Dark Matter Real: Studies Confirm, Modifying Gravity Can't Work

[PeterDonis]

I think it's pefectly reasonable to use the determination of whether this galaxy has dark matter or not to inform one's opinion on what the correct theory of gravity is.

But how can you make that determination without assuming some theory of gravity? You have to measure things like the velocity dispersion, brightness, rotation curve, etc. and compare them with a theoretical prediction in order to determine whether the matter you can see can account for the observed properties, which is what you need to do to determine if there is dark matter present.

 

[Ken G]

One issue is whether X or Y piece of evidence (e.g. DF2 or the Bullet cluster) proves that dark matter phenomena is the only possible reality and that a modified gravity explanation isn't possible, attacking the entire modified gravity paradigm. Assertions like that can be disproven by a single example of a modified gravity explanation of the same evidence.

What I can agree with is that no single observation should ever be used to say "this proves theory X is the only possibility." Indeed, no single astronomical observation (as opposed to, say, a single well-controlled and reproducible experiment) should ever be taken without considering the larger context of the full body of astronomical data, because no astronomy observations are well controlled experiments. So I agree we should not say "DF2 proves dark matter is real." Indeed, I find it odd that DF2 does not have a high M/L due to low star-formation efficiency, even if it has a low M due to a lack of dark matter.

So when we look at DF2, we should only say "how does this observation inform what we have already seen everywhere else", never "what does this observation all by itself tell us." But as we look at the body of observations, we should never juggle different versions of our theories as we attempt to explain one or another, we should always select one single theory, and see how well it does for all the observations. We may not require perfect agreement in all cases, but we must also never tolerate a complete failure in any. If we get a complete failure of a single theory in regard to some observation, we must either be able to doubt the observation was interpreted correctly, or we must look at modifying the theory. We don't have to abandon the theory, but it is of no value to swap in other theories at that point. Calling these things "MOND" seems to suggest they are all the same theory, but of course that is not true at all, they are different theories that postulate new force effects instead of new particles. Similarly, different dark matter particles are also different dark matter theories.

So when we ask what DF2 tells us about dark matter vs. MOND, we must ask what it tells us along with everything else we know, and we must look at particular dark matter theories vs. particular MOND theories. If you need one MOND in one context and a different one in a different context, then what is missing is the demonstration that the combination of both would fit both contexts. Until that argument can be made, the theory is simply not working. Similarly, if dark matter theorists are using combinations of warm and dark matter, they are producing a single theory that includes both, not one theory for one set of situations and another for another. IF warm dark matter succeeds in one context, it is of no value if it messes up the success of cold dark matter in some other context.

So keeping the focus on DF2 in the context with all else we know about M/L in galaxies, we find that DF2 has the M/L we associate with Milky Way baryons in any situation where Newtonian gravity is used to characterize M. We could try to explain that by saying that DF2 has only baryons, and those baryons have the same star-formation efficiency as the Milky Way. That doesn't seem entirely satisfactory to me, as I'd expect reduced star-formation efficiency, but I could be wrong. Or, we could adopt a MOND that has an external field effect, and say the external field effect lowers the Newtonian-equivalent M (i.e., not the actual M, the M inferred from Newtonian gravity). But for that to be satisfactory to me, that external field effect must not mess up other things, in particular cosmology (as there the external fields pervasive over the history of the universe have spanned a very wide scale of strengths and should give a clear signal of any such MOND effect).

So yes, taken in a vacuum, DF2 might actually challenge dark matter theories more than it challenges some particular MOND theories that include an external field effect. But the real issue is, what happens when we take everything else we've seen, including cosmology, and then add DF2 into the mix, and ask, what theories still survive with only small modifications? If we think it's dark matter, our challenge is to answer, why is the baryonic M/L the same in DF2 as in the Milky Way? If we think it's MOND, our challenge is to answer, what would the external field effect that we need for DF2 do to the history of the expansion of the universe?

 

[sysprog]

If observations falsify the theory it means it is indeed a good theory, just not the one that describes reality.

Hmm -- maybe I should take a whiff of dephlogisticated air, and ponder that.

 

[RobertArvanitis]

Two recent studies have found galaxies with little or no apparent dark matter, indicating modifying gravity can't work.
https://iopscience.iop.org/article/10.3847/2041-8213/ab0e8chttps://iopscience.iop.org/article/10.3847/2041-8213/ab0d92/meta

Your topic line is appropriate but the phrase "dark matter is real" seems like click-bait.
Humans operate at the meter level. We can hypothesize from 10^-30 to 10^+30 scales, but it is only by inference.
All our models, from Aristotle to the Standard, don't really fix things, but simple make new and more subtle errors.

 

[RUTA]

These new results are perfectly compatible with my idea which argues we don't need dark matter (DM):

The Missing Mass Problem as a Manifestation of GR Contextuality

Abstract: In Newtonian gravity, mass is an intrinsic property of matter while in general relativity (GR), mass is a contextual property of matter, i.e., matter can simultaneously possesses two different values of mass when it is responsible for two different spatiotemporal geometries. Herein, we explore the possibility that the astrophysical missing mass attributed to non-baryonic dark matter (DM) actually obtains because we have been assuming the Newtonian view of mass rather than the GR view. Since an exact GR solution for realistic astrophysical situations is not feasible, we explore GR-motivated ansatzes relating proper mass and dynamic mass for one and the same baryonic matter, as justified by GR contextuality. We consider four GR alternatives and find that the GR ansatz motivated by metric perturbation theory works well in fitting galactic rotation curves (THINGS data), the mass profiles of X-ray clusters (ROSAT and ASCA data) and the angular power spectrum of the cosmic microwave background (CMB, Planck 2015 data) without DM. We compare our galactic rotation curve fits to modified Newtonian dynamics (MOND), Burkett halo DM and Navarro-Frenk-White (NFW) halo DM. We compare our X-ray cluster mass profile fits to metric skew-tensor gravity (MSTG) and core-modified NFW DM. We compare our CMB angular power spectrum fit to scalar-tensor-vector gravity (STVG) and ΛCDM. Overall, we find our fits to be comparable to those of MOND, MSTG, STVG, ΛCDM, Burkett, and NFW. We present and discuss correlations and trends for the best fit values of our fitting parameters. For the most part, the correlations are consistent with well-established results at all scales, which is perhaps surprising given the simple functional form of the GR ansatz.

This is a longer version of the Gravity Research Foundation essay that won Honorable Mention and was published in IJMPD last year. The bottom line is that we can fit DM phenomena without DM and without modifying GR. There is another option, i.e., composite/adjoined GR solutions, consistent with the complex (real world) matter distributions we are dealing with. Essentially, the extrinsic curvature at the interface between the two solutions allows the matter to have different values of mass in the two different geometries. In GR, mass is a geometric consequence of matter, so when you have two different geometries associated with the same matter, you can have two values of mass for the same matter.

 

[PeterDonis]

matter can simultaneously possesses two different values of mass when it is responsible for two different spatiotemporal geometries

How is this possible in GR? I'm not aware of any solution of the Einstein Field Equation that has this property.

 

[RUTA]

How is this possible in GR? I'm not aware of any solution of the Einstein Field Equation that has this property.

Here is an Am. J. Phys. article I published explaining the computational details: http://users.etown.edu/s/STUCKEYM/AJP1994.pdf

There is a graph in the arXiv paper in post #40 showing the possible disparity.

If you've derived the Schwarzschild solution, for example, you know just what I'm talking about when I say mass is a geometric property in GR.

 

[PeterDonis]

If you've derived the Schwarzschild solution, for example, you know just what I'm talking about when I say mass is a geometric property in GR.

Yes, I know what that means. I'm just very confused by the phrase "two different spatiotemporal geometries". At any given event in a spacetime, the geometry is one thing; it can't be two different things.

Skimming through the papers, it appears that what you actually mean is that you have a spacetime with two different regions, one containing matter and a vacuum region surrounding it, and those regions have two different geometries. Of course this is common in GR solutions that are used for real-world problems. But that just pushes the confusion back to the phrase "two different values of mass".

In the vacuum region, there is only one mass, and it's a constant. And since we, observing a distant object, are in the vacuum region with respect to that object, we observe one mass. It is true that if we were somewhere inside the object, i.e., in the matter region, we could observe a different mass, because mass is not a constant there: it depends on position. But I don't see what relevance that has to the mass we, far distant in the vacuum region, observe.

 

[Buckethead]

I cringe a little when it is assumed that MOND and DM are the only hypotheses out there to explain the phenomenon which give rise to such hypotheses. This leads to comments such as "This shows that MOND can't work so DM must be the correct hypothesis" But there are other hypotheses. For example this one by Kohkichi Konno et .al:

https://arxiv.org/pdf/0807.0679.pdf
There is also the hypotheses of Emergent Gravity written by Erik Verlinde:

https://arxiv.org/abs/1611.02269
The paper by Kohkichi Konno is particularly interesting as it refers to a frame dragging effect. The part of this paper that grabed my attention was in the introduction:

"The long range feature of frame-dragging effect under the Chern-Simon gravity well explains the flat rotation curves of galaxies which is a central evidence of dark matter. "

The reason this is so fascinating to me is that it seems to imply that if there is no frame dragging, then there should be no appearance of excessive mass in a galaxy. So perhaps the absence of DM in DF4/DF2 may be related to this.

 

[RUTA]

Yes, I know what that means. I'm just very confused by the phrase "two different spatiotemporal geometries". At any given event in a spacetime, the geometry is one thing; it can't be two different things.

Skimming through the papers, it appears that what you actually mean is that you have a spacetime with two different regions, one containing matter and a vacuum region surrounding it, and those regions have two different geometries. Of course this is common in GR solutions that are used for real-world problems. But that just pushes the confusion back to the phrase "two different values of mass".

In the vacuum region, there is only one mass, and it's a constant. And since we, observing a distant object, are in the vacuum region with respect to that object, we observe one mass. It is true that if we were somewhere inside the object, i.e., in the matter region, we could observe a different mass, because mass is not a constant there: it depends on position. But I don't see what relevance that has to the mass we, far distant in the vacuum region, observe.

The mass of the matter in the FLRW ball surrounded by the Schwarzschild vacuum region is an integrated and conserved total. Are you familiar with the FLRW solution? It does not equal the mass in the Schwarzschild metric unless the FLRW spatial geometry is flat. Therefore, the two different techniques for measuring the mass of one and the same matter can yield two different, correct results in GR (contextuality). That is precisely what is going on with DM phenomena -- we obtain two different values of mass for the same matter using two different measurement techniques -- one a local technique for obtaining M/L and the other a global technique, e.g., rotation curves.

 

[PeterDonis]

The mass of the matter in the FLRW ball surrounded by the Schwarzschild vacuum region is an integrated and conserved total.

What is the integral? (I know it's in the papers you cited, but if we're going to discuss it here we should make sure we're looking at the same one and can quote it here.)

Are you familiar with the FLRW solution?

Of course.

It does not equal the mass in the Schwarzschild metric unless the FLRW spatial geometry is flat.

I have no doubt you can exhibit an integral taken over an FLRW region surrounded by Schwarzschild vacuum that gives this result. I'm not so sure what that would mean physically.

we obtain two different values of mass for the same matter using two different measurement techniques -- one a local technique for obtaining M/L and the other a global technique, e.g., rotation curves.

What "local technique" can we possibly use? We aren't in the distant galaxy. We're here. Our observations of M/L are just as "global" as our observations of rotation curves.

 

[RUTA]

What is the integral? (I know it's in the papers you cited, but if we're going to discuss it here we should make sure we're looking at the same one and can quote it here.)

Look at Eq. (12) in the AJP paper and keep in mind ρa³ is a constant.

I have no doubt you can exhibit an integral taken over an FLRW region surrounded by Schwarzschild vacuum that gives this result. I'm not so sure what that would mean physically.

It means mass is not an intrinsic property of matter as we generally suppose.

What "local technique" can we possibly use? We aren't in the distant galaxy. We're here. Our observations of M/L are just as "global" as our observations of rotation curves.

We might determine M in M/L for the Sun using the rotational parameters of the planets. That is a "local" determination of M. [Local versus global determinations for each of the three types of data fits -- galactic, cluster, cosmological -- are explained in the paper.] If you suppose that the local value of M is an intrinsic property of the matter, then of course this value of M should simply contribute with all other locally determined M values to determine the galactic rotation curve. That's what doesn't work. So, we infer the existence of unseen matter when we should simply acknowledge that what is missing is mass, not necessarily matter.

 

[PeterDonis]

Look at Eq. (12) in the AJP paper and keep in mind ρa3 is a constant.

This is just integrating the density over the volume. So the lack of "equality" in the closed and open models is simply because space in those models is not Euclidean, so the amount of volume over a given range of $\chi$ is not the same as it would be in Euclidean space. (In the flat model, it is, so naturally you get equality in the flat model.)

However, the real question is what "mass" corresponds to the "dynamic mass" $M$ measured in the vacuum Schwarzschild region outside the FLRW matter region. As far as I can tell, the answer to that is that the integrals given in Eq. (12) in the paper do. But you labeled those integrals as $M_p$, "proper mass". So it seems to me that either $M_p = M$ for all these models (proper mass = dynamic mass) or you have mislabeled the integrals.

Further, stars and galaxies are not modeled by FLRW spacetimes, since they aren't expanding or contracting, they're stationary (to a good approximation). So I don't see the relevance of any of these integrals to the question of whether there is "missing mass" in galaxies that is not visible.

We might determine M in M/L for the Sun using the rotational parameters of the planets. That is a "local" determination of M.

But we can't do this for stars in other galaxies since we can't even see planets orbiting them, much less get accurate measurements of their orbital parameters, so the fact that if we could do it, we would have to then apply a correction when we summed up all these local M values to get a total for the galaxy is irrelevant.

 

[PeterDonis]

we infer the existence of unseen matter when we should simply acknowledge that what is missing is mass, not necessarily matter.

Apart from the other questions I've been asking, it seems to me that the correction you are saying should be applied is in the wrong direction.

Basically, what I understand you to be saying is that we are totaling up "local" measurements of mass for all the visible matter in a galaxy, and then using the total to infer what the rotation curve should be. But the actual rotation curve is not the one that is inferred; it is the one we would expect if there were more mass present besides what we can see.

In your terminology, we are totaling up the proper mass of the stars, and finding that the total is smaller than the dynamic mass required to account for the rotation curve. So if there is some correction that should be applied to how we total the proper mass, its effect would need to be to make the total of proper mass larger. But it seems like the correction you are talking about would make the total of proper mass smaller, which would make the problem worse, not better.

 

[RUTA]

Apart from the other questions I've been asking, it seems to me that the correction you are saying should be applied is in the wrong direction.

If that were true, how would you explain the data fits? Read the IJMPD paper carefully.

 

[PeterDonis]

If that were true, how would you explain the data fits?

I don't know; I haven't dug into the details of how you did the data fits, because I'm still trying to build a basic understanding of the theoretical claim you're making. It seems to me that if my theoretical understanding is wrong, it should be easy for you to correct it on theoretical grounds.

 

[RUTA]

I don't know; I haven't dug into the details of how you did the data fits, because I'm still trying to build a basic understanding of the theoretical claim you're making. It seems to me that if my theoretical understanding is wrong, it should be easy for you to correct it on theoretical grounds.

Read Sections 1 and 2 of the IJMPD paper (in my arXiv link) carefully, the answers to all your questions are there in conceptual detail. The answer to this specific question then starts after Eq (7) with “Suppose that the Schwarzschild vacuum surrounding the FLRW dust ball ... .”

 

[PeterDonis]

The answer to this specific question then starts after Eq (7) with “Suppose that the Schwarzschild vacuum surrounding the FLRW dust ball ... .”

But, as I've already pointed out, galaxies are not modeled by FLRW spacetimes, because they're not expanding or contracting, they're stationary. An idealized model for a galaxy, if you want to keep the spherical symmetry and the "dust" aspect (zero pressure, which seems reasonable for a galaxy since the individual stars have no appreciable interaction other than gravity), would be a rotating dust. I don't know of any exact solution of this type (the van Stockum dust has density increasing from the rotation axis, which is the opposite of what happens in a galaxy), but I would expect that one could be approximated numerically.

Leaving that aside, your model, schematically, is "galaxy surrounded by Schwarzschild vacuum region surrounded by expanding FLRW dust region", and we on Earth are supposed to be in the expanding FLRW dust region. But we are talking about distance scales (roughly 20 Mpc) which are well below the scale at which the universe is well modeled by FLRW dust. It seems to me that a realistic model for observing a galaxy at that distance would have us on Earth in the Schwarzschild vacuum region (essentially "at infinity").

But even leaving that aside, I don't understand the following claim in your paper (from p. 5, the paragraph starting with the sentence you quoted): "observers in the surrounding FLRW dust (global context) will obtain the “globally determined” proper mass $M_p$ for the collapsed dust ball while observers in the Schwarzschild vacuum (local context) will obtain the “locally determined” dynamic mass $M$ for the collapsed dust ball".

Observers in the Schwarzschild region will measure the mass $M$ that appears in the metric for that region (which is the same everywhere in the region), and they will of course interpret this physically as the mass of the galaxy that is surrounded by the Schwarzschild region. Earlier in the paper (Eq. 1 and the surrounding discussion) you showed that (for the non-flat cases) the Schwarzschild mass $M$ is not the same as the proper mass $M_p$ of the FLRW region, which here is the FLRW region inside the Schwarzschild region, i.e., the FLRW region that describes the galaxy. But now you're claiming that, somehow, observers in the FLRW region outside the Schwarzschild region will measure $M_p$ instead of $M$? That doesn't make sense. Observers in the outside FLRW region will observe the entire region to the interior of theirs (Schwarzschild plus interior FLRW) as having mass $M$ (the Schwarzschild geometry mass). What else could they possibly observe?

 

[PeterDonis]

It does not equal the mass in the Schwarzschild metric unless the FLRW spatial geometry is flat.

Just to make sure I understand how this conclusion is derived for the closed case (since that appears to be the case of interest for this particular discussion), from my understanding of the Oppenheimer-Snyder model of stellar collapse, the following would seem to be true:

(1) At the instant when the collapse starts, we have a spherically symmetric region containing matter with a boundary at some areal radius $R_0$, surrounded by Schwarzschild vacuum. At this instant, all of the matter in the matter region is at rest. The geometry of the matter region is a portion of a closed FLRW geometry at the instant of maximum expansion.

(2) At the above instant, the Schwarzschild region has a mass $M$ (a property of its geometry), and the proper mass of the matter region, obtained by integrating its (constant) density over its volume, is $M_p > M$. The reason $M_p$ is greater than $M$ is that space inside the matter region is not Euclidean; its proper volume is greater than $4 \pi R_0^3 / 3$, so we have $M_p > 4 \pi \rho R_0^3 / 3$. But the Schwarzschild mass $M$ is obtained from the integral $\int_0^{R_0} 4 \pi \rho r^2 dr$, which obviously gives $M = 4 \pi \rho R_0^3 / 3$, hence $M_p > M$.

(3) Both $M_p$ and $M$ are constant in time, so their values at the instant the collapse starts will remain their values throughout the collapse.

 

[PeterDonis]

from my understanding of the Oppenheimer-Snyder model of stellar collapse, the following would seem to be true

Assuming that what I wrote there is correct, the next obvious question is, which mass determines the rotation curve of the "galaxy" that occupies the matter region--$M$ or $M_p$?

Here is a simple argument for why it must be $M$. Suppose I have an object in a circular free-fall orbit in the Schwarzschild region just outside $R_0$. The orbital parameters are determined by $M$. Now suppose I move the object into the matter region just inside $R_0$. There should be no significant difference in all of those parameters, because the metric has to be the same on both "sides" of the boundary (more precisely, the limits as you approach the boundary from each side must be equal). But that means the circular free-fall orbit just inside $R_0$ must be essentially the same as the circular free-fall orbit just outside--which in turn means the rotation curve (which is just the velocity of a free-fall orbit) in the matter region must be determined by $M$, since if it were determined by $M_p$ there would be a discontinuous jump at the boundary.

Am I missing something?

 

[RUTA]

Assuming that what I wrote there is correct, the next obvious question is, which mass determines the rotation curve of the "galaxy" that occupies the matter region--$M$ or $M_p$?

Here is a simple argument for why it must be $M$. Suppose I have an object in a circular free-fall orbit in the Schwarzschild region just outside $R_0$. The orbital parameters are determined by $M$. Now suppose I move the object into the matter region just inside $R_0$. There should be no significant difference in all of those parameters, because the metric has to be the same on both "sides" of the boundary (more precisely, the limits as you approach the boundary from each side must be equal). But that means the circular free-fall orbit just inside $R_0$ must be essentially the same as the circular free-fall orbit just outside--which in turn means the rotation curve (which is just the velocity of a free-fall orbit) in the matter region must be determined by $M$, since if it were determined by $M_p$ there would be a discontinuous jump at the boundary.

Am I missing something?

First, we are supposing there are two different techniques for measuring the mass of the matter responsible for the geometry in the two different solutions (regions of spacetime), which is the case in astrophysics. If this wasn't the case, there would not be different values of mass to begin with, i.e., no missing mass problem. For example, we measure the radiation emitted by the galactic gas and we must infer a mass based on that data. How do we do that? From the physics we know about that type of gas as measured in the our labs. The same is true for stars using the HR diagram. Our claim is that the physics is only in apparent discord because we're assuming that the mass of the matter determined in one context is the same as its mass in any and all contexts. We know this is false already, e.g., neutron mass in the nucleus versus the neutron mass out of the nucleus. Likewise with electrons in solid state physics calculations. Here again is the beginning of that paragraph:

Suppose that the Schwarzschild vacuum surrounding the FLRW dust ball in our example above is itself surrounded by the remaining FLRW dust, i.e., the ball of FLRW dust has collapsed out of its FLRW cosmological context and is now separated from that cosmological context by the Schwarzschild vacuum. The spacetime geometry of the surrounding FLRW dust will be unaffected by the intervening Schwarzschild vacuum, so observers in the surrounding FLRW dust (global context) will obtain the “globally determined” proper mass M_p for the collapsed dust ball while observers in the Schwarzschild vacuum (local context) will obtain the “locally determined” dynamic mass M for the collapsed dust ball. Thus, stellar mass-to-luminosity ratios would be based on M and would not give the proper mass M_p required to explain galactic RC’s.

As we point out earlier, binding energy is not a big enough effect to account for the missing mass, but what about the disparity between mass in adjoined GR solutions? That brings me to my second point, i.e., our example is heuristic and we use it only to show that adjoined/complex/realistic GR solutions can indeed harbor large enough mass discrepancies to account for the missing mass. As we write at the end of that paragraph:

Of course, this is an idealization and the actual situation in a galaxy would be far more complicated since such a nested solution would have to marry up with other such nested solutions. Indeed, the surrounding FLRW dust may not even remain, having coalesced into other bodies and clouds. And, as we will see, the largest contribution to the correction of galactic dynamic mass is not from the stellar disk or bulge, but the gas. Again, given the complexity of GR, no such exact solution can be expected, so some approximation method for obtaining the proper mass from the dynamic mass must be motivated and checked for efficacy against astrophysical data.

We did that work two years ago and I had intended to develop the idea last summer, but Bub gave me a more interesting problem to work on instead. I solved his problem, wrote two papers, and now I'm busy trying to sell that (attending conferences and making videos). Someday I hope to finish this missing mass idea :-)

 

[PeterDonis]

we are supposing there are two different techniques for measuring the mass of the matter responsible for the geometry in the two different solutions (regions of spacetime), which is the case in astrophysics.

Why would an observer in the external FLRW region be able to use these techniques while an observer in the Schwarzschild region can't?

The same is true for stars using the HR diagram.

Let's take this as an example. The HR diagram let's us infer absolute luminosity from the spectrum, and then we infer mass from absolute luminosity using a known mass-luminosity relationship for the type of star. And you are saying this mass would be the "proper mass" of the star, i.e., the mass that would be observed by an observer in the same local region as the star (like us measuring the mass of our own Sun based on our observations of the orbital parameters of the planets).

So now we total up the proper masses of all the stars in the galaxy using this method, giving a result $M_p$. And you are saying this will be larger than the "dynamic mass" $M$ due to those stars, which could be measured, for example, by putting a test object in orbit around the entire galaxy and measuring its orbital parameters.

Do I have this correct? If so, it just leads me right back to a previous issue I raised: the "missing mass" problem is that the mass that we infer from rotation curves is larger than the mass $M_p$ we obtain by observing the luminous matter in the galaxy and inferring its mass by methods such as the HR diagram and mass-luminosity relationship. But by your arguments, the "dynamic mass" $M$ is smaller than the mass $M_p$. There is no "alternative mass" you have given that is larger than the mass $M_p$, and larger is what would be needed to remove the missing mass problem. So, again, it seems like any corrections available for the reasons you give would be in the wrong direction: they would make the problem worse, not better.

Our claim is that the physics is only in apparent discord because we're assuming that the mass of the matter determined in one context is the same as its mass in any and all contexts. We know this is false already, e.g., neutron mass in the nucleus versus the neutron mass out of the nucleus.

And all of these examples have a simple explanation: binding energy. But you say binding energy can't fix the problem:

As we point out earlier, binding energy is not a big enough effect to account for the missing mass

And not only that, but binding energy gives a "correction" in the wrong direction, as noted above: it makes the dynamic mass $M$ smaller than the proper mass $M_p$. But fixing the missing mass problem requires a correction that makes the mass larger, not smaller.

what about the disparity between mass in adjoined GR solutions?

The only example you give is the one that models a galaxy using an FLRW spacetime, which I don't think is a good model for the reasons I've already given. But once more, the correction you obtain even from this model is in the wrong direction, as noted above.

 

[RUTA]

Why would an observer in the external FLRW region be able to use these techniques while an observer in the Schwarzschild region can't?

The observers in the FLRW region have an entirely different spacetime geometry as a result of the entirely different mass for one and the same matter.

Let's take this as an example. The HR diagram let's us infer absolute luminosity from the spectrum, and then we infer mass from absolute luminosity using a known mass-luminosity relationship for the type of star. And you are saying this mass would be the "proper mass" of the star, i.e., the mass that would be observed by an observer in the same local region as the star (like us measuring the mass of our own Sun based on our observations of the orbital parameters of the planets).

Other way around, this is the dynamic mass M, so everything you say afterwards is a non-sequitur.

And all of these examples have a simple explanation: binding energy. But you say binding energy can't fix the problem

Right, it's too small of an effect.

And not only that, but binding energy gives a "correction" in the wrong direction, as noted above: it makes the dynamic mass $M$ smaller than the proper mass $M_p$. But fixing the missing mass problem requires a correction that makes the mass larger, not smaller.

The condensed (bound) mass (dynamic mass) is smaller than the free mass (proper mass) for the neutron example. Did you read footnote 1?

1. Typically, “dynamical mass” and “luminous mass” are the terms used with dynamical mass larger than luminous mass. Our terminology is following the GR convention.

The only example you give is the one that models a galaxy using an FLRW spacetime, which I don't think is a good model for the reasons I've already given. But once more, the correction you obtain even from this model is in the wrong direction, as noted above.

I think I see why you're getting this backwards. Here is a quote from p. 24:

The average ratio of total proper mass to total dynamic mass in the THINGS data was 4.19 ± 0.81. This is consistent with a dark matter fraction of 79% in galaxies found using microlensing ([83] and references therein).

Read the two excerpts from post #56 again carefully. Maybe you didn't read footnote 1 and notice the use of scare quotes around "proper mass" and "dynamic mass¹" there:

... the “proper mass” M_p of the matter, as measured locally in the matter, can be different than the “dynamic mass¹” M in the Schwarzschild metric ...

The terminology can be tricky because the dynamic mass of the Schwarzschild metric is larger than the proper (bound) mass in that usage (per Wald). But, it's the other way around for the FLRW dust surrounding the Schwarzschild vacuum. The problem is "proper mass" in GR means "as measured in the local frame of reference." You can see why that's going to be confusing with my use of "locally measured" for "dynamic mass." The proper mass of the FLRW dust surrounding the Schwarzschild vacuum is measured in the local frame of reference for the FLRW observers, but I'm calling it the "globally determined proper mass" in the astrophysical contexts. I've tried writing it both ways and either way always ends up confusing someone.

 

[PeterDonis]

The condensed (bound) mass (dynamic mass) is smaller than the free mass (proper mass) for the neutron example.

Yes, I understand that.

"proper mass" in GR means "as measured in the local frame of reference."

Yes, and I gave an example of that: the "proper mass" for a star that is inferred from its spectral class and luminosity is the same as the mass that an observer in the same solar system as that star would measure from orbital parameters. But if we add up all the "proper masses" for the stars in a galaxy, we get a total that is larger than the "dynamic mass" for the galaxy as a whole, that would be measured by putting an object in orbit about the entire galaxy and measuring its orbital parameters.

The terminology can be tricky

I'm not drawing inferences from your terminology. I'm drawing inferences from the physics of the examples you use. I would be just as happy to not use your terminology at all, since, as you admit, no matter how you use it it's going to confuse somebody.

The proper mass of the FLRW dust surrounding the Schwarzschild vacuum is measured in the local frame of reference for the FLRW observers

Of the FLRW dust surrounding the Schwarzschild vacuum, yes. But that's not the proper mass of the galaxy; the galaxy is inside the Schwarzschild vacuum, not outside it. There is no way observers in the FLRW region outside the Schwarzschild vacuum region can be in the "local frame of reference" of a galaxy that is inside the Schwarzschild vacuum region.

 

[PeterDonis]

The terminology can be tricky because the dynamic mass of the Schwarzschild metric is larger than the proper (bound) mass in that usage (per Wald).

I think this is backwards. You are defining the dynamic mass $M$ as the mass that appears in the Schwarzschild metric, and the proper mass $M_p$ as the mass you get when you measure locally in the matter. But this means $M_p > M$; the difference $M_p - M$ is the binding energy of the system.

What particular Chapter/section of Wald are you referring to?

Note, btw, that a similar argument applies for the case I analyzed in post #54: a contracting FLRW matter region surrounded by Schwarzschild vacuum. I showed why $M_p > M$ for this case (this analysis, as far as I know, agrees with Eq. 1 in your arxiv paper). But the matter region in this case is also a bound system: the matter in this region does not have enough energy to escape to infinity. And the difference $M_p - M$ has a straightforward physical interpretation as the binding energy: the energy that would need to be added to the matter region to allow the matter to just escape to infinity. So in this respect, I don't see any difference between the "FLRW inside Schwarzschild vacuum" case and the "stationary matter region inside Schwarzschild vacuum" case.

 

[RUTA]

Yes, and I gave an example of that: the "proper mass" for a star that is inferred from its spectral class and luminosity is the same as the mass that an observer in the same solar system as that star would measure from orbital parameters. But if we add up all the "proper masses" for the stars in a galaxy, we get a total that is larger than the "dynamic mass" for the galaxy as a whole, that would be measured by putting an object in orbit about the entire galaxy and measuring its orbital parameters.

No, it's just the opposite. As a star orbits the galactic center (call it Star A), the mass inside its galactic orbit needed for the star to maintain that orbit is the proper mass (globally determined mass). If you determine the mass of a star from objects in orbit around it or mass of a gas from laboratory experiments, that is dynamic mass (locally determined mass). When you add up the locally determined dynamic mass of the stars and gas inside the orbital radius of Star A, it is less than the proper mass. That is why some people believe there is non-baryonic dark matter in galaxies (and clusters and ... ).

Of the FLRW dust surrounding the Schwarzschild vacuum, yes. But that's not the proper mass of the galaxy; the galaxy is inside the Schwarzschild vacuum, not outside it. There is no way observers in the FLRW region outside the Schwarzschild vacuum region can be in the "local frame of reference" of a galaxy that is inside the Schwarzschild vacuum region.

Keep in mind the FLRW-Schwarzschild adjoined spacetime is only an analogy. In that analogy, the orbit of Star A is determined by the larger proper mass inside its galactic orbit, just as the geometry of the FLRW solution outside the Schwarzschild vacuum annulus is determined by the proper mass of the FLRW dust ball inside the Schwarzschild annulus. The dynamic mass of the FLRW dust ball per the Schwarzschild geometry is analogous to the locally determined dynamic mass of all the gas and stars inside the orbital radius of Star A.

 

[PeterDonis]

As a star orbits the galactic center (call it Star A), the mass inside its galactic orbit needed for the star to maintain that orbit is the proper mass (globally determined mass). If you determine the mass of a star from objects in orbit around it or mass of a gas from laboratory experiments, that is dynamic mass (locally determined mass)

Ok, at this point I simply refuse to use your terminology since it is only obfuscating the physics.

When you say "the mass inside its galactic orbit needed for the star to maintain that orbit", I assume you mean the mass that is needed to account for the rotation curve, correct? And that mass is obtained by using a Newtonian formula. I'm going to call this mass $M_{RC}$ ("RC" for "rotation curve").

We can't observe objects in orbit around individual stars in other galaxies, so any definition of "mass" that uses this is irrelevant to this discussion.

We can infer a mass from observations of a galaxy's spectrum and luminosity by using the H-R diagram and mass-luminosity relations for various types of stars, and properties of radiation from gas. I'm going to call this mass $M_{L}$ ("L" for "luminosity").

The "missing mass" problem is that, for most galaxies, $M_{RC} > M_L$, and usually by a fairly large margin.

One obvious correction that could be made is to use GR instead of Newtonian gravity. As I understand it, this would somewhat reduce $M_{RC}$, since in GR the orbital velocity due to a given mass is somewhat higher than in Newtonian gravity (how much higher depends on how compact the mass is and how close the orbit is to the center). I don't know if this is what you are getting at with the "binding energy" correction.

I still am unable to understand what other correction you are proposing, or why it should be there. Can you explain that without using any of your obfuscating terminology, and using the definitions for $M_{RC}$ and $M_L$ that I gave above?

 

[PeterDonis]

the geometry of the FLRW solution outside the Schwarzschild vacuum annulus is determined by the proper mass of the FLRW dust ball inside the Schwarzschild annulus.

Um, what? How does this work? These two regions are disconnected, and if any mass were going to affect the FLRW dust ball outside the Schwarzschild vacuum region, it would be the Schwarzschild mass $M$ of that region.

 

[ohwilleke]

Or, we could adopt a MOND that has an external field effect, and say the external field effect lowers the Newtonian-equivalent M (i.e., not the actual M, the M inferred from Newtonian gravity). But for that to be satisfactory to me, that external field effect must not mess up other things, in particular cosmology (as there the external fields pervasive over the history of the universe have spanned a very wide scale of strengths and should give a clear signal of any such MOND effect).

So yes, taken in a vacuum, DF2 might actually challenge dark matter theories more than it challenges some particular MOND theories that include an external field effect. But the real issue is, what happens when we take everything else we've seen, including cosmology, and then add DF2 into the mix, and ask, what theories still survive with only small modifications? If we think it's dark matter, our challenge is to answer, why is the baryonic M/L the same in DF2 as in the Milky Way? If we think it's MOND, our challenge is to answer, what would the external field effect that we need for DF2 do to the history of the expansion of the universe?

It is worth recalling that the external field effect means that in that field, you have Newtonian gravity (or more precisely, no MOND effects). The question at the cosmology level (which really hasn't been worked out yet, mostly for lack of attention), is how modified gravity fills the role that dark matter did in cosmology. In so far as the external field effect matters it is in how can there be places not subject to the external field effect in the early universe, so that modified gravity can substitute for dark matter, not what effect the external field effect has itself.

 

[RUTA]

Um, what? How does this work? These two regions are disconnected, and if any mass were going to affect the FLRW dust ball outside the Schwarzschild vacuum region, it would be the Schwarzschild mass $M$ of that region.

Yes, the FLRW solution behaves as if the rest of the FLRW solution is connected directly to it. You can see this from the calculation in the AJP paper. Also, you can imagine expanding the FLRW ball of dust (the proper mass remains constant) so that the Schwarzschild vacuum annulus shrinks to zero. Does this change the geometry of the exterior FLRW solution? No.

 

[RUTA]

When you say "the mass inside its galactic orbit needed for the star to maintain that orbit", I assume you mean the mass that is needed to account for the rotation curve, correct? And that mass is obtained by using a Newtonian formula. I'm going to call this mass $M_{RC}$ ("RC" for "rotation curve").

Sounds good.

We can't observe objects in orbit around individual stars in other galaxies, so any definition of "mass" that uses this is irrelevant to this discussion.

We can infer a mass from observations of a galaxy's spectrum and luminosity by using the H-R diagram and mass-luminosity relations for various types of stars, and properties of radiation from gas. I'm going to call this mass $M_{L}$ ("L" for "luminosity").

Any means used to obtain the mass of stars for the H-R diagram (to include orbital kinematics) is local (within our galaxy). We then simply infer it is the same for the same types of stars in other galaxies (actually, there's just one M/L for all stars in the THINGS data). I never said we used this method directly for stars in other galaxies. I'll keep your terminology.

The "missing mass" problem is that, for most galaxies, $M_{RC} > M_L$, and usually by a fairly large margin.

For the THINGS galactic rotation data the ratio is 4.19 +/- 0.81.

One obvious correction that could be made is to use GR instead of Newtonian gravity. As I understand it, this would somewhat reduce $M_{RC}$, since in GR the orbital velocity due to a given mass is somewhat higher than in Newtonian gravity (how much higher depends on how compact the mass is and how close the orbit is to the center). I don't know if this is what you are getting at with the "binding energy" correction.

No, the binding energy example is shown just to make the point that mass is contextual.

I still am unable to understand what other correction you are proposing, or why it should be there. Can you explain that without using any of your obfuscating terminology, and using the definitions for $M_{RC}$ and $M_L$ that I gave above?

It's very simple. The mass of the matter interior to Star A's orbit determined by "the H-R diagram and mass-luminosity relations for various types of stars, and properties of radiation from gas" (what you're calling $M_L$) does not equal the mass of that matter per Star A's orbit (what you're calling $M_{RC}$). We are proposing that the difference exists because we should be modeling the galaxy by a compound GR solution. In compound GR solutions, the mass of the matter can have significantly different values (as we showed in the FLRW-Schwarzschild example). How do you use this fact to get from $M_L$ to $M_{RC}$? There's no way to know unless you could actually produce a multiply compound GR solution that accurately models the complexity of the galactic matter distribution. That isn't going to happen. So, we argued for and tested some ansatzes. The results are displayed against several alternatives (MOND, STVG, MSTG, NFW, Burkett, and $\Lambda$CDM) and discussed in the paper.

 

[PeterDonis]

the FLRW solution behaves as if the rest of the FLRW solution is connected directly to it.

This can't be right, because the two FLRW solutions have different geometries. The inner one is a portion of a collapsing closed FLRW geometry. The outer one is an expanding FLRW geometry with a spherically symmetric region removed. In our actual universe, the expanding FLRW geometry is spatially flat, so even leaving out the difference between expanding and collapsing, its spatial slices are Euclidean, whereas the spatial slices of the inner FLRW geometry are not. But even if we look at the case where the outer expanding FLRW geometry is closed, it has a much, much larger radius of curvature than the inner collapsing FLRW geometry, so its spatial geometry is still not the same.

You can see this from the calculation in the AJP paper.

Which particular equations in the AJP paper do you think support your claim?

 

[RUTA]

This can't be right, because the two FLRW solutions have different geometries. The inner one is a portion of a collapsing closed FLRW geometry. The outer one is an expanding FLRW geometry with a spherically symmetric region removed. In our actual universe, the expanding FLRW geometry is spatially flat, so even leaving out the difference between expanding and collapsing, its spatial slices are Euclidean, whereas the spatial slices of the inner FLRW geometry are not. But even if we look at the case where the outer expanding FLRW geometry is closed, it has a much, much larger radius of curvature than the inner collapsing FLRW geometry, so its spatial geometry is still not the same.

Which particular equations in the AJP paper do you think support your claim?

The AJP paper shows that the Schwarzschild vacuum can surround the FLRW ball or be surrounded by the FLRW solution. And, the FLRW solution is time symmetric (expanding free fall or collapsing free fall are both solutions), so just put the Schwarzschild solution between the two versions of the FLRW solution and marry them up at the two boundaries as shown in the AJP paper.

 

[PeterDonis]

Any means used to obtain the mass of stars for the H-R diagram (to include orbital kinematics) is local (within our galaxy). We then simply infer it is the same for the same types of stars in other galaxies (actually, there's just one M/L for all stars in the THINGS data).

Yes, agreed.

I never said we used this method directly for stars in other galaxies.

I never said we did either. As you say, we can't; we can only infer that the relationships we can directly measure for stars in our galaxy also hold for similar stars in other galaxies. That's what I was assuming we did in order to obtain $M_L$.

We are proposing that the difference exists because we should be modeling the galaxy by a compound GR solution. In compound GR solutions, the mass of the matter can have significantly different values

Sorry, I'm refusing to use your terminology, so just saying "the mass of the matter can have significantly different values" means nothing. You need to specify how "the mass of the matter" is being measured. I specified two ways. So you need to explain to me how your "compound GR solution" gives rise to a correction to one (or both) of those ways that makes the two values, $M_{RC}$ and $M_L$, come out the same.

How do you use this fact to get from MLM_L to MRCM_{RC}? There's no way to know unless you could actually produce a multiply compound GR solution that accurately models the complexity of the galactic matter distribution. That isn't going to happen. So, we argued for and tested some ansatzes.

And I'm asking you to explain, for any or all of those ansatzes, how the ansatz gives rise to a correction to either $M_{RC}$ (the mass we infer from the rotation curve) or $M_L$ (the mass we infer from luminosity data), or both, that could make them come out the same, without using your terminology. How should the process of obtaining $M_{RC}$ from the observed rotation curve (which we assume is fixed by observation), or the process of obtaining $M_L$ from the observed luminosity data (which we assume is fixed by observation), or both, be changed? I understand that you don't have an exact solution or a numerical model that accurately captures an actual galaxy; but I'm not asking the above question with regard to an exact solution for an actual galaxy. I'm asking it for any one, or all, of your ansatzes, and you should be able to answer the question for those since you picked them precisely in order to illustrate how such corrections could arise.

 

[PeterDonis]

The AJP paper shows that the Schwarzschild vacuum can surround the FLRW ball or be surrounded by the FLRW solution. And, the FLRW solution is time symmetric (expanding free fall or collapsing free fall are both solutions), so just put the Schwarzschild solution between the two versions of the FLRW solution and marry them up at the two boundaries as shown in the AJP paper.

This doesn't prove your claim, because the two FLRW geometries are different, as I've already pointed out. Saying "the FLRW solution is time symmetric" doesn't change that, because we're not talking about a single FLRW solution and its time reverse, we're talking about two different FLRW solutions, with different spatial geometries, so they aren't time reverses of each other.