Dr.Rodal -Frobnicat seems far more nimble than me at this stuff, but since you ask for some mathematical critique, I think I owe you a stab.There are three frames involved: frame 0, where the object is initially at rest. Frame 1, where it has acquired its initial velocity, and frame 2, where it has acquired its final velocity.The total mass-energy is thus m0c^2, and does not vary between frames: in fact I think your equationm1V1/gamma1 = m2V2/gamma2 should be replaced by the conservation of the norm of the energy momentum vector, vis:m0^2c^4 =gamma1^2*m1^2*c^4-gamma1^2*m1^2*v1^2 =gamma2^2*m2^2*c^4-gamma2^2*m2^2*v2^2[I may or may not have this right, but there will definitely be mass-energy terms mixed with the momentum terms.] I'd take a stab that this is the kind of covariant formulation Frobnicat eluded to above. I won't comment on later parts of the analysis, since if my comment is correct the rest would not follow.R.

.../...where, in the above equation and in the ones to follow, m is the rest mass m=m_{o}, the mass of an object in its rest frame. Also, as already discussed here: http://forum.nasaspaceflight.com/index.php?topic=39214.msg1488362#msg1488362 the InitialVelocity must be measured with respect to the same frame where the InitialMass of the object was measured. This is an acceleration problem, hence the frame where the rest mass is measured is a privileged, non-inertial frame. If other frames of reference are used, not only the Initial Velocity will be different, but the Initial Mass will be different too, if measured in any frame other than the object's initial frame of reference to measure its mass. .../...

...I don't see how you can start with a singleton lump closed system that's supposed to respect conservation of energy and conservation of mass in the framework of SR...

Dr Rodal, you insist that you are here doing a study of a closed system composed of a single lump of mass, i.e. that there is no separation (separation in 2 parts from a single part, as would be the case for action/reaction) nor joining ("melting" of 2 parts in a single part, as would be the case for an inelastic collision/aggregation) nor bounce (2 parts exchanging momentum but still being separate before and after such interaction). Do I understand correctly your premise ?...All right so we are not using so called relativistic mass m_{rel}, and by avoiding the traps of m_{rel} we are conforming to prescriptions of modern physics teaching. Fine with me. ...

...m^{2} c^{4} = E^{2} - p^{2} c^{2} ....

Hi,I did not follow the whole conversation during last few days. Why is a variable mass related to any force toward the small end of the cavity? If there IS a lower or higher mass related to the EM field inside the resonator what is that meaning? The smaller diameter side of the cavity itself have had a lower mass because it consists of less volume of copper as the larger side. Also the earth gravity is almost homogeneous over the size of the cavity. How it can generate a thrust in shifting the center of the mass of itself? A slightly other force composition in relation to the gravity field around ok but thrust generation? IMHO Only a negative mass value would explain thrust generation against the background gravity field at all (repulsive energy). The direction of this force would be against the attractive gravity of the biggest mass nearby: the earth it self.I think I have to study your the last posts.

..Quote...In order to have a non-dimensional scalar field φ of values around unity, in expression(1) the constant G0 representing Newton gravitational constant is included...Based on this statement in section 2. of the paper there is a (weak)coupling to the gravity and if so there is also coupling on the earth gravity field for this scalar. And yes gravity is a very weak force in comparison to the other forces inclusive electromagnetism. I read the paper one or two years ago and it's an impressive idea. I have to read it again to follow your statements but its hard stuff and it will take a while.

...In order to have a non-dimensional scalar field φ of values around unity, in expression(1) the constant G0 representing Newton gravitational constant is included...

4) Minotti's paper predicts, that for copper wall thickness ~1 mm, the thicker the copper (as long as significantly greater than the skin depth), the greater the force.

Ok, trying to wrap what's left of my mind around this:Quote4) Minotti's paper predicts, that for copper wall thickness ~1 mm, the thicker the copper (as long as significantly greater than the skin depth), the greater the force.So, say you have two EM Drive units that are identical, except one has 'skin depth' of 1 mm and the other has 'skin depth' of say 3 mm. According to this theory, the second device should perform significantly better. Is that correct? If so, this appears to be something within the capabilities of our DIY crowd.But...1 - would the increased weight of the device with the thicker skin offset the thrust measurements? (I suspect I am missing something glaringly obvious here.)2 - Does the entire skin need to be thicker, or just the end plates?

Assuming a cavity with thin walls (but much thicker than the penetration depth ,in order to the boundary conditions used to be correct) of mass surface density ...There are no details in the literature as to the precise dimensions of the cavitiesused in the experiments, so that an example roughly similar to the overall dimensionreported and with the proportions observed in the published photographs will be used.Assuming a wall of thickness 1 mm, and a copper mass density of 8.9 × 103 kg/m3, wehave = 8.9 kg/m2.

.../...Variable mass, implying the need for negative mass to self-accelerate, addresses both conservation of momentum and it also addresses conservation of energy.Energy is conserved, and such a propulsion device is not a free-energy machine, because the greater the speed, the lower the mass. More on that later...(The practical problem of course is that up to now, nobody has found experimental evidence of negative mass )

.../...4) The EM Drive is a closed system, in which case the only way I see to conserve momentum-energy for acceleration of the EM Drive is to have creation of negative mass-energy in the EM Drive.../...

Dr Rodal, thanks for answering my comments about your work around "negative mass" creation and conservation of momentum. It was clear from the equations that you considered only conservation of momentum and didn't consider conservation of total energy, but from early posts (on the main EM drive thread) I had the impression that you were assuming some compatibility with conservation of energy, for instance (bold added by me for emphasis) :Quote from: Rodal on 02/03/2016 01:27 PM.../...Variable mass, implying the need for negative mass to self-accelerate, addresses both conservation of momentum and it also addresses conservation of energy.Energy is conserved, and such a propulsion device is not a free-energy machine, because the greater the speed, the lower the mass. More on that later...(The practical problem of course is that up to now, nobody has found experimental evidence of negative mass )

...I see two different problems. Possibility of negative mass, that is Weak Energy Condition breaking, is one thing. But modification of rest (invariant) mass in a closed system without a balanced counterpart in changes in kinetic energy is another one, it looks more like conservation of (total) energy breaking, whether the change appears as + or - mass. As an example of the difference between those 2 hypothesis : we can apply usual SR for conservation of energy and momentum when considering a single particle of mass m1>0 splitting into two particles of mass m2>m1 and m3<0. The hypothesis of the existence of particle of mass<0 doesn't change the equations. But your approach seems to ignore one equation (conservation of energy) and hence gives the system a degree of freedom absent of this initial example. Negative mass deltas is then not specifically implicated, and indeed your solution space also shows positive (unbalanced) delta mass, i.e. actually either positive or negative total energy evolutions.Your latest answer clarifies this as citing a scalar field or supplementary spatial dimension as required to make sense of such mass variation... I would have a hard time following in detail Minotti (or you following Minotti) on such topic outside of usual SR application, but maybe could understand a few words of how, one way or another, energy is conserved in the end to say that the approach "conserves momentum-energy" ?...

Is the concept of a time-dependent, variable "rest-mass" (mass that would be measured at rest), allowed by the following theories?1) Special Relativity: NO. The rest mass cannot be a function of the time coordinate, as Lorentz covariance would not be preserved. Variation of rest mass cannot be due to kinematic (velocity or position) evolution....

Is the concept of a time-dependent, variable "rest-mass" (mass that would be measured at rest), allowed by the following theories?...2) Einstein's General Relativity (GR): It appears as NO. It appears that a time-dependent variable mass would give rise to a time-dependent Energy Stress tensor, solely due to the mass variability with time, which is not consistent with Einstein's GR theory of gravitation. Also it appears that Relativity's Energy-Momentum equation:m^{2} c^{4} = E^{2} - p^{2} c^{2} may prevent general time-dependent variable mass.Note: I need to review how does Woodward accommodate variable negative mass-energy in his theory (which I understand he states is consistent with Einstein's General Relativity), as it appears to me that there should be an issue with the time-dependent Energy-Stress tensor and with the Energy-Momentum equation in such a theory....

Quote from: frobnicat on 02/10/2016 03:31 PMDr Rodal, thanks for answering my comments about your work around "negative mass" creation and conservation of momentum. It was clear from the equations that you considered only conservation of momentum and didn't consider conservation of total energy, but from early posts (on the main EM drive thread) I had the impression that you were assuming some compatibility with conservation of energy, for instance (bold added by me for emphasis) :Quote from: Rodal on 02/03/2016 01:27 PM.../...Variable mass, implying the need for negative mass to self-accelerate, addresses both conservation of momentum and it also addresses conservation of energy.Energy is conserved, and such a propulsion device is not a free-energy machine, because the greater the speed, the lower the mass. More on that later...(The practical problem of course is that up to now, nobody has found experimental evidence of negative mass )1) The above statement that negative mass addresses the conservation of energy problem is correct in the specific instance of constant zero initial lumped mass.

Specifically, the initial condition of lumped rest mass=0, staying constant (for example as a result of equal magnitude negative mass as positive mass) answers all your previous posts regarding conservation of energy, and it does so trivially, as the kinetic energy is zero for zero mass, and you cannot use the EM Drive as a generator if it has effective zero inertial mass. (Effectively I know that even photons, with zero rest mass have energy, so if you want you can throw an expression with the Plank constant there, and replace zero kinetic energy with very very small kinetic energy).

If my memory is correct, in your consideration of energy conservation you never considered that the rest mass could be zero.

You assumed (unstated) that the rest mass was greater than zero.

The concept which you addressed in your energy conservation statements, the EM Drive, has been discussed by Dr. White and by Dr. Minotti as involving negative energy-mass, so one should discuss the consequences of such negative energy-mass in considerations of energy conservation, instead of ignoring it, and assuming as in your considerations, that the energy-mass was positive, and constant. In other words, your energy considerations for the EM Drive, ignore the premise of these authors.

2) The argument that <<nobody has found experimental evidence of negative mass >> is a non-starter in this discussion ...

... because: eminent physicists like Kip Thorne, Hawking and others have discussed negative mass (to stabilize wormholes for example), so there is no "shame" in theoretically considering negative mass. As to experimental evidence, whether Casimir effect and other types of negative energy can indeed be considered experimental evidence of negative energy is up for discussion, but again eminent physicists (for example in discussion of stabilization of wormholes posit the Casimir energy as the means for the negative energy).

And again, in your considerations of conservation of energy you are dealing with a concept where some authors (Dr. White and Dr. Minotti) explicitly state that they are considering negative energy !

Therefore, since your conservation of energy considerations ignore negative energy-mass, your considerations of energy conservation seem to be inapplicable to the concepts advanced by Dr. White and Dr. Minotti.

Quote from: frobnicat on 02/10/2016 03:31 PM...I see two different problems. Possibility of negative mass, that is Weak Energy Condition breaking, is one thing. But modification of rest (invariant) mass in a closed system without a balanced counterpart in changes in kinetic energy is another one, it looks more like conservation of (total) energy breaking, whether the change appears as + or - mass. As an example of the difference between those 2 hypothesis : we can apply usual SR for conservation of energy and momentum when considering a single particle of mass m1>0 splitting into two particles of mass m2>m1 and m3<0. The hypothesis of the existence of particle of mass<0 doesn't change the equations. But your approach seems to ignore one equation (conservation of energy) and hence gives the system a degree of freedom absent of this initial example. Negative mass deltas is then not specifically implicated, and indeed your solution space also shows positive (unbalanced) delta mass, i.e. actually either positive or negative total energy evolutions.Your latest answer clarifies this as citing a scalar field or supplementary spatial dimension as required to make sense of such mass variation... I would have a hard time following in detail Minotti (or you following Minotti) on such topic outside of usual SR application, but maybe could understand a few words of how, one way or another, energy is conserved in the end to say that the approach "conserves momentum-energy" ?...In the above post: http://forum.nasaspaceflight.com/index.php?topic=39214.msg1489632#msg1489632, I already addressed the fact that I think that variable rest mass is incompatible with Special Relativity, and so it is perplexing why you are bringing Special Relativity (assuming this is what you mean by "SR") into the picture again, as if demanding that Special Relativity should be obeyed.I thought this was clear:Quote from: Rodal on 02/09/2016 05:30 PMIs the concept of a time-dependent, variable "rest-mass" (mass that would be measured at rest), allowed by the following theories?1) Special Relativity: NO. The rest mass cannot be a function of the time coordinate, as Lorentz covariance would not be preserved. Variation of rest mass cannot be due to kinematic (velocity or position) evolution....

As to conservation of energy, I thought that this was also clear:Quote from: Rodal on 02/09/2016 05:30 PMIs the concept of a time-dependent, variable "rest-mass" (mass that would be measured at rest), allowed by the following theories?...2) Einstein's General Relativity (GR): It appears as NO. It appears that a time-dependent variable mass would give rise to a time-dependent Energy Stress tensor, solely due to the mass variability with time, which is not consistent with Einstein's GR theory of gravitation. Also it appears that Relativity's Energy-Momentum equation:m^{2} c^{4} = E^{2} - p^{2} c^{2} may prevent general time-dependent variable mass.Note: I need to review how does Woodward accommodate variable negative mass-energy in his theory (which I understand he states is consistent with Einstein's General Relativity), as it appears to me that there should be an issue with the time-dependent Energy-Stress tensor and with the Energy-Momentum equation in such a theory....Also it is perplexing why you keep bringing up conservation of energy for a variable mass problem in a closed system, when it was stated repeatedly that any such variable mass in a closed system to make sense it would involve creation (or destruction) of negative energy-mass.

As to how variable mass can be addressed in scalar-tensor theories of gravitation, this has already been done prior to my posting. I'll try to find the references... In your discussion of conservation of energy in the EM Drive: you disregard the fact that authors like Dr. White and Dr. Minotti posit a solution that involves negative energy-mass, and instead you insist in considerations of energy conservation that involve the assumption of constant positive energy-mass, an assumption in contradiction with the assumptions of the authors of the concept (EM Drive) you are addressing in your consideration.Rather than insisting on obeying Special Relativity and constant energy-mass, when discussing the EM Drive concept, it seems to me that it is better for me (and you too) to address the fact that the authors (Dr. White and Dr. Minotti) posit negative energy-mass, rather than disregarding the author's assumptions.

...This is a lot of trouble just for asserting that it could conceivably be a closed system when, given the phenomenology of the considered equations, all indicates that it is open....