Newtonian limit difficulties of General Relativity (linear regime)
Juan R.
I am working in relationship between General Relativity and Newton
theory.
My work consists of two parts. The first part addresses the common
claim that geodesic equation of General Relativity reduces to Newton law
of motion in the linear limit, i.e. g_ab = \eta_ab + h_ab when ||h|| is
small enough to ignore quadratic and higher orders.
Take for example Carroll lecture notes available online:
http://lanl.arXiv.org/abs/gr-qc/9712019v1
Carroll starts from geodesic equation (4.9) and derives equation (4.19),
which he identifies with Newton equation (4.4) after using (4.20).
During the derivation Carroll uses the linear constraint (4.13).
A more rigorous analysis does not support Carroll conclusions.
Carroll is not computing the linear geodesic equation of motion but
'inventing' a non-geometrical equation [see below].
The equation of motion in the linear limit is a = 0. Therefore, the
equation of motion of General Relativity does not coincide with that
from Newtonian gravity in the linear regime.
For both the zeroth and the linear regimes of General Relativity bodies
have to move on straight lines.
A way to see this is expanding the geodesic equation in a perturbative
series and retaining terms up to linear order on expansion parameter
lambda,
<a^\mu> + \lambda \delta a^\mu
=
\lambda \delta \Gamma_{\rho\sigma}^\mu <u^\rho> <u^\sigma>
Carroll is assuming that left hand side may be approximated by
<a^\mu> + \lambda \delta a^\mu
and the right hand side by
\lambda \delta \Gamma_{\rho\sigma}^\mu c^2
See his (4.19).
But by geometrical requirements (Carroll and textbooks are not checking
any geometrical consistency check)
\delta a^\mu = \delta \Gamma_{\rho\sigma}^\mu = 0
and the consistent result is a = 0 in the linear regime and bodies have
to move on straight lines.
Another way to see this is deriving motion from D{T_ab} = 0.
In the linear regime, it reduces to \partial{T_ab} = 0, and like in the
special relativity case, this implies bodies move in straight lines.
If one denotes the geodesic equation using next simplified notation
a = \Gamma vv
The linearized, L[], geodesic equation reads
L[a] = L[\Gamma vv] = L[\Gamma] Z[vv]
where Z[] denotes the zeroth order application.
But textbook and online lectures as that by Carroll are really computing
Z[a] = L[\Gamma] Z[vv]
Carroll is applying different limits to left and hand sides of the
original equation therein my claim he is 'inventing' the final equation
rather than deriving it by mathematical steps.
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Juan R.
> Carroll is assuming that left hand side may be approximated by
>
> <a^\mu> + \lambda \delta a^\mu
>
> and the right hand side by
>
> \lambda \delta \Gamma_{\rho\sigma}^\mu c^2
>
> See his (4.19).
Mistake. I did mean that Carroll is assuming that left hand side may be
approximated by
<a^\mu>
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Juan R.
As said in original message, conventional derivations of the Newtonian
limit of General Relativity (GR) are not correct.
If one takes consistently the linear limit of GR, one rigorously obtains
the final equation
a = 0
instead the incorrect equation showed in textbooks and lecture notes.
I have discussed this part of my work with an expert on curved spacetime
equations of motion, Eric Poisson [1]. Eric confirms that a = 0 in the
linear regime of GR:
(\blockquote
Since the energy-momentum tensor is already of first-order, in the
linearized theory the conservation equations must be written down with
the Minkowski metric, and this implies that the matter cannot have
gravitational interactions. Or as you point out, particles would have to
move on straight lines.
)
One would imagine that keeping up to second order terms, the acceleration
will be nonzero. Yes, but the correspondence with Newtonian limit is again
broken.
The argument using above perturbative series expansion of the geodesic
equation turns difficult due non linearity
\lambda^2 /= \lambda\lambda
However the conclusion is the same: no consistent Newtonian limit.
A more easy argument follows from computing the spatial element of line up
to second order in h
d\sigma^2 = \gamma_ij dx^i dx^j
where, as usual, \gamma_ij = g_ij + \gamma_i\gamma_j
For a Schwarzschild metric \gamma_i = \gamma_j = 0 and the spatial metric
is
\gamma_ij = g_ij = 1 - {2\phi \over c^2} + {2\phi \over c^2}^2
But for the Newtonian theory
\gamma_ij = 1
Indeed some derivations of Newtonian limit i revised assume that
g_00 /= -1
*and*
g_ij = 1
which is geometrically impossible because for the Schwarzschild metric
g_00 = (-1 / g_ij)
Thus again the derivations are invalid. Again General Relativity does not
reduce to Newtonian gravity.
The problem, as you may imagine, is in the geometric approach.
I have revised the Newtonian limit on two alternative theories of gravity,
Feynman field theory of gravity (FTG) and relativistic action at a
distance (AAAD). In both cases one obtains in the linear regime:
a = -\grad \phi
*and*
d\sigma^2 = dx^i dx^j
This clear advantage over General Relativity may be traced to the
nongeometrical formulation. In those alternative theories gravitation is a
real force as is electromagnetism.
My analysis of the field theory of gravity (FTG) has been checked by a
current expertise on FTG, professor Y. Baryshev.
He also confirms Newtonian limit to be ill-defined in GR.
Now a more interesting question would be asked. Could we modify General
Relativity to give the correct Newtonian limit? The response is negative.
No metric theory of gravity contains a Newtonian limit and thus can be
consistently quantized. Geometry may be abandoned as a foundation for
physics and merely drawn as an analogy (as Steven Weinberg stated in his
book /Gravitation and Cosmology/).
But this is second part of my research and will be discussed in a
separated post.
[1] http://relativity.livingreviews.org/Articles/lrr-2004-6/
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Oh No
>"Juan R." González-Álvarez wrote on Thu, 01 May 2008 13:25:06 -0600:
>
>As said in original message, conventional derivations of the Newtonian
>limit of General Relativity (GR) are not correct.
>
>If one takes consistently the linear limit of GR, one rigorously
>obtains the final equation
>
>a = 0
>
>instead the incorrect equation showed in textbooks and lecture notes.
>
>I have discussed this part of my work with an expert on curved
>spacetime equations of motion, Eric Poisson [1]. Eric confirms that a =
>0 in the linear regime of GR:
>
>(\blockquote
> Since the energy-momentum tensor is already of first-order, in the
> linearized theory the conservation equations must be written down with
> the Minkowski metric, and this implies that the matter cannot have
> gravitational interactions. Or as you point out, particles would have
>to
> move on straight lines.
>)
I am not sure what you mean by a=0, but it seems that the expert is
explaining to you that that the Newtonian limit is not the same as the
linear limit. Rather than claim that the text books have got it wrong,
you study what the Newtonian limit actually is.
Regards
--
Charles Francis
moderator sci.physics.foundations.
charles (dot) e (dot) h (dot) francis (at) googlemail.com (remove spaces and
braces)
Juan R.
>>I have discussed this part of my work with an expert on curved spacetime
>>equations of motion, Eric Poisson [1]. Eric confirms that a = 0 in the
>>linear regime of GR:
>>
>>(\blockquote
>> Since the energy-momentum tensor is already of first-order, in the
>> linearized theory the conservation equations must be written down with
>> the Minkowski metric, and this implies that the matter cannot have
>> gravitational interactions. Or as you point out, particles would have
>> to
>> move on straight lines.
>>)
>
> I am not sure what you mean by a=0,
Do not know that zero acceleration mean?
> but it seems that the expert is
> explaining to you that that the Newtonian limit is not the same as the
> linear limit. Rather than claim that the text books have got it wrong,
> you study what the Newtonian limit actually is.
He had no problem to accept my finding about weak fields. Why do you have
one?
Poisson is simply confirming i said in my start message. E.g. I said
(\blockquote
Another way to see this is deriving motion from D{T_ab} = 0.
In the linear regime, it reduces to \partial{T_ab} = 0, and like in the
special relativity case, this implies bodies move in straight lines.
)
Textbooks state that in the weak field limit (linear regime) bodies move
according to equation
a = -\grad \phi
But that is not true, because the *correct* weak field equation (linear
regime) is
a = 0
The mistake in usual textbook 'derivations' has been traced and
discussed. E.g. if one check details of Carroll lecture notes one can
notice he is computing
Z[a] = L[\Gamma] Z[vv]
and this gives the equation of motion
a = -\grad \phi
But Carroll would compute instead the linear limit of the geodesic
L[a] = L[\Gamma] Z[vv]
and then would really obtain
a = 0
by *geometrical* requirements.
Or, in words, in the linearized theory particles would have to move on
straight lines.
--
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Oh No
>
>>>I have discussed this part of my work with an expert on curved spacetime
>>>equations of motion, Eric Poisson [1]. Eric confirms that a = 0 in the
>>>linear regime of GR:
>>>
>>>(\blockquote
>>> Since the energy-momentum tensor is already of first-order, in the
>>> linearized theory the conservation equations must be written down with
>>> the Minkowski metric, and this implies that the matter cannot have
>>> gravitational interactions. Or as you point out, particles would have
>>> to
>>> move on straight lines.
>>>)
>>
>> I am not sure what you mean by a=0,
>
>Do not know that zero acceleration mean?
a can stand for anything you want it to. If you want it to stand for
acceleration, you have to say so.
>
>> but it seems that the expert is
>> explaining to you that that the Newtonian limit is not the same as the
>> linear limit. Rather than claim that the text books have got it wrong,
>> you study what the Newtonian limit actually is.
>
>He had no problem to accept my finding about weak fields. Why do you have
>one?
Perhaps you did not make clear to him that you were confusing the linear
limit with the weak field limit.
>
>Poisson is simply confirming i said in my start message.
He is not confirming that the text books have it wrong.
>E.g. I said
>
>(\blockquote
> Another way to see this is deriving motion from D{T_ab} = 0.
> In the linear regime, it reduces to \partial{T_ab} = 0, and like in the
> special relativity case, this implies bodies move in straight lines.
>)
>
>Textbooks state that in the weak field limit (linear regime) bodies move
>according to equation
>
>a = -\grad \phi
>
>But that is not true,
It is true in the weak field limit. The weak field limit is *defined* as
the regime in which this is true.
> because the *correct* weak field equation (linear
>regime) is
>
and it is not the same as your linear regime.
>
>The mistake in usual textbook 'derivations' has been traced and
>discussed. E.g. if one check details of Carroll lecture notes one can
>notice he is computing
>
>Z[a] = L[\Gamma] Z[vv]
>
>and this gives the equation of motion
>
>a = -\grad \phi
>
>But Carroll would compute instead the linear limit of the geodesic
>
>L[a] = L[\Gamma] Z[vv]
>
>and then would really obtain
>
>a = 0
>
>by *geometrical* requirements.
>
>Or, in words, in the linearized theory particles would have to move on
>straight lines.
We know very well that particles move on locally straight lines, i.e.
geodesics.
Juan R.
>>He had no problem to accept my finding about weak fields. Why do you
>>have one?
>
> Perhaps you did not make clear to him that you were confusing the linear
> limit with the weak field limit.
If you have no serious argument then I will reply this irrelevant and
false one. I will copy and paste fragments of my communication with him:
"Several textbooks state that linearized General Relativity"
(\blockquote
But the condition \partial^\mu T_{\mu\nu} = 0 implies that test
bodies move on geodesics of the flat metric n_{\mu\nu}; i.e, if one stay
consistently within the linear approximation, one predicts that test
bodies are unaffected by gravity.
)
"... specific issue of linearized equations of motion."
"I have tried to derive that test bodies are unaffected by gravity within
the linear approximation ..."
"I would be glad if you could revise my argument about linearized
geodesics, ..."
"Up to linear order (n=1) the geodesic equation is..."
"Introducing this constraint in the linearized geodesic equation
one predicts that within the linear approximation ..."
"in the LHS of the linearized geodesic whereas taking a low velocity
limit on the RHS, yielding the final equation..."
"However, those authors did not notice that the linear constraint implies"
You seem unable to read even the title of this thread (the title i choose
for this thread says *linear regime*).
But I am rather sure he and other three people who i contacted were able
to understand i was revising the *linear regime* of geometric GR.
>>Poisson is simply confirming i said in my start message.
>
> He is not confirming that the text books have it wrong.
He confirmed that the equation (4.19)
on
http://lanl.arXiv.org/abs/gr-qc/9712019v1
is not consistent with linear constraint (4.13) applied to (4.9)
He confirmed that if you apply condition (4.13) the geodesic equation
(4.9) reduces to
a = 0
This is also confirmed by Wald. And other three experts who read my work
found no error.
>>Textbooks state that in the weak field limit (linear regime) bodies move
>>according to equation
>>
>>a = -\grad \phi
>>
>>But that is not true,
>
> It is true in the weak field limit. The weak field limit is *defined* as
> the regime in which this is true.
In *my statement*, which you are replying, one can read a clever "(linear
regime)".
>> because the *correct* weak field equation (linear
>>regime) is
>>
> and it is not the same as your linear regime.
It seems you cannot read. The condition of weakness is usually defined as
equation (4.13) in all 'derivations' i have checked (a total of three).
It is equation (4.13) i have rigorously applied to derive the equation of
motion
a = 0.
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Oh No
>
>>>He had no problem to accept my finding about weak fields. Why do you
>>>have one?
>>
>> Perhaps you did not make clear to him that you were confusing the linear
>> limit with the weak field limit.
>
>If you have no serious argument then I will reply this irrelevant and
>false one. I will copy and paste fragments of my communication with him:
It is not much help. Incoherent and does not make your case.
>
>>>Poisson is simply confirming i said in my start message.
>>
>> He is not confirming that the text books have it wrong.
>
>He confirmed that the equation (4.19)
>
>on
>
>http://lanl.arXiv.org/abs/gr-qc/9712019v1
>
>is not consistent with linear constraint (4.13) applied to (4.9)
>
>He confirmed that if you apply condition (4.13) the geodesic equation
>(4.9) reduces to
>
>a = 0
The geodesic equation always reduces to a=0 with a suitable choice of
coordinates or definition of a. This is not inconsistent with equation
4.19 in a different choice of coordinates. There is nothing wrong with
the derivation of 4.19 from 4.13 in the reference you give.
Juan R.
> It is not much help. Incoherent and does not make your case.
It would be if you read them. Fragments contain the terms "linear" or
"linearized" about ten times. That would be enough to understand i am
computing the linear limit of the geodesic equation of motion.
The title of this thread (chosen by me of course) also says "linear
regime". Four authors of a total of four understood i was speaking about
linearized GR. That seems to indicate i am writing fine and your
accusation had no basis.
>>He confirmed that if you apply condition (4.13) the geodesic equation
>>(4.9) reduces to
>>
>>a = 0
>
> The geodesic equation always reduces to a=0 with a suitable choice of
> coordinates or definition of a. This is not inconsistent with equation
> 4.19 in a different choice of coordinates. There is nothing wrong with
> the derivation of 4.19 from 4.13 in the reference you give.
Two recognized expertise on GR (Wald and Poisson) confirm my point that
derivation is not correct at all. And others three authors (including
expertise on Newtonian limits of GR) have found no serious mistake in my
paper. This is enough 'review' for me.
Still I will explain once more for *you*.
I am not writing the geodesic equation of motion with a /= 0 and then
applying a change of coordinates to another system where a = 0. You
misread me again.
I start from the geodesic equation of motion (4.9).
Then I apply *consistently* the weak field condition (4.13) to (4.9) and
*rigorously* I obtain the equation of motion
a = 0
which, of course, hold in *any* system of coordinates verifying the
linear constraint.
I.e. if (a = 0) in one system of coordinates then a = 0 in other if and
only if the constraint (4.13) holds in both.
This thought is so obvious that i did not emphasized that before. Nobody
more misunderstood the equivalence principle in the way you are doing.
Poisson reply to me is so clever i reproduce it
(\blockquote
the linearized theory [...] implies that the matter cannot have
gravitational interactions. Or as you point out, particles would have to
move on straight lines.
)
With Robert Wald emphasizing the same
(\blockquote
if one stay consistently within the linear approximation, one predicts
that test bodies are unaffected by gravity.
)
That was also my prediction: in the linear regime (a = 0), test bodies
are unaffected by gravity and move on straight lines
----------------------o
As pointed before, and carefully discussed in start message, the
derivation of (4.19) from (4.13) in the reference given is *inconsistent*
(note also Wald remark about being *consistency*). The derivation is
inconsistent because the correct computation gives the final equation is
(a = 0) instead the equation (4.19).
The mistake on the derivation given in the reference has been also
pointed.
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Oh No
>
>> It is not much help. Incoherent and does not make your case.
>
>It would be if you read them. Fragments contain the terms "linear" or
>"linearized" about ten times. That would be enough to understand i am
>computing the linear limit of the geodesic equation of motion.
You keep saying that. But the limit you are calculating is still not the
Newtonian limit, and nor is it the weak field limit, so you have no
basis on which to say the text books are wrong.
Peter
> Thus spake Juan R. González-Álvarez <juanR...@canonicalscience.com>
> >Oh No wrote on Wed, 07 May 2008 08:55:12 -0600:
> >
> >> It is not much help. Incoherent and does not make your case.
> >
> >It would be if you read them. Fragments contain the terms "linear" or
> >"linearized" about ten times. That would be enough to understand i am
> >computing the linear limit of the geodesic equation of motion.
>
> You keep saying that. But the limit you are calculating is still not the
> Newtonian limit, and nor is it the weak field limit, so you have no
> basis on which to say the text books are wrong.
>
> Regards
Hello *,
May be it's a good idea that you first agree about the notions of limits used.
IMHO, GTR is *essentially* *non*-linear, because both the interacting bodies
and the mediating agens carry mass. This makes
force ~ mass^2
Hence, within a linear theory,
force = 0
The Newtonian limit is analogous to electrostatics, ie,
force ~ mass * field
Hope this helps,
Peter
Oh No
so far as I can tell, Juan is talking of proper acceleration, which is
always 0, but taking the limits which define the acceleration in a
roundabout way so as to make a trivial result seem obscure. The weak
field limit refers to acceleration in particular coordinates.
Juan R.
> Thus spake Juan R. González-Álvarez <juanR...@canonicalscience.com>
>>Oh No wrote on Wed, 07 May 2008 08:55:12 -0600:
>>
>>> It is not much help. Incoherent and does not make your case.
>>
>>It would be if you read them. Fragments contain the terms "linear" or
>>"linearized" about ten times. That would be enough to understand i am
>>computing the linear limit of the geodesic equation of motion.
>
> You keep saying that. But the limit you are calculating is still not the
> Newtonian limit, and nor is it the weak field limit, so you have no
> basis on which to say the text books are wrong.
I misunderstood the whole issue once again.
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Juan R.
> It would also help to have a proper definition of acceleration :-). In
> so far as I can tell, Juan is talking of proper acceleration,
Juan is not.
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Juan R.
> Hello *,
>
> May be it's a good idea that you first agree about the notions of limits
> used.
I did clear since very start i am speaking about linear limit [#], i.e.
g_ab = \eta_ab + h_ab
with |h| << 1 so that terms squared and higher order on h are neglected.
> IMHO, GTR is *essentially* *non*-linear, because both the interacting
> bodies and the mediating agens carry mass. This makes
>
> force ~ mass^2
>
> Hence, within a linear theory,
>
> force = 0
I do not like to speak about forces in geometric GR. But the conclusion
in any case is that if one applies the linear metric constraint the
resulting equation is
a = 0
Or in words (Robert Wald)
(\blockquote
if one stay *consistently* within the linear approximation, one predicts
that test bodies are unaffected by gravity.
)
[#] This is also clear from title i choose for the post.
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Oh No
>
>> Hello *,
>>
>> May be it's a good idea that you first agree about the notions of limits
>> used.
>
>I did clear since very start i am speaking about linear limit [#], i.e.
>
>g_ab = \eta_ab + h_ab
>
>with |h| << 1 so that terms squared and higher order on h are neglected.
It is not clear what this means. In general h is a function of position.
One might define, for example, spherical coordinates and neglect terms
in h of order r^2. But then h_00=r would still be a field in which
acceleration was always away from the origin. Nor do you seem to mean
cartesian coordinates, because e.g. h_00=x would give a constant
acceleration.
Juan R.
> Thus spake Juan R. González-Álvarez <juanR...@canonicalscience.com>
>>Peter wrote on Thu, 08 May 2008 09:43:56 -0600:
>>
>>> Hello *,
>>>
>>> May be it's a good idea that you first agree about the notions of
>>> limits used.
>>
>>I did clear since very start i am speaking about linear limit [#], i.e.
>>
>>g_ab = \eta_ab + h_ab
>>
>>with |h| << 1 so that terms squared and higher order on h are neglected.
>
> It is not clear what this means.
It is discussed in any textbook.
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Oh No
>Oh No wrote on Thu, 08 May 2008 14:58:17 -0600:
>
>> Thus spake Juan R. González-Álvarez <juanR...@canonicalscience.com>
>>>Peter wrote on Thu, 08 May 2008 09:43:56 -0600:
>>>
>>>> Hello *,
>>>>
>>>> May be it's a good idea that you first agree about the notions of
>>>> limits used.
>>>
>>>I did clear since very start i am speaking about linear limit [#], i.e.
>>>
>>>g_ab = \eta_ab + h_ab
>>>
>>>with |h| << 1 so that terms squared and higher order on h are neglected.
>>
>> It is not clear what this means.
>
>It is discussed in any textbook.
>
Eric Gisse
> Thus spake Juan R. González-Álvarez <juanREM...@canonicalscience.com>
>
> >Peter wrote on Thu, 08 May 2008 09:43:56 -0600:
>
> >> Hello *,
>
> >> May be it's a good idea that you first agree about the notions of limits
> >> used.
>
> >I did clear since very start i am speaking about linear limit [#], i.e.
>
> >g_ab = \eta_ab + h_ab
>
> >with |h| << 1 so that terms squared and higher order on h are neglected.
>
> It is not clear what this means. In general h is a function of position.
> One might define, for example, spherical coordinates and neglect terms
> in h of order r^2. But then h_00=r would still be a field in which
> acceleration was always away from the origin. Nor do you seem to mean
> cartesian coordinates, because e.g. h_00=x would give a constant
> acceleration.
I'll summarize.
The topic under discussion is called the weak field limit, or
linearized GR.
The basic assumption is that the metric tensor [g_uv] is predominantly
Minkowski space [flat space, \eta_uv] with a small perturbation [h_uv]
imposing itself on Minkowski space.
The inverse is defined as g^uv = \eta^uv - h^uv, with the explicit
neglection of terms of O(h^2) and higher. The condition |h| << 1 isn't
really covariant, but it really sets the theme since there are gauge
choices to be made when deriving the linearized field equations.
>From there, the metric is fed into the Riemann tensor to form the
Ricci tensor & scalar, and then the field equations. All terms of
O(h^2) and higher are tossed. It really is discussed in every GR text
I have seen - you do have Wald, don't you?
His whole beef is with the way the conservation law for the stress-
energy tensor rolls up after applying everything and tossing higher
order terms. Which makes him believe that somehow that the weak field
limit is inconsistent & unapplicable or whatever. Its' silly and
pedantic, on top of being well-discussed in every comprehensive
relativity text I've seen.
Peter
<juanR...@canonicalscience.com> writes:
> I am working in relationship between General Relativity and Newton
> theory.
I'm interested in the gravito-electromagnetic limit, do you know something
about that?
Thank you,
Peter
Eric Gisse
In the linearized theory of GR, the 3-force on a test particle takes a
form analogous to the Lorentz force law.
Peter
> Peter wrote:
Thank you. Why the gravito-electromagnetic eqs. differ a little bit from the
microscopic Maxwell(-Lorentz) eqs.? Nonlinearity?
Best wishes,
Peter
Eric Gisse
I'll copy parts the discussion from Carroll's _Spacetime_and_Geometry_
since a text description is just going to waste everyone's time for a
week.
In the linearized theory of general relativity under the assumption of
Minkowski space plus a small perturbation [g_uv = \eta_uv + h_uv], the
metric can a certain form.
ds^2 = - ( 1 + 2 \phi) dt^2 + w_i (dt dx^i + dx^i dt) + [(1- 2\psi)
delta_ij + 2 s_ij) dx^i dx^j
Under that form of the perturbation, the 3-force dp^i / dt on a test
particle with energy E is given as:
dp^i / dt = - E [ @_i \phi + @_0 w_i + 2( @_[i w_j] + @_0 h_ij) v^j +
( @_(j h_k)i - 1/2 @_i h_jk) v^j v^k
What a mess.
Introduce two 3-vector fields defined as:
G^i = - @_i \phi - @_0 w_i
H^i = epsilon^ijk @_j w_k
The 3-force becomes:
dp^i / dt = E [ G^i + (v x H)^i - 2(@_0 h_ij) v^j - ( @_(j h_k)i - 1/2
@_i h_jk) v^j v^k ]
This is the linearized GR analog of the Lorentz force law. Obviously
it is different because of the linear and quadratic couplings between
the perturbation and the velocity of the particle, but you can see why
there is an analogy.
I should have said this before - and did, but thought it was too
obvious:
There is abso-goddamn-lutely no electromagnetism present. None
whatsoever. This is a purely gravitational effect - the fields phi,
psi, h, etc, are simply various components of the perturbation
decomposed into more useful forms.
>
> Best wishes,
> Peter
Oh No
I think you mean gravitomagnetic. In the weak field limit the equations
of gtr are equivalent to a classical gravitational potential on a
Minkowski background. This is parallel to the Coulomb potential. One can
Lorentz transform the gravitational force, in exactly the same way as
one transforms the Coulomb force.
http://www.teleconnection.info/rqg/IntroductionToTensors#Electromagnetic
Force.
Thus one finds a gravitomagnetic force, analogous to the magnetic force
in classical e.m. I think the answer to your question is that the
equations differ because Maxwell's equations differ a little bit from
the equations of Newtonian gravity, but another answer is that the weak
field limit is not strictly covariant, whereas the equations of e.m.
are.
Ken S. Tucker
> Peter wrote:
> > Eric Gisse <jowr...@gmail.com> writes:
>
> > > Peter wrote:
>
> > > > "Juan R." =?iso-8859-1?q?Gonz=E1lez-=C1lvarez?=
>
> > > > > I am working in relationship between General Relativity and Newton
> > > > > theory.
>
> > > > I'm interested in the gravito-electromagnetic limit, do you know
> > > > something about that?
>
> > > > Thank you,
> > > > Peter
>
> > > In the linearized theory of GR, the 3-force on a test particle takes a
> > > form analogous to the Lorentz force law.
>
> > Thank you. Why the gravito-electromagnetic eqs. differ a little bit from the
> > microscopic Maxwell(-Lorentz) eqs.? Nonlinearity?
Edward G. Harris (Am.J.Phys. 59(5), May 1991) has
a good article entitled, " Analogy between general relativity
and electromagnetism for slowly moving particles in weak
gravitational fields".
Some years ago, I was invited to NASA's MSFC in Alabama
to examine and opine on their "antigravity" experiments,
based on Ning Li's theory which you can pick-up here,
http://en.wikipedia.org/wiki/Ning_Li_(physicist)
The upshot is her paper(s) were accepted by PRD, however
after considerable expenditure and very careful experiments
by NASA, no gravitomagnetic field effect could be detected.
In addition, a related effect sometimes referred to as "frame
dragging" was explored by NASA at considerable expense,
by satellite "Gravity Probe B", and it too was unable to
verify a "gravitomagnetic" type of effect that was thought to
be due to Earth's rotational effect on spacetime.
I (we) concluded in the 90's the effect was not detectable,
and informed NASA of that, and NASA wanted to subsequently
cancel GP-B, however for political and IMHO good scientific
reasons it was launched anyway. I figure that even a null
result is good science, just as the null result of the MMX
led to the abandonment of the aether theory became one
of the most famous experiments in history.
> I'll copy parts the discussion from Carroll's _Spacetime_and_Geometry_
> since a text description is just going to waste everyone's time for a
> week.
You (Eric) could have provided a link, to save time,
then follow-up with commentary.
...
> What a mess.
Ok :-), you won't mind if I snip.
...
Regards
Ken S. Tucker
Peter
> Thus spake Peter <end...@dekasges.de>
> >Eric Gisse <jow...@gmail.com> writes:
> >
> >> Peter wrote:
> >
> >> > "Juan R." =?iso-8859-1?q?Gonz=E1lez-=C1lvarez?=
> >> > <juanR...@canonicalscience.com> writes:
> >> >
> >> > > I am working in relationship between General Relativity and Newton
> >> > > theory.
> >> > I'm interested in the gravito-electromagnetic limit, do you know
> >> > something about that?
> >> >
> >> > Thank you,
> >> > Peter
> >> In the linearized theory of GR, the 3-force on a test particle takes a
> >> form analogous to the Lorentz force law.
> >Thank you. Why the gravito-electromagnetic eqs. differ a little bit from
> >the microscopic Maxwell(-Lorentz) eqs.? Nonlinearity?
> I think you mean gravitomagnetic.
both
> In the weak field limit the equations
> of gtr are equivalent to a classical gravitational potential on a
> Minkowski background. This is parallel to the Coulomb potential. One can
> Lorentz transform the gravitational force, in exactly the same way as
> one transforms the Coulomb force.
>
> http://www.teleconnection.info/rqg/
IntroductionToTensors#ElectromagneticForce.
This is well known, indeed
> Thus one finds a gravitomagnetic force, analogous to the magnetic force
> in classical e.m.
O. Heaviside, A gravitational and electromagnetic analogy. Part I, The
Electrician 31 (1893) 281-282, Part II, Ibid. 359; Reprints: Jefimenko,
Causality, Electromagnetic Induction, and Gravitation, ²2000; http://
www.as.wvu.edu/coll03/phys/www/OJ/Heavisid.htm
> I think the answer to your question is that the
> equations differ because Maxwell's equations differ a little bit from
> the equations of Newtonian gravity
??
> , but another answer is that the weak
> field limit is not strictly covariant, whereas the equations of e.m.
> are.
Yes - the question was, where this difference stems from?
Best wishes,
Peter
PS: On killfile.org, I cannot reply to your posting about "your" QFT; it
seems to me that Dirac has given another representation; I would like to
recommend you to include anyons in your section about multi-particle states
(a good start may be Laughlin's Nobel speech)
Oh No
>Oh No <No...@charlesfrancis.wanadoo.co.uk> writes:
>
>> Thus spake Peter <end...@dekasges.de>
>> >Eric Gisse <jow...@gmail.com> writes:
>> >
>> >> Peter wrote:
>> >
>> >> > "Juan R." =?iso-8859-1?q?Gonz=E1lez-=C1lvarez?=
>> >> > <juanR...@canonicalscience.com> writes:
>> >> >
>> >> > > I am working in relationship between General Relativity and Newton
>> >> > > theory.
>
>
>> >> > I'm interested in the gravito-electromagnetic limit, do you know
>> >> > something about that?
>> >> >
>> >> > Thank you,
>> >> > Peter
>
>
>> >> In the linearized theory of GR, the 3-force on a test particle takes a
>> >> form analogous to the Lorentz force law.
>
>
>> >Thank you. Why the gravito-electromagnetic eqs. differ a little bit from
>> >the microscopic Maxwell(-Lorentz) eqs.? Nonlinearity?
>
>
>> I think you mean gravitomagnetic.
>
>both
Then I don't know what you mean
>
>> In the weak field limit the equations
>> of gtr are equivalent to a classical gravitational potential on a
>> Minkowski background. This is parallel to the Coulomb potential. One can
>> Lorentz transform the gravitational force, in exactly the same way as
>> one transforms the Coulomb force.
>>
>> http://www.teleconnection.info/rqg/
>IntroductionToTensors#ElectromagneticForce.
>
>This is well known, indeed
>
yes.
>> , but another answer is that the weak
>> field limit is not strictly covariant, whereas the equations of e.m.
>> are.
>
>Yes - the question was, where this difference stems from?
It stems from the approximations involved in the weak field limit.
>
>PS: On killfile.org, I cannot reply to your posting about "your" QFT; it
>seems to me that Dirac has given another representation;
Please elucidate. I think this view of Dirac's was widely known. He gave
many lectures on it in his later years.
Peter
> > Peter wrote:
> > > Thank you. Why the gravito-electromagnetic eqs. differ a little bit
> > > from the microscopic Maxwell(-Lorentz) eqs.? Nonlinearity?
> Edward G. Harris (Am.J.Phys. 59(5), May 1991) has
> a good article entitled, " Analogy between general relativity
> and electromagnetism for slowly moving particles in weak
> gravitational fields".
Can you send me a copy? Thanks a lot!
> Some years ago, I was invited to NASA's MSFC in Alabama
> to examine and opine on their "antigravity" experiments,
> based on Ning Li's theory which you can pick-up here,
> http://en.wikipedia.org/wiki/Ning_Li_(physicist)
> The upshot is her paper(s) were accepted by PRD, however
> after considerable expenditure and very careful experiments
> by NASA, no gravitomagnetic field effect could be detected.
>
> In addition, a related effect sometimes referred to as "frame
> dragging" was explored by NASA at considerable expense,
> by satellite "Gravity Probe B", and it too was unable to
> verify a "gravitomagnetic" type of effect that was thought to
> be due to Earth's rotational effect on spacetime.
>
> I (we) concluded in the 90's the effect was not detectable,
> and informed NASA of that, and NASA wanted to subsequently
> cancel GP-B, however for political and IMHO good scientific
> reasons it was launched anyway. I figure that even a null
> result is good science, just as the null result of the MMX
> led to the abandonment of the aether theory became one
> of the most famous experiments in history.
Are there measurable effect in Einstein's field equations that vanishe in the
gravito-electromagnetic approximation?
Thank you,
Peter
Ken S. Tucker
On May 28, 3:07 am, Peter <end...@dekasges.de> wrote:
> "Ken S. Tucker" <dynam...@vianet.on.ca> writes:
>
> > > Peter wrote:
> > > > Thank you. Why the gravito-electromagnetic eqs. differ a little bit
> > > > from the microscopic Maxwell(-Lorentz) eqs.? Nonlinearity?
> > Edward G. Harris (Am.J.Phys. 59(5), May 1991) has
> > a good article entitled, " Analogy between general relativity
> > and electromagnetism for slowly moving particles in weak
> > gravitational fields".
>
> Can you send me a copy? Thanks a lot!
I think it's available here,
http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000059000005000421000001&idtype=cvips&gifs=yes
otherwise I'll snail you my photocopy.
My photocopy is from UBC, and is a bit poor.
Harris's use of the word "Analogy" is appropriate.
> > Some years ago, I was invited to NASA's MSFC in Alabama
> > to examine and opine on their "antigravity" experiments,
> > based on Ning Li's theory which you can pick-up here,
> >http://en.wikipedia.org/wiki/Ning_Li_(physicist)
> > The upshot is her paper(s) were accepted by PRD, however
> > after considerable expenditure and very careful experiments
> > by NASA, no gravitomagnetic field effect could be detected.
>
> > In addition, a related effect sometimes referred to as "frame
> > dragging" was explored by NASA at considerable expense,
> > by satellite "Gravity Probe B", and it too was unable to
> > verify a "gravitomagnetic" type of effect that was thought to
> > be due to Earth's rotational effect on spacetime.
>
> > I (we) concluded in the 90's the effect was not detectable,
> > and informed NASA of that, and NASA wanted to subsequently
> > cancel GP-B, however for political and IMHO good scientific
> > reasons it was launched anyway. I figure that even a null
> > result is good science, just as the null result of the MMX
> > led to the abandonment of the aether theory became one
> > of the most famous experiments in history.
>
> Are there measurable effect in Einstein's field equations that vanishe in the
> gravito-electromagnetic approximation?
The gravitomagnetic effect (IMO) appears as a
figment of a CS by using mathematical assumptions,
without a good physics foundation.
IOW's people seem have read things into equations
that really aren't there.
> Thank you,
> Peter
Your welcome Peter.
Let me know what I can do to help.
Ken
Peter
> Hi Peter.
> > > > > Thank you. Why the gravito-electromagnetic eqs. differ a little bit
> > > > > from the microscopic Maxwell(-Lorentz) eqs.? Nonlinearity?
> > > Edward G. Harris (Am.J.Phys. 59(5), May 1991) has
> > > a good article entitled, " Analogy between general relativity
> > > and electromagnetism for slowly moving particles in weak
> > > gravitational fields".
> > Can you send me a copy? Thanks a lot!
> I think it's available here,
>
>http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS00
>0059000005000421000001&idtype=cvips&gifs=yes
> otherwise I'll snail you my photocopy.
> My photocopy is from UBC, and is a bit poor.
This would be great, because one has to be a member of that society to get
the paper for free
> Harris's use of the word "Analogy" is appropriate.
This is a crucial point for me. The microscopic Maxwell eqs. are derivable
through a straightforward extension of Helmholtz's analysis between forces
and energies. In this sense, it is not by chance that Newton's and Coulomb's
force laws are isomorphic. I was not yet able, however, to get this
difference of a factor of 2 between the ordinary and the gravito-magnetic
field this way.
If I have nothing overlooked (what may well be - recently, I have found, that
I had overlooked the scalar potential in my general ansatz for Gauss' law, -
after adding it, I have obtained the Yukawa potential, too), then, both sets
are more mathematically than physically similar.
Of course, this mathematical similarity has physical consequences, such as
the transversality of freely propagating gravitational waves (within this
approximation).
> > > Some years ago, I was invited to NASA's MSFC in Alabama
> > > to examine and opine on their "antigravity" experiments,
> > > based on Ning Li's theory which you can pick-up here,
> > >http://en.wikipedia.org/wiki/Ning_Li_(physicist)
> > > The upshot is her paper(s) were accepted by PRD, however
> > > after considerable expenditure and very careful experiments
> > > by NASA, no gravitomagnetic field effect could be detected.
> >
> > > In addition, a related effect sometimes referred to as "frame
> > > dragging" was explored by NASA at considerable expense,
> > > by satellite "Gravity Probe B", and it too was unable to
> > > verify a "gravitomagnetic" type of effect that was thought to
> > > be due to Earth's rotational effect on spacetime.
> >
> > > I (we) concluded in the 90's the effect was not detectable,
> > > and informed NASA of that, and NASA wanted to subsequently
> > > cancel GP-B, however for political and IMHO good scientific
> > > reasons it was launched anyway. I figure that even a null
> > > result is good science, just as the null result of the MMX
> > > led to the abandonment of the aether theory became one
> > > of the most famous experiments in history.
I see
> > Are there measurable effect in Einstein's field equations that vanishe in
> > the gravito-electromagnetic approximation?
> The gravitomagnetic effect (IMO) appears as a
> figment of a CS by using mathematical assumptions,
> without a good physics foundation.
> IOW's people seem have read things into equations
> that really aren't there.
What does it yield for, say, Mercury's perihelion rotation?
Thank you, once more,
Peter
Ken S. Tucker
> "Ken S. Tucker" <dynam...@vianet.on.ca> writes:
>
> > Hi Peter.
> > > > > > Thank you. Why the gravito-electromagnetic eqs. differ a little bit
> > > > > > from the microscopic Maxwell(-Lorentz) eqs.? Nonlinearity?
> > > > Edward G. Harris (Am.J.Phys. 59(5), May 1991) has
> > > > a good article entitled, " Analogy between general relativity
> > > > and electromagnetism for slowly moving particles in weak
> > > > gravitational fields".
> > > Can you send me a copy? Thanks a lot!
> > I think it's available here,
>
> >0059000005000421000001&idtype=cvips&gifs=yes
> > otherwise I'll snail you my photocopy.
> > My photocopy is from UBC, and is a bit poor.
>
> This would be great, because one has to be a member of that society to get
> the paper for free
LOL, I'll snail you my copy. Wife tell's me postage
costs $can 1.60, but I'm a cheap-skate so I intend
to use only $1.59 worth of stamps, so it will arrive
on your doorstep with postage due!
> > Harris's use of the word "Analogy" is appropriate.
>
> This is a crucial point for me. The microscopic Maxwell eqs. are derivable
> through a straightforward extension of Helmholtz's analysis between forces
> and energies. In this sense, it is not by chance that Newton's and Coulomb's
> force laws are isomorphic. I was not yet able, however, to get this
> difference of a factor of 2 between the ordinary and the gravito-magnetic
> field this way.
Well let's review Ampere's Force law,
http://en.wikipedia.org/wiki/Amp%C3%A8re%27s_force_law
where the relative current directions make a big
difference, (creating opposite magnetic force), so
you have a +1 or -1 force, depending on relative
current, (per unit length of wire).
Next, lets examine gravitation, with two water pipes
side by side and water (mass current) moving in the
pipes, in either relative direction, such as,
=====> pipe 1
<===== pipe 2
=====> pipe 1
=====> pipe 2'
Is the gravitational effects between pipe1 and pipe2
different from pipe1 and pipe2' if the only change is
the relative direction of water flow (mass current)?
Is that a reasonable question?
I'd like to work that gedanken if you want.
> If I have nothing overlooked (what may well be - recently, I have found, that
> I had overlooked the scalar potential in my general ansatz for Gauss' law, -
> after adding it, I have obtained the Yukawa potential, too), then, both sets
> are more mathematically than physically similar.
Yeah, that's why I need to evaluate mathematics
with gedankens in a developement.
> Of course, this mathematical similarity has physical consequences, such as
> the transversality of freely propagating gravitational waves (within this
> approximation).
Yeah that's a deep linkage. If GP-b and other similiar
experiments null, then based on current classical GR
theory, LIGO will likely null.
The perihelion rotations are a fairly straightforward
result from SS (Schwarzschild), based on a static
g-field, nothing fancy there. The SS can be deduced
from the gravitational "red-shift" with a bit of effort.
> Thank you, once more,
> Peter
My pleasure Peter, your insights are like a good cup
of stimulating coffee for me.
Regards
Ken S. Tucker
Juan R.
[Resubmitting to spf by Peter advice]
If you mean gravitomagnetism you may find useful this link
http://en.wikipedia.org/wiki/Gravitomagnetism
and several pdf linked below.
The geometric proof on absence of Newtonian limits for geometric GR only
uses the weak field condition, therefore is common for both Newtonian
limit and gravitomagnetism (where velocities are not so small).
However, a correct gravitomagnetic limit can be derived from both field
theory (FTG) and DPI theories.
The field equations obtained from FTG and DPI are equivalent to Wikipedia
article and give same gravitomagnetic effects.
The important differences are in the equation of motion which has not
geodesic interpretation and in the spatial geometry.
The complete proof of absence of Newtonian limits for any metric theory
of gravity (this includes GR, scalar generalizations, gauge theory, and
string branes approaches) uses *gravitational dualism*.
Electromagnetic dualism was introduced because it was showed that Maxwell
electrodynamics lacks a rigorous Coulomb limit. See:
1996: Phys. Rev. E 53, 5373. Chubykalo, Andrew E; Smirnov-Rueda, Roman.
1997: Phys. Rev. E 55, 3793. Chubykalo, Andrew E; Smirnov-Rueda,
Roman.
1998: Phys. Rev. E 57, 3683. Chubykalo, Andrew E; Smirnov-Rueda, Roman.
1999: Int. J. of Mod. Phys. A 14(24), 3789. Chubykalo, Andrew E; Vlaev,
Stoyan J.
In
1996: Phys. Rev. E 53, 5373. Chubykalo, Andrew E; Smirnov-Rueda, Roman.
authors generalized the Maxwell equations to yield the correct Coulomn
limit. Dualism implies that gravitomagnetic equations may be also
generalized.
For instance the first GEM equation change to (Chubykalo and Smirnov-
Rueda notation)
\nabla E_0(R(t)) = -4 \pi G \rho(R(t))
the RHS of the third changes to
-1/c \partial B^*(r,t) \partial t
And similar modifications. See cited papers for notation, detailed
discussion, and modified field equations in the case of EM.
--
Center for CANONICAL |SCIENCE)
http://canonicalscience.org