I think that there is no difference between usual and negative masses in the sense of their behavior in the separated and isolated close systems, namely isolated system of the negative masses behaves as usual one in any sense. The negative and usual masses, in turn, repulse each another.

By the way, is the question related with the following, let say, problem in your construction: how the black hole consists of the negative mass particles has an attractive gravitation force?

The inside of the black hole is a causally disconected patch of space-time. No information from whats inside can affect the exterior. You could have negative masses with antigravitational interactions inside, but because space deformations/ gravitational waves cant exit the black hole, an external observer will never notice that interaction.

In other words, the gravitational field "remembers" the spacetime deformation before opening that causally disconected region, that is, it remembers the original mass of the star (you may ask yourself where the mass of the black hole exists or how can it attract objects far away if no info can exit the black hole, for a better undestanding of the subject). More technically it remembers the space-time deformation the mass of the star made before the collapse. Those negative mass interactions would only affect the inside of the black hole. From the exterior, the spacetime deformation and thus, the "force" felt by external test particles should remain the same as the case of the inital star. The electromagnetic field works the same way.

It is kind unclear to me. Namely, how you get positive mass of black hole from the negative volume density? Black holes are created from the usual stars due the collapse, so how there the negative mass is coming?

Black holes do have gravitational mass seen from outside. Not only the inital stars have positive mass, but we can observe the attractive interactions between a black hole and surronding stars. The question relies on what happens at the inside. In that region, where GR physics predicts a physical singularity (an indication of a wrong prediction?), the possibility of negative masses should be considered. What if a switch to negative energies (and masses) occur at the event horizon? An external observer would still observe the surrondings of the black hole, and for him, the black hole would seem to have positive and gravitational mass.

This is the point. I can calculate mass parameters of collapsing to black hole star, but basing on the positive density of the internal layers of the star. Imagine that nothing is passing the horizon, what is the mass sign of the matter in interior hole's region which remain there after the collapse?

I believe in order to have matter inside a black hole, it must have crossed the event horizon, right? As you said before, can`t you make the sign of M dependent on a value of r, for example de schwarzschild radius?

If nothing is passing the horizon, that is the static solution. That static solution would be geometrically the same in the exterior as a black hole (it approaches flat space for far away points), and in the interior, it should have a connection or bridge (like an Einstein-Rosen bridge) in the event horizon connecting to a closed space region, that is, a De Sitter Space. That De Sitter space should also harbor the event horizon of that connection. I tried to work out that space with the time-reversed de Sitter-Schwarzschild metric, but connecting both metrics is over my abilities.

Jean Pierre Petit made something similar by connecting two flat spaces by a bridge (like an Einstein-Rosen bridge) considering that a time transformation takes places and that implies a change in the signature of mass.

As far as I remember , the matter inside a newborn black hole is from the star's matter and not from outside. After that the hole increases it's mass but initially nothing is coming in. In general I also think that the negative mass must be located in some separated region, the question only what is the region, i some sense my note is about.

I have doubts about the collapse of the inital star. But I believe that the schwarzschild radius first appears at the inside of the star, as a small radius, and while the star increases its mass, it also decreases its radius. When the radius of the star meets the schwarzschild radius, we can talk about the black hole. Would you consider then that the matter of inital star hasn`t cross any horizon?

If you want to explore further regions of space regarding black holes, you might want to take a look at the maximally extended solution of a black hole and the penrose diagram for it. Those are the I, II, III and IV regions of the Kruskal Szekeres coordinates. Its interesting to relate some of these regions, like the white hole region, because that is a "time reversed black hole", and as Jean Pierre Petit proposes, a reverse in time implies a change in the signature of masses.

The radius depends on the star's mass, so it is different for each object, but any way there is always some mass inside the sphere, it is not pointlike.

Yes, my note is about, full discrete symmetries "keep" the regions inside the Schw. solution relating the regions of the solution. This is why I think that mass changes sign when you pass from region to region but not when you come in the black hole interior region from an outside.

I recommend this article, which suggests that two different spaces of that maximally extended solution could be indeed the same space. (Fig. 2)

Radial motion into an Einstein-Rosen bridge

Nikodem J. Poplawski

https://arxiv.org/pdf/0902.1994.pdf

The discussion seems to have drifted but I'll address the original question. I think of this in three steps.

First, how would light be deflected passing a negative mass in the Schwarzschild Metric? I think GR predicts that it would be deflected away from the mass since to first order, the angle is proportional to the mass, not its square.

Second, I think any small positive test mass would follow a similar (though not null) geodesic, being deflected away from the negative mass, since the curvature is dominated by the larger mass.

Third I think the same would be true of a small negative test mass since for a sufficiently small value, the path is independent of the mass.

George Dishman That is what Bonnor (1989) predicts regarding negative masses in GR. Negative gravitational fields and repulsive interactions, for black holes also. The only problem addressed by Bondi for this approach is the runaway motion between a positive and a negative mass.

Curiosly, a negative mass black hole looks like a time-reversed black hole (a white hole), which is an hypothetical repulsive object in which no particle or information can enter to its event horizon from outside.

All of that is related to black holes created by negative masses. The question drifted to other interesting posibilities, such as the reverse from positive masses to negative masses ocurring at the event horizon. Since no information (no space deformation either) can exit the black hole, I believe it is possible to describe an attractive black hole at the exterior, with an interior of repulsive interactions.

Dear Manuel,

MUP: The only problem addressed by Bondi for this approach is the runaway motion between a positive and a negative mass.

That depends on whether you see it as a problem, it is the basis of the Alcubierre Metric.

MUP: All of that is related to black holes ...

Your original question doesn't mention black holes. The laws inside a BH would be the same as outside between masses that have fallen through the event horizon (while they lasted).

George Dishman We all know that the runaway motion is used by the Alcubierre metric, and even James Farnes used it to describe his cosmological model in an attempt to explain dark matter and dark energy together.

But the runaway motion still creates paradoxes like perpetual motion machines and infinite sources of energy, thats why Bondi rightly suggested that negative masses shouldnt exist, and Bonnor only considered in his research the hypothetical case of a universe made up entirely on negative masses.

Jean Pierre Petit (this question is posted on one of his projects) solves the runaway problem by considering a bimetric model in his Janus cosmological model. He considers that a change from positive masses to negative masses occur when crossing the event horizon of a black hole (that is why we are referring to blak holes in this question).

Furthermore, there are other ways to solve the runaway motion problem. I am currently studying the use of the fact that the inside of a black hole is causally disconnected from the outside. In this case, if you consider that there are only negative masses existing inside the black hole, and only positive masses existing outside the black hole, no runaway motion occurs (just like in Petit's model, it is the universe who prohibits this problematic interaction).

MUP: I am currently studying the use of the fact that the inside of a black hole is causally disconnected from the outside.

That's not entirely true, events inside cannot affect those outside but outside events do affect the interior, stuff falls in!

George Dishman Absolutely correct. Thats why in the lasts anwers to this question, I suggested Prof. Bondarenko that those masses that fall inside the black hole should suffer a switch to negative masses.

No runaway motion would occur because no mutual interaction between particles inside and outside the black hole take place.

This event would happen a the event horizon, and Jean Pierre Petit explains it by considering that a time transformation occurs at the horizon. He quotes Souriau to explain that a time transformation switches the sign of masses, but you can also demonstrate it by the interpretation of Feynman Stueckelberg about the negative energy particle solutions of the Dirac equation as I did in my research.

Manuel hi,

thank you for the reference, it is interesting paper. As far as I understood, to the first sight, there is different kind of black hole discussed there, kind of exotic. For the usual black hole, normal one, there is positive mass inside from the beginning, as I wrote above, so, again, what you will do with that? Also, what kind of motivation for the introducing of negative mass in an interior of black hole? Namely, the main point to have the negative masses outside the black holes, does it?

Sergey Bondarenko A normal black hole is considered to contain positive mass in its interior, because the inital mass of the star is positive.

But if you consider that any mass crossing the schwarzschild radius switches its sign, a collapsing star would cross that radius at the moment of the collapse. The mass of the inital star falls into the event horizon while the event horizon grows, it does not suddenly appear around the star. That`s how you could consider negative masses in the interior. The motivation for that consideration is to solve the gravitational singularity, because negative masses should interact antigravitationally (repulsive interaction), so no singularities could form. By this, black holes could be fully explained with GR and without the need of a theory of quantum gravity, and no runaway motion takes place ever.

I know most researchers study negative mass black holes in an attempt to explain dark matter. But as you said, if you consider negative masses outside black holes, you should consider a negative mass region outside it. If you consider the electromagnetism-like interaction between masses, there is no runaway motion. But that is an arbitrary choice! That is my original question, why choose that one instead of the one that GR predicts?

Furthermore, Jean Pierre Petit`s approach tries to explain dark matter, among other things, by considering only negative masses in the "interior" of the black hole and electromagnetism-like interactions. In his model, positive masses that cross the event horizon switch their sign and they are bounced into the other metric of his bimetric model.

I assumed in the calculations that the interactions are not similar to electromagnetism, the same masses attract (whatever their sign) , different sign masses repulse (whatever their sign as well) and I assume that the inertial masses are always positive (whatever the sign of gravitational masses). In this case the negative masses do not solve the singularity problem, but as far as I understand in general it is not bad that this problem exists. Namely from the very basic quantum principles you got the minimal length and time scales, and that may be (to my opinion must be) related to teh problem of singularity in classical GR.

Based on those assumptions, Bonnor in Negative Mass in General Relativity 1989 discussed that the easiest way to consider negative masses in GR is to consider that all masses (rest, inertial, active, passive..) are either negative or positive. I dont remember exactly the problems that implies considering that intertial masses are always positive and at the same time, another mass can be negative, but you can check it on this paper.

My best whishes on the research

As far as I remember, there is no problem to take the inertial mass always positive. Anyway thank you, and good luck to you as well.

I have checked it and Bonnor only states that for the equivalence principle to hold, inertial mass should be equal to passive gravitational mass.

I do not think that this statement in the paper of Bonnor is correct. The inertial mass is a coefficient in the second Newton law, in the negative mass world there is two minus signs in the r.h.s of the equation of the motion in the gravitational field (passive and active masses), therefore overall sign in the r.h.s of the equation is the same as in the world of positive masses.

Namely, must be equal absolute values of the masses.

Interesting... I might have to read more about this. There is no doubt this topic is fascinating.

Hi Manuel,

I recently wrote a paper on precisely this subject, which was published in Mod Phys Lett A back in January. See https://arxiv.org/abs/1904.07943

The extension to general relativity is included in the longer version of the paper here: Article On the existence of exotic matter in classical Newtonian mec...

Best wishes,

Sabbir

Sabbir Rahman I understand that the prediction of general relativity is A and B interactions on your paper, just like Bonnor and Bondi predict, right?

Yes, that is correct. However in the longer paper I argue that C and D must also exist in GR.

Although I mentioned that two metrics are therefore needed, in reality only one metric is needed;- particles of class C and D just accelerate as if they have negative mass rather than positive mass in the geodetic equation (which requires that GR be generalised to allow maximal violation of the principle of equivalence).

You may find these slides interesting/helpful:

Data Exotic Matter in Classical Newtonian Mechanics and General R...