Mixed modified-gravity--modified-inertia MOND theories

summary

The gist

The gist: Mixed modified-gravity–modified-inertia MOND theories explore families of nonrelativistic MOND theories that modify both gravity and inertia, showing how their combined effects yield

In short

The research explores nonrelativistic MOND theories that modify both gravity and inertia simultaneously through a parameter $\eta$. These mixed theories combine to produce primary MOND predictions, such as rotation curves and mass-asymptotic-speed relations. The key finding is that neither gravity modification nor inertia modification alone yields MOND phenomenology; the combination is necessary.

Key concepts

Scaling Dimension ($\eta$): This parameter, ranging from 0 to 1, dictates how the theory scales under spacetime transformations in the Deep-MOND Limit (DML). It represents a unifying scaling dimension for all terms in the modified action. A value of $\eta=0$ corresponds to pure modified gravity, while $\eta=1$ corresponds to pure modified inertia.
Deep-MOND Limit (DML): This is the limit where the theory becomes spacetime scale invariant (SI). In this limit, the equations of motion must remain unchanged when coordinates $(t, r)$ are scaled by a factor $\lambda$. The combined action of mixed MG-MI theories ensures this invariance through specific scaling properties.
Mass-Asymptotic-Speed Relation (MASR): This is a predicted universal relationship for the asymptotic rotational speed ($V_{\infty}$) in DML systems. It links the asymptotic velocity to the characteristic mass scale $M_{A0}$ and theory parameters ($\gamma$, $\zeta$, $\eta$). Achieving this relation requires specific constraints on normalization constants, such as setting $\gamma = \zeta$ when defining $a_0$.
Mixed MG-MI Theories: These theories modify both the gravitational action (pure MG) and the inertial action (pure MI). The paper demonstrates that while pure modifications fail to produce MOND effects, their combination successfully generates primary MOND predictions like asymptotic flatness and virial relations.

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This episode discusses

The paper

Mixed modified-gravity--modified-inertia MOND theories · Read on arXiv

Mordehai Milgrom

Weizmann Institute

It is plausible that in a more fundamental theory of MOND both gravity and inertia will differ from their Newtonian versions. Such theories, if they embody the basic tenets of MOND, will share its primary predictions with existing theories, which modify either gravity or inertia. But they are expected to differ in their structure and in their secondary predictions. Here, I discuss families of nonrelativistic MOND theories that modify both the kinetic and the gravitational actions of Newtonian dynamics. Each family is spanned by one parameter 0 η 1, which is the scaling dimension of the deep-MOND limit of all terms in the action. Pure modified gravity (η=0) and pure modified inertia (η=1) are the end cases in each family. I speak of families, since the modified action terms themselves are not strictly specified, but are treated in rather general terms. These theories are artificial in that they are not based on a single unifying idea that dictates the modification of the two components. But they are still useful in pinpointing some idiosyncracies of such theories. For example, neither of the two modified terms, by itself, leads to MOND phenomenology; only their combined effect does. I show how the two combine to make some of the primary MOND predictions, such as asymptotic flatness of rotation curves, the mass-asymptotic-speed relation, and the deep-MOND, virial, mass-velocity-dispersion (M-σ) relation. Another idiosyncracy of such theories is the emergence of different interpolating functions in the kinetic equations of motion and in the gravitational field equations, each relevant under different circumstances, and neither of which have the standard deep-MOND limit. Such examples alert us to the wide range of theories that can lead to MOND phenomenology.

Transcript

Introduction to the show: ident: Astrophysics Radio. Generated commentary on the latest astrophysics papers.

Vera: We're looking at the paper: "Mixed modified-gravity--modified-inertia MOND theories".

Jocelyn: The gist: Mixed modified-gravity–modified-inertia MOND theories explore families of nonrelativistic MOND theories that modify both gravity and inertia,

Vera: First, who's behind it and why it matters.

Title and authors: Vera: So we're looking at this paper today called "Mixed modified-gravity--modified-inertia MOND theories." It's exploring families of nonrelativistic MOND theories where they modify both gravity and inertia, showing how that combination gives us the standard MOND predictions while letting them differ on other things.

Jocelyn: What that means for us is that we have these modified theories, and each one is defined by this parameter eta from zero to one. Eta basically sets the scaling dimension of the deep-MOND limit across all terms in the action.

Subrahmanyan: That means at zero, eta equals zero gives you a pure modified gravity theory, and at one, eta equals one gives you a pure modified inertia theory. It's like those are the two extreme ends of this family that they're looking at <ref:2610.21699#pg1>.

Vera: And the interesting part is that these theories aren't based on one single idea dictating how both gravity and inertia get modified, which is a bit unusual for MOND research.

Jocelyn: Exactly. The paper says they are treated in rather general terms because there isn't one single unifying concept guiding the modification of those two components <ref:2610.21699#pg1>.

Subrahmanyan: And this generality leads them to study mixed theories where a single term in the action is responsible for getting that deep-MOND behavior and making the transition from Newtonian gravity to the deep-MOND limit <ref:2610.21699#pg2>.

Vera: That combined effect of those two terms is what generates some primary MOND predictions, like the asymptotic flatness of rotation curves and that mass-asymptotic-speed relation <ref:2610.21699#pg2>.

Jocelyn: The paper points out that neither modified term by itself produces MOND phenomenology; it's only when they are combined that you get those results <ref:2610.21699#pg3>.

Subrahmanyan: The deep-MOND limit, the DML, requires the theory to become spacetime scale invariant in that limit. This means all the equations of motion in that limit must be invariant under a transformation like (t, r) to lambda(t, r).

Vera: They achieve this by requiring the DML action to have a specific scaling dimension of negative eta. That ensures all terms scale correctly under spacetime scaling <ref:2610.21699#pg3>.

Jocelyn: And because they have this two-parameter scaling symmetry in equation sixteen the resulting equations of motion turn out to be spacetime invariant <ref:2610.21699#pg3>.

Subrahmanyan: When we look at the virial relations, those general equations reduce to something like (two plus eta) times the mass times some quantity equals the dot product of position and the gradient of the potential in that DML <ref:2610.21699#pg2>.

Vera: Which simplifies further to (two plus eta) times the mass times some quantity equals something involving r dot gradient phi tilde in standardized quantities <ref:2610.21699#pg3>.

Jocelyn: They predict a universal DML MOND relation of sigma to MA zero over xi, where that xi is a dimensionless number that can depend on the theory <ref:2610.21699#pg3>.

Subrahmanyan: For a proper choice of sigma, they argue that this xi has a value of order one for all DML systems <ref:2610.21699#pg3>.

Vera: Thinking about the asymptotic behavior of the potential in the DML, they get expressions like phi p m equals negative gamma over eta times MA zero to the power of one over two times r M over r for eta not equal to zero <ref:2610.21699#pg3>.

Jocelyn: For circular orbits, the kinetic side simplifies into a functional relation between acceleration a and g phi <ref:2610.21699#pg3>.

Subrahmanyan: This leads to the prediction of a constant asymptotic rotational speed V infinity equals gamma over zeta to the power of one over two times one plus eta times MA zero to the power of four divided by four <ref:2610.21699#pg3>.

Vera: That's what they call the mass-asymptotic-speed relation, or MASR <ref:2610.21699#pg3>.

Jocelyn: So, looking at the numbers and caveats, this is how they connect those actions to observable things like rotation curves <ref:2610.21699#pg3>.

Subrahmanyan: But one thing the paper flags is that this whole structure relies on a specific normalization of A zero in equation thirty-eight to ensure that the predicted universal MOND mass-asymptotic-speed relation holds for all DML systems <ref:2610.21699#pg3>.

Vera: That suggests we need to check if the constraint gamma equals zeta when defining that normalization of A zero <ref:2610.21699#pg3>.

Jocelyn: And then there's the question about the kinetic interpolation function mu k x for circular orbits, does it show behavior consistent with a pure modified inertia theory or mixed theories?

Subrahmanyan: Specifically, they look at whether that function tends to unity for high accelerations in the DML, which is what equation thirty-six requires <ref:2610.21699#pg3>.

Vera: If it does tend to unity, that would lean toward a pure modified inertia scenario rather than a mixed one <ref:2610.21699#pg3>.

Jocelyn: So, what does this mean for someone who only listens to the show? It suggests that while these mixed theories share those primary MOND predictions, the secondary effects are where they really start to diverge <ref:2610.21699#pg3>.

Subrahmanyan: That's right, so we have a wide range from pure modified gravity to pure modified inertia spanning eta from zero to one in this paper on mixed modified-gravity--modified-inertia MOND theories <ref:2610.21699#pg1>.

Vera: We’ve seen how these two components modify the dynamics, and now we're looking at what those modifications imply for structure formation on a larger scale <ref:2610.21699#pg3>.

Jocelyn: It makes you wonder how these differing secondary predictions might show up in the external field effect or solar system tests <ref:2610.21699#pg3>.

Subrahmanyan: That's where a lot of the future work needs to focus, because connecting these NR theories to relativistic ones is still a big step <ref:2610.21699#pg3>.

Vera: We’ll be looking at those implications in our next segment.

The paper's summary: Vera: So basically, this paper is about these mixed modified gravity and modified inertia theories where they both change gravity and how things move through spacetime.

Jocelyn: Right, they’re looking at a whole family of these theories defined by a parameter eta that goes from zero to one.

Subrahmanyan: That means at zero it’s pure modified gravity, and at one it’s pure modified inertia. It maps out the whole spectrum between those two extremes.

Vera: And the main thing they found is that these two components working together—gravity and inertia modification—is what actually generates those primary MOND predictions we see in things like rotation curves or mass-asymptotic-speed relations.

Jocelyn: Exactly, because neither gravity alone nor inertia alone produces those effects on its own; it’s the combination that does the heavy lifting for MOND phenomenology.

Subrahmanyan: The paper spends a lot of time on the deep-MOND limit, that DML where everything becomes spacetime scale invariant.

Vera: They show how they need both actions to have that same scaling dimension, which is negative eta under spacetime scaling. That’s the key requirement for symmetry there.

Jocelyn: And because they have this two-parameter scaling symmetry in equation sixteen, the equations of motion end up being scale invariant in that limit.

Subrahmanyan: When you look at those virial relations, they simplify down to something like two plus eta times mass times some quantity equals the dot product of position and the gradient of the potential.

Vera: That simplifies further to (two plus eta) times mass times some quantity equals something involving r dot gradient phi tilde in standardized quantities.

Jocelyn: They predict a universal DML MOND relation, sigma to MA zero over xi, where that xi is just a dimensionless number that can change depending on the specific theory you pick.

Subrahmanyan: But they argue that for a proper choice of sigma, this xi ends up being order one for all DML systems.

Vera: They also look at the asymptotic behavior of the potential in the DML, and they get expressions like phi p m equals negative gamma over eta times MA zero to the power of one over two times r M over r when eta is not zero.

Jocelyn: For circular orbits specifically, it simplifies down to a functional relation between acceleration a and g phi.

Vera: That’s what they call the mass-asymptotic-speed relation, or MASR, and it’s a constant value for that specific set of parameters.

Jocelyn: So, looking at the numbers and the caveats, this is how they connect those actions to observable things like rotation curves.

Subrahmanyan: But one thing they flag is that all this structure relies on a specific normalization of A zero in equation thirty-eight to make sure that universal MOND mass-asymptotic-speed relation holds for every DML system.

Vera: That suggests we need to check if the constraint gamma equals zeta when defining that normalization of A zero.

Jocelyn: And then there’s the question about the kinetic interpolation function mu k x for circular orbits, does it show behavior consistent with a pure modified inertia theory or mixed theories?

Subrahmanyan: Specifically, they look at whether that function tends to unity for high accelerations in the DML, which is what equation thirty-six requires.

Vera: If it does tend to unity, that would lean toward a pure modified inertia scenario rather than a mixed one.

Jocelyn: So, what this means for someone who only listens to the show is that while these mixed theories share those primary MOND predictions, the secondary effects are where they really start to diverge.

Subrahmanyan: That’s right, so we have a wide range from pure modified gravity to pure modified inertia spanning eta from zero to one in this paper on mixed modified-gravity--modified-inertia MOND theories.

Vera: We’ve seen how these two components modify the dynamics, and now we're looking at what those modifications imply for structure formation on a larger scale.

Jocelyn: It makes you wonder how these differing secondary predictions might show up in the external field effect or solar system tests.

Subrahmanyan: That's where a lot of the future work needs to focus, because connecting these NR theories to relativistic ones is still a big step.

The paper's improvements: Vera: So we're looking at what the authors suggest as improvements for these mixed modified gravity and modified inertia theories today.

Jocelyn: Right, they’re proposing ways to make these theoretical models more robust and better connected to reality.

Subrahmanyan: They are suggesting that a more fundamental understanding of MOND might require modification of both gravity and inertia, rather than just one component.

Vera: The paper points out that existing theories often modify only one part, but this mixed approach shares those primary MOND predictions while allowing them to differ on secondary things.

Jocelyn: That means these theories can have different predictions for things like the external field effect or solar system tests, which is a big difference.

Subrahmanyan: They are suggesting that these mixed theories might be more physically realistic because they allow for that dual modification.

Vera: The authors are also looking at how to connect these nonrelativistic theories to their relativistic counterparts, which is where the real challenge lies.

Jocelyn: They need a better bridge there, because the connection between NR and relativistic MOND is still a big step in this research area.

Subrahmanyan: And they’re exploring how these models might impact structure formation on larger scales, which is where we see the biggest cosmic picture.

Vera: The implication here is that if we can figure out how these mixed theories behave under different conditions, it could help us understand the large scale distribution of matter.

Jocelyn: It suggests that we need to keep looking at these kinds of dual modifications in our observational follow-up programs with pulsars and other time-domain sources.

Subrahmanyan: We’re hoping that by exploring these mixed families, we can find constraints from cosmological observations that either confirm or challenge this two-component modification idea.

Conclusion: Vera: So to wrap up, this paper on "Mixed modified-gravity--modified-inertia MOND theories" explores how combining gravity and inertia modifications can produce primary MOND predictions while allowing for different secondary results.

Jocelyn: Exactly, it shows that these mixed theories share the core MOND results but branch out in things like the external field effect or solar system tests.

Subrahmanyan: The implication is that a more fundamental understanding of MOND might actually involve modifying both gravity and inertia, unlike current models that only change one aspect.

Vera: It suggests we need to look for observational evidence that can distinguish between pure gravity modifications and pure inertia modifications in the real world.

Jocelyn: And they are pointing toward the need to connect these nonrelativistic theories with their relativistic versions, which is still a big hurdle in this area.

Subrahmanyan: That connection is crucial because it’s how we can see if this dual modification idea actually holds up when we look at larger cosmic structure formation.

Vera: So, the paper gives us a wide range from pure modified gravity to pure modified inertia spanning that eta parameter from zero to one.

Jocelyn: It really shows the diversity in secondary predictions that these mixed theories can have while still hitting those key MOND targets.

Subrahmanyan: We’re hoping that by exploring these mixed families, we can find constraints from cosmological observations that either confirm or challenge this two-component modification idea.

Vera: So, if you're driving and thinking about the sky, remember the paper is "Mixed modified-gravity--modified-inertia MOND theories."

Jocelyn: That’s right. Next time we’ll be looking at how these kinds of dual modifications might show up in structure formation on a larger scale.

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