Birkhoff's theory of gravitation

[cmaj10@yahoo.com]

I would like to know what makes Einstein's curved space-time theory of
gravitation more acceptable than Birkhoff's flat space-time theory of
gravitation?

Birkhoff's theory can be found here:

http://www.pubmedcentral.gov/pagerender.fcgi?artid=1078697&pageindex=1

It predicts all of Einstein's effects, including the advance of
Mercury's perihelion. Surely, the claim that these effects are observed
can't be used to claim that curved space-time is verified, because we
can say the same thing about Birkhoff's flat space-time. But that's
what we nonetheless keep hearing.

Weyl said that he preferred Einstein's theory because it makes inertia
and gravitation the same thing, and he considered this fact a "radical
explanation". But it's obvious that it's no explanation at all: the
principle of equivalence merely postulates a well-known empirical fact.
Since when is a postulate an explanation?

Chris

[tessel@um.bot]

On Sun, 2 Apr 2006 cmaj10@yahoo.com wrote:

> I would like to know what makes Einstein's curved space-time theory of
> gravitation more acceptable than Birkhoff's flat space-time theory of
> gravitation?

This is addressed by MTW, section 39.1:

"Birkhoff's (1943) theory predicts the same redshift, perihelion shift,
deflection, and time delay as general relativity. But it requires that
the pressure inside gravitating bodies equal the total density of
mass-energy, p=rho; and, as a consequence, it demands that sound waves
travel with the speed of light. Of course, this prediction disagrees
violently with experiment. Therefore, Birkhoff's theory is not viable."

I trust this answers your first question. I didn't understand your second
question, but note that there are several "principle of equivalence"
discussed in the literature, e.g. by Dicke. You didn't give a citation
for Weyl's comment but I would guess it predates work by Dicke and others
sorting out some of the different ways this phrase has been used.

"T. Essel"

[Jay R. Yablon]

>
> "Birkhoff's (1943) theory predicts the same redshift, perihelion shift,
> deflection, and time delay as general relativity. But it requires that
> the pressure inside gravitating bodies equal the total density of
> mass-energy, p=rho; and, as a consequence, it demands that sound waves
> travel with the speed of light. Of course, this prediction disagrees
> violently with experiment. Therefore, Birkhoff's theory is not viable."
>
I have a question about this:

Of course the speed of sound is different for different media, But I am
wondering what we know about the speed of sound might be under certain
circumstances as detailed below:

For example, looking at MTW section 39.1 as well as their page 132, I
believe the p and rho you mention above come from the perfect fluid tensor
T^uv = (rho + p) U^u U^v + g^uv p

If p=rho is what makes sound travel at light speed, would this mean, for
example, that a relationship like p=(1/7)rho would mean that sound travels
at 1/7 the speed of light in whatever perfect fluid this relation applies
to? That is, is the it multiplier factor (here 1/7, for example) which
determines the ratio of sound to light propagation, which is why p= 1 x rho
would mean that the speed of sound for such a "stiff matter" equals the
speed of light?

And, have there been any material "fluids" observed in which the speed of
sound is determined to be a significant fraction (say 1/7) of the speed of
light?

Related to this, does anyone know the *fastest* speed of sound that has ever
been observed in *any* material medium, and if so, what is it?

Thanks.

Jay.

[cmaj10@yahoo.com]

tessel@um.bot wrote:
> On Sun, 2 Apr 2006 cmaj10@yahoo.com wrote:
>
> > I would like to know what makes Einstein's curved space-time theory of
> > gravitation more acceptable than Birkhoff's flat space-time theory of
> > gravitation?
>
> This is addressed by MTW, section 39.1:
>
> "Birkhoff's (1943) theory predicts the same redshift, perihelion shift,
> deflection, and time delay as general relativity. But it requires that
> the pressure inside gravitating bodies equal the total density of
> mass-energy, p=rho; and, as a consequence, it demands that sound waves
> travel with the speed of light. Of course, this prediction disagrees
> violently with experiment. Therefore, Birkhoff's theory is not viable."
>
> I trust this answers your first question.

Thanks! If that's all there is against this theory then I'm not ready
to throw the baby with the water. We have to prepare for the impending
null result of the GP-B experiment, after all. Pardon my bias ;-).

> I didn't understand your second
> question, but note that there are several "principle of equivalence"
> discussed in the literature, e.g. by Dicke.

I don't know how any clearer I could be. A postulate is an ad-hoc rule
of the game, it cannot supply any explanation. Some postulates are
better than others if they are inspired by empirical observation. But
they still have no explanatory power.

> You didn't give a citation
> for Weyl's comment but I would guess it predates work by Dicke and others
> sorting out some of the different ways this phrase has been used.

It turns out that I mistakenly provided the link to Weyl's paper. So
here it is again:

http://www.pubmedcentral.gov/pagerender.fcgi?artid=1078697&pageindex=1

"Communicated July 20, 1944

Whereas electric charge is not universally proportional to the inertial
mass of bodies their gravity is. This fundamental fact, supported both
by daily experience and the most refined experiments, led Einstein to
the conception that inertia and gravitation are one (principle of
equivalence) -and thus to his theory of general relativity. The main
reason for my and many others' belief in that theory is the radical
explanation it affords for the fact just mentioned. Any theory which
breaks up the unity of inertia and gravitation, as Birkhoff's recent
theory of gravitation in a flat world' does, throws us back into the
position before Einstein where we had to accept the identity of mass and
weight without understanding it. Is there any reason for such a
withdrawal?" [H. Weyl]

And here is G. Birkhoff's first paper:

"Matter, Electricity and Gravitation in Flat Space-Time"
Proc Natl Acad Sci U S A. 1943 August; 29(8): 231-239.
http://www.pubmedcentral.gov/articlerender.fcgi?artid=1078600

A. Barajas' paper:
Proc Natl Acad Sci U S A. 1944 Mar 15;30(3):54-57.
"Birkhoff's Theory of Gravitation and Einstein's Theory for Weak
Fields."
http://www.pubmedcentral.gov/articlerender.fcgi?artid=1078668

And here is G. Birkhoff's second paper:

"Flat Space-Time and Gravitation"
Proc Natl Acad Sci U S A. 1944 Oct 15;30(10):324-334. No abstract
available.
http://www.pubmedcentral.gov/pagerender.fcgi?tool=pmcentrez&pageindex=1&artid=1078721

Barajas criticizes Weyl regarding his statement on the principle of
equivalence, among other things.

Chris

> "T. Essel"

[carlip-nospam@physics.ucdavis.edu]

tessel@um.bot wrote:
> On Sun, 2 Apr 2006 cmaj10@yahoo.com wrote:

>> I would like to know what makes Einstein's curved space-time theory of
>> gravitation more acceptable than Birkhoff's flat space-time theory of
>> gravitation?

> This is addressed by MTW, section 39.1:

> "Birkhoff's (1943) theory predicts the same redshift, perihelion shift,
> deflection, and time delay as general relativity. But it requires that
> the pressure inside gravitating bodies equal the total density of
> mass-energy, p=rho; and, as a consequence, it demands that sound waves
> travel with the speed of light. Of course, this prediction disagrees
> violently with experiment. Therefore, Birkhoff's theory is not viable."

It's been a while since I've looked at this, but if my memory is right,
then apart from this objection:

1. Birkhoff's theory is linear. In particular, this means that the
gravitational field does not itself carry gravitational energy.
This is incompatible with observations of the Moon's orbit, which
show that the gravitational binding energy of the Earth and Moon
contribute to gravitational mass.

2. While Birkhoff's theory allows gravitational waves, these do not
carry energy. It's hard to see how the theory can explain the
observed decay of binary pulsar systems, which in general relativity
is explained (quantitatively, and to a very high accuracy) as being
due to loss of energy due to gravitational radiation.

Steve Carlip

[tessel@um.bot]

On Mon, 3 Apr 2006, Jay R. Yablon wrote:

> For example, looking at MTW section 39.1 as well as their page 132, I
> believe the p and rho you mention above come from the perfect fluid
> tensor T^uv = (rho + p) U^u U^v + g^uv p

Yes, and this is also written out by Birkhoff 1943.

> If p=rho is what makes sound travel at light speed, would this mean, for
> example, that a relationship like p=(1/7)rho would mean that sound travels
> at 1/7 the speed of light in whatever perfect fluid this relation applies
> to? That is, is the it multiplier factor (here 1/7, for example) which
> determines the ratio of sound to light propagation, which is why p= 1 x rho
> would mean that the speed of sound for such a "stiff matter" equals the
> speed of light?

No, it's not that simple, and in fact it's a bit more complicated than
some physicists seem to realize (to judge from their papers). The
relation often used is that in a material with EOS giving p as function
of rho, dp/drho = (v_sound)^2, but see gr-qc/0103065. Such
considerations may account for the discrepancy between p=rho (often
called the EOS for "stiff fluid" in the literature) and p=rho/2, which
is given by Birkhoff 1943 as a possible EOS for a perfect fluid in which
the speed of sound equals the speed of light. Birkhoff seems to imply
that he had some rationale for considering a theory in which all
gravitating matter is required to have this property (he adopts this as
a hypothesis), but this is not explained in his 1943 paper, and I'm not
inclined to chase down references to try to figure out what he may have
had in mind. Especially since elsewhere in the paper he seems to
suggest that his principle motivation was to come up with a
mathematically simple theory, rather than one which would be useful for
physicists.

"T. Essel"

[brian a m stuckless]

Jay R. Yablon wrote: > > >
> > "Birkhoff's (1943) theory predicts the same redshift, perihelion
> > shift, deflection, and time delay as general relativity. But it
> > requires that the pressure inside gravitating bodies equal the
> > total density of mass-energy, p=rho; and, as a consequence, it
> > demands that sound waves travel with the speed of light. Of
> > course, this prediction disagrees violently with experiment.
> > Therefore, Birkhoff's theory is not viable."
> >
> I have a question about this:
>
> Of course the speed of sound is different for different media,
> But I am wondering what we know about the speed of sound might
> be under certain circumstances as detailed below:
>
> For example, looking at MTW section 39.1 as well as their page
> 132, I believe the p and rho you mention above come from the
> perfect fluid tensor T^uv = (rho + p) U^u U^v + g^uv p
>
> If p=rho is what makes sound travel at light speed, would this
> mean, for example, that a relationship like p=(1/7)rho would
> mean that sound travels at 1/7 the speed of light in whatever
> perfect fluid this relation applies to? That is, is the it
> multiplier factor (here 1/7, for example) which determines the
> ratio of sound to light propagation, which is why p= 1 x rho
> would mean that the speed of sound for such a "stiff matter"
> equals the speed of light?
>
> And, have there been any material "fluids" observed in which
> the speed of sound is determined to be a significant fraction
> (say 1/7) of the speed of light?
>
> Related to this, does anyone know the *fastest* speed of sound
> that has ever been observed in *any* material medium, and if so,
> what is it? > > Thanks. > > Jay.
Re: Birkhoff's theory of gravitation

$$ Mathematically, phoTons are indistinguishable from phoNons.
$$ Sincerely, ```Brian A M Stuckless, Ph.T (Tivity).

[Daryl McCullough]

cmaj10@yahoo.com says...

>Weyl said that he preferred Einstein's theory because it makes inertia
>and gravitation the same thing, and he considered this fact a "radical
>explanation". But it's obvious that it's no explanation at all: the
>principle of equivalence merely postulates a well-known empirical fact.
>Since when is a postulate an explanation?

If you reduce a mysterious phenomenon to a well-understood phenomenon,
then you've (at least partially) explained it. The mysterious phenomenon
is the observed proportionality of gravitational and inertial mass. The
well-understood phenomenon is fictitious forces that arise from the use
of a curvilinear coordinate system.

Newton's laws say that

mass * acceleration = force_external

In good old inertial Cartesian coordinates, the acceleration
is simply

acceleration = (d^2/dt^2) position

where position is the position vector. However, in curvilinear
or accelerated coordinates, the acceleration has some additional
terms. So we can write

mass * [ acceleration_c + acceleration_nc ] = force_external

where acceleration_c is the inertial cartesian part of the
acceleration, and acceleration_nc is the non-cartesian, non-inertial
correction terms. We can move the non-cartesian correction terms
to the right side to get:

mass * acceleration_c = force_external - mass * acceleration_nc
= force_external + force_fictitious

where force_fictitious is just - mass * acceleration_nc. So
we can "pretend" that we are still using inertial Cartesian
coordinates, as long as we include the fictitious forces.
These fictitious forces are such effects as the Coriolis
force, the Centrifugal force, the inertial forces felt in
an accelerating vehicle.

Note that the fictitious forces are *necessarily* proportional
to inertial mass, since they are just due to the term
mass * acceleration_nc.

Okay, so classical physics has no problem understanding why
fictious forces are always proportional to inertial mass. Einstein's
contribution to explaining why gravitational mass is proportional
to inertial mass is to assume that

Gravity is a *fictitious* force due to the use of
noninertial and/or curvilinear coordinates.

So it is an explanation, in that it explains one mystery
(the proportionality of inertial and gravitational masses)
in terms of a well-understood phenomenon (fictitious forces
are always proportional to inertial mass).

--
Daryl McCullough
Ithaca, NY

--
NewsGuy.Com 30Gb $9.95 Carry Forward and On Demand Bandwidth

[tessel@um.bot]

On Mon, 3 Apr 2006, Jay R. Yablon wrote:

> For example, looking at MTW section 39.1 as well as their page 132, I
> believe the p and rho you mention above come from the perfect fluid
> tensor T^uv = (rho + p) U^u U^v + g^uv p

Yes, and this is also written out by Birkhoff 1943.

> If p=rho is what makes sound travel at light speed, would this mean, for
> example, that a relationship like p=(1/7)rho would mean that sound travels
> at 1/7 the speed of light in whatever perfect fluid this relation applies
> to? That is, is the it multiplier factor (here 1/7, for example) which
> determines the ratio of sound to light propagation, which is why p= 1 x rho
> would mean that the speed of sound for such a "stiff matter" equals the
> speed of light?

No, it's not that simple, and in fact it's a bit more complicated than
some physicists seem to realize (to judge from their papers). The
relation often used is that in a material with EOS giving p as function
of rho, dp/drho = (v_sound)^2, but see gr-qc/0103065. Such
considerations may account for the discrepancy between p=rho (often
called the EOS for "stiff fluid" in the literature) and p=rho/2, which
is given by Birkhoff 1943 as a possible EOS for a perfect fluid in which
the speed of sound equals the speed of light. Birkhoff seems to imply
that he had some rationale for considering a theory in which all
gravitating matter is required to have this property (he adopts this as
a hypothesis), but this is not explained in his 1943 paper, and I'm not
inclined to chase down references to try to figure out what he may have
had in mind. Especially since elsewhere in the paper he seems to
suggest that his principle motivation was to come up with a
mathematically simple theory, rather than one which would be useful for
physicists.

"T. Essel"

[Jay R. Yablon]

>> For example, looking at MTW section 39.1 as well as their page 132, I
>> believe the p and rho you mention above come from the perfect fluid
>> tensor T^uv = (rho + p) U^u U^v + g^uv p
>
> Yes, and this is also written out by Birkhoff 1943.
>
>> If p=rho is what makes sound travel at light speed, would this mean, for
>> example, that a relationship like p=(1/7)rho would mean that sound
>> travels
>> at 1/7 the speed of light in whatever perfect fluid this relation applies
>> to? That is, is the it multiplier factor (here 1/7, for example) which
>> determines the ratio of sound to light propagation, which is why p= 1 x
>> rho
>> would mean that the speed of sound for such a "stiff matter" equals the
>> speed of light?
>
> No, it's not that simple, and in fact it's a bit more complicated than
> some physicists seem to realize (to judge from their papers). The
> relation often used is that in a material with EOS giving p as function
> of rho, dp/drho = (v_sound)^2, but see gr-qc/0103065.

If dp/drho = (v_sound)^2, and I assume the "d" is a partial derivative, then
the equation just before (6.2) in
http://home.nycap.rr.com/jry/Papers/Draft%20Lorentz%20Force%20Paper.pdf
would seem to tell us that (v_sound)^2 = (gamma-1/gamma), where gamma is the
barotropic parameter p=(gamma-1)mu. So, actually, the "1/7" example I used
earlier would have v_sound = 1/8.

In gr-qc/0103065, they do not assume an a priori equation of state, and for
similar reasons I am not assuming an a prior equation of state either. How
can we simply assert that in a real physical situation, the pressure of a
fluid will everywhere be a constant multiple of the rest mass density? Much
better to leave in gamma as a field to be determined by the field equations.
I have never seen a weather report showing the earth's atmosphere to be 30
psi everywhere, and one would suspect that this is a very limiting
assumption to apply to a perfect fluid. Also, the speed of sound will not
be constant in a given fluid; local variations in the fluid of any sort
should be expected to produce local variations in the sound speed as well.
More simply put, I agree with the authors of gr-qc/0103065 about not
specifying an equation of state but letting the field equations do it for
us.

Jay.

[Oh No]

Thus spake cmaj10@yahoo.com
>I would like to know what makes Einstein's curved space-time theory of
>gravitation more acceptable than Birkhoff's flat space-time theory of
>gravitation?
>
>Birkhoff's theory can be found here:
>
>http://www.pubmedcentral.gov/pagerender.fcgi?artid=1078697&pageindex=1
>
>It predicts all of Einstein's effects, including the advance of
>Mercury's perihelion. Surely, the claim that these effects are observed
>can't be used to claim that curved space-time is verified, because we
>can say the same thing about Birkhoff's flat space-time. But that's
>what we nonetheless keep hearing.
>

>Weyl said that he preferred Einstein's theory because it makes inertia
>and gravitation the same thing, and he considered this fact a "radical
>explanation". But it's obvious that it's no explanation at all: the
>principle of equivalence merely postulates a well-known empirical fact.
>Since when is a postulate an explanation?

I think it is a very good explanation. Not only is it a well known
empirical fact, but it reduces gravity to a statement about geometry. In
other words gravity is not an impressed force, and it needs no
explanation in terms of the action of one body on another. If one wants
to understand it any more deeply than that, one should look at how we
actually go about measuring things, as Einstein did in the 1905 sr
paper, and try to understand why one geometry (the one we have) is more
appropriate than some other geometry which we might find easier to think
about. One certainly won't get any answers by assuming flat space, for
which there is no prior empirical evidence, and then assuming an active
force for which there is also no prior empirical evidence. That would be
quite an unscientific way to proceed.



Regards

--
Charles Francis
substitute charles for NotI to email

[Homo Lykos]

<carlip-nospam@physics.ucdavis.edu> schrieb:
>
> 1. Birkhoff's theory is linear. In particular, this means that the
> gravitational field does not itself carry gravitational energy.
> This is incompatible with observations of the Moon's orbit, which
> show that the gravitational binding energy of the Earth and Moon
> contribute to gravitational mass.

How certain is this? Can you give some refernces to your strong
"moon-orbit-statement".

Homo Lykos

[Dirk Bruere]

Oh No wrote:
> Thus spake cmaj10@yahoo.com
>
>>I would like to know what makes Einstein's curved space-time theory of
>>gravitation more acceptable than Birkhoff's flat space-time theory of
>>gravitation?
>>
>>Birkhoff's theory can be found here:
>>
>>http://www.pubmedcentral.gov/pagerender.fcgi?artid=1078697&pageindex=1
>>
>>It predicts all of Einstein's effects, including the advance of
>>Mercury's perihelion. Surely, the claim that these effects are observed
>>can't be used to claim that curved space-time is verified, because we
>>can say the same thing about Birkhoff's flat space-time. But that's
>>what we nonetheless keep hearing.
>>
>
>
>>Weyl said that he preferred Einstein's theory because it makes inertia
>>and gravitation the same thing, and he considered this fact a "radical
>>explanation". But it's obvious that it's no explanation at all: the
>>principle of equivalence merely postulates a well-known empirical fact.
>>Since when is a postulate an explanation?
>
>
> I think it is a very good explanation. Not only is it a well known
> empirical fact, but it reduces gravity to a statement about geometry. In
> other words gravity is not an impressed force, and it needs no
> explanation in terms of the action of one body on another. If one wants
> to understand it any more deeply than that, one should look at how we
> actually go about measuring things, as Einstein did in the 1905 sr
> paper, and try to understand why one geometry (the one we have) is more
> appropriate than some other geometry which we might find easier to think
> about. One certainly won't get any answers by assuming flat space, for
> which there is no prior empirical evidence, and then assuming an active
> force for which there is also no prior empirical evidence. That would be
> quite an unscientific way to proceed.

So is the existing 'method'.
All you seem to be arguing for is 'a neat concept' that provides a
pretty picture of 'what is REALLY happening'. However, that is
philosophy - not science. If two mathematical systems provide identical
answers from radically different pictures then I don't see that science
can say anything about the pictures themselves. That's a matter of
aesthetics.

Dirk

[markwh04@yahoo.com]

Daryl McCullough wrote:
> cmaj10@yahoo.com says...
> >Weyl said that he preferred Einstein's theory because it makes inertia
> >and gravitation the same thing, and he considered this fact a "radical
> >explanation".
> If you reduce a mysterious phenomenon to a well-understood phenomenon,
> then you've (at least partially) explained it.
[...]
> Okay, so classical physics has no problem understanding why
> fictious forces are always proportional to inertial mass. Einstein's
> contribution to explaining why gravitational mass is proportional
> to inertial mass is to assume that

The "contribution" is independent of the distinction from classical vs.
relativistic physics and, in fact, has nothing per se to do with it.

What the transition from flat spacetime to curved spacetime does is
accommodate the equivalence principle in geometric form, providing a
geometric interpretation along the lines of subsuming free fall and
inertial motion under a single umbrella.

This feature is orthogonal to the separate (and independent) innovation
that leads from Galilean Relativity to Poincare' Relativity and would
(for example) be equally applicable in Newtonian Physics to yield a
Galilean Relativistic theory of general relativity.

This is, in fact, a development that was initially carried out by
Cartan as part and parcel of his discovery of the independence of
affine structure from metric. I've also discussed the curved spacetime
formulation of Newtonian physics here briefly in a recent article here.

[carlip-nospam@physics.ucdavis.edu]

Homo Lykos <lykos@lykos.ch> wrote:
> <carlip-nospam@physics.ucdavis.edu> schrieb:

>> 1. Birkhoff's theory is linear. In particular, this means that the
>> gravitational field does not itself carry gravitational energy.
>> This is incompatible with observations of the Moon's orbit, which
>> show that the gravitational binding energy of the Earth and Moon
>> contribute to gravitational mass.

> How certain is this? Can you give some refernces to your strong
> "moon-orbit-statement".

See, for example, Williams et al., arxiv.org/abs/gr-qc/0507083,
section 3 (especially sections 3.2 and 3.3).

Steve Carlip

[Homo Lykos]

<carlip-nospam@physics.ucdavis.edu> schrieb:
> Homo Lykos <lykos@lykos.ch> wrote:
>> <carlip-nospam@physics.ucdavis.edu> schrieb:
>
>>> 1. Birkhoff's theory is linear. In particular, this means that the
>>> gravitational field does not itself carry gravitational energy.
>>> This is incompatible with observations of the Moon's orbit, which
>>> show that the gravitational binding energy of the Earth and Moon
>>> contribute to gravitational mass.
>
>> How certain is this? Can you give some refernces to your strong
>> "moon-orbit-statement".
>
> See, for example, Williams et al., arxiv.org/abs/gr-qc/0507083,
> section 3 (especially sections 3.2 and 3.3).

Thankyou for your interesting reference. But as I can see you find in it no
proof for your strong "moon-orbit-statement": These measurements are only
sensitive on the difference of the binding energy of the Earth and Moon if
eta = 4 beta - gamma - 3 is not 0 as it is in GR
*and if* there exists a contibution of the binding energy to gravitational
mass.

This means: If there exists no such contribution you can't find also any
descrepancy with these measurements.


Homo Lykos

[carlip-nospam@physics.ucdavis.edu]

Homo Lykos <lykos@lykos.ch> wrote:
> <carlip-nospam@physics.ucdavis.edu> schrieb:
>> Homo Lykos <lykos@lykos.ch> wrote:
>>> <carlip-nospam@physics.ucdavis.edu> schrieb:

>>>> 1. Birkhoff's theory is linear. In particular, this means that the
>>>> gravitational field does not itself carry gravitational energy.
>>>> This is incompatible with observations of the Moon's orbit, which
>>>> show that the gravitational binding energy of the Earth and Moon
>>>> contribute to gravitational mass.

>>> How certain is this? Can you give some refernces to your strong
>>> "moon-orbit-statement".

>> See, for example, Williams et al., arxiv.org/abs/gr-qc/0507083,
>> section 3 (especially sections 3.2 and 3.3).

> Thankyou for your interesting reference. But as I can see you find in it no
> proof for your strong "moon-orbit-statement": These measurements are only
> sensitive on the difference of the binding energy of the Earth and Moon if
> eta = 4 beta - gamma - 3 is not 0 as it is in GR
> *and if* there exists a contibution of the binding energy to gravitational
> mass.

> This means: If there exists no such contribution you can't find also any
> descrepancy with these measurements.

You are wrong.

The Lunar Laser Ranging experiments allow us to compare the contribution
of gravitational binding energy to gravitational mass with the contribution
of gravitational binding energy to inertial mass. The results show that
the two are equal, to within less than a tenth of a percent. As long as
you accept the *special* relativistic relation of energy and inertial mass,
E=mc^2, the observations show that gravitational energy also contributes
an amount E/c^2 to gravitational mass. If you want a simpler explanation,
try chapter 7 of Will's book, _Was Einstein Right?_

You can, I suppose, argue that we should throw out special relativity, and
reject the claim that all energy contributes to inertial mass. That would
require throwing out conservation of energy, and I suspect conservation of
momentum as well. Is this what you are suggesting?

Steve Carlip

[Homo Lykos]

<carlip-nospam@physics.ucdavis.edu> schrieb:
> Homo Lykos <lykos@lykos.ch> wrote:
>> <carlip-nospam@physics.ucdavis.edu> schrieb:
>>> Homo Lykos <lykos@lykos.ch> wrote:
>>>> <carlip-nospam@physics.ucdavis.edu> schrieb:
>
>>>>> 1. Birkhoff's theory is linear. In particular, this means that the
>>>>> gravitational field does not itself carry gravitational energy.
>>>>> This is incompatible with observations of the Moon's orbit, which
>>>>> show that the gravitational binding energy of the Earth and Moon
>>>>> contribute to gravitational mass.
>
>>>> How certain is this? Can you give some refernces to your strong
>>>> "moon-orbit-statement".
>
>>> See, for example, Williams et al., arxiv.org/abs/gr-qc/0507083,
>>> section 3 (especially sections 3.2 and 3.3).
>
>> Thankyou for your interesting reference. But as I can see you find in it
>> no proof for your strong "moon-orbit-statement": These measurements are
>> only sensitive on the difference of the binding energy of the Earth and
>> Moon if
eta = 4 beta - gamma - 3 is not 0 as it is in GR
>> *and if* there exists a contibution of the binding energy to
>> gravitational mass.
>
>> This means: If there exists no such contribution you can't find also any
>> descrepancy with these measurements.
>
> You are wrong.

I don't think so.

>
> The Lunar Laser Ranging experiments allow us to compare the contribution
> of gravitational binding energy to gravitational mass with the
> contribution of gravitational binding energy to inertial mass. The
> results show that the two are equal, to within less than a tenth of a
> percent. As long as you accept the *special* relativistic relation of
> energy and inertial mass, E=mc^2, the observations show that gravitational
> energy also contributes an amount E/c^2 to gravitational mass.

Yes, if you assume that it is a statement of special relativity, that
gravitational binding energy contributes to inertial mass E/c^2.

>
> You can, I suppose, argue that we should throw out special relativity, and
> reject the claim that all energy contributes to inertial mass.

I think, gravitational binding energy is a special case, because it implies
nonlinearity.

> That would
> require throwing out conservation of energy, and I suspect conservation of
> momentum as well. Is this what you are suggesting?

In the context of your strong "moon-orbit-statement": yes. But so long as
violations of conservation of energy and eventual further violations are
not too strong to become incompatible with until now known measurements,
that's no problem.

More generally "suggesting" is too strong: Until now I'm only asking, if
there exists any direct experimental evidence for the nonlinearity of
gravity.

Homo Lykos