A Principle Explanation of the “Mysteries” of Modern Physics

[PeterDonis]

Correct.

Ok. One thing about that example that makes it inappropriate for modeling something like galaxy rotation curves is that the "interior" FRW spacetime region cannot be stationary, whereas to model something like a galaxy, you would need an "interior" spacetime region that was stationary. This would also be true for modeling an individual star in a galaxy, but a stationary model for a star is easy: a spherically symmetric blob of matter in hydrostatic equilibrium with a constant surface area. A galaxy is not a continuous distribution of matter, although a really rough approximation could perhaps model it as such; but a better model would be a system of objects orbiting their common center of mass under their mutual gravity. I don't know how much models of that sort have been constructed in the literature; the only one that I can bring to mind at the moment is the one in a paper by Einstein in the 1930s where he was trying to prove that black holes were impossible (of course he didn't use the term "black hole" since it hadn't been invented yet) by showing that no such stationary system of mutually orbiting objects could have an "areal radius" smaller than $3 G M / c^2$, where $M$ is the externally measured mass of the system. A galaxy of course has a much larger "areal radius" so that wouldn't be an issue for such a model.

 

[PeterDonis]

Have you done the fits for these data using MOND, various modified gravity theories, and the different DM models?

I'm not approaching this from the perspective of trying to do empirical curve fits. Your basic contention is that, if we properly model something like a galaxy using GR instead of a Newtonian approximation, we can explain the discrepancy between the mass inferred from observed galaxy rotation curves and the mass inferred from observed total luminous objects in the galaxy without having to use dark matter. I'm trying to figure out if I agree that some kind of GR model could be constructed that would do that.

You're not exhibiting any such model in the paper; you're just using an ansatz that "looks reasonable" to you and doing empirical curve fitting with it. To me that's backwards. First you would need to construct a GR model--an actual spacetime geometry--that was a viable simplified model for something like a galaxy (i.e., stationary, which, as I have pointed out, the "interior" FRW region in the example you give is not), although obviously it would not be able to capture all the details of a real galaxy. Then you would need to show that this model exhibits the effect you are looking for--that there is a discrepancy between the mass inferred from rotation curves in the model and the mass inferred from observed total luminous objects in the model--and that the size of the effect is of the right rough order of magnitude. Only once you have done that would it be justified, IMO, to extract an ansatz from such a model and use it for empirical curve fitting.

What I'm trying to do for myself is the first two steps I just described: to see if I think there could be a simplified GR model that exhibits the effect in question of the right rough order of magnitude, based on your general description of a difference between "proper mass" and "dynamic mass", but a model whose "interior region" is stationary.

 

[RUTA]

All of these are really good points, Peter, and I agree with them. The place I would look first would be interior solutions for the rotating Kerr solution exterior. I would look at that numerically, since no such interior solutions are know as yet. I would do that exercise to develop a feel for how hydrodynamical cases marry up to vacuum exteriors. Doable, but nontrivial :-)

If you look at what else is being done in this area, you'll see for example the dark matter models are nothing more than searches for functional fitting forms (in that case, a search for the distribution of dark matter). And, as we point out in one of our papers, our ansatz is just as motivated as MOND's. I think modified GR is better theoretically, but even there one can ask, why those particular additions to the Lagrangian? The bottom line is always the same, because they work to fit the data. Of course, if you deviate from GR, then you lose its divergence-free nature, i.e., you violate local conservation laws (which they readily admit). That's why I wanted to find something in GR proper.

 

[PeterDonis]

The place I would look first would be interior solutions for the rotating Kerr solution exterior.

I'm actually starting with something simpler, a Schwarzschild exterior around a spherically symmetric matter distribution. To add angular momentum to the system, which is the primary reason for using a Kerr exterior, it might be sufficient to just add small correction terms to the interior and exterior metric, rather than trying to use the full-blown Kerr exterior metric, which, as you note, is significantly more complicated. That is how, for example, an experiment like Gravity Probe B is analyzed, as I understand it.

However, I'm not actually convinced that it is necessary to add angular momentum to the system, because the effects that doing that would be required to account for, such as the Lense-Thirring precession that Gravity Probe B was testing, are much too small to be what you are looking for. If the effect you are looking for is actually present in a GR model, it should be present in a model in which the angular momentum of the gravitating system overall can be ignored. So a Schwarzschild exterior with a stationary interior matter distribution should be enough.

dark matter models are nothing more than searches for functional fitting forms

Yes, but what is being fitted in that case is simply a distribution for the stress-energy tensor, which is already a free parameter in GR. In other words, the assumption is that the actual stress-energy tensor distribution is different from the one that would be inferred solely from the observed distribution of luminous matter, and the fitting is done to see how much different the actual stress-energy tensor distribution has to be to account for the observed rotation curves, using standard assumptions about the effects of spacetime geometry.

That's not what you're doing; you're assuming that the stress-energy tensor distribution is fixed by the distribution of luminous matter, and proposing that the spacetime geometry created by that stress-energy tensor distribution will have effects that differ from the standard assumptions, and that this effects will include a mismatch between the mass inferred from rotation curves and the mass inferred from the distribution of luminous matter. There are no free parameters to fit in such a model. The "fitting" you are doing is based on assumptions about what effects will be present in such a model, without actually constructing it to see if those assumptions are correct.

if you deviate from GR, then you lose its divergence-free nature, i.e., you violate local conservation laws (which they readily admit). That's why I wanted to find something in GR proper.

I agree that this is a very good reason to want to find a model that works within standard GR.

 

[RUTA]

But, there is a tacit assumption of the DM models that is just as speculative because they have no candidates for what they're placing in the stress-energy tensor. It's easy to wave your hand and say, "Well, someday maybe we'll discover a missing particle." But, the properties that particle would have to possesses are highly dubious. See this article by Sean Carroll.

 

[RUTA]

BTW, Peter, if you'd like another pair of eyes on your paper before submitting it, send me the arXiv link when you get it done. I'm VERY interested in what you find!

 

[PeterDonis]

there is a tacit assumption of the DM models that is just as speculative because they have no candidates for what they're placing in the stress-energy tensor

Yes, that's true. DM models have to assume that there is some non-baryonic kind of matter that will give rise to the stress-energy tensor they need, even though we have not found any such kind of matter in any experiments.

 

[PeterDonis]

if you'd like another pair of eyes on your paper before submitting it, send me the arXiv link when you get it done

I wasn't referring to any potential paper I am working on; I'm not an academic. If I find the time to make any calculations along the lines I was describing, I will post them here.

 

[PeterDonis]

Btw, @RUTA, if you are able, it would be helpful if you could comment on whether the understanding of your papers (refs. 23 and 24 in the Insights article) that I have described in the following thread is correct:

https://www.physicsforums.com/threads/is-there-a-simple-dark-matter-solution-rooted-in-gr.994526/

I ask because it seems to me that one of the papers (ref. 24, the one we have been discussing here) is using standard GR, while the other (ref. 23) is not--it is proposing a model in which the GR assumption of spacetime as a continuous manifold is only an approximation. If I am correct about that, discussion of those two papers should be in separate threads; the thread I linked to above, which is in the Beyond the Standard Model forum, would be appropriate for ref. 23, since it is proposing a model that goes beyond standard GR, but discussion of ref. 24, if it were to go anywhere other than this thread, should properly be in the relativity forum, since that paper is using standard GR.

 

[ohwilleke]

Yes, that's true. DM models have to assume that there is some non-baryonic kind of matter that will give rise to the stress-energy tensor they need, even though we have not found any such kind of matter in any experiments.

Equally important, DM particle model makers are increasingly concluding that they need either a self-interaction force (SIDM) or a 5th force between DM and ordinary matter, to make the distributions of DM inferred fit to the properties of the BSM DM particle. True collisionless cold dark matter, or warm dark matter particle doesn't, produce halos of the shapes observed from observation and doesn't mimic features in the baryonic matter distributions in a galaxy or cluster in the way that is observed.

So, one needs not just a new DM particle, but also a new force mediated by another new dark sector particle.

Once you need a new force anyway, the benefit of a DM particle theory over a modification of an existing force disappears, in terms of an Occam's Razor type analysis.

 

[RUTA]

I wasn't referring to any potential paper I am working on; I'm not an academic. If I find the time to make any calculations along the lines I was describing, I will post them here.

If you do find something that's not suitable for PF (since it's not been properly refereed), please notify me! Let me say specifically what I hope someday to have the time to explore.

The way the momentum-energy content of the matter-occupied region of spacetime affects the geometry of the vacuum region surrounding it is via the coupling between regions as expressed in the extrinsic curvature K on the spatial hypersurface boundary. The goal would be to find the cumulative functional form for nested embeddings, i.e., K1 to K2 to ... . How does the mass M in a vacuum geometry vary from "shell" to "shell" as a function of the K's? You can see why I was considering a Kerr solution, since I don't have hydrodynamic support and I don't want radially expanding or collapsing shells.

 

[PeterDonis]

The way the momentum-energy content of the matter-occupied region of spacetime affects the geometry of the vacuum region surrounding it is via the coupling between regions as expressed in the extrinsic curvature K on the spatial hypersurface boundary.

This is one way of viewing the connection, but not the only one. A drawback of viewing it this way is that the extrinsic curvature you describe depends on how you slice up the spacetime into spacelike slices. In the simple examples you discuss, there is a "preferred" slicing given, roughly speaking, by the "rest frame" of the central body in the asymptotically flat vacuum region. But there won't always be any way to pick out a slicing from any symmetries in the problem.

Also, if we're talking about something like galaxy rotation curves, what we're really interested in is how the stress-energy in the interior region affects the geometry in the interior region. The rotation curves we measure for galaxies are not measurements of objects outside the galaxy orbiting it; they are measurements of objects inside the galaxy, responding to the local spacetime geometry in the galaxy's interior. The geometry in the exterior vacuum region only comes into play to the extent it affects the trajectories of the light rays we see coming from the galaxy, and that effect is going to be small, and is not the kind of effect you're looking for in any case.

I don't have hydrodynamic support and I don't want radially expanding or collapsing shells.

Yes, that's indeed a problem, but it's a problem in the interior region; you need a stationary blob of matter that is not supported by hydrostatic equilibrium. That doesn't necessarily require the exterior region to be Kerr; in principle the total angular momentum of the whole blob could be zero, with various individual pieces of matter in the blob orbiting in different planes so their individual orbital angular momenta end up cancelling. (Or, more realistically, the total overall angular momentum could be very small compared to other parameters, so it could be ignored or approximated by small corrections to the zero total angular momentum case.) But in the interior, of course, each individual piece of matter has to be in a geodesic orbit about the overall center of mass, since there's no other way for the system to be stationary.

 

[RUTA]

I'm referring to the extrinsic curvature because that's the method I used to join solutions (metric on surface and K on surface are equal). It's obvious how that will map to empirical situations, so the invariance is not an issue.

 

[RUTA]

The mass inferred from galactic orbital kinematics (orbital mass) needs to grow with orbital radius faster than the luminous mass of the matter inside the orbital radius (proper mass). That's why I'm thinking about "shells" or "rings" of adjoined solutions. Since we know such adjoined solutions allow for larger orbital mass than proper mass, my intuition tells me that the "extra mass" is encoded in K. So, I'm simply building up those differences in the K's at the boundaries.

 

[RUTA]

I'm not taking about the effect of geometry on the light rays.

 

[PeterDonis]

we know such adjoined solutions allow for larger orbital mass than proper mass

Isn't the difference the opposite? The "orbital mass" is what you are calling "dynamic mass" in the paper, and it is smaller than the "proper mass".

 

[PeterDonis]

I'm not taking about the effect of geometry on the light rays.

Ok, good. I didn't think so, but thanks for confirming.

 

[RUTA]

Isn't the difference the opposite? The "orbital mass" is what you are calling "dynamic mass" in the paper, and it is smaller than the "proper mass".

The orbital mass obtained from galactic rotation curve data is larger than the locally-determined proper mass (obtained from mass-luminosity ratios for example). That's the "missing mass" problem.

 

[PeterDonis]

The orbital mass obtained from galactic rotation curve data is larger than the locally-determined proper mass (obtained from mass-luminosity ratios for example).

I understand what the actual data says. But the statement of yours that I quoted in post #51 does not seem correct as a description of the effect that is present in the model. In the model, "dynamic mass" is smaller than "proper mass", not larger. So if the "dynamic mass" in the model is supposed to correspond to the orbital mass obtained from rotation curve data, and the "proper mass" in the model is supposed to correspond to the mass obtained from luminosity data, then the model is obviously wrong, since the model says "dynamic mass" should be smaller than "proper mass" but the actual data says "dynamic mass" is larger than "proper mass".

So either the model is wrong or I've misunderstood how the "dynamic mass" and "proper mass" in the model are supposed to correspond to the "orbital mass" (from rotation curves) and the "proper mass" (from luminosity) in the data.

 

[RUTA]

I understand what the actual data says. But the statement of yours that I quoted in post #51 does not seem correct as a description of the effect that is present in the model. In the model, "dynamic mass" is smaller than "proper mass", not larger. So if the "dynamic mass" in the model is supposed to correspond to the orbital mass obtained from rotation curve data, and the "proper mass" in the model is supposed to correspond to the mass obtained from luminosity data, then the model is obviously wrong, since the model says "dynamic mass" should be smaller than "proper mass" but the actual data says "dynamic mass" is larger than "proper mass".

So either the model is wrong or I've misunderstood how the "dynamic mass" and "proper mass" in the model are supposed to correspond to the "orbital mass" (from rotation curves) and the "proper mass" (from luminosity) in the data.

I went back and looked at the paper and you're right, we flipped the terms there from what I said above. I was using the term "dynamical mass" as in astronomy where it corresponds to "orbital mass" (the larger mass). In the paper, the term "dynamical mass" corresponds to what we were going to take as the mass obtained from the mass-luminosity relationship, i.e., the "local" value, since that's how one ultimately obtains the ML relationship. Thus, the terms are flipped. See on p. 5 starting with "Suppose that the Schwarzschild vacuum surrounding the FLRW dust ball in our example above is itself surrounded ... ."

 

[PeterDonis]

the terms are flipped. See on p. 5

Yes, I see that part, but I don't think it's correct. On p. 2, the relationship between proper mass and dynamic mass is given by:

$$
dM_p = \left( 1 - \frac{2 G M}{c^2 r} \right)^{- 1/2} dM
$$

where $M_p$ is proper mass and $M$ is dynamic mass. This formula clearly says that proper mass is locally measured and dynamic mass is externally measured, and the text accompanying the formula agrees with that.

However, on p. 5, in the text you refer to, "proper mass" $M_p$ is now claimed to be "globally determined" and to be the mass that would be measured by an observer in the surrounding FRW region. That is inconsistent with the formula and text on p.2, and also with the standard GR treatment of the spacetime geometry the paper is describing.

In the text on p. 5, you are describing a collapsing FRW region, which I'll call the "interior region" (which, to properly model something like a galaxy, should really be a stationary region containing matter, as I have commented before, but making that change would not affect what I am about to say), surrounded by a Schwarzschild vacuum region, surrounded by an expanding FRW region, which I'll call the "exterior universe". The interior region and Schwarzschild vacuum region together I will call the "bubble".

In the Schwarzschild vacuum region, the mass of the interior region, as measured by orbital dynamics of objects in the Schwarzschild vacuum region, is $M$. The text on p. 5 agrees with that.

However, the mass of the interior region as measured by an observer in the exterior universe, will not be $M_p$. An observer in the exterior universe cannot even measure the mass of the interior region directly, using orbital dynamics, because any such orbit will be affected by the stress-energy in the exterior universe that is closer to the bubble than the orbit itself. And if we imagine correcting such a measurement to subtract out the mass in the exterior universe that is affecting the orbit, the remainder will be $M$, not $M_p$.

The simplest way to see this is to observe that the function $m(r)$, which gives the "mass inside radius $r$" ("mass" meaning the mass measured by orbital dynamics) as a function of the areal radius $r$ centered on the bubble, must be continuous, and its value in the Schwarzschild vacuum region is $M$. Call the areal radius of the exterior boundary of the bubble $R_0$. Then we have $m(R_0) = M$. Now consider $m(R_0 + dr)$, the value of $m(r)$ just a little way into the exterior universe. This value, by continuity, must be $M + dM$ for $dM$ infinitesimal. But the paper's claim would require it to be $M_p + dM$, where $M_p - M$ is not infinitesimal. So the paper's claim is inconsistent with continuity of $m(r)$.

 

[RUTA]

Yes, I see that part, but I don't think it's correct. On p. 2, the relationship between proper mass and dynamic mass is given by:

$$
dM_p = \left( 1 - \frac{2 G M}{c^2 r} \right)^{- 1/2} dM
$$

where $M_p$ is proper mass and $M$ is dynamic mass. This formula clearly says that proper mass is locally measured and dynamic mass is externally measured, and the text accompanying the formula agrees with that.

However, on p. 5, in the text you refer to, "proper mass" $M_p$ is now claimed to be "globally determined" and to be the mass that would be measured by an observer in the surrounding FRW region. That is inconsistent with the formula and text on p.2, and also with the standard GR treatment of the spacetime geometry the paper is describing.

In the text on p. 5, you are describing a collapsing FRW region, which I'll call the "interior region" (which, to properly model something like a galaxy, should really be a stationary region containing matter, as I have commented before, but making that change would not affect what I am about to say), surrounded by a Schwarzschild vacuum region, surrounded by an expanding FRW region, which I'll call the "exterior universe". The interior region and Schwarzschild vacuum region together I will call the "bubble".

In the Schwarzschild vacuum region, the mass of the interior region, as measured by orbital dynamics of objects in the Schwarzschild vacuum region, is $M$. The text on p. 5 agrees with that.

However, the mass of the interior region as measured by an observer in the exterior universe, will not be $M_p$. An observer in the exterior universe cannot even measure the mass of the interior region directly, using orbital dynamics, because any such orbit will be affected by the stress-energy in the exterior universe that is closer to the bubble than the orbit itself. And if we imagine correcting such a measurement to subtract out the mass in the exterior universe that is affecting the orbit, the remainder will be $M$, not $M_p$.

The simplest way to see this is to observe that the function $m(r)$, which gives the "mass inside radius $r$" ("mass" meaning the mass measured by orbital dynamics) as a function of the areal radius $r$ centered on the bubble, must be continuous, and its value in the Schwarzschild vacuum region is $M$. Call the areal radius of the exterior boundary of the bubble $R_0$. Then we have $m(R_0) = M$. Now consider $m(R_0 + dr)$, the value of $m(r)$ just a little way into the exterior universe. This value, by continuity, must be $M + dM$ for $dM$ infinitesimal. But the paper's claim would require it to be $M_p + dM$, where $M_p - M$ is not infinitesimal. So the paper's claim is inconsistent with continuity of $m(r)$.

Sorry for the delay, I'm working on another paper now, let me get back to you with my thinking on this :-)

 

[Jimster41]

Correct.

In order that the DM fits are compelling, we would need to derive theoretical predictions for the fitting factors currently found empirically (for galactic rotation curves, galactic cluster mass profiles, CMB anisotropies) using contributions from those boundary terms. Again, that's just a simplification, but no one is ever going to solve Einstein's equations for a real galaxy. What we need to do is at least motivate the fitting factors via other measurements (luminosity, temperature, etc.). Then check the theoretical (approximated) predictions for the fitting factors against those obtained empirically. The work done to date was simply to find out whether or not the inverse square law functional form is reasonable (the answer there is clearly affirmative), so we know what we're looking for in the GR formalism. Have you done the fits for these data using MOND, various modified gravity theories, and the different DM models? If so, you'll see that our result is on par with all of those (I did all those and showed the comparisons in our papers). Anyway, finding theoretical predictions for the fitting factors should be possible, but I've been working on other questions in foundations that I find more interesting :-)

What I find more interesting than finishing the "no-DM-GR-is-correct model" is showing how the whole of physics is coherent, contrary to popular belief. And, I found a big piece of that by answering Bub's question, "Why the Tsirelson bound?" So, I've been busy these past two years working on the consequences of that answer.

So this T bound says there is some cutoff that makes the classical world have zero QM (neutral monistic) magic (no long-distance-large-object/ensemble-non-local ...ness), is that roughly right?

Is it related to “decoherence” which I sort of interpret as the Gaussian noise canceling effect of Lots of long distance large object/ensemble non-localness - which seems plausible but statistical and therefore unsatisfying (FYI that was a joke)... or is it somehow a clean unavoidable deduction?

And if it seemingly analytic and clean could that be due to the fact all Alice and Bob cases are toys (I.e they pretend there are these bounds on the lab to begin with)? Alternatively could it be that there are something more like Tsirelson “gaps” or troughs (waves) recurrence etc?

I know that’s a lot of question, so, just say I’m looking forward to learning about that one. Also, to me it bears on the discussion you and Peter were having about observes “inside the mass” vs “orbiting the mass”

 

[RUTA]

Here is the paper I was working on "Einstein's missed opportunity to rid us of 'spooky actions at a distance'". I'm posting the link here since it's relevant to this Insight. Hopefully I can get back to the pending questions above tomorrow.

 

[RUTA]

So this T bound says there is some cutoff that makes the classical world have zero QM (neutral monistic) magic (no long-distance-large-object/ensemble-non-local ...ness), is that roughly right?

Is it related to “decoherence” which I sort of interpret as the Gaussian noise canceling effect of Lots of long distance large object/ensemble non-localness - which seems plausible but statistical and therefore unsatisfying (FYI that was a joke)... or is it somehow a clean unavoidable deduction?

And if it seemingly analytic and clean could that be due to the fact all Alice and Bob cases are toys (I.e they pretend there are these bounds on the lab to begin with)? Alternatively could it be that there are something more like Tsirelson “gaps” or troughs (waves) recurrence etc?

I know that’s a lot of question, so, just say I’m looking forward to learning about that one. Also, to me it bears on the discussion you and Peter were having about observes “inside the mass” vs “orbiting the mass”

The Tsirelson bound is the most QM can violate the Bell inequality known as the CHSH inequality. Classical physics says the CHSH quantity must reside between $\pm 2$, but the Bell states give $\pm 2 \sqrt{2}$ (the Tsirelson bound). Superquantum correlations respect no-superluminal-signaling and give a CHSH quantity of 4. So, quantum information theorists want to know "Why the Tsirelson bound?" That is, why doesn't Nature produce superquantum correlations? Our answer is "conservation per NPRF." Of classical, QM, and superquantum, only QM satisfies this constraint.

 

[RUTA]

Yes, I see that part, but I don't think it's correct. On p. 2, the relationship between proper mass and dynamic mass is given by:

$$
dM_p = \left( 1 - \frac{2 G M}{c^2 r} \right)^{- 1/2} dM
$$

where $M_p$ is proper mass and $M$ is dynamic mass. This formula clearly says that proper mass is locally measured and dynamic mass is externally measured, and the text accompanying the formula agrees with that.

However, on p. 5, in the text you refer to, "proper mass" $M_p$ is now claimed to be "globally determined" and to be the mass that would be measured by an observer in the surrounding FRW region. That is inconsistent with the formula and text on p.2, and also with the standard GR treatment of the spacetime geometry the paper is describing.

In the text on p. 5, you are describing a collapsing FRW region, which I'll call the "interior region" (which, to properly model something like a galaxy, should really be a stationary region containing matter, as I have commented before, but making that change would not affect what I am about to say), surrounded by a Schwarzschild vacuum region, surrounded by an expanding FRW region, which I'll call the "exterior universe". The interior region and Schwarzschild vacuum region together I will call the "bubble".

In the Schwarzschild vacuum region, the mass of the interior region, as measured by orbital dynamics of objects in the Schwarzschild vacuum region, is $M$. The text on p. 5 agrees with that.

However, the mass of the interior region as measured by an observer in the exterior universe, will not be $M_p$. An observer in the exterior universe cannot even measure the mass of the interior region directly, using orbital dynamics, because any such orbit will be affected by the stress-energy in the exterior universe that is closer to the bubble than the orbit itself. And if we imagine correcting such a measurement to subtract out the mass in the exterior universe that is affecting the orbit, the remainder will be $M$, not $M_p$.

The simplest way to see this is to observe that the function $m(r)$, which gives the "mass inside radius $r$" ("mass" meaning the mass measured by orbital dynamics) as a function of the areal radius $r$ centered on the bubble, must be continuous, and its value in the Schwarzschild vacuum region is $M$. Call the areal radius of the exterior boundary of the bubble $R_0$. Then we have $m(R_0) = M$. Now consider $m(R_0 + dr)$, the value of $m(r)$ just a little way into the exterior universe. This value, by continuity, must be $M + dM$ for $dM$ infinitesimal. But the paper's claim would require it to be $M_p + dM$, where $M_p - M$ is not infinitesimal. So the paper's claim is inconsistent with continuity of $m(r)$.

Yes, there is a formal inconsistency exactly as you point out. The reason for that is we have discrete objects (stars) separated by light years modeled by a continuum for data collection and curve fitting. I'm thinking of the discrete objects for the use of varying boundaries for varying mass values while modeling the effect in continuum fashion for comparison with the data (which has to be collected that way obviously).

On another note, the mass we obtain from atomic/molecular spectra (ultimately responsible for the mass-luminosity relationship and mass of orbiting gas) is the "locally measured interior mass," since the spectra depend on the mass of the atoms/molecules as would be obtained in a lab on Earth.

One more note and I have to run. Note that the missing mass seems large (as large as a factor of 10 increase), but in terms of spatial curvature on galactic scales, it's tiny. It's in the paper, but I think the spatial curvature for galactic mass densities is on the order of $10^{-45}m^{-2}$. So, a change by a factor of 10 one way or another isn't very big in terms of GR.

 

[PeterDonis]

there is a formal inconsistency exactly as you point out

I don't think it's a formal inconsistency; I think it's a physical error.

The reason for that is we have discrete objects (stars) separated by light years modeled by a continuum

I don't see how that helps any, but perhaps I'm misunderstanding your argument. Let me try to frame a question that might help to elucidate what your argument is.

We on Earth observe some distant galaxy, and we see a discrepancy between two methods of estimating that galaxy's mass:

Method #1: Measure the rotation curves and use that to estimate the mass from the appropriate dynamical equations. This gives us a mass which I'll call $M_R$.

Method #2: Measure the aggregate luminosity and use that to estimate the mass using the appropriate relationships between mass and luminosity for stars. This gives us a mass which I'll call $M_L$.

The discrepancy is that we find $M_R > M_L$ by some significant factor.

Now the question, in two parts:

(1) Which of the two observations above, $M_R$ or $M_L$, do you think is affected by whatever source of error your paper is describing, and which your alternative method of analysis in the paper claims to fix?

(2) How does your alternative method of analysis fix the error? That is: if your answer to #1 is that $M_R$ as estimated by standard methods is larger than it should be, how does your method make $M_R$ smaller so it matches $M_L$? Or, if your answer to #1 is that $M_L$ as estimated by standard methods is smaller than it should be, how does your method make $M_L$ larger so it matches $M_R$?

 

[PeterDonis]

Now the question, in two parts:

(1) Which of the two observations above, $M_R$ or $M_L$, do you think is affected by whatever source of error your paper is describing, and which your alternative method of analysis in the paper claims to fix?

(2) How does your alternative method of analysis fix the error? That is: if your answer to #1 is that $M_R$ as estimated by standard methods is larger than it should be, how does your method make $M_R$ smaller so it matches ? Or, if your answer to #1 is that $M_L$ as estimated by standard methods is smaller than it should be, how does your method make $M_L$ larger so it matches ?

Perhaps it will also help if I give what I take to be the answers to these questions that would be given by a proponent of (A) dark matter, and (B) MOND.

(A1) $M_R$ is correct, but $M_L$ is too small because it only counts luminous matter.
(A2) There is other matter present, dark matter, which is not luminous and so can't be counted that way. When we add in the mass of dark matter, we get $M_L + M_D = M_R$, which fixes the discrepancy.

(B1) $M_L$ is correct, but $M_R$ is too large because it is estimated using the wrong dynamical equations.
(B2) MOND changes the dynamical equations so that $M_R$ is smaller and matches $M_L$, which fixes the discrepancy.

 

[ohwilleke]

FWIW, my read on the paper (and RUTA can correct me if I'm wrong) is that B1 and B2 are the answers here as well, although not so much "the wrong dynamical equations" as the wrong operationalization of the right dynamical equations, causing non-Newtonian, non-linear GR effects to be disregarded because they naively seem insignificant in any particular instance where they are considered. But the GR deviations from Newtonian approximations actually add up to something that is a big deal in a system like a galaxy, or a galaxy cluster, or the immediate post-Big Bang universe as a whole, because the insignificant effects don't cancel out.

 

[PeterDonis]

my read on the paper (and RUTA can correct me if I'm wrong) is that B1 and B2 are the answers here as well, although not so much "the wrong dynamical equations" as the wrong operationalization of the right dynamical equations

This was my initial read as well, based on the usage of the terms "proper mass" and "dynamical mass" -- basically, that the non-Newtonian effects mean that the "proper mass", which includes corrections for things like spatial curvature, is a better thing to plug into the dynamical equations than the "dynamical mass", which does not include those corrections. However, that can't be the right answer because, as I noted in an earlier post, the correction is in the wrong direction: the corrections due to things like including spatial curvature make the estimate for $M_R$ larger, not smaller.

 

[RUTA]

I don't think it's a formal inconsistency; I think it's a physical error.
I don't see how that helps any, but perhaps I'm misunderstanding your argument. Let me try to frame a question that might help to elucidate what your argument is.

We on Earth observe some distant galaxy, and we see a discrepancy between two methods of estimating that galaxy's mass:

Method #1: Measure the rotation curves and use that to estimate the mass from the appropriate dynamical equations. This gives us a mass which I'll call $M_R$.

Method #2: Measure the aggregate luminosity and use that to estimate the mass using the appropriate relationships between mass and luminosity for stars. This gives us a mass which I'll call $M_L$.

The discrepancy is that we find $M_R > M_L$ by some significant factor.

Now the question, in two parts:

(1) Which of the two observations above, $M_R$ or $M_L$, do you think is affected by whatever source of error your paper is describing, and which your alternative method of analysis in the paper claims to fix?

(2) How does your alternative method of analysis fix the error? That is: if your answer to #1 is that $M_R$ as estimated by standard methods is larger than it should be, how does your method make $M_R$ smaller so it matches $M_L$? Or, if your answer to #1 is that $M_L$ as estimated by standard methods is smaller than it should be, how does your method make $M_L$ larger so it matches $M_R$?

There is no error, they are two different measurements of mass for one and the same matter, as allowed per GR. You can map them one to the other as I showed in the paper.

 

[PeterDonis]

There is no error

What I am calling the "error" there is the discrepancy between $M_R$ and $M_L$ when using standard methods to estimate both. So there is an error. I am trying to understand how your proposed method resolves it. See below.

they are two different measurements of mass for one and the same matter, as allowed per GR. You can map them one to the other as I showed in the paper.

This doesn't answer my question, because I can't make what you're describing here give an answer that resolves the discrepancy. I don't understand how to relate what you are saying in the paper to what I am calling $M_R$ and $M_L$, or how what you are describing in the paper corrects the discrepancy between the two. The only sense I can make of "two different measurements of mass for one and the same matter" gives an answer that makes the discrepancy worse, not better, for the reason I gave in post #65 (it doesn't change $M_L$ and it makes $M_R$ larger, not smaller). So I'm confused. I am hoping that if you give explicit answers to the two questions I gave, it will help to resolve, or at least reduce, my confusion.

 

[PeterDonis]

The only sense I can make of "two different measurements of mass for one and the same matter" gives an answer that makes the discrepancy worse, not better, for the reason I gave in post #65 (it doesn't change $M_L$ and it makes $M_R$ larger, not smaller).

Actually, on thinking this over some more, I think I have misstated this a bit. Including non-Newtonian effects of the kind implied by "two different measurements of mass for one and the same matter" (which means correcting for things like spatial curvature) should not change either $M_L$ or $M_R$. What it should change is our calculation of how much proper spatial volume the mass occupies, i.e., it should change our estimate of the average density (more precisely, proper density, the density measured by an observer locally inside the galaxy) of the matter in a galaxy, as compared to a Newtonian calculation (our estimate of average proper density should be reduced--more spatial volume for the same mass). But it should not change the mass we infer from rotation curves or luminosity at all, because those estimates are independent of our estimate of the average density.

 

[PeterDonis]

Including non-Newtonian effects of the kind implied by "two different measurements of mass for one and the same matter" (which means correcting for things like spatial curvature) should not change either $M_L$ or $M_R$.

On further thought, this way of putting it, while it is correct, might not really get at the issue I'm trying to describe. So let me try it another way.

Suppose we know $M_R$ for some galaxy but we don't know $M_L$ (say we can't see the galaxy directly because it's obscured by a dust cloud, but we've been sent rotation curve data for the galaxy by some alien civilization that isn't behind the dust cloud). What value for $M_L$ would we predict from our knowledge of $M_R$? Or, to put it another way that makes the logic clearer, what value for total luminosity for the galaxy would we predict from our knowledge of $M_R$, if we assume that all the matter in the galaxy is luminous (stars) and has similar properties to stars in our own galaxy?

The Newtonian prediction is simple: we expect $M_L = M_R$, so we just apply some expected distribution of stellar luminosities, plug those into an integral for total luminosity, and use known mass-luminosity relationships to apply the appropriate weighting factors in the integral so that the total mass, given our assumed distribution of stellar luminosities (which we can translate into an assumed distribution of stellar masses) comes out to $M_L$. This kind of calculation, since it gives an answer for total luminosity that is significantly larger than the total luminosity we actually observe, is the basic argument used by proponents of dark matter: there must be non-luminous matter present to make up the total mass that is needed to account for the observed rotation curves.

Since in actual fact we observe $M_L < M_R$ by some significant factor, if we want to avoid dark matter (and if we also want to avoid modifying our theory of gravity, which is what the paper under discussion wants to do--it wants to find a solution within GR), we need to find some non-Newtonian effect that would modify the above prediction procedure. The first obvious non-Newtonian effect to consider is gravitational binding energy: in Newtonian gravity, binding energy doesn't affect mass, but in GR, it does. However, this effect is in the wrong direction: it leads us to expect $M_L > M_R$ in the above prediction procedure (i.e., it leads us to expect a larger total luminosity than what we would infer by assuming $M_L = M_R$ in the above procedure), because gravitational binding energy is negative; the mass measured "from the outside" by something like rotation curves is smaller than the mass measured "from the inside", by an observer locally that is next to some particular star, and that locally measured mass is what is correlated with the luminosity in our known mass-luminosity relationships. This kind of effect is what I was thinking of in post #65, but I incorrectly mixed it up with space curvature.

A second non-Newtonian effect to consider is space curvature; but as I noted in my previous post (quoted above), that doesn't affect either $M_R$ or $M_L$ in the above prediction procedure (i.e., it doesn't affect the total luminosity we would infer from rotation curve data). It just affects the proper volume we assign to the galaxy, which means it affects the average density we expect a local observer, inside the galaxy, to measure.

So neither of those non-Newtonian effects can account for our observation that $M_L < M_R$. And I am not aware of any other non-Newtonian effect that would be significant in this scenario, nor does the paper appear to me to suggest one; it only suggests one of the above two, neither of which work.

 

[RUTA]

Sorry for the confusion, Peter, let me try one more time to clarify what we did

When you adjoin the FRW and Schwarzschild solutions, the comparative mass can be equal, larger, or smaller, and the matter can be surrounded by vacuum or vice-versa, as I explain in the corresponding AJP paper. That fact alone is used heuristically as follows.

$M_R$ is inferred from orbital speeds (or other data in the cases of mass profiles of X-ray clusters and anisotropies in the angular power spectrum of the cosmic microwave background) by assuming Newtonian gravity supplies the centripetal acceleration for UCM of the orbiting stars and gas. When the orbital speeds are measured (via redshifts) we find $M_R$ is much larger than $M_L$. So, we are considering a correction to Newtonian gravity for a situation that otherwise doesn't seem to warrant it (very small spatial curvature). The correction we proposed is simply to replace $M_L$ in the Newtonian acceleration with a corrected value, since the mass of the matter responsible for the acceleration is being measured in two different ways and GR allows for simultaneously differing mass values for one and the same matter. We found a functional form for correcting $M_L$ that rivals or beats all competitors across three different astronomical matter distributions -- galactic, galactic cluster, and cosmological. That is interesting because no other approach works as well across all three distributions. The rivals we compared are:

1. Two different DM distribution models (Burkett and NFW). The functional forms for these DM distributions are not based on any knowledge of the physics for these hypothetical and unlikely (as shown by Carroll) particles.
2. Correction to Newtonian physics (MOND). The ad hoc change of Newtonian acceleration at large scales. There is a relativistic counterpart for this now, but it's ugly and does not satisfy local conservation (not divergence-free).
3. Two different corrections to GR (MSTG and STVG). These are also otherwise unmotivated and do not satisfy local conservation.

Again, none of the rivals match our fits using our simple functional form for the correction of $M_L$ across all three matter distributions. Keep in mind that we're not talking about a new phenomenon here. The missing mass phenomenon was introduced in the 1930's (e.g., Zwicky, F: On the masses of nebulae and clusters of nebulae. The Astrophysical Journal 86, 217-246 (1937)). So, after 80+ years these are still our best guesses.

Conclusion, maybe it's reasonable to consider the idea from GR that matter can simultaneously possesses different values of mass based on spatially different measurement contexts. We already know it's true for different temporal contexts, e.g., free neutron mass greater than bound neutron mass. So, is this such a stretch?

That's the best I can do, Peter. Hope it suffices.