H(r-r_0) has derivative 0.5*\delta(r_0). Only for the "unphysical" value r_0=0 (no step, since there are no nonnegative r0 some term that diverges with k. This corresponds to the infinity in your first approximation since there, no spherical integration is performed, that for r_0=0 "eats up the divergence" due to the r^2 factor in the measure. (Remember r*\delta(r)=0)
Thesis Reduced Total Energy Requirements for the Natario Warp Drive...
Spherical waves are illegitimate. they do not describe free particles after scattering.
If you use them, then there are no diffraction from crystals. Look my lecture 4
http://www.academia.edu/2081578/Lecture_4._Scattering_of_scalar_waves_in_a_3-dimensional_ideal_crystal
The idea of my question is to discuss the contribution of Loup and Rocha. I think that it is questionable regarding the point raised in my question.
Dear V. Ignatovich, does your answer refer to the use of a spherical measure by Loup and Rocha?
See also the following link
https://www.researchgate.net/post/How_to_make_this_engine
Dear Herbert Holmer, do not care about Loup and Rocha. I only poited out a contradiction in using spherical waves in scattering theories.
Sorry, dear Herbert Homeier for my incorrect spelling and errors. Correctly: I do not care, I pointed out.
I think I don't understand why this is a problem. Wouldn't the same be true for any use of polar or spherrical coordinates?
Perhaps in equation 38 the integration constant implicitly associated with indefinite integrals has been ignored. When you perform the indicated integration, you implicitly exchange limit and integral which means you must add an unknown constant depending on k. Since this constant could easily be infinite in the limit, the right hand side seems to be without meaning.
No you don't have complete freedom to choose the constant. It can be a function of anything except the integration variable. So since you don't know it it can indeed be a function of anyother constant, k.. So one is reduced to evaluating both sides. On the left the limit comes to zero since k/(1+k^2r^2) = 1/(1/k+kr^2). The r integral of zero is zero. On the right side the arctan dosn't go to zero. This isn't an equation.
Incidently it does turn out to be an equation if r=0, in which case the integral is meaning less.
The question is whether a spherical bubble of radius zero (which would correspond to a Heaviside function in r with "step" at r=0) would be OK for a warp drive.
That I don't understand. If r is radius, it has a built in step at r=0 because we don't use negative r values. And this is nothing new.
That is exactly the point. In the limit k to infinity, the warp bubble shrinks to radius 0 for the "second Heaviside", cf. section 4 of the article. But that may not be sufficient for the warp drive. Thus, the physical question regarding low enough energy densities may rule out sufficiently large warp bubbles. Finiteness of the energy densities in the limit k to infinity might not be sufficient for obtaining a large enough warp bubble for finite k.
This attaches special significance to the spherical coordinate system because this step dissappears on switching to cartesian coordinates. Physics has to be independent of coordinate transformations.
The jump at r=0 is an artifact of the spherical coordinate system. It doesn't occur in Cartesian coordinates. No physical effect can be dependent on a particular coordinate system.
well, the argument in the manuscript was for the spherical case. Before I do not see a proof for it in Cartesian coordinates (valid mathematics :) ), I am not convinced.
Not sure what you're after. Isn't it obvious that in Cartesian coordinates there's no jump?
I simply would like to see a proof that in cartesian coordinates the following can be ruled out:
"That is exactly the point. In the limit k to infinity, the warp bubble shrinks to radius 0 for the "second Heaviside", cf. section 4 of the article. But that may not be sufficient for the warp drive. Thus, the physical question regarding low enough energy densities may rule out sufficiently large warp bubbles. Finiteness of the energy densities in the limit k to infinity might not be sufficient for obtaining a large enough warp bubble for finite k."
That is, will there be enough space for the warp bubble if you use cartesian coordinates to prove it besides the required low enough energy density? I think that one has to _very_ careful with the maths used.
But the derivation you want to use is defective all by itself. It involves the exchange of a limit and an indefinite integral.
Well, that might be a further point where one has to be rigorous. I am quite content that I do not have the burden to review the paper ...
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