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Let's re-capitulate the versions of Mach's Principle listed by Samuel and Bondi:
Mach0: The universe, as represented by the average motion of distant galaxies, does not appear to rotate relative to local inertial frames.
Goedel's metric violates this
Mach7: If you take away all matter, there is no more space.
This is violated by all vacuum solutions, and is particularly violated by Ozsváth and Schücking's because it contains no singularities behind which matter can hide
PS: The discussion above is purely within General Relativity. When you talk about your EM ZPF you are discussing a different theory. In General Relativity there is no such thing as Quantum Vacuum. Quantum Mechanics is separate from General Relativity. If you consider the EM ZPF to have a ground state with non-zero-density or a fluctuating density around this ground state, then according to General Relativity, the EM ZPF is not a vacuum. According to General Relativity a vacuum has constant zero density. This is what is meant by Ozsváth and Schücking's solution: it is a solution for a universe with constant zero density.
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"constant zero density", are you referring to Mass Density or Energy Density?
An EM field has zero rest mass, which could be interpreted as zero "mass" density, but it has non-zero energy density. The QV has non-zero Spectral Energy Density and we know this to be true "everywhere". So there is no "vacuum solution" that agrees with QED.
It seems to be, both GR and Mach's principle are flawed and outdated, given what we know about the QV. If there is no hope to modify GR without calling it a "different theory", then I have to admit, I profess to disagree with GR. "Nothing can't be curved!"

Kinetic energy is not frame invariant, it is a function of velocity which is obviously not frame invariant. It is also not a linear function of velocity, so differences in kinetic energy also cannot be frame invariant.
The change in KE and work done can be frame invarient as follows.
Acceleration causes a change in velocity, which is frame invarient. All frames observe the same change in velocity, the same dV.
Using the dV, the change in KE can be calculated as dKE = m dV^2 / 2. Work done is then equal to dKE.I am guessing you haven't studied calculus, that is the incorrect way to take a derivative.
The correct answer is dKe/dV = m*v, which is a function of velocity, not difference in velocity, so by the third or fourth different method, it is still not frame invariant.
Please note, that at this point further "proofs" from you are pointless unless you directly point out something wrong with all of the proofs I have provided. My proofs include directly calculating kinetic energy using the equation for kinetic energy, and I also linked you to a first principles derivation of this equation.
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What about r = 0, that is to say R = Rs ?
It has been shown (reference) that a change of coordinates can eliminate this "true" singularity, because such singularity is not really "true" it is caused by a wrong choice of local topology.
The "magical" change of variable is to take:
r = Rs (1 + Log ch ρ)
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Unfortunately they (http://www.jp-petit.org/papers/cosmo/2015-ModPhysLettA.pdf) don't achieve this simply by a "magical" change in variables in a conventional metric cosmology.
There is a heavy price to pay.
It is not just topology of conventional spacetime manifolds that others were unable to see.
Instead the way they manage to eliminate the singularity is by using a very unconventional "Janus" Cosmological Model of the universe composed by positive and negative energy (and mass if they own) particles, respectively described separate metrics g(+)µν and g(−)µν , solutions of a coupled field equation system. https://forum.nasaspaceflight.com/index.php?topic=43501.msg1709505#msg1709505
Two distincts things there.
The authors indeed propose both a cosmological model involving negative mass; and a black hole model where the central singularity is eliminated with a change of variable. Both models are of course compatible (i.e. when used together) but one or the other can also exist without the other one being true.
Yes, the "diabolo topology" strongly militates for a connection of two distinct spacetimes. But it could also describe the connection of two distant regions of the same spacetime. Although the second proposition may seem problematic at first due to the lack of any observable white fountain in the universe, this is not such a problem anymore when the process is considered as ephemeral, as radial frame-dragging implies a finite (and quite fast) process.
Both the cancellation of the central singularity (using the proper change of variable) and the finite-time transit (due to radial frame-dragging) are disconnected from the Janus cosmological model roots and could still be used in other frameworks (such as the brane cosmology you and others cited many times in these threads).
So again, the change of variable can be particularly interpreted as a mass inversion process in a finite time transit through an orbifold acting as a one-way membrane triggering CPT symmetry between two spacetimes in the framework of the Janus cosmological model, after Sakharov-Souriau-Petit. But such black hole model with no central singularity is not restricted to this particular cosmological model. They just fit well together and the paper you and I are referring to (Cancellation of the central singularity of the Schwarzschild solution with natural mass inversion process indeed makes use of the two models in an interweaved way, as implied by its title).QuoteTheir model implies injection of negative energy (and mass if they own) particles in spacetime, considered as a manifold plus two metrics . They also had to assume that particles of opposite masses do not interact neither by electromagnetic forces nor strong or weak forces.
This is because gravity is different in essence (Hinted already. But you do not agree. Although there may be different interpretations?). Gravity is spacetime curvature. Gravity is not mediated by particles following geodesics of spacetime, like photons. Consider models of gravity where one (not two!) manifold includes several metrics. Not a problem for mathematicians. Let's say there are only two metrics: in physics this is a particular form of bimetric gravity (also called "bigravity"). Negative energy photons and positive energy photons follow null-geodesics of their own metric, that preclude any electromagnetic interaction. This is different for gravity. Being part of the same universe, but not the same metric, positive and negative masses induce spacetime curvature that is felt by both species:
There you can understand that a mass is not intrinsically "negative" as a mass always induces a positive curvature in its own metric. The negative nature of gravity is also a relativity concept, as a mass is felt as being "negative" only from the point of view of an observer in the opposite metric (a hypothetical red observer made of "negative" mass in the image above would feel the blue Earth, although invisible, as being a concentration of negative mass, from his own point of view). This phenomenon can be described as "conjugate curvatures" or "conjugate geometries": Positive mass here, induces negative mass there.
But the 2D didactic image of a flat bedsheet or wire mesh being distorted is incomplete, as geodesics are different when a positive or a negative curvature is considered. Positive mass locally induces positive curvature in spacetime, i.e. where the sum of angles of a triangle > 180°); although negative mass locally induces negative curvature in spacetime where the sum of angles of a triangle < 180° in such a hyperbolic space the shape of a horse saddle:
So keeping in mind that in the Janus model, positive mass there induces negative curvature in the opposite metric, therefore some matter (in yellow below) flowing from one metric to the other (in purple) through a bridge located at the center of a destabilized neutron star (Schwarzschild black hole) can be represented in this 2D animation showing the evolution of conjugate curvatures through the process:
More explanations there to prevent this message from becoming too long.
Again, you see a mass is never intrinsically negative. Topology produces only apparent and relative negative curvatures. Instead of introducing "negative mass in cosmology" the Janus model explains the nature of negative spacetime curvature as being a relative concept between two metrics, so everything follows from topology. And is still described by field equations.QuoteThere is no physical evidence for such a cosmological unconventional model composed by positive and negative energy (and mass if they own) particles, respectively described separate metrics g(+)µν and g(−)µν
So, they replace one problem deep inside the event horizon of a black hole (the singularity at r=0) with another problem (2 separate metrics, and existence of negative energy) for the whole naked Universe, -which we can see- for which we have no cosmological experimental evidence of existence (aside from a very narrow context: an interpretation of the Casimir effect).
The existence of singularities are a problem.
The introduction of negative energy in cosmology is not a problem: it is rather the non-observation of theoretically possible negative energy states that is a problem.
Although being theoretically predicted, scientists are satisfied deciding that negative mass does not exist in physics, not only because it is not observed directly, but because of the preposterous Runaway motion paradox arising in general relativity, as described by Hermann Bondi in 1957. It is easier that way, otherwise too much problem.
But the complete Poincaré group describes negative energy and negative mass particles as being real, they are simply the counterpart of all known positive energy particles: electrons, protons, neutrons, photons, neutrinos, etc. but with a negative energy. Discarding this possibility on the altar of Occam's razor is too easy. On the contrary, WIMPs superparticles following from supersymmetry (SuSy) should now be detected already. 40 years AWOL but still considered as "almost around the corner" and "the most plausible explanation for dark matter".
As for the lack of "evidence", your mileage may vary. Any respectable theory has to:
1- be mathematically sound
2- be in agreement with observations
3- make successful predictions
4- be falsifiable with experiments
If a theory agrees with these four points, it should be considered and evaluated accordingly.
I am guessing you haven't studied calculus, that is the incorrect way to take a derivative.
The correct answer is dKe/dV = m*v, which is a function of velocity, not difference in velocity, so by the third or fourth different method, it is still not frame invariant.
Please note, that at this point further "proofs" from you are pointless unless you directly point out something wrong with all of the proofs I have provided. My proofs include directly calculating kinetic energy using the equation for kinetic energy, and I also linked you to a first principles derivation of this equation.
Hi Meberbs,
Change in velocity = final velocity after acceleration - initial velocity before acceleration is frame invarient. Agree?
Therefore the work done to cause the velocity change and associated KE change of the accelerated mass is J = m dV^2 / 2, which is frame invarient.
I am guessing you haven't studied calculus, that is the incorrect way to take a derivative.
The correct answer is dKe/dV = m*v, which is a function of velocity, not difference in velocity, so by the third or fourth different method, it is still not frame invariant.
Please note, that at this point further "proofs" from you are pointless unless you directly point out something wrong with all of the proofs I have provided. My proofs include directly calculating kinetic energy using the equation for kinetic energy, and I also linked you to a first principles derivation of this equation.
Hi Meberbs,
Change in velocity = final velocity after acceleration - initial velocity before acceleration is frame invarient. Agree?
Therefore the work done to cause the velocity change and associated KE change of the accelerated mass is J = m dV^2 / 2, which is frame invarient.You are using an equation that both me and Bob012345 explained was wrong from different perspectives (I interpreted your use of variables like dv to be differentials, since that is standard notation. Bob012345 explained it from the perspective that you intended to be writing Δv = change in velocity as you just defined it, which probably was your intention. Note the first post in this thread has a link to help with writing math symbols and Greek letters)
From either perspective, it appears that you did not read our posts since you are still using the equation that we demonstrated was wrong.
Kinetic energy is not frame invariant, it is a function of velocity which is obviously not frame invariant. It is also not a linear function of velocity, so differences in kinetic energy also cannot be frame invariant.
The change in KE and work done can be frame invarient as follows.
Acceleration causes a change in velocity, which is frame invarient. All frames observe the same change in velocity, the same dV.
Using the dV, the change in KE can be calculated as dKE = m dV^2 / 2. Work done is then equal to dKE.I am guessing you haven't studied calculus, that is the incorrect way to take a derivative.
The correct answer is dKe/dV = m*v, which is a function of velocity, not difference in velocity, so by the third or fourth different method, it is still not frame invariant.
Please note, that at this point further "proofs" from you are pointless unless you directly point out something wrong with all of the proofs I have provided. My proofs include directly calculating kinetic energy using the equation for kinetic energy, and I also linked you to a first principles derivation of this equation.
Hi Meberbs,
Change in velocity = final velocity after acceleration - initial velocity before acceleration is frame invarient. Agree?
Therefore the work done to cause the velocity change and associated KE change of the accelerated mass is J = m dV^2 / 2, which is frame invarient.
All types of energy gravitate (electromagnetic energy gravitates: energy density, Maxwell stress tensor and Poynting vector are all in the stress-energy tensor and hence they count as E/c^2 , as pressure also does). The only energy not included is the gravitational potential energy.
Yes, given what is known about Quantum Mechanics a unified theory with GR is necessary to understand the Big Bang, what is inside Black Holes, probably to understand Dark Energy and many other things
The calculation of the work needed to be done on the mass to achieve the dV change is correct.
Do you mean;
d(V2) or (dV)2 ?
The way you write it is not conventional, and it is not at all obvious to be one or the other. Parenthesis do amazing things!
The calculation of the work needed to be done on the mass to achieve the dV change is correct.
Regardless of how many times you repeat the same thing, it does not become true. Your calculation is NOT correct, as has been shown many times. The work needed to be done to accelerate a mass by dv equals the change in kinetic energy, which is "KE after" minus "KE before", i.e. m*(v0+dv)^2/2 - m*v0^2/2. This cannot be approximated by m*(dv)^2/2 if v0 is non-zero.
The calculation of the work needed to be done on the mass to achieve the dV change is correct.
So far I know of 4 P-P drive types
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All feature the same effect. Propulsion without needing to exhaust mass. if you still desire to believe none of the 4 work, maybe tell Dr. Rodal the MEGA drive doesn't work.
As I see it, there are 2 ways to calc the work done by P-P drives when they accelerate mass:
1) Work = (N^2 t^2) / 2 m.
2) Work = change in KE via change in velocity.
I prefer the 1st equation as it doesn't need to know velocity.
35,000 x (1,000^2) / 2 = 17.5 x10^9 Joules of work was done by the P-P MEGA drive on the ship's mass to achieve the 1km/sec velocity change.
Using an initial velocity of another rest frame, will generate a different KE change and work done in every rest frame. As there is no fuel mass and no potential energy change of the fuel mass, nor exhaust mass nor mass exhaust velocity to adjust the KE change, how to make each frame see the same change in KE?
The last gravitational and luminal observations of merging neutron stars falsified bigravity (https://arxiv.org/abs/1710.06394 & https://arxiv.org/abs/1703.07785).
Q5. Why doesn’t the MEGA drive result in free energy, and therefore used for energy
generation?
The MEGA drive does not violate energy conservation. The kinetic energy comes from the
gravitational field, not from the electrical power applied to the device. There is a large
gravitational potential in the universe we are tapping into to gain kinetic energy of the
device. If we take a small amount of energy, practically no loss will be noticed by the
whole universe. There are far more efficient ways of extracting energy, for example, from
nuclear or solar power. Trying to extract energy from gravitation via the Mach effect is very
inefficient.
The benefit of Mach effect propulsion is to avoid carrying propellant for long space
missions, particularly for interstellar missions.
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The whole issue reminds me of a quote from, “Thinking fast and slow“, by Daniel Kahneman
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When you say with certainty that, no acceleration can occur without interacting with something outside the frustum, you are saying you know everything there is to know. If your argument had been phrased as, this is what I believe, rather than as a certainty that implies we already know all there is to know, it would have been better, the way I read your comments.
Daniel Kanehman has a Nobel Memorial Prize in Economics, "the dismal science" so called for many reasons (including the fact that one cannot have well-controlled experiments in Economics), including that this prize has the dubious distinction that it has been conferred on the same year to people having diametrically different economic opinions.
What I am saying is not (like you incorrectly claim) that I know everything (I never said that), but that I am confident enough to place a financial bet with you that it will turn out as I wrote concerning conservation of momentum, internal and external fields. Unfortunately there is no option market to place such a bet at the moment, as I think that it would be a good financially rewarding bet to place(of course it would be a matter of what would be the odds in such a market...)
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There are precisely 0 drives demonstrated with propellantless propulsion that have been conclusively demonstrated. I am also fairly certain that Rodal has never claimed that the MEGA drive works via true propellantless propulsion.
Using an initial velocity of another rest frame, will generate a different KE change and work done in every rest frame. As there is no fuel mass and no potential energy change of the fuel mass, nor exhaust mass nor mass exhaust velocity to adjust the KE change, how to make each frame see the same change in KE?You are finally starting to see the problem. There is no way to make a true propellantless propulsion obey conservation of energy, since the same work will generate a different kinetic energy in every frame, and there is no propellant to balance this. Your repeated attempts to do so simply result in you using equations that simply give wrong and inconsistent answers.
The last gravitational and luminal observations of merging neutron stars falsified bigravity (https://arxiv.org/abs/1710.06394 & https://arxiv.org/abs/1703.07785).
Let's make it clear that there are two kinds of bimetric gravity theories. "Bigravity" in these papers is about massive gravitons; whereas the "negative mass bigravity" I talked previously is about two coupled metrics with conjugate geometries in a single Riemannian manifold.
There are precisely 0 drives demonstrated with propellantless propulsion that have been conclusively demonstrated. I am also fairly certain that Rodal has never claimed that the MEGA drive works via true propellantless propulsion.
I'd say some have been shown fairly convincingly if not yet conclusively. But what exactly constitutes a 'true' PP drive? Are you basically saying a 'true' PP drive violates physics or that if it can be explained by an interaction with the universe or some such thing, it's not really a PP drive. Is that part of your basic operating definition? What would you call a working EMdrive, a pseudo, a virtual or an effective PP?
Using an initial velocity of another rest frame, will generate a different KE change and work done in every rest frame. As there is no fuel mass and no potential energy change of the fuel mass, nor exhaust mass nor mass exhaust velocity to adjust the KE change, how to make each frame see the same change in KE?You are finally starting to see the problem. There is no way to make a true propellantless propulsion obey conservation of energy, since the same work will generate a different kinetic energy in every frame, and there is no propellant to balance this. Your repeated attempts to do so simply result in you using equations that simply give wrong and inconsistent answers.
In answer to both, there is another way. The total change in KE if you include the 'exhaust', however that is interpreted, is invariant but the change in ship KE depends on observer if you include both parts I showed. The 'exhaust' comes from realizing that any force truly operating from within a reference frame is acting like that reference frame has infinite inertia. You can model the situation by assuming the EMDrive conserves momentum with a really really big mass, the reference frame it's in, and let that mass go to infinity in the limit which balances everything out for all observers.
Propellantless propulsion by definition is something accelerating without pushing on anything else. This by definition violates Newton's third law and conservation of momentum, and it is simple to go from there and show it also violates conservation of energy.
I can't answer your question of what to call a working emDrive without knowing how it works. It is much more likely that a working emDrive pushes on something unknown than violates conservation of momentum. It is much more likely than either of those that there is no such thing as a working emDrive, and it is just an experimental artifact.