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#20 by Rodal on 13 Jan, 2016 19:10
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I verified numerically what I wrote above, using the exact solution for a truncated cone in terms of spherical Bessel and associated Legendre functions, using Wolfram Mathematica,
Numerical verification analysis details
NASA test without dielectric insert
Input
bigDiameter = (11.01 inch)*(2.54 cm/inch)*(1 m/(100 cm));
smallDiameter = (6.25 inch)*(2.54 cm/inch)*(1 m/(100 cm));
axialLength = (9 inch)*(2.54 cm/inch)*(1 m/(100 cm));
Material: Copper alloy 101; resistivity = 1.71*10^(-8) ohm meter
Exact solution output
TE012 natural frequency = 2.16467 GHz
TE012 skin depth = 1.41457 micrometers
TE012 Q = 78,642.4
10x larger
Input
bigDiameter = (110.1 inch)*(2.54 cm/inch)*(1 m/(100 cm));
smallDiameter = (62.5 inch)*(2.54 cm/inch)*(1 m/(100 cm));
axialLength = (90 inch)*(2.54 cm/inch)*(1 m/(100 cm));
Material: Copper alloy 101; resistivity = 1.71*10^(-8) ohm meter
Exact solution output
TE012 natural frequency = 0.216467 GHz
TE012 skin depth = 4.43121 micrometers
TE012 Q = 251,049.
frequency scaling: (2.1646723144342628`*^9/2.1646723144342667`*^8)/10 =1.
Q scaling: (78642.44767279371`/251049.34868706256`)*Sqrt[10] = 0.990599
1/10 of original size
Input
bigDiameter = (1.101 inch)*(2.54 cm/inch)*(1 m/(100 cm));
smallDiameter = (0.625 inch)*(2.54 cm/inch)*(1 m/(100 cm));
axialLength = (0.9 inch)*(2.54 cm/inch)*(1 m/(100 cm));
Material: Copper alloy 101; resistivity = 1.71*10^(-8) ohm meter
Exact solution output
TE012 natural frequency = 21.6467 GHz
TE012 skin depth = 0.443121 micrometers
TE012 Q = 25,104.9
frequency scaling: (2.1646723144342628`*^9/2.164672314434267`*^10)*10 =1.
Q scaling: (78642.44767279371`/25104.934868706456`)/Sqrt[10] = 0.990599
We confirm:
when using the exact solution for resonance of a frustum of a cone, for constant resistivity and magnetic permeability of the interior wall of the cavity and for constant geometrical ratios, constant medium properties μr,εr, and for the same mode shape TE012:
* the frequency scales (exactly) like the inverse of any geometrical dimension
* therefore the skin depth scales (exactly) like the square root of any geometrical dimension
* the quality of resonance (Q) scales approximately like the square root of any geometrical dimension, within 1% accuracy
The 1% error is due to this approximation:
(∫ElectromagneticEnergy dV/ ∫ ElectromagneticEnergy dA) ~
~ (ElectromagneticEnergy/ElectromagneticEnergy) (∫dV/ ∫ dA )
~ InteriorVolume/InteriorSurfaceArea
~ π R2L/(2 π R (R+L) )
~ R/(2(1+R/L))
approximating the behavior of the electromagnetic mode shape as being almost constant throughout the cavity (this approximation is pretty good for a low mode like TE012 but is expected to degrade if one considers higher modes) -
#21 by SeeShells on 13 Jan, 2016 22:14
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VERY NICE!
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#22 by DaCunha on 14 Jan, 2016 12:22
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Highly interesting. So it is proven that the dielectric insert is essential.
But this makes it difficult to understand the optical path length changes.
Why should there be any significant change in path length, if the effect can be attributed to inertial mass fluctuations between the cavity ends?
Has there been any computation of the equivalent mass distribution that would, according to General Relativity , make for the path length change measured by Eagleworks ?
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#23 by Rodal on 15 Jan, 2016 18:55
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...
These are great news.
I came to nearly the same conclusion some time last year(Q=79011). I never post it, at least I am not sure about the formula (found an approximation in an cern paper about cavities if my memory is correct) and my implementation. No -3dB bandwidth needed for the calculation, its mode,volume, conductivity dependent.
Based on this the Q at larger volumes is in general (mode dependent) bigger than for smaller volume. I think more energy can be stored in larger volumes.
If I try to use to divide all dimensions by a factor of 10, I get a 10 times higher resonant frequency (good so far) but a Q of only 24985. Could you so kind to check this please, I can feel something may still wrong with this calculation although the number for the original dimensions fits yours very well.
Please notice that when reducing the size to 1/10 of the original size, you calculate a Q=24,985 down from the original Q=79,011
So that the scaling you calculate is:
X-Ray calculated Q scaling: (79,011/24,985)/Sqrt[10] = 1.0000
which goes exactly like the square root of the dimension, instead of my calculation (here: https://forum.nasaspaceflight.com/index.php?topic=39214.msg1474351#msg1474351 ) using the exact solution:
Rodal calculated Q scaling: (78642.44767279371`/25104.934868706456`)/Sqrt[10] = 0.990599
showing that the exact solution differs by 1% from the approximate rule of Q scaling like the square root.
I justify the 1% difference between the exact solution for Q and the approximation involved in the scaling calculations for Q, as due to the approximation of the energy integral in my discussion of Q scaling:QuoteThe 1% error is due to this approximation:
(∫ElectromagneticEnergy dV/ ∫ ElectromagneticEnergy dA) ~
~ (ElectromagneticEnergy/ElectromagneticEnergy) (∫dV/ ∫ dA )
~ InteriorVolume/InteriorSurfaceArea
~ π R2L/(2 π R (R+L) )
~ R/(2(1+R/L))
approximating the behavior of the electromagnetic mode shape as being almost constant throughout the cavity (this approximation is pretty good for a low mode like TE012 but is expected to degrade if one considers higher modes)
QUESTION to X-Ray; are you approximating the energy integral calculation in your Q calculation as above, and is that why your calculation results in perfect scaling of Q going like the square root of the dimension ? -
#24 by X_RaY on 15 Jan, 2016 19:23
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It scales like the square root because it is the square root...
These are great news. I came to nearly the same conclusion some time last year(Q=79011). I never post it, at least I am not sure about the formula (found an approximation in an cern paper about cavities if my memory is correct) and my implementation. No -3dB bandwidth needed for the calculation, its mode,volume, conductivity dependent.
Based on this the Q at larger volumes is in general (mode dependent) bigger than for smaller volume. I think more energy can be stored in larger volumes.
If I try to use to divide all dimensions by a factor of 10, I get a 10 times higher resonant frequency (good so far) but a Q of only 24985. Could you so kind to check this please, I can feel something may still wrong with this calculation although the number for the original dimensions fits yours very well.
Please notice that when reducing the size to 1/10 of the original size, you calculate a Q=24,985 down from the original Q=79,011
So that the scaling you calculate is:
X-Ray calculated Q scaling: (79,011/24,985)/Sqrt[10] = 1.0000
which goes exactly like the square root of the dimension, instead of my calculation (here: https://forum.nasaspaceflight.com/index.php?topic=39214.msg1474351#msg1474351 ) using the exact solution:
Rodal calculated Q scaling: (78642.44767279371`/25104.934868706456`)/Sqrt[10] = 0.990599
showing that the exact solution differs by 1% from the approximate rule of Q scaling like the square root.
I justify the 1% difference between the exact solution for Q and the approximation involved in the scaling calculations for Q, as due to the approximation of the energy integral in my discussion of Q scaling:QuoteThe 1% error is due to this approximation:
(∫ElectromagneticEnergy dV/ ∫ ElectromagneticEnergy dA) ~
~ (ElectromagneticEnergy/ElectromagneticEnergy) (∫dV/ ∫ dA )
~ InteriorVolume/InteriorSurfaceArea
~ π R2L/(2 π R (R+L) )
~ R/(2(1+R/L))
approximating the behavior of the electromagnetic mode shape as being almost constant throughout the cavity (this approximation is pretty good for a low mode like TE012 but is expected to degrade if one considers higher modes)
QUESTION to X-Ray; are you approximating the energy integral calculation in your Q calculation as above, and is that why your calculation results in perfect scaling of Q going like the square root of the dimension ?
And yes for higher quantum number the calculated values will be problematic, not ad hoc obviously but in relation to field simulations.
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#25 by Rodal on 15 Jan, 2016 19:30
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...
These are great news. I came to nearly the same conclusion some time last year(Q=79011). I never post it, at least I am not sure about the formula (found an approximation in an cern paper about cavities if my memory is correct) and my implementation. No -3dB bandwidth needed for the calculation, its mode,volume, conductivity dependent.
Based on this the Q at larger volumes is in general (mode dependent) bigger than for smaller volume. I think more energy can be stored in larger volumes.
If I try to use to divide all dimensions by a factor of 10, I get a 10 times higher resonant frequency (good so far) but a Q of only 24985. Could you so kind to check this please, I can feel something may still wrong with this calculation although the number for the original dimensions fits yours very well.
Please notice that when reducing the size to 1/10 of the original size, you calculate a Q=24,985 down from the original Q=79,011
So that the scaling you calculate is:
X-Ray calculated Q scaling: (79,011/24,985)/Sqrt[10] = 1.0000
which goes exactly like the square root of the dimension, instead of my calculation (here: https://forum.nasaspaceflight.com/index.php?topic=39214.msg1474351#msg1474351 ) using the exact solution:
Rodal calculated Q scaling: (78642.44767279371`/25104.934868706456`)/Sqrt[10] = 0.990599
showing that the exact solution differs by 1% from the approximate rule of Q scaling like the square root.
I justify the 1% difference between the exact solution for Q and the approximation involved in the scaling calculations for Q, as due to the approximation of the energy integral in my discussion of Q scaling:QuoteThe 1% error is due to this approximation:
(∫ElectromagneticEnergy dV/ ∫ ElectromagneticEnergy dA) ~
~ (ElectromagneticEnergy/ElectromagneticEnergy) (∫dV/ ∫ dA )
~ InteriorVolume/InteriorSurfaceArea
~ π R2L/(2 π R (R+L) )
~ R/(2(1+R/L))
approximating the behavior of the electromagnetic mode shape as being almost constant throughout the cavity (this approximation is pretty good for a low mode like TE012 but is expected to degrade if one considers higher modes)
QUESTION to X-Ray; are you approximating the energy integral calculation in your Q calculation as above, and is that why your calculation results in perfect scaling of Q going like the square root of the dimension ?
Yes, that shows that your scaling factor is exactly Sqrt[L], for example for scaling by a factor of 10, Sqrt[10] = 3.16227.
Your formula for Q uses the same approximation I used in the scaling !
Should be fine for low modes particular for TE01 modes (constant in m, and only a small variation in n).
Thank you so much for sharing your calculation and making this clear (No secret magic )
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#26 by Rodal on 16 Jan, 2016 20:07
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EDITED: work under review. Will post more results
________________________________________________
As Tajmar disclosed that the tested fustrum of a cone had spherical ends (in a number of e-mails with Flux_Capacitor), and he did not provide a drawing clarifying what is the technical meaning of the dimensions he provided, it is not possible to have a unique interpretation of the dimensions of his resonant cavity: there are several possible interpretations of "height" of a frustum of a cone with spherical ends.
I will show results for two likely interpretations to what Tajmar refers as "height" of the fustrum of a cone with spherical ends:
A) The lateral length of the conical walls of the fustrum of a cone, from the small end to the big end
B) The length (from the small end to the big end) measured perpendicular to the lines defining the small and big diameters of the frustum of a cone.
All cases will assume that the small radius and big radius are the correct internal dimensions of the diameters divided by 2:
bigR = 0.0541 meter;
smallR = 0.0385 meter;
where the height given in the latest version of his AIAA paper, is assumed to be 1/2 the internal height so that the actual height is assumed twice that value (2*0.0686 meter)
This is the geometry defining the spherical radii r1, r2 and the halfconeangle "θ"
A1) Height assumed to mean the lateral length of the conical walls of the frustum of a cone, from the small end to the big end, where the height given in the latest version of his AIAA paper, (2*0.0686 meter) is assumed to be the internal height. Then, we have:
bigR = 0.0541 meter;
smallR = 0.0385 meter;
axialLength = 2*0.0686 meter
halfAngleConeRadians = ArcSin[(bigR - smallR)/axialLength];
halfAngleConeDegrees = (180/Pi)*halfAngleConeRadians
= 6.5288 degrees
r1 = axialLength/((bigR/smallR) - 1)
= 0.338603 meter
r2 = axialLength/(1 - (smallR/bigR))
= 0.475803 meter
(r2 - r1)/2 = 0.0686 meter (as given)
EXACT SOLUTION
first natural frequency (mode shape TM010 (*)) = 2.36611 GHz
Q (resistivity = 1.71*10^(-8) ohm-meter (*Material: Copper alloy 101*)) =34515.3
second natural frequency (mode shape TM011 (*)) = 2.85355 GHz
_________________________________________________________________
B1) Height assumed to mean the length (from the small end to the big end) measured perpendicular to the lines defining the small and big diameters of the frustum of a cone, where the height given in the latest version of his AIAA paper, (2*0.0686 meter) is assumed to be the internal height. Then, we have:
bigR = 0.0541 meter;
smallR = 0.0385 meter;
axialLength = 2*0.0686 meter
halfAngleConeRadians = ArcTan[(bigR - smallR)/axialLength];
halfAngleConeDegrees = (180/Pi)*halfAngleConeRadians
= 6.48682 degrees
r1 = smallR /Sin[halfAngleConeRadians]
= 0.340784 meter
r2 = bigR /Sin[halfAngleConeRadians]
= 0.478868 meter
(r2 - r1)/2 = 0.069042 meter
EXACT SOLUTION
first natural frequency (mode shape TM010 (*)) = 2.35096 GHz
Q (resistivity = 1.71*10^(-8) ohm-meter (*Material: Copper alloy 101*)) = 34626.3
second natural frequency (mode shape TM011 (*)) = 2.83528 GHz
_________________________________________________________________
(*) The first mode shape in a truncated cone is NOT constant in the longitudinal direction. We label it as TM010 here following the convention in these threads of calling the mode shape closest to the one in a cylindrical cavity, but it should be understood that TM010 electromagnetic fields vary in the longitudinal direction
(**) The theoretical Q, for perfect coupling should have been a little less than 34,000. Since Tajmar's test had an awful small Q (48.8 in ambient conditions and 20 in partial vacuum), Tajmar's test had horribly bad coupling ! No doubt due to the way that they coupled the huge waveguide into the small cavity.
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#27 by X_RaY on 16 Jan, 2016 21:00
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Nice analysis! Seems you put a lot of work into it.
It would be much easier if the paper would be correct at least for the used dimensions of the truncated cone by the group itself.
I like to remember, the first version of the paper was completely inconsistent in this point. After the first contact they corrected the radii and I am a little surprised now, obviously not all the dimensions were evaluated and corrected exactly.
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#28 by Rodal on 02 Feb, 2016 19:36
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UNDER CONSTRUCTION
CONSERVATION OF RELATIVISTIC MOMENTUM FOR REACTION-LESS PROPULSION THROUGH VARIABLE INERTIAL MASS
A) Minotti shows the EM Drive force to be due to a gravitomagnetic General Relativity effect (coupling of
a 4-dimensional version of Kaluza-Klein's unified field theory of gravitation and electromagnetism, built around the idea of a fifth dimension beyond the usual four of space and time coupled to an external scalar field ψ, which in turn couples to matter),
(Fernando O. Minotti, Scalar-tensor theories and asymmetric resonant cavities, Grav. & Cosmol. 19 (2013) 201, http://arxiv.org/abs/1302.5690)
Minotti states that the weak energy condition (the condition that demands that the mass should be greater than zero) (https://en.wikipedia.org/wiki/Energy_condition#Weak_energy_condition), is violated for the EM Drive in Minotti's theory.
B) Minotti also references Lobo and Visser's paper
(Francisco S. N. Lobo, Matt Visser, Fundamental limitations on "warp drive" spacetimes, Class.Quant.Grav. 21 (2004) 5871-5892, http://arxiv.org/abs/gr-qc/0406083)
that states that the weak energy condition (requiring positive mass) is also violated in other models of propellant-less (reaction-less) forms of proposed space-propulsion.
C) McCulloch,
(M. E. McCulloch, "Can the Emdrive Be Explained by Quantised Inertia?", PROGRESS IN PHYSICS Issue 1, Volume 11, (January 2015) (http://www.ptep-online.com/index_files/2015/PP-40-15.PDF) and "Testing quantised inertia on the emdrive", EPL (Europhysics Letters), Volume 111, Number 6, 1 October 2015)
also proposes that the EM Drive self-accelerates because radio frequency photons at the larger end have higher inertial mass, and therefore to conserve momentum in its reference frame, the cavity must move towards the narrow end.
This motivated me to analyze conservation of momentum for the EM Drive (or any such resonant cavity proposed for reaction-less propulsion) analyzed as a lumped-mass that is able to change its inertial mass. Thus, conservation of momentum of the EM Drive under these theories, would be satisfied, when duly taking into consideration the change in mass.
Here, I define momentum, using the relativistic definition of momentum
(https://en.wikipedia.org/wiki/Momentum#Relativistic_mechanics, https://en.wikipedia.org/wiki/Mass_in_special_relativity#The_relativistic_energy-momentum_equation, and
https://en.wikipedia.org/wiki/Tests_of_relativistic_energy_and_momentum),
which leads to the following equation:
I then define the following dimensionless variables:
dimensionless change in mass
dimensionless change in velocity
dimensionless initial velocity
which allow me to express the conservation of relativistic momentum in terms of these dimensionless variables. The equation (resulting from conservation of momentum) gives the following dimensionless change in mass:
I then calculate the dimensionless change in mass as a function of the other two variables: 1) dimensionless change in velocity and 2) dimensionless initial velocity. I plot the results using Wolfram Mathematica.
______________________________________________________________________
Results and discussion
1) Acceleration, with deltaV/InitialVelocity ranging from 0 to 2 and with InitialVelocity ranging from 0 to 10% of the speed of light
For this range we see the deltaMass/InitialMass to be practically independent of the magnitude of the InitialVelocity/c ratio. Acceleration implies a negative change in mass (decrease in inertial mass) from 0 (for zero change in velocity) to a decrease in mass of 60% of the initial mass for an increase in deltaV/InitialVelocity from 0 to 2.
2) Acceleration, with deltaV/InitialVelocity ranging from 0 to 2 and with InitialVelocity ranging from 0 to the speed of light
For this range we see the deltaMass/InitialMass to strongly depend on the magnitude of the InitialVelocity/c ratio, for initial velocities exceeding 15% of the speed of light. Acceleration implies a negative change in mass (decrease in inertial mass) from 0 (for zero change in velocity) to a decrease in mass approaching 100% of the initial mass (at which point the magnitude of the negative mass is equal to the initial mass), for an increase in deltaV/InitialVelocity from 0 to 2.
A frontier is formed (for deltaMass/InitialMass= - 1), at speeds being a sizeable fraction of the speed of light, for which it is not longer possible to accelerate.
3) Acceleration, with deltaV/InitialVelocity ranging from 0 to 50 and with InitialVelocity ranging from 0 to 40% of the speed of light
For this range we see the deltaMass/InitialMass to strongly depend on the magnitude of the InitialVelocity/c ratio, for initial velocities exceeding 1.5% of the speed of light. Acceleration implies a negative change in mass (decrease in inertial mass) from 0 (for zero change in velocity) to a decrease in mass approaching 100% of the initial mass (at which point the magnitude of the negative mass is equal to the initial mass).
A frontier is formed (for deltaMass/InitialMass= - 1), at speeds being a % of the speed of light, for which it is not longer possible to accelerate.
4) Acceleration, with deltaV/InitialVelocity ranging from 0 to 500 and with InitialVelocity ranging from 0 to 1% of the speed of light
For this range we see the deltaMass/InitialMass to strongly depend on the magnitude of the InitialVelocity/c ratio, for initial velocities exceeding 0.15% of the speed of light. Acceleration implies a negative change in mass (decrease in inertial mass) from 96% (for small change in velocity) to a decrease in mass approaching 100% of the initial mass (at which point the magnitude of the negative mass is equal to the initial mass).
A frontier is formed (for deltaMass/InitialMass= - 1), even at speeds being a small fraction of the speed of light, for which it is not longer possible to accelerate.
5) Based on the above plots we see that such a mode of space propulsion (reaction-less propulsion by variable mass) is quite limited on the speeds and changes in speed that it would be able to achieve.
6) Deceleration
For curiosity's sake we display what it would be like to decelerate by changing inertial mass. Deceleration would be achieved by an internal increase in mass. The needed increase in mass approaches infinity for speeds approaching the speed of light, or for deltaV/InitialVelocity approaching -100%
7) I also show a plot that includes the deceleration and acceleration ranges in the same plot.
Notes:
1) No warping of spacetime is considered in the analysis, only a reactionless variable mass is considered.
2) Forward (Robert Forward, "Negative matter propulsion", Journal of Propulsion and Power, Vol. 6, No. 1 (1990), http://arc.aiaa.org/doi/abs/10.2514/3.23219?journalCode=jpp), and Bondi, have used similar expressions when discussing momentum conservation (https://en.wikipedia.org/wiki/Negative_mass#Runaway_motion), but they only consider the case of two bodies, one with identical absolute value of mass: one body with mass +m and another one with mass -m instead of the case being discussed here of continuous variability in mass.
3) The equations presented are frame-indifferent, but one of the variables chosen to present the results graphically, is not frame indifferent: deltaV/InitialVelocity.DeltaV is obviously frame-indifferent, being a difference of velocities. But the speed of light is clearly the only frame-indifferent speed to non-dimensionalize all variables, instead of using the initial velocity to non-dimensionalize the deltaV.
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#29 by Rodal on 03 Feb, 2016 21:49
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UNDER CONSTRUCTION
CONSERVATION OF RELATIVISTIC MOMENTUM FOR REACTION-LESS PROPULSION THROUGH VARIABLE INERTIAL MASS
It is better to express the solution solely in terms of frame-indifferent variables, as much as possible, which can readily be accomplished for the dimensionless form of the deltaV variable with the following change of variables, as follows:
deltaV/InitialVelocity =( deltaV/c ) / (InitialVelocity/c)
deltaV is obviously frame-indifferent, being a difference of velocities. The speed of light is clearly the only frame-indifferent speed to non-dimensionalize all variables.
The term (InitialVelocity/c) appears naturally in the Lorentz (γ) factor of relativity https://en.wikipedia.org/wiki/Lorentz_factor
Actually, one can substitute this variable (InitialVelocity / c) with γ:
InitialVelocity / c = √(1-(1/γ)2)
Therefore, we define a new dimensionless deltaV variable, by dividing deltaV by the speed of light instead of the initial speed:
we express the equation for the dimensionless change in mass only in terms of the above variable deltaV/c and the previously defined variable InitialVelocity/c:
and the dimensionless change in mass :
Since conservation of momentum means that the relativistic momentum for the initial and final configurations are equal:
this equality becomes, when expressed in terms of the above-mentioned variables, the following expression:
and using this equation, one can plot the results as a function of these terms:
deltaMass/InitialMass = function (deltaV/c , InitialVelocity/c)
everything becomes more clear. The reason why there is a frontier becomes clear.
Very interesting that small increases in speed mean much less need of negative mass
Small increase in speed (10^-6 deltaV/c) requires very small negative mass
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#30 by frobnicat on 04 Feb, 2016 23:43
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.../...
and using this equation, one can plot the results in completely frame-indifferent terms:
deltaMass/InitialMass = function (deltaV/c , InitialVelocity/c)
.../...
It is not clear to me in your argument why InitialVelocity/c would qualify as "frame-indifferent". Different inertial observers could agree on deltaV/c but see different values for InitialVelocity/c, and hence predict different outcome deltaMass/InitialMass. The other way around, measuring a certain deltaMass/InitialMass and a certain deltaV/c would imply one peculiar InitialVelocity/c that would hold for one privileged inertial observer and not for the others. Am I misunderstanding something ?
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#31 by RERT on 05 Feb, 2016 08:39
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On a similar vein to frobnicat's post above: in the EM Drive thread you noted that deceleration requires significantly different negative mass creation than acceleration from rest.
But consider this: a) use your device to accelerate something to some velocity b) turn it off, so that the device now moves at constant velocity c) move your frame of reference to that inertial frame, so that the device is once again at rest d) rotate the device 180 degrees then switch it back on, accelerating in that new inertial frame (and decelerating in the original frame).
It is extremely hard to see how acceleration from rest in the opposite direction can require any different consumption of anything than in the original direction.
R.
[modified to make more sense after posting] -
#32 by Rodal on 05 Feb, 2016 18:21
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Frobnicat and Rert
Thank you for your comments, they are much appreciated. The post was labeled UNDER CONSTRUCTION when you posted them.
I need to get the time to finish my post, and then to carefully address the comments. I will answer your questions once the "UNDER CONSTRUCTION" label is removed.
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#33 by Rodal on 05 Feb, 2016 19:59
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On a similar vein to frobnicat's post above: in the EM Drive thread you noted that deceleration requires significantly different negative mass creation than acceleration from rest.
Although I need to have the time to finish my post, and remove the "UNDER CONSTRUCTION" label, these are my present thoughts
But consider this: a) use your device to accelerate something to some velocity b) turn it off, so that the device now moves at constant velocity c) move your frame of reference to that inertial frame, so that the device is once again at rest d) rotate the device 180 degrees then switch it back on, accelerating in that new inertial frame (and decelerating in the original frame).
It is extremely hard to see how acceleration from rest in the opposite direction can require any different consumption of anything than in the original direction.
R.
[modified to make more sense after posting]
1) Please go over carefully, and in more detail the mathematics of your statements (actually I would appreciate seeing equations), regarding conservation of momentum, particularly this one: <<d) rotate the device 180 degrees >>. Non-inertial systems are not equivalent, the space–time symmetries do not hold anymore as it was the case for inertial systems. Therefore, scientists living inside a box that is being rotated (or otherwise accelerated) can measure their frame motion or acceleration by observing the inertial forces on physical objects inside the box. In a rotating frame, one space direction is superior to all other directions, the axis of rotation which breaks the isotropy of space. In the case of our earth as a closed system, experiments like the Foucault pendulum can demonstrate the rotation of the earth, or by using gyrocompasses which can exploit the rotation of the earth to find the direction of true north during navigation.
2) Take a look at the mathematical solution. The solution function is continuous, for negative values of deltaV/c it is double-valued. The asymmetry you are addressing in your response arises because you are arbitrarily taking into account only one of the possible values for negative deltaV/c. Mathematically and for consistency you should instead take into account all possible values of a multi-valued function, when addressing symmetry of a multi-valued function.
Since the solution is multi-valued I need the time to further examine it in both directions, and assess its physical significance (if any, because the notion of negative mass and energy is anything but intuitive! ).
Negative Mass-energy is prevented by the Weak Energy Condition (WEC), so this solution may not be physically possible if WEC holds. And concerning the Casimir effect, as I have explained several times in previous threads, Profl. Jaffe at MIT, and others think that the Casimir effect (and other effects that some view as "negative energy") can perfectly well be explained without resorting to the notion of negative energy. So, I would not be too surprised if a negative mass-energy solution looks strange...but I will look at it further -
#34 by Rodal on 05 Feb, 2016 23:41
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You are correct. "Frame Indifferent" only applies to deltaV/c..../...
and using this equation, one can plot the results in completely frame-indifferent terms:
deltaMass/InitialMass = function (deltaV/c , InitialVelocity/c)
.../...
It is not clear to me in your argument why InitialVelocity/c would qualify as "frame-indifferent". Different inertial observers could agree on deltaV/c but see different values for InitialVelocity/c, and hence predict different outcome deltaMass/InitialMass. The other way around, measuring a certain deltaMass/InitialMass and a certain deltaV/c would imply one peculiar InitialVelocity/c that would hold for one privileged inertial observer and not for the others. Am I misunderstanding something ?
The InitialMass is the mass of the object in the object's rest frame.
Every observer will agree on which frame is the rest frame. Therefore, InitialVelocity is the the velocity of the object in its reference frame, the same frame used for the mass.
This frame 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. The InitialVelocity/c term is also in the gamma factor, and is clearly intimately associated with the mass of the object.
We are discussing an acceleration problem, after all, where the speed of the same object changes, and therefore we have to deal necessarily with a non-inertial frame in regards to the InitialMass and the Initial Velocity associated with the same frame used to measure the rest mass.
Still, expressing deltaV/c as a frame-indifferent variable helps to better understand the graphical output.
The solution makes sense for InitialVelocity/c = 0 (γ=1) (it agrees with Bondi's momentum equation for negative mass in that case). (H. Bondi, Negative Mass in General Relativity, Rev. Mod. Phys. 29, 423;1 July 1957)
It makes sense for InitialVelocity/c = 1 (γ=∞) , from the point of view that the only way to reach the speed of light, InitialVelocity/c = 1 (γ=∞), is with zero mass (deltaMass/InitialMass=-1).
It makes sense that there is a frontier (deltaMass/InitialMass=-1) between deltaV/c and InitialVelocity/c and that this frontier is a straight line, since deltaV/c + InitialVelocity/c = FinalVelocity/c ≤ 1 and hence the frontier is specified by the linear constraint equation deltaV/c + InitialVelocity/c = 1, giving:
InitialVelocity/c = 1 - deltaV/c
At the frontier, InitialVelocity/c equals a constant (unity) minus a frame indifferent variable (deltaV/c).
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#35 by frobnicat on 06 Feb, 2016 00:26
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It is extremely hard to see how acceleration from rest in the opposite direction can require any different consumption of anything than in the original direction.
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2) Take a look at the mathematical solution. The solution function is continuous, for negative values of deltaV/c it is double-valued. The asymmetry you are addressing in your response arises because you are arbitrarily taking into account only one of the possible values for negative deltaV/c. Mathematically and for consistency you should instead take into account all possible values of a multi-valued function, when addressing symmetry of a multi-valued function.
Since the solution is multi-valued I need the time to further examine it in both directions, and assess its physical significance (if any, because the notion of negative mass and energy is anything but intuitive! ).
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Following from the relativistic momentum conservation (on a single axis) I get the same expression as what you wrote for deltaMassBar as a function of deltavc and vbarc. From the initial equality (momentum conservation) to this last equality there is no need to take a square root of the equation, nor solve for solutions of second degree polynomial : so why do you say the function is double-valued, and specifically for negative deltavc ?
For instance with vbarc=1/2 and deltavc=-1/4 the expression unequivocally gives deltaMassBar=sqrt(5)-1 > 0
What would be an other solution ?
I understand this is under construction
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#36 by Rodal on 06 Feb, 2016 00:36
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.../...
It is extremely hard to see how acceleration from rest in the opposite direction can require any different consumption of anything than in the original direction.
.../...
.../...
2) Take a look at the mathematical solution. The solution function is continuous, for negative values of deltaV/c it is double-valued. The asymmetry you are addressing in your response arises because you are arbitrarily taking into account only one of the possible values for negative deltaV/c. Mathematically and for consistency you should instead take into account all possible values of a multi-valued function, when addressing symmetry of a multi-valued function.
Since the solution is multi-valued I need the time to further examine it in both directions, and assess its physical significance (if any, because the notion of negative mass and energy is anything but intuitive! ).
.../...
Following from the relativistic momentum conservation (on a single axis) I get the same expression as what you wrote for deltaMassBar as a function of deltavc and vbarc. From the initial equality (momentum conservation) to this last equality there is no need to take a square root of the equation, nor solve for solutions of second degree polynomial : so why do you say the function is double-valued, and specifically for negative deltavc ?
For instance with vbarc=1/2 and deltavc=-1/4 the expression unequivocally gives deltaMassBar=sqrt(5)-1 > 0
What would be an other solution ?
I understand this is under construction
For acceleration (deltaV/c>0) there is only one way to satisfy conservation of momentum: creation of negative mass.
which is a single manifold occupying only half of the area defined by deltaV/c and InitialV/c.
In contrast,
there are two possible ways to achieve negative values of deltaV/c (associated with two different values of InitialVelocity/c):
1) with negative mass creation, such that deltaMassBar is negative. You have to create this negative mass out of thin air, since it is production of a negative quantity, I can see Rert calling this as "consumption"
2) with positive mass creation, such that deltaMassBar is positive (I would call this positive mass creation "production" instead of consumption.)
Rert is arbitrarily only following choice #1 and ignoring #2, since he referring to "consumption" of something in:QuoteIt is extremely hard to see how acceleration from rest in the opposite direction can require any different consumption of anything than in the original direction.
I would not consider creation of positive mass as "consumption". Actually it is the opposite of consumption.
Neither he, nor I, stated any constraint that choices of deltaV/c and InitialVelocity/c were to be constrained to manifold #1 instead of being able to use manifold #2. So, when discussing deceleration, both options #1 and #2 have to be considered.
The terminology used could have been improved from "multivalued" to multi-folded sheets or something like that, but again this is under construction and the answers are under construction 
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#37 by frobnicat on 06 Feb, 2016 01:42
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I see. So actually deltaMass<0 for deltaV<0 only if deltaV<-initialV. In the common sense (relative to an implicit ground) this is more than deceleration, this is deceleration followed by acceleration in the opposite direction (and going through an infinite in the equation...)
Sticking to a "reasonable" deceleration of magnitude such that the vehicle doesn't goes to a halt (it still goes in the same direction, only slower) that is for -initialV<deltaV<0 (assuming initialV>0) then deltaMass>0 always holds. It doesn't seem "arbitrary" when discussing deceleration (in the common sense) to "choose" such deltaV that -initialV<deltaV<0. For instance an unpowered ground vehicle that can only brake (only dissipate energy) can't brake so hard as to get a deltaV in excess of -initialV.
I can't help but have the impression that from the start your approach defines an implicit ground... In Newtonian / Galilean relativity terms :
m1v1=m2v2 is not covariant (unless v2=v1 and m2=m1)
m1(v1+vc) ≠ m2(v2+vc)
I'd say that m1v1=m2v2 (and v2≠v1) hides a term for an implicit mass at 0 velocity (either left hand or right hand, I'll choose left hand) :
m1v1+(m2-m1)0=m2v2
which is now fully covariant :
m1(v1+vc)+(m2-m1)(0+vc)=m2(v2+vc)
See what I mean ? A non covariant equation seems ill defined, unphysical from start.
Obviously, again sticking to Newtonian mechanics for sake of simplicity, a closed system that respects both momentum and energy conservation just goes
m1v1=m2v2
˝m1v1˛=˝m2v2˛
=> v1=v2 and m1=m2
unless m1=m2=0 for instance as the result of the vehicle being composed of same amount of positive and negative mass (or of being non existent ?)
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#38 by Rodal on 06 Feb, 2016 01:59
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Yes, the vehicle would have to start, at zero speed with a negative mass equal (in absolute magnitude) to its positive mass (*) . But if creation of negative mass is allowed (in other words if one can violate the Weak Energy Condition), all this means is that the necessary amount of negative mass, for acceleration from a complete stop starts at 100% (equal in absolute magnitude to the amount of positive mass) and to accelerate further (until speeds of about ~1/2 of c) the amount of negative mass needed to be created for successive deltaV increases, diminishes with time.
After all, in Bondi's and Forward's drive concepts, one starts with a full 100% of negative mass (in the diametral drive, with 2 masses, one having 100% positive mass and the other mass having 100% negative mass).
But conversely (as per Shawyer's/TheTraveller's argument that it has been found out in experiments that the EM Drive needs to be motivated 
), the vehicle could be accelerated to a relative slow speed by some other conventional means, at which point (for relatively small speeds) the amount of negative mass necessary is much smaller, as shown in the following graph:
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(*) Excellent point about showing covariance, which is dependent on the initial mass. You are correct the equations as written implied positive mass=-negative mass as initial condition, just like Bondi and Forward (that's why I had referenced that point)...The solution makes sense for InitialVelocity/c = 0 (γ=1) (it agrees with Bondi's momentum equation for negative mass in that case). (H. Bondi, Negative Mass in General Relativity, Rev. Mod. Phys. 29, 423;1 July 1957)
...2) Forward (Robert Forward, "Negative matter propulsion", Journal of Propulsion and Power, Vol. 6, No. 1 (1990), http://arc.aiaa.org/doi/abs/10.2514/3.23219?journalCode=jpp), and Bondi, have used similar expressions when discussing momentum conservation (https://en.wikipedia.org/wiki/Negative_mass#Runaway_motion), but they only consider the case of two bodies, one with identical absolute value of mass: one body with mass +m and another one with mass -m instead of the case being discussed here of continuous variability in mass.
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#39 by Rodal on 07 Feb, 2016 20:18
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This very interesting quote is from Geoffrey Landis, then associated with the NASA Lewis Research Center (nowadays NASA Glenn):
GEOFFREY A. LANDIS "Comment on 'Negative matter propulsion'" Journal of Propulsion and Power, Vol. 7, No. 2 (1991), pp. 304-304. doi: 10.2514/3.23327Quote from: GEOFFREY A. LANDISIt is interesting to note that, in the spaceship consisting of
positive and negative mass elements discussed by Forward, as
the total mass of the spaceship approaches zero (M _ - M+ =~ 0)
the Brownian motion of the ship due to impact of various particles
will buffet it around at increasingly large velocities. Even
in a perfect vacuum, photons of cosmic background radiation
will become important if the mass is low enough. At
M _ ~ M+. the mass of the ship equals zero and any impact will
apparently send it moving off at the speed or light. (Actually
M will never precisely equal zero, as the ship will be constantly
absorbing and emitting thermal photons.)
A particle hitting a zero mass spaceship would, of course,
actually hit either the positive or negative mass portion. In a
ship consisting of nearly equal amounts of positive and
negative mass, the center of mass can move faster than either
of the constituent masses and will do so whenever the distance
between the two masses changes. Unless the ship is allowed to
come apart the true motion of the ship must eventually recon·
cile with the motion of the center or mass. This occurs due to
the force on the link connecting the masses. The force on the
link will cause the masses to move as described by Forward, so
that even a small initial impulse will cause very large change in
velocity if the positive and negative masses are nearly equal.
This fact is of use in propulsion: a very nearly zero mass spaceship
could be propelled by a flashlight.
The point being that if one were able to produce an amount of negative mass equal in absolute magnitude as the positive mass, then one might as well use (instead of the hypothetical EM Drive creating negative mass) the propulsion concept of Bondi/Forward since as they stated, and remarked by Landis, with M _ ~ M+ the Bondi/Forward pair of masses will self-accelerate to high speeds.
My contribution to this is to show that:
* for InitialVelocity/c >0 the amount of negative mass required quickly diminishes, such that small acceleration can result for very small amounts of negative mass (compared to the magnitude of the positive mass)
As shown in the above post.
https://en.wikipedia.org/wiki/Hermann_Bondi
I came to nearly the same conclusion some time last year(Q=79011). I never post it, at least I am not sure about the formula (found an approximation in an cern paper about cavities if my memory is correct) and my implementation. No -3dB bandwidth needed for the calculation, its mode,volume, conductivity dependent.
