection
has been reduced to an SO(3,1) connection. (Hence, one sees modern
formalisms, like the Penrose-Newman formalism formulated entirely in
terms of the structure coefficients of the frame 1-forms and the
(related) connection coefficients; with no involvement of the metric
whatsoever).
Historically, it might have first appeared in the context of a purely
metrical Riemannian geometry in 1915 -- but that was because nobody
knew about the more general Riemann-Cartan geometries, where it
properly belongs. The Einstein-Hilbert action is simply
S = integral K theta^a ^ theta^b ^ Omega_{ab}
where theta^a is the connection 1-form, Omega_{ab} the connection 2-
forms and K a constant.
There is no metric here, other than the constant frame metric eta_{ab}
used to lower one of the indices of Omega^a_b. It plays no role in the
dynamics. The Euler-Lagrange equations only give dynamics for theta
and Omega.
Now, for Riemann-Cartan geometry in a Galilean signature, I know there
are problems with formulating a consistent Lagrangian dynamics that
captures the Newtonian law of gravity. Part of your very objection was
what was being alluded to in the incompleteness article (which,
ironically, by your reply above you're tacitly endorsing, without your
even realising it!)
But none of this has any bearing on the question being raised here.
juanR...@canonicalscience.com
> "Juan R." <juanR...@canonicalscience.com>
wrote in
> message news:pan.2009.05...@canonicalscience.com...
>> Rock Brentwood wrote on Wed, 13 May 2009 07:17:34 +0000:
>>
(...)
>> One central issue of my message (deleted in your response) was the
>> correction to Harry's incorrect claim (also corrected in spf).
>
> Not only you apparently confuse me with Einstein but I already corrected
> his imprecise formulation (in the part that you snipped!). And there was
> a kind of red herring by you as your summary obscured my clarification:
>
> Whereas in 1911 Einstein only accounted for gravitational time dilation,
> GRT takes into account both time dilation and gravitational length
> contraction; the calculation shows that this results in twice the
> bending.
I sniped the part where you changed the terms
Newtonian field --> time dilation
curved space --> gravitational length contraction
to repeat the *same* mistake corrected for days in spf!
In Einstein 1911 scalar theory, time dilation was given by "phi".
Time dilation in GR is *not* given by a "phi" factor. The entire 2phi
factor in the Einstein 1916 calculation for light signals can be
obtained exclusively from the time dilation formulae in GR.
Thus, it is not true that a factor "phi" is due to time dilation and
another factor "phi" is due to length contraction as you claim now.
Regards
Juan R.
Rock Brentwood wrote on Wed, 20 May 2009 07:30:29 +0200:
> On May 18, 4:57Â pm, "Juan R." González-Álvarez:
(...)
>> The correction is given (by definition) by a *spatial* metric, a
>> *temporal* metric and the gradient of a scalar potential phi (which is
>> also introduced in the theory).
>
> The equation for gravity is just R_{mn} = k T_{mn} in Newton-Cartan
> gravity. There is no scalar potential here.
Those equations do not fix the equation of motion in NC [1], neither give
the Poisson equation defining the scalar potential unless one introduces a
rotational postulate [2].
> There are no "spatial" and "temporal" metrics
[...]
> the identification of them as "spatial" and "temporal" is a red-herring.
The name is appropiate: the spatial metric h_ab gives the spatial element
of line.
>> [#] Recall GR is a metric theory.
>
> No. It's a theory of connections on a Riemann-Cartan geometry.
GR is a metric theory and NC is certain limit of GR.
REFERENCES
[1]
Section II in Exactly Soluble Sector of Quantum Gravity 1997: Phys. Rev. D
56, 4844-4877, Christian, J.
[2]
Section 3 in http://arxiv.org/abs/gr-qc/9604054
Juan R.
> "Juan R. González-Álvarez" <juanR...@canonicalscience.com> wrote in
> message news:pan.2009.05...@canonicalscience.com...
>> Rock Brentwood wrote on Wed, 13 May 2009 07:17:34 +0000:
>>
>>> On May 8, 4:32Â pm, "Juan R." González-Álvarez
(...)
>> One central issue of my message (deleted in your response) was the
>> correction to Harry's incorrect claim (also corrected in spf).
>
> Not only you apparently confuse me with Einstein but I already corrected
> his imprecise formulation (in the part that you snipped!). And there was
> a kind of red herring by you as your summary obscured my clarification:
>
> Whereas in 1911 Einstein only accounted for gravitational time dilation,
> GRT takes into account both time dilation and gravitational length
> contraction; the calculation shows that this results in twice the
> bending.
I sniped the part where you changed the terms
Newtonian field --> time dilation
curved space --> gravitational length contraction
to repeat the same mistake corrected for days in spf!
In Einstein 1911 scalar theory, time dilation was given by "phi".
Time dilation in GR is *not* given by a "phi" factor. The entire 2phi
factor in the Einstein 1916 calculation can be obtained exclusively from
the time dilation formulae in GR.
Thus, it is not true that a factor "phi" is due to time dilation and
another factor "phi" is due to length contraction as you claim.
Regards
harry
message news:pan.2009.05...@canonicalscience.com...
>
> wrote in
>> message news:pan.2009.05...@canonicalscience.com...
>>> Rock Brentwood wrote on Wed, 13 May 2009 07:17:34 +0000:
>
>
>>> One central issue of my message (deleted in your response) was the
>>> correction to Harry's incorrect claim (also corrected in spf).
>>
>> Not only you apparently confuse me with Einstein but I already corrected
>> his imprecise formulation (in the part that you snipped!). And there was
>> a kind of red herring by you as your summary obscured my clarification:
>>
>> Whereas in 1911 Einstein only accounted for gravitational time dilation,
>> GRT takes into account both time dilation and gravitational length
>> contraction; the calculation shows that this results in twice the
>> bending.
>
>
> Newtonian field --> time dilation
>
> curved space --> gravitational length contraction
>
No, I merely don't falsify citations; in reality I was the first to
criticize Einstein's "Newtonian field"! Apart of that, "curved space" of
GRT is just another word for "gravitational length contraction" - such
expressions are not part of Newtonian theory.
> In Einstein 1911 scalar theory, time dilation was given by "phi".
>
> Time dilation in GR is *not* given by a "phi" factor. The entire 2phi
> factor in the Einstein 1916 calculation for light signals can be
> obtained exclusively from the time dilation formulae in GR.
That is erroneous: clock frequency does not have directionality and GR's
time dilation corresponds to that of Einstein 1911. My comment was an
addition to that of "of 1001 nights" who already explained that time
dilation and spatial curvature each contribute half the bending. This is
also clearly explained in Bazin and Schiffer's "Introduction to GRT",
who remarked that the "physical argument" of time dilation and length
contraction which leads to the double bending corresponds to the fact
that, using the Schwartzschild metric, "the first-order terms in m of
g_00 and g_11 contribute equal amounts to phi".
> Thus, it is not true that a factor "phi" is due to time dilation and
> another factor "phi" is due to length contraction as you claim now.
I did not suggest such a thing; phi is double the 1911 value!
Regards,
Harald
harry
message news:pan.2009.05...@canonicalscience.com...
>
> wrote in
>> message news:pan.2009.05...@canonicalscience.com...
>>> Rock Brentwood wrote on Wed, 13 May 2009 07:17:34 +0000:
>
>
>>> One central issue of my message (deleted in your response) was the
>>> correction to Harry's incorrect claim (also corrected in spf).
>>
>> Not only you apparently confuse me with Einstein but I already corrected
>> his imprecise formulation (in the part that you snipped!). And there was
>> a kind of red herring by you as your summary obscured my clarification:
>>
>> Whereas in 1911 Einstein only accounted for gravitational time dilation,
>> GRT takes into account both time dilation and gravitational length
>> contraction; the calculation shows that this results in twice the
>> bending.
>
harry
message news:pan.2009.05...@canonicalscience.com...
>
> wrote in
>> message news:pan.2009.05...@canonicalscience.com...
>>> Rock Brentwood wrote on Wed, 13 May 2009 07:17:34 +0000:
>
>
>>> One central issue of my message (deleted in your response) was the
>>> correction to Harry's incorrect claim (also corrected in spf).
>>
>> Not only you apparently confuse me with Einstein but I already corrected
>> his imprecise formulation (in the part that you snipped!). And there was
>> a kind of red herring by you as your summary obscured my clarification:
>>
>> Whereas in 1911 Einstein only accounted for gravitational time dilation,
>> GRT takes into account both time dilation and gravitational length
>> contraction; the calculation shows that this results in twice the
>> bending.
>
Juan R.
> neither give the Poisson equation defining the scalar
> potential unless one introduces a rotational postulate [2].
Sorry I cited them in the inverse!
Those equations do not fix the equation of motion in NC [2],
neither give the Poisson equation defining the scalar
potential unless one introduces a rotational postulate [1].