From: intuitionist1@gmail.com   
      
   On Friday, May 4, 2018 at 10:59:21 AM UTC+3, Steven Carlip wrote:   
   > But your argument is wrong, too.  According to the equivalence   
   > principle, the trajectory of an object in a gravitational field   
   > is independent of its mass.  In a vacuum, a hammer falls with the   
   > same acceleration as a feather; a negative mass hammer would do   
   > the same.  It's simply not true that a negative mass particle   
   > would be repelled by a black hole, in either general relativity   
   > or Newtonian gravity.   
   >   
   > It *is* true that a particle, of positive or negative mass, would   
   > be repelled by a negative mass black hole.  But that's irrelevant   
   > to Hawking radiation, and it's probably irrelevant to everything,   
   > since there's no reason to believe that negative mass black holes   
   > can exist.   
   >   
   > Steve Carlip   
      
   Hi Steve,   
      
   I was just browsing the group and came across this message where you   
   make some statements that I think need some kind of clarification.   
      
   I think we need to be careful when we talk about positive and negative   
   mass even in Newtonian mechanics, as there are three different types,   
   namely:   
      
     (i) inertial mass,   
    (ii) active gravitational mass , and   
   (iii) passive gravitational mass   
      
   For normal matter, these are all positive and of the same magnitude. But   
   let's suppose we allow for the existence of exotic types of matter for   
   which the sign of each of these types of mass can be either positive or   
   negative (let's assume for simplicity that the magnitude of each of each   
   of the types of mass is the same for a given particle, so that only the   
   sign changes for each mass type).   
      
   Then if we let the magnitude of a given particle's mass be 'm', and let   
   the sign of the inertial, active and passive masses be I, A and P   
   respectively, i.e. each can be +1 or -1.   
      
   Then there can in principle be 8 particle types each labelled by the   
   triple (I,A,P), namely of type (+,+,+), (+,+,-), (+,-,+), (+,-,-),   
   (-,+,+), (-,+,-), (-,-,+), and (---).   
      
   Now, suppose there are two particles of absolute mass m1, m2 and of type   
   (I1,A1,P1) and (I2,A2,P2). Then according to Newton's law of   
   gravitation, the force on particle 1 due to particle 2 will be:   
      
   F12 = I1 m1 a1 = G P1 m1 A2 m2 / |r12|^2 in the direction r12   
      
   where r12 is the displacement from particle 1 to particle 2. The   
   direction of acceleration of particle 1 will therefore depend upon the   
   produce I1 P1 A2, and in particular, a particle of type (I,A,P) will   
   have precisely the same interactions as a particle of type (-I,A,-P),   
   and so we can simplify the situation by defining 4 'classes' of   
   particle, namely 'A', 'B', 'C' and 'D' as follows:   
      
   Class A: (+,+,+) or (-,+,-)   
   Class B: (+,+,-) or (-,+,+)   
   Class C: (+,-,+) or (-,-,-)   
   Class D: (+,-,-) or (-,-,+)   
      
   I have used 'A' twice here but hopefully it will be clear from the   
   context whether I am talking about the sign of the active mass or the   
   particle class.   
      
   Note that we can, if we wish, restrict ourselves only to (i) particles   
   with positive inertial mass, OR (ii) particles with equal active and   
   passive gravitational masses, and still have one particle type from each   
   class.   
      
   We can then produce a little table of all the possible interactions   
   between the different particle types as follows:   
      
     A B C D   
   A + + - -   
   B - - + +   
   C + + - -   
   D - - + +   
      
   The rows refer to the class of particle 1, and the columns refer to the   
   class of particle 2, and the sign in the table is positive if particle 1   
   is attracted to particle 2, and negative if it is repelled by it. Thus   
   for example, particles of class A are are attracted to particles of   
   class B but particles of class B are repelled by particles of class A.   
   [Of course this particular case means that the two particles will   
   accelerate off to infinity together if there were no other particles   
   around, and I am aware that some physicists have found this   
   objectionable, but let us be open-minded here and allow for that   
   possibility, because interesting things can happen if the universe is   
   closed and/or there are lots of particles around of different types].   
      
   Now, it is clear from the table that if there are lots of particles of   
   class A (i.e. normal matter, or exotic matter of type (-,+,-)) then   
   these will cluster and eventually condense to form black holes.   
      
   This will also be true of particles of class D (i.e. matter of type   
   (+,-,-) which I think is what you are referring to as 'negative mass'   
   particles, or exotic matter of type (-,-,+)).   
      
   I am aware that some physicists have claimed that some of these   
   interactions do not conserve momentum or do not satisfy Newton's third   
   law, or defy Einstein's principle of equivalence, etc, but I personally   
   would be very interested to know whether either Newtonian gravitation or   
   general relativity would actually become 'broken' if any or all of these   
   particle types were to exist, or, if indeed they are, whether they could   
   not be generalised in some way to allow for each of these particle   
   types.   
      
   With regards to momentum or energy non-conservation, it is clear that if   
   the momentum of a particle is defined as the product of the inertial   
   mass and the velocity, i.e. p=Imv, then this will not be conserved for   
   some of these interactions. However, there is a conserved momentum-like   
   quantity and energy-like quantity which is conserved by all of the   
   interactions, namely:   
      
   Momentum: p = IAP*mv   
   Kinetic energy: E = (1/2)*IAP*mv^2   
      
   I.e. by including the sign factor IAP in the definition of a particle's   
   momentum and energy, then all of the interactions above conserve both,   
   and thus cannot be ruled out as unphysical priori on this basis alone.   
      
   This leads to one of the points I would like to make;- If we have a   
   massive body formed by condensation of particles of class A (e.g. normal   
   matter), then there *can* in principle be matter that is repelled by the   
   corresponding field. In particular, it is true for particles of class D   
   (i.e. those which I think you are referring to as having 'negative   
   mass'), and this contradicts the general statement that you have made   
   above.   
      
   Similarly, if we have a massive body formed by condensation of particles   
   of class D (i.e of negative mass'), then these do not repel all particle   
   types - they actually attract other particles of class D (i.e. of   
   negative mass), again contradicting the statement you have made above.   
      
   Some kind of clarification, then, of your statements would be   
   appreciated.   
      
   Also, I do not believe that any issues would arise if all 8 types of   
   particle were to exist in Newtonian mechanics. My question then is this:   
      
   Is there any reason why all 8 types of particle cannot exist in (if   
   necessary, a suitably generalised version of) classical general   
      
   [continued in next message]   
      
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