I recently came across this news item: Factoring in gravitomagnetism could do away with dark matter By disregarding general relativistic corrections to Newtonian gravity arising from mass currents, … Ludwig asserts [standard] models also miss significant modifications to [galaxy] rotational curves … because of an effect in general relativity not present in Newton’s theory of gravity — frame-dragging … Ludwig presents a new model for the rotational curves of galaxies which is in agreement with previous efforts involving general relativity. … even though the effects of gravitomagnetic fields are weak, factoring them into models alleviates the difference between theories of gravity and observed rotational curves — eliminating the need for dark matter.

FWIW, pressure is not conventionally about momentum. Say you are sitting on a chair. The chair is under pressure, but it and you are both motionless - and so have no momentum. Pressure is defined in terms of force, not momentum.

A better definition of pressure is something like the dot product of the momentum transferred through a surface with an area vector of the surface (pointing to the same side of the surface that the momentum is going to, to get the signs correct). E.g. the diagonal space components of the stress-energy tensor.

If you have a ball of stars moving isotropically and held together by gravity, and put inside that ball a magic star-reflecting mirror that reflected stars, the mirror would on average be hit the same amount from each side, and stay in place on average, and the statistical properties of the ball of stars would be pretty much the same as if it weren't there. Same as if the ball of stars were a ball of ideal gas. But, the mirror would obviously experience substantial force (i.e. pressure) from each side. This pressure doesn't go away just because the mirror isn't there.

In the case of the self-gravitating ball of stars, this star-motion pressure would obviously balance the gravitational force. However, the gravitational force isn't normally counted as pressure, in the same way that we don't say that the center of the Earth is at zero pressure even though the the gravitational force within the Earth is balanced by the repulsive forces between the atoms.

Pressure is defined as being force per unit area. You could apply the concept to gravity-only systems - if you don't mind the force being negative. The assumption of zero pressure could then refer to when attractive gravitational forces and repulsive forces (e.g. due to radiation pressure) are exactly in balance. Put it like that and it does seem like a bit of a strange assumption.

Idiosyncratic motion of objects is pressure. Just as a ball of ideal gas could be held up from collapsing to a point by the pressure of the motion of its particles, a ball of stars can be held up by the the motion of the stars, which is also pressure.

Normally, I, like you, would not refer to the galactic bulge as being held up by "pressure" - I'd prefer to talk about motion directly. However, the idiosyncratic motion of the stars that you mention have been thrown out of the galactic plane is pressure.

In the context of the set of equations quoted by Robin in the post, assuming zero pressure amounts to assuming zero idiosyncratic motion, and thus assuming zero stars "thrown out of the galactic plane".

Zero pressure doesn't sound like too bad an assumption. Of course, there is radiation pressure - which could be significant near the middle of a galaxy, but models have to make some assumptions. Anyway, gravity alone can likely create a mild nuclear bulge in galaxies - without invoking the concept of positive pressure. The mechanism would be: stars get close, interact via gravity and are thrown out of the galactic plane. I think you can even see this in simulations - such as the NIHAO galaxy simulations.

Assuming that Robin's writeup of Ludwig's argument is accurate, it seems the difference is that Lisi actually estimated the numbers directly, and that Ludwig, instead of actually estimating the numbers directly, made an indirect argument based on equilibrium conditions that relied on a false assumption.

I'm confused, and maybe you can set me straight. I apologize for the wall of text, but want to make sure my question --- which, stemming from confusion, is not very well-posed --- is clear.

Lisi's quick calculation and the original paper both seem to based on the same simplified model. The paper: ignores pressure, assumes all mass is at z=0. The paper then estimates the size of GM effects, finds them to be of the same order as GE effects. There are details there, but there's a big central assertion: in a simple model of a galaxy with a pressureless gas rotating azimuthally, etc etc, GM ~ GE.

Lisi's calculation appears to be applicable to just such a model. The complexities of the actual distribution and behavior of matter in a galaxy don't appear. "Here's the rough distribution of mass and velocity at z=0, here's the relative size of GM vs GE." And yet he finds GE >>> GM, where Ludwig finds them comparable.

I'm worried that might not be very clear, so let me try to rephrase. Shown Ludwig's model of a galaxy, I might try to do a consistency check --- a quick back-of-the-envelope calculation that GM effects are indeed important. But Lisi's note looks like exactly the right estimate to do, not for a real galaxy, but for Ludwig's simplified galaxy! So why do they come to such different conclusions?

I hope it's clear that this is a separate question from "is pressureless a good model". It's a bad model (I quite like your demonstration of this), but that doesn't resolve why two different calculations in the same bad model come to such radically different answers.

We could argue, "well, Ludwig did a more careful calculation with more bells and whistles", but then it has to be the case that some part of Lisi's calculation was very sensitive to a parameter which was only imprecisely estimated (like the distribution rho(z), or T, or...). I can't see what that would be.

(It occurs to me that the answer could just be "pressure is created by gravitational interactions, so pressureless models aren't self-consistent"... but that seems like a too-easy cop-out and I probably missed the real answer.)

Separate question: are you aware of any papers discussing GM effects in the context of the bullet cluster? I've searched a bit and haven't found any, but you wrote about that recently...

When I read the post I immediately spotted the error of thinking that collisions were needed for pressure. However, I then (without thinking about it) made the assumption that Ludwig was concerned about pressure only from the standpoint of the gravitational effects of the pressure, and thus dismissed this error as unimportant and continued reading, because (thinking in terms of stars in the disk itself) the idiosyncratic motion should indeed be small relative to the azimuthal motion.

When I encountered the discussion of motion of objects not in the disk, I considered the failure to consider the non-azimuthal motion as a separate and more relevant error, since I had lumped the pressure assumption into the "gravitational" bin and not the "dynamical" bin.

Then I looked at how the pressure was actually being used in the equations, and of course it is being used in the "dynamical" equation 2.2 and not in the "gravitational" equations 2.3. So, it is correct to consider the zero pressure assumption to be where Ludwig went wrong.

> So similarly the usual picture of galaxies “held up” by momentum is actually a picture of a non-zero pressure, a pressure highest near the center and declining away from it, and a pressure strong enough to counter gravity and “hold up” the average density of stars near the North pole. So the pressure is not near zero, even though collisions are very rare.

A quick way to check whether the magnitude of the density is in the right range is to have a look at the Jeans length of a dust cloud the size of a galaxy.

About the note 5p it infers that one should carefully read the cited papers before introducing more bias. In particular paper 3 with results from Gaia data (data are independent from any model!). The very simple calculations shown above introduce more confusion without an appropriate context. GR is a complicate theory, not just a pN correction. Probably the 15 referees have not made the same mistake.

Sure. More specifically, star-star collisions being very rare does not justify the idea that velocities are azimuthal. Forces perpendicular to the galactic plane depend on star-star interactions, not star-star collisions. Gravity is a long-range force, so stars can interact at a considerable distances. Assuming otherwise seems mistaken.

There is no reason to argue that something holds up the north pole of that galaxy. It could consist of objects in eliptical orbits about the centre of the galaxy. They would just not be orbiting in the galactic plane.

Galaxy Of Error: One doesn't need to do detailed calculations. Gravity is r⁻². The additional force needed to explain galaxy rotation curves has to be negligible at solar system distances but must dominate gravity at galactic distances, so it must decrease less quickly with distance; in most MOND theories it is r⁻¹. But magnetism and gravitomagnetism are r⁻³; they decrease more quickly with distance. Thus they just can't work for this. They scale the wrong way.

FWIW, pressure is not conventionally about momentum. Say you are sitting on a chair. The chair is under pressure, but it and you are both motionless - and so have no momentum. Pressure is defined in terms of force, not momentum.

A better definition of pressure is something like the dot product of the momentum transferred through a surface with an area vector of the surface (pointing to the same side of the surface that the momentum is going to, to get the signs correct). E.g. the diagonal space components of the stress-energy tensor.

If you have a ball of stars moving isotropically and held together by gravity, and put inside that ball a magic star-reflecting mirror that reflected stars, the mirror would on average be hit the same amount from each side, and stay in place on average, and the statistical properties of the ball of stars would be pretty much the same as if it weren't there. Same as if the ball of stars were a ball of ideal gas. But, the mirror would obviously experience substantial force (i.e. pressure) from each side. This pressure doesn't go away just because the mirror isn't there.

In the case of the self-gravitating ball of stars, this star-motion pressure would obviously balance the gravitational force. However, the gravitational force isn't normally counted as pressure, in the same way that we don't say that the center of the Earth is at zero pressure even though the the gravitational force within the Earth is balanced by the repulsive forces between the atoms.

Pressure is defined as being force per unit area. You could apply the concept to gravity-only systems - if you don't mind the force being negative. The assumption of zero pressure could then refer to when attractive gravitational forces and repulsive forces (e.g. due to radiation pressure) are exactly in balance. Put it like that and it does seem like a bit of a strange assumption.

Idiosyncratic motion of objects is pressure. Just as a ball of ideal gas could be held up from collapsing to a point by the pressure of the motion of its particles, a ball of stars can be held up by the the motion of the stars, which is also pressure.

Normally, I, like you, would not refer to the galactic bulge as being held up by "pressure" - I'd prefer to talk about motion directly. However, the idiosyncratic motion of the stars that you mention have been thrown out of the galactic plane is pressure.

In the context of the set of equations quoted by Robin in the post, assuming zero pressure amounts to assuming zero idiosyncratic motion, and thus assuming zero stars "thrown out of the galactic plane".

Zero pressure doesn't sound like too bad an assumption. Of course, there is radiation pressure - which could be significant near the middle of a galaxy, but models have to make some assumptions. Anyway, gravity alone can likely create a mild nuclear bulge in galaxies - without invoking the concept of positive pressure. The mechanism would be: stars get close, interact via gravity and are thrown out of the galactic plane. I think you can even see this in simulations - such as the NIHAO galaxy simulations.

Assuming that Robin's writeup of Ludwig's argument is accurate, it seems the difference is that Lisi actually estimated the numbers directly, and that Ludwig, instead of actually estimating the numbers directly, made an indirect argument based on equilibrium conditions that relied on a false assumption.

I'm confused, and maybe you can set me straight. I apologize for the wall of text, but want to make sure my question --- which, stemming from confusion, is not very well-posed --- is clear.

Lisi's quick calculation and the original paper both seem to based on the same simplified model. The paper: ignores pressure, assumes all mass is at z=0. The paper then estimates the size of GM effects, finds them to be of the same order as GE effects. There are details there, but there's a big central assertion: in a simple model of a galaxy with a pressureless gas rotating azimuthally, etc etc, GM ~ GE.

Lisi's calculation appears to be applicable to just such a model. The complexities of the actual distribution and behavior of matter in a galaxy don't appear. "Here's the rough distribution of mass and velocity at z=0, here's the relative size of GM vs GE." And yet he finds GE >>> GM, where Ludwig finds them comparable.

I'm worried that might not be very clear, so let me try to rephrase. Shown Ludwig's model of a galaxy, I might try to do a consistency check --- a quick back-of-the-envelope calculation that GM effects are indeed important. But Lisi's note looks like exactly the right estimate to do, not for a real galaxy, but for Ludwig's simplified galaxy! So why do they come to such different conclusions?

I hope it's clear that this is a separate question from "is pressureless a good model". It's a bad model (I quite like your demonstration of this), but that doesn't resolve why two different calculations in the same bad model come to such radically different answers.

We could argue, "well, Ludwig did a more careful calculation with more bells and whistles", but then it has to be the case that some part of Lisi's calculation was very sensitive to a parameter which was only imprecisely estimated (like the distribution rho(z), or T, or...). I can't see what that would be.

(It occurs to me that the answer could just be "pressure is created by gravitational interactions, so pressureless models aren't self-consistent"... but that seems like a too-easy cop-out and I probably missed the real answer.)

Separate question: are you aware of any papers discussing GM effects in the context of the bullet cluster? I've searched a bit and haven't found any, but you wrote about that recently...

I initially had a similar reaction to Tim.

When I read the post I immediately spotted the error of thinking that collisions were needed for pressure. However, I then (without thinking about it) made the assumption that Ludwig was concerned about pressure only from the standpoint of the gravitational effects of the pressure, and thus dismissed this error as unimportant and continued reading, because (thinking in terms of stars in the disk itself) the idiosyncratic motion should indeed be small relative to the azimuthal motion.

When I encountered the discussion of motion of objects not in the disk, I considered the failure to consider the non-azimuthal motion as a separate and more relevant error, since I had lumped the pressure assumption into the "gravitational" bin and not the "dynamical" bin.

Then I looked at how the pressure was actually being used in the equations, and of course it is being used in the "dynamical" equation 2.2 and not in the "gravitational" equations 2.3. So, it is correct to consider the zero pressure assumption to be where Ludwig went wrong.

N.B. I have not actually read Ludwig's paper.

No, that's as mathematically solid as saying that -3 is not greater than -2. You can disagree, but you'll be wrong.

> So similarly the usual picture of galaxies “held up” by momentum is actually a picture of a non-zero pressure, a pressure highest near the center and declining away from it, and a pressure strong enough to counter gravity and “hold up” the average density of stars near the North pole. So the pressure is not near zero, even though collisions are very rare.

A quick way to check whether the magnitude of the density is in the right range is to have a look at the Jeans length of a dust cloud the size of a galaxy.

https://en.wikipedia.org/wi...

About the note 5p it infers that one should carefully read the cited papers before introducing more bias. In particular paper 3 with results from Gaia data (data are independent from any model!). The very simple calculations shown above introduce more confusion without an appropriate context. GR is a complicate theory, not just a pN correction. Probably the 15 referees have not made the same mistake.

Sure. More specifically, star-star collisions being very rare does not justify the idea that velocities are azimuthal. Forces perpendicular to the galactic plane depend on star-star interactions, not star-star collisions. Gravity is a long-range force, so stars can interact at a considerable distances. Assuming otherwise seems mistaken.

Did you even read the post?

There is no reason to argue that something holds up the north pole of that galaxy. It could consist of objects in eliptical orbits about the centre of the galaxy. They would just not be orbiting in the galactic plane.

That's a heuristic argument, not a rock solid proof. So it has to give way to more detailed calculations.

Galaxy Of Error: One doesn't need to do detailed calculations. Gravity is r⁻². The additional force needed to explain galaxy rotation curves has to be negligible at solar system distances but must dominate gravity at galactic distances, so it must decrease less quickly with distance; in most MOND theories it is r⁻¹. But magnetism and gravitomagnetism are r⁻³; they decrease more quickly with distance. Thus they just can't work for this. They scale the wrong way.