Particle with negative mass inside a black hole horizon

[timmdeeg]

[Moderator's note: Spin off from previous thread due to topic change.]

The concept of "negative mass" described in the link you gave has nothing whatever to do with virtual particles and is not the same concept as the concept of "virtual particles having negative energy"

Just as an aside and not related to the OP, would a real particle with negative mass inside the event horizon follow the runaway motion? Would it be ejected?

 

[PeterDonis]

a real particle with negative mass

There is no such thing.

 

[pervect]

Hypothetical particles of small negative mass would follow the same test-particle geodesics as particles of small positive mass.

However, I think there may be some serious stability issues with particles of negative mass, especially if they interact with particles of positive mass. I can't quite recall what I may have read on the topic, or how reliable it was.

 

[PeterDonis]

Hypothetical particles of small negative mass

I don't think any claims should be made about such hypothetical particles in the context of GR without a reference that gives consistent math describing them in that context. I'm not aware of any such reference.

 

[timmdeeg]

Hypothetical particles of small negative mass would follow the same test-particle geodesics as particles of small positive mass.

However, I think there may be some serious stability issues with particles of negative mass, especially if they interact with particles of positive mass. I can't quite recall what I may have read on the topic, or how reliable it was.

https://www.researchgate.net/publication/263659720_Solving_the_negative_mass_paradox

Abstract
In 1957 H.Bondi showed that the introduction of negative masses in the universe goes with a preposterous phenomenon : the so-called runaway effect : when a positive mass encounters a negative mass, it escapes, and the second runs after it. ... Masses with same signs attract each other through Newton’s law. Masses with opposite signs repel each other through anti-Newton’s law.

After some search I found this hint regarding runaway.

 

[PeterDonis]

I found this hint

This model is referenced in the Wikipedia article on negative mass, but it's important to recognize what it is not.

It is not a model of negative mass within GR; this model has two metrics and two field equations, not one of each, so it is a different theory from GR.

It is not a model of negative mass particles moving in a GR spacetime. It is a model with two different kinds of continuous fluid, a "positive mass" fluid and a "negative mass" fluid, filling the universe. It attempts to derive two different sets of geodesics from the two metrics, but it's not clear to me how the two sets of geodesics are supposed to correspond to motions of test particles; the paper does not seem to derive this, but just handwaves it.

It is not a model of negative mass particles moving in a black hole spacetime. It's not even clear whether analogues of black hole solutions exist in this model; the paper does not treat this at all. It is solely concerned with cosmology.

 

[pervect]

I don't think any claims should be made about such hypothetical particles in the context of GR without a reference that gives consistent math describing them in that context. I'm not aware of any such reference.

On the elementary level, I thought it was well known that in both Newtonian theory and in GR, that if a positive mass fell "down", a negative mass would also fall down. Invoking the equivalence principle demonstrates this. For instance, consider having both a positive and negative mass follow an inertial trajectory on Einstein's elevator. THen in an inertial frame, both masses would stay stationary, in the accelerating frame they would both "fall" in the same direction.

In Newtonian theory, the force on the negative mass changes sign (GMm/r^2) switches sign when m goes from positive to negative, but the accleartion, F/m =- GM/R^2 does not change sign when m changes sign.

I do believe there may be some difficulty with proving that a negative test mass follows a geodesic directly from Einstein's equation. I'm not aware of the details. I vaguely recall seeing references to papers on this issue, and I suspect that might be your concern.

It seems to me that worst case we might need to make an auxiliary assumption of the geodesic motion for test masses rather than deriving it directly from Einstein's equations. I don't see any problem with making such an assumption, but it's not impossible that I've overlooked something. However, I am not aware of any proof that it is known to be inconsistent to make such an assumption. Are you aware of any such proof?

I believe there were some thermodynamic stability issues, regarding a model of an ideal gas composed of negatie mass particles. But I don't recall where I read this anymore.

 

[PeterDonis]

I do believe there may be some difficulty with proving that a negative test mass follows a geodesic directly from Einstein's equation.

The geodesic equation in GR does not involve the mass of the test particle at all; it can be factored out in the process of deriving the equation. (As you note, this is just a manifestation of the fact that GR obeys the equivalence principle.) The only restriction is that the mass must be nonzero (more precisely, that deriving the geodesic equation for timelike geodesics involves a slightly different procedure than doing it for null geodesics). The sign of the mass plays no role at all.

That being the case, even to define the "mass" of a test particle following a timelike geodesic involves some subtleties. By definition, the test particle does not affect the spacetime curvature at all, so there is no way to define a "mass" from its stress-energy since it has zero stress-energy by definition. The fact that its worldline is timelike doesn't help, because once you have made a choice of which direction is the "future" direction on the worldline, the sign of the particle's 4-momentum, which formally would depend on the sign of its mass, is AFAIK a convention (whether you want a 4-momentum pointing in the future direction to be positive or negative), not a physical quantity; therefore the sign of its mass is a convention as well. AFAIK no physical observable depends on the sign.

Another issue is the fact that in GR, unlike in Newtonian physics, there is no separate concept of "inertial" vs. "gravitational" mass (more precisely, "passive gravitational" mass, since as noted above, a test particle has zero "active" gravitational mass by definition since it is not a source of gravity). So the idea of switching the sign of one for a test particle while leaving the sign of the other the same isn't even well-defined in GR.

The concept of negative active gravitational mass is well-defined in GR: the simplest example is Schwarzschild spacetime with $M$ negative in the line element. Geodesics in this spacetime "fall" away from the center instead of falling towards it, and it has no horizon and a timelike curvature singularity at $r = 0$. But, as above, this is true regardless of the mass of test particles. Also, this solution is, AFAIK, not considered physically reasonable by any physicists working in the field.

 

[timmdeeg]

It is not a model of negative mass particles moving in a black hole spacetime. It's not even clear whether analogues of black hole solutions exist in this model; the paper does not treat this at all. It is solely concerned with cosmology.

I wasn't aware of that, thanks for clarifying.

Apart of being physically not reasonable what would happen to a negative mass m (not a test particle) within a black hole with M positive? I failed to extract that from your discussion with @pervect .

A quick search:

Can negative mass be considered in General Relativity?

We show, through Newtonian approximation, that shifting to a bimetric model of the Universe based on a suitable system of coupled field equations, removes the preposterous runaway effect and gives different interaction laws, between positive and negative masses, that bring new insights into the alternative VLS interpretations previously proposed by several authors, and strengthening their assumptions.

According to that - removes the preposterous runaway effect and gives different interaction laws, between positive and negative masses - seems to leave it open how a hypothetical negative mass would move within the horizon. How do you think about that?

 

[PeterDonis]

what would happen to a negative mass m (not a test particle) within a black hole with M positive?

If the negative mass is not a test particle, then you need to tell me what solution of the Einstein Field Equation describes such a thing. AFAIK there isn't one, not even in numerical simulation.

A quick search

This is the same "bimetric" model that I already covered in post #6. (Different paper, but same model.)

 

[timmdeeg]

If the negative mass is not a test particle, then you need to tell me what solution of the Einstein Field Equation describes such a thing. AFAIK there isn't one, not even in numerical simulation.

A clear answer, thanks.

 

[PeterDonis]

If the negative mass is not a test particle, then you need to tell me what solution of the Einstein Field Equation describes such a thing. AFAIK there isn't one, not even in numerical simulation.

Just to elaborate a bit more on this: in one of my Insights articles [1], I derive (following numerous textbook derivations) the relevant components of the Einstein Field Equation for a static, spherically symmetric spacetime. A non-test-particle object with negative mass, assuming such an object is stable, would have to be described by such a solution with the mass function $m(r)$ negative.

However, as you will see if you read the Insights article, the mass function $m(r)$ only depends on the energy density $\rho$, i.e., the component $T_{00}$ of the stress-energy tensor in the coordinates used in the article. It does not depend on any other stress-energy tensor components. So the only way for $m(r)$ to be negative is for $\rho$ to be negative. But no form of stress-energy either known or hypothesized has this property.

Note carefully that $\rho$ being negative is not the same as "exotic matter", which has positive energy density $\rho$ but negative pressure (either radial or tangential or both)--dark energy aka a cosmological constant is the simplest case of this (where $p = - \rho$). But this cannot make the mass function $m(r)$ negative; objects made of such exotic matter still have positive mass. They just have other counterintuitive properties due to the negative pressure. (For one thing, such an object can only exist in hydrostatic equilibrium if the pressure is not isotropic somewhere inside them. But that's a matter for a separate discussion.)

[1] https://www.physicsforums.com/insig...-in-a-static-spherically-symmetric-spacetime/

 

[PAllen]

It is worth adding that to derive that the motion of small body follows a timelike geodesic in the limit of becoming a test body, directly from the field equations, requires the assumption of the dominant energy condition. More precisely, the dominant energy condition is both necessary and sufficient for timelike geodesic motion to follow from the field equations for the small body limit. Further, it is known that an isolated body of matter violating the dominant energy condition (even slightly) can follow a spacelike trajectory without violating the field equations.

Since all energy conditions are at least somewhat suspect in the context of quantum gravity, Wald and Gralla found that, if instead of energy conditions, one simply assumes (as an axiom) that a small body must follow a timelike path, then geodesic motion follows in the limit. However, they noted there is no way to arrive at the requirement for timelike motion from the field equations without the assumption of the dominant energy condition.

 

[PeterDonis]

It is worth adding that to derive that the motion of small body follows a timelike geodesic in the limit of becoming a test body, directly from the field equations, requires the assumption of the dominant energy condition.

This paper seems relevant (and contains references that are also relevant, but I haven't been able to find those earlier papers on arxiv):

https://arxiv.org/pdf/0806.3293.pdf

For clarity, "dominant energy condition" here refers to the stress-energy that makes up the "small body".

 

[PeterDonis]

For clarity, "dominant energy condition" here refers to the stress-energy that makes up the "small body".

Also, to be clear, the dominant energy condition is a much stricter condition than simply requiring that the energy density $\rho$ be positive.

 

[PAllen]

This paper seems relevant (and contains references that are also relevant, but I haven't been able to find those earlier papers on arxiv):

https://arxiv.org/pdf/0806.3293.pdf

For clarity, "dominant energy condition" here refers to the stress-energy that makes up the "small body".

Yes, this is the key paper I was referring to. It references all the important earlier papers (e.g. those with Geroch as co-author). In times past, I was able to find open access to the Geroch papers.

Here is one reference on the possibility of spacelike motion without assumption of dominant energy condition (actually, this paper makes several refinements of dominant energy condition, which I lump together, for simplicity):

https://arxiv.org/abs/1106.2336

 

[PAllen]

However, as you will see if you read the Insights article, the mass function $m(r)$ only depends on the energy density $\rho$, i.e., the component $T_{00}$ of the stress-energy tensor in the coordinates used in the article. It does not depend on any other stress-energy tensor components. So the only way for $m(r)$ to be negative is for $\rho$ to be negative. But no form of stress-energy either known or hypothesized has this property.

I thought Casimir effect, in a limited way, demonstrated negative energy density.

 

[PeterDonis]

Here is one reference on the possibility of spacelike motion without assumption of dominant energy condition

Very interesting paper. The only note I would add is that the conditions (3) and (4) on p. 2 of the paper simply describe how we model isolated objects in GR: as "world tubes" containing some kind of stress-energy whose support is limited to the world tube, and which must be nonzero somewhere in order to have an object at all. The paper then investigates what additional assumptions are needed to show that the world tube is timelike (and those assumptions turn out to be strong enough that we expect certain kinds of stress-energy likely to occur physically, such as some quantum field configurations, to violate them).

(Also, condition (2) in the paper is automatically satisfied for any solution of the Einstein Field Equation; I'm not sure why the paper doesn't mention that--unless I missed something.)

 

[PAllen]

Very interesting paper. The only note I would add is that the conditions (3) and (4) on p. 2 of the paper simply describe how we model isolated objects in GR: as "world tubes" containing some kind of stress-energy whose support is limited to the world tube, and which must be nonzero somewhere in order to have an object at all. The paper then investigates what additional assumptions are needed to show that the world tube is timelike (and those assumptions turn out to be strong enough that we expect certain kinds of stress-energy likely to occur physically, such as some quantum field configurations, to violate them).

(Also, condition (2) in the paper is automatically satisfied for any solution of the Einstein Field Equation; I'm not sure why the paper doesn't mention that--unless I missed something.)

Those are restatements of a prior result by Geroch and Jang, so perhaps blame them. More plausibly, they are just being stated for emphasis, rather than a more abstract statement that implies them.

 

[timmdeeg]

I thought Casimir effect, in a limited way, demonstrated negative energy density.

I think it demonstrates that there is less energy density between the plates compared to outside. But are we sure that the energy density outside is zero? How about the zero point energy?

 

[timmdeeg]

However, as you will see if you read the Insights article, the mass function $m(r)$ only depends on the energy density $\rho$, i.e., the component $T_{00}$ of the stress-energy tensor in the coordinates used in the article. It does not depend on any other stress-energy tensor components. So the only way for $m(r)$ to be negative is for $\rho$ to be negative. But no form of stress-energy either known or hypothesized has this property.

This is very enlightening! So not even in a thought experiment one should speculate about a particle with negativ mass in this context.

It follows from the respective integral in your insights article that pressure isn't included in the mass function $m(r)$ but I have no clue to understand why. So I will look at your previous series of articles on “Does Gravity Gravitate” which you mentioned. Not sure if the technical barrier isn't too high though.

 

[PeterDonis]

I thought Casimir effect, in a limited way, demonstrated negative energy density.

No, it doesn't. It just demonstrates that the vacuum state in a space between two conductors is different from the vacuum state outside them. The mass of the device displaying the Casimir effect is still positive.

 

[PeterDonis]

How about the zero point energy?

In QFT in the absence of gravity, i.e., in flat spacetime, the choice of the "zero point" of energy is arbitrary; only differences in energy matter. The QFT version of "no negative energy anywhere" is "the Hamiltonian is bounded below". That is true for the Casimir effect.

In QFT with gravity present, i.e., in curved spacetime, we expect quantum "zero point energy" to have a gravitational effect, so the choice of "zero point" of energy is no longer arbitrary. The "zero point energy" from QFT should appear in the Einstein Field Equation as a cosmological constant (or dark energy) term. However, when we calculate what QFT seems to say the magnitude of this term should be, we get a result some 120 orders of magnitude larger than the cosmological constant/dark energy we actually observe. This is one of the main unsolved problems that a quantum theory of gravity is expected to resolve.

 

[pervect]

Certainly negative non-test mass has been proposed as a hypothesis in the literature, in particular
"Negative mass in general relativity", H Bondi - Reviews of Modern Physics, 1957, along with an associated metric. I believe it was just the Schwarzschild metric with a negative mass function, but it's been a long time since I've looked at Bondi's paper.

If you insist on all the details, it's much harder to tell what happens when you drop a mass - negative or positive - into a black hole, especially if you insist on going the exact route with einstein's field equations. Baring the stability issues , I have no reason to believe that it's any more - or less -complicated to find the mass of a black hole when you drop a positive non-test mass into it as when you drop a negative non-test mass in it. The first question is to define what one means by mass more precisely - ADM and Bondi masses in an asymptotically flat space-time would be the two main candidates. Since gravitational radiation will in general be emitted, the two masses will differ, as the ADM mass includes gravitational radiation and the Bondi mass doesn't. However, if the mass of the infalling object is much less than that of the black hole, the amount of gravitational radiation is negligible. I think Chris Hillman had a short post on the topic of the total amount of gravitational wave energy radiated by a non-test mass free-fall from infinity into a black hole back in the day, but again I've forgotten the details.

The stability issues I think are the most vexing and would demand the most caution.

My memory isn't what it used to be, so it'd be worth checking out the Bondi paper to see what he said, and to do a literature search in general on the topic of negative masses.

 

[PAllen]

@pervect , the issue is that negative mass means violation of the dominant energy condition. Without the dominant energy condition, there is no way to derive geodesic (or even timelike) motion from the field equations. A big contribution of the Gralla and Wald research program is to find that if you add timelike motion as an independent axiom on top of the field equations, then geodesic motion follows wholly independent of the stress energy tensor of the body. Thus one can say: if the motion of a body is assumed to be timelike, then a small body of negative mass must follow the same geodesic as a small body of positive mass.

 

[PeterDonis]

I believe it was just the Schwarzschild metric with a negative mass function

That's my understanding as well.