Newtonian Limit of GR: Experimentally Constraining k in the Einstein Equation

[smallphi]

According to the official textbook version, the coefficient of proportionality k in the Einstein equation:

$$R_{\mu\nu} - \frac{R}{2} \, g_{\mu \nu} = k \, T_{\mu \nu}$$

is fixed by requiring the equation boils down to Poisson equation for some kind of self gravitating fluid:

$$\nabla^2 \phi = 4 \pi G \rho$$



Experimentally, all we have tested about the Newtonian limit of gravity is the Schwartzshild solution of Einstein equations -we haven't measured how an actual fluid clumps under its own gravity. So based on the available experimental data all we can require is the Schwartzschild solution. Unfortunately, the Schwartzschild solution has a zero energy-momentum tensor so it is independent of the constant k that multiplies that tensor in the Einstein equation. Thus, I conclude our experimental data does not constraint the constant k.

Where exactly am I supposedly wrong?

 

[pervect]

Experimentally, all we have tested about the Newtonian limit of gravity is the Schwartzshild solution of Einstein equations -we haven't measured how an actual fluid clumps under its own gravity.

The Sun (and the Earth too, not that it matters that much to the overall gravitational field in the solar system) IS a big lump of fluid as far as Einstein's equation's go. I'm not sure what sense you think the Sun "isn't an actual fluid"? There may be some fine print in the defintion of "fluid" that's concerning you which might be worth pinning down in more detail.

In any event, I think our measurements of G are ultimately based on Eotvos type experiments. I.e. we use Eotvos experiments to determine G, the value of G is then used to determine the mass of the Sun. We don't really determine the mass of the sun directly, we determine it from our knowledge of G - IIRC. (Flame me if I've remembed wrong, but politely, please! :-)).

[add]for instance, google finds http://flux.aps.org/meetings/YR00/APR00/abs/S5400.html

Going back to the title of this thread, "the Newtonian limit", PPN theory gives a coherent approximation scheme for small-scale gravity experiments, like Eotvos experiments. The PPN theory is good enough for most solar system applications as well - it's even better at smaller scales. PPN is slightly better than the Newtonian limit, if you give PPN a lobotomy you can turn it into the true Newtonian limit.

I think the point is that there is only one value of k in the EFE which has the correct Newtonian limit.

 

[smallphi]

My point was we measure the gravitational effect of sun or planet OUTSIDE of it, so this is the Schwartzschild solution whose Newtonian limit is the Poisson equation for mass density ZERO (also known as Laplace equation).

So we have experimental data for the Laplace case outside the gravitating body:

$$\nabla^2 \phi = 0$$

but not for the general Poisson case:

$$\nabla^2 \phi = 4 \pi G \rho,\,\, \rho \ne 0$$

cause we never measured INSIDE the gravitating body where rho is not zero.

The problem is Laplace is linked to Schwartzschield which doesn't restrict the proportionality constant in Einstein equation at all.

 

[pervect]

Suppose you have a region where $\phi$ is nonzero, surrounded by a vacuum region where $\phi=0$

As soon as you know that $$\nabla^2 \phi = 0$$ in the vacuum region, you know by Gauss's theorem that [/tex]_S \oint (\nabla \phi) \cdot dA[/tex] for any surface S which completely enclosed the non-vaccum region.

The surface integral gives you a way of assigning a specific charge (or in this case, mass) to the non-vacuum region.

I don't understand why you think you need anything more than this.

 

[pervect]

Suppose you have a region where $\phi$ is nonzero, surrounded by a vacuum region where $\phi=0$

As soon as you know that $$\nabla^2 \phi = 0$$ in the vacuum region, you know by Gauss's theorem that $$\oint_S (\nabla \phi) \cdot dA$$ is constant, i.e. is numerically the same for any surface S which completely encloses the non-vaccum region.

The surface integral gives you a way of assigning a specific charge (or in this case, mass) to the non-vacuum region.

I don't understand why you think you need anything more than this.

 

[Wallace]

I think with smallphi is trying to say (correct me if I'm wrong, and allowing me to paraphrase) is that in the derivation of the Schwarzschild metric, there is a parameter that is clearly interpretable as the 'mass' at the central point, however there is no way to a priori determine the correct scaling between this parameter and mass in correct units. In any derivation of the metric there is always a final step that sets this parameter by reference to the Newtonian result.

It's somewhat unsettling that we can't just get GR to work on its own, without needing to scale the solution via the Newtonian limit.

 

[smallphi]

Yes, there is an integration constant in the Schwarzschild solution which we choose to call GM. This constant is not connected to the proportionality coefficient k in Einstein eq. in any way because k was multiplied by the zero energy momentum tensor and dropped from consideration. Moreover, the Einstein eq. does not play any role in our factorizing this constaint into G times M. It's our knowing of G from Newtonian physics that allows us to assign M , the 'gravitational mass at infinity' as they call it, to the gravitating body. The punch line is that demanding the Newtonian limit $$\nabla^2 \phi = 0$$ which corresponds to Schwarzschild in spherical symmetry case, does NOT fix the constant in Einstein equation because $$\nabla^2 \phi = 0$$ does NOT contain G at all! Unfortunately, I don't think we have experimental data that allows us to demand the more encompassing Newtonian limit $$\nabla^2 \phi = 4 \pi G \rho \, \, , \rho \ne 0$$ which does contain G and does fix the constant in Einstein eq.

G from Newtonian physics is measured by the faint attraction between two balls on a torsion balance according to Wikipedia and my understanding. Interpreting the orbits in Solar system, we always measure the GM product and then assign M, knowing the value of G from the torsion balance terrestrial experiments. That putting by hand process doesn't have anything to do with the constant k in Einstein eq. because the constant k does not play any part in Schwartzschild (or in any other vacuum solution) - it neither fixes GM nor factorizes it.

 

[Chris Hillman]

Please avoid immediately jumping to extreme conclusions!

This is the second time this week that I have noticed someone leaping to the most extreme possible conclusion (that gtr/cosmology textbooks are fundamentally erroneous) on the basis of simple misconceptions. The unusual twist here is that the OP is (without realizing it) trying to argue that Newtonian gravitation is "fundamentally flawed"! [sic]

This has nothing to do with "fluids"; the OP appears to be confused about the setup in Newtonian gravitation (in the field theory version in which the field equation is the Poisson equation). I think the basic problem here is that the OP lacks essential background in the basic theory of the Poisson and Laplace equations; see for example Ronald B. Guenther and John W. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations, Dover reprint, and E. C. Zachmanoglou and Dale W. Thoe, Introduction to Partial Differential Equations with Applications, Dover reprint. Note in particular that inferring behavior in the interior of some neighborhood (diffeomorphic to a ball) from behavior on the exterior is a fundamental topic in this theory! (Pervect tried to make the same point in other words.)

To state the obvious: you certainly can't understand the Newtonian limit of gtr if you don't understand the classical non-relativistic gravitational field theory known as "Newtonian gravitation" (although the Poisson equation came along after Newton's death), and you can't understand this unless you understand the theory of the Poisson and Laplace equations!

It's somewhat unsettling that we can't just get GR to work on its own, without needing to scale the solution via the Newtonian limit.

You didn't define what you mean by "work" or "own", but under any reasonable interpretation I think your remark is seriously misleading, since there is no problem whatever here.

It should be clear that any viable gravitation theory will agree with Newtonian gravitation in weak-field slow motion circumstances, simply because we know on the basis of extensive experience and testing that the latter theory works very well in such circumstances. So there's nothing disturbing about the textbook procedure! But if by historical accident, gtr had been discovered before Newton's theory was discovered as a useful approximation to gtr which is valid in many circumstances which is easier to work with when it is valid to use it, then it should be clear that the constant could have been determined without first knowing Newton's theory. (I think pervect was getting at this point among others in his Post #3.)

You may be interested in http://arxiv.org/abs/0712.0019" recent pre-print
...
The paper I linked to argues that the weak field GR result for a smooth lump of mass is not the same as the Newtonian one. While they agree that the weak-field results agree for point masses, or systems with the mass highly concentrated and surrounded by vacuum (stars, planets, black-holes), they argue that this is not the case for diffuse bodies, such as galaxies and clusters of galaxies, where the gravitating mass is spread out over the whole region of interest.

I think you may have missed a key point: it seems to me that they are discussing a local versus infinitesimal "level of structure" issue; compare this with the determination of the multiplicative constant in the EFE using the Newtonian limit. (The discussion in various places in MTW, Gravitation, Freeman, 1973, should help advanced students to understand my point.)

There are (at least) three levels of structure which students of manifolds often confuse:

• jet spaces (generalization of tangent spaces, which treat linear approximations, to quadratic approximations, etc.); this is an "infinitesimal structure",
• local neighborhoods,
• global structure, such as topology and global conformal structure.

These distinctions underlie the additional structure we obtain by endowing a smooth manifold with (for example) a Riemannian or Lorentzian metric tensor. See for example John M. Lee, Introduction to Smooth Manifolds, Springer, for some good discussion of levels of structure in the theory of smooth manifolds.

Just to make things even more confusing until recently, many careless writers in the physics literature referred to infinitesimal structure as "local" [sic], which contradicts more modern usage which follows mathematical usage in which "local" refers to "local neighborhood" (in the theory of manifolds; mathematical usage suffers from similar confusion when we turn to localization in ring theory, but let's not get into that!).

Hope this helps! There is a lot to be confused about, and too few textbooks even attempt to clarify the crucial issue of levels of structure, so it's not surprising that misconceptions about levels of structure seem to arise here so often.

 

[smallphi]

Also, it should be clear that any viable gravitation theory will agree with Newtonian gravitation in weak-field slow motion circumstances, since we know the latter works very well in such circumstances.

We know Newtonian gravitation works very well outside the gravitating body where $$\rho = 0$$ so we probed experimentally ONLY in that region. What is the gravitational potential inside the gravitating body, we never measured and can't claim to know. The solution outside does NOT fix the solution inside - I know enough from differential equation theory to know that. I'm not aware of any experiment performed inside the gravitating fluid/body where $$\rho \ne 0$$. So the textbooks impose a Newtonian limit, $$\nabla^2 \phi = 4 \pi G \rho \, \, , \rho \ne 0$$, which has no experimental verification - we just think it must be true assuming there is superposition principle for the fields of the elementary gravitational masses comprising $$\rho \ne 0$$. How do we know the superposition principle holds for gravitational field inside the mass configuration, have we measured it ? All our experiments on gravitational superposition (the first thing that comes to mind is the planets influencing each other on their orbits around the sun) are outside the sources of the fields. Superposition of gravity for planets is one thing, superposition of gravity from two atoms in the environment of many other atoms could be totally different unless there is an experiment verifying it. If someone thinks I'm wrong you have to cite the experiment that measures inside, that's the point of this discussion.

 

[Wallace]

You didn't define what you mean by "work" or "own", but under any reasonable interpretation I think your remark is seriously misleading, since there is no problem whatever here.

I agree that there isn't a major problem, I'm not suggesting there is anything fundamentally wrong with GR, I was trying to point out that GR is calibrated via the Newtonian limit, rather than reducing to it (in numerical terms) a priori. That's not a controversial statement, as you clearly agree, but it is something that many people reading may not have realized or been aware of.

It should be clear that any viable gravitation theory will agree with Newtonian gravitation in weak-field slow motion circumstances, simply because we know on the basis of extensive experience and testing that the latter theory works very well in such circumstances. So there's nothing disturbing about the textbook procedure! But if by historical accident, gtr had been discovered before Newton's theory was discovered as a useful approximation to gtr which is valid in many circumstances which is easier to work with when it is valid to use it, then it should be clear that the constant could have been determined without first knowing Newton's theory. (I think pervect was getting at this point among others in his Post #3.)

I agree, in so far as this applies to compact systems in which most of the region of interest is considered a vacuum. If we had come across GR first, we could have realized that it reduced to a simple 1/r^2 linear law for most mass ranges we come across, and the constants could have been calibrated from there.

I think you may have missed a key point: it seems to me that they are discussing a local versus infinitesimal "level of structure" issue; compare this with the determination of the multiplicative constant in the EFE using the Newtonian limit. (The discussion in various places in MTW, Gravitation, Freeman, 1973, should help advanced students to understand my point.)

There are (at least) three levels of structure which students of manifolds often confuse:

• jet spaces (generalization of tangent spaces, which treat linear approximations, to quadratic approximations, etc.); this is an "infinitesimal structure",
• local neighborhoods,
• global structure, such as topology and global conformal structure.

These distinctions underlie the additional structure we obtain by endowing a smooth manifold with (for example) a Riemannian or Lorentzian metric tensor. See for example John M. Lee, Introduction to Smooth Manifolds, Springer, for some good discussion of levels of structure in the theory of smooth manifolds.

Just to make things even more confusing until recently, many careless writers in the physics literature referred to infinitesimal structure as "local" [sic], which contradicts more modern usage which follows mathematical usage in which "local" refers to "local neighborhood" (in the theory of manifolds; mathematical usage suffers from similar confusion when we turn to localization in ring theory, but let's not get into that!).

Hope this helps! There is a lot to be confused about, and too few textbooks even attempt to clarify the crucial issue of levels of structure, so it's not surprising that misconceptions about levels of structure seem to arise here so often.

Can you elaborate on how this applies to the Cooperstock and Tieu paper on the internal and external observers view of the velocity of the collapsing dust? As far as I can see there point boils down to a fairly simple one, namely that in a system in which we would normally consider the Newtonian approximation to be adequate, they find significant deviation from the expected Newtonian behavior when treating the problem with a complete GR analysis. I haven't gone through the full details of their analysis, but for arguments sake, if there solution to the EFE is correct, do you think this is a significant result?