We? I speak tensor calculus fine.
Someone already referenced a couple of them early in this thread.
No, that's not even close to what that image is about. That image isn't even about acceleration of the ship. Even if the ship is at constant warp speed, it doesn't change that image which is NOT about a gravity dipole. Its a proposed way to imagine the geometry of a spacelike hypersurface through the ship for constant warp speed.
No.
Consider for the moment the following warp drive in cylindrical coordinates appropriate for a remote observer with respect to which the ship is at rest when β is zero,
ds² = dct² - (dZ-βfdct)² - dZ² - r²dφ²
just so that you can compare to what you can look up.
I find it more helpful working with warp drive to express things in coordinates more appropriate for the perspective of ship frame observers. After all, you want it to be the ship forming the stress energy tensor to engage the warp drive. So do the transformation
z=Z - ∫βdct
where β is a function of time that you are doing an anti derivative with respect to.
Also define a new function g of
g = 1-f
The new coordinate expression for warp drive appropriate for the perspective of on board observers becomes
ds² = dct² - (dz+βgdct)² - dz² - r²dφ²
Now you can take β to be any function of time that is zero when the ship isn't at warp and g is any function of (z,r) that has boundary conditions that
g → 0 as z²+r² → 0
and
g → 1 as z²+r² → ∞ .
Define ρ
ρ² = z²+r²
You can now calculate the Einstein tensor to get the ship frame energy density and find it is
T⁰⁰ = - β²(c⁴/32ⲡG)(dg/dρ)²(r/ρ)²
This describes the energy density according to the ship frame throughout ALL space. Notice that there is no energy dipole. No positive energy on one side of the ship with negative on the other. No gravity dipole. And since this is valid for any function of time for β, it remains that case even when it is "accelerating". The negative energy density for Alcubierre's warp drive is ringed around its sides. What determines the direction that it goes into warp toward is actually the T¹⁰ , T⁰¹ terms, at its sides.
On the one hand, it’s cool that you’re posting here in a thread that's about your own work. Everyone here would like to understand it more clearly.
But on the other hand you’re not really making an effort to be understood. And the abstracts of these two papers from 1947 and 1948 that Dejan cited earlier in this thread make no mention of anything discernibly applicable to the gravitational field propulsion principle that you’re describing:
Papapetrou, A. (1948): "A static solution of the equations of the gravitational field for an arbitrary charge distribution", Proc. Roy. Irish Acad. A 51 191.
"ABSTRACT: A stationary axially symmetric exterior electrovacuum solution of the Einstein-Maxwell field equations was obtained. An interior solution for rotating charged dust with vanishing Lorentz force was also obtained. The two spacetimes are separated by a boundary which is a surface layer with surface stress-energy tensor and surface electric 4-current. The layer is the spherical surface bounding the charged matter. It was further shown, that all the exterior physical quantities vanished at the asymptotic spatial infinity where spacetime was shown to be flat. There are two different sets of junction conditions: the electromagnetic junction conditions, which were expressed in the traditional 3-dimensional form of classical electromagnetic theory; and the considerably more complicated gravitational junction conditions. It was shown that both—the electromagnetic and gravitational junction conditions—were satisfied. The mass, charge and angular momentum were determined from the metric. Exact analytical formulae for the dipole moment and gyromagnetic ratio were also derived. The conditions, under which the latter formulae gave Blackett’s empirical result for rotating stars, were investigated."
A. Papapetrou, “A Static Solution of the Equations of the Gravitational Field for an Arbitrary Charge-Distribution,” Proceedings of the Royal Irish Academy, Vol. 51, 1945-1948, p. 191. - Open Access Library
And:
Majumdar, S D (1947). "A Class of Exact Solutions of Einstein's Field Equations". Physical Review. 72 (5): 390–398.
Phys. Rev. 72, 390 (1947) - A Class of Exact Solutions of Einstein's Field Equations
And then you go out of your way to argue that the spatial contraction at the front of an Alcubierre warp bubble, and the spatial expansion at the rear of the bubble, don't constitute a gravitational dipole. But the primary source material suggests otherwise.
For example, when we look at Alubierre’s paper:
He employs the same kind of isometric diagram that we saw in Dr. Harold White’s paper – here we see that Alcubierre labeled Fig. 1 “Expansion of the normal volume elements” where the volume elements are contracted in the front of the craft and expanded behind the craft:
And in this section (below) he even explicitly describes how “the spacecraft will be pushed away from the Earth and pulled towards a distant star by spacetime itself.” That certainly sounds like an antigravitational field behind the craft akin to the dark energy effect, and a gravitational field ahead of the craft:
And in White's paper he explicitly labels the diagram with “Space Contraction” in front of the ship and “Space Expansion” at the rear of the ship...strange that didn't label them as "spacelike hypersurfaces":
So how is that
not equivalent to an antigravitational field at the rear of the bubble...and a positive gravitational field at the forward side of the warp bubble? It's the presence of an antigravitational field throughout the cosmos that expands the space between the galaxy clusters, and the presence of a positive gravitational field that contracts the space between more proximal bodies, so the spacetime regions on opposite sides of the warp bubble sure look like positive and negative poles of gravitation to me. Frankly I'm bewildered that anyone doesn't see it that way.
These regions also correspond to the York time so White labels the same diagram in this way, with an explanation right above it: “The region directly in front of the spacecraft experiences the most contraction of space, while the region directly behind the spacecraft experiences the most expansion of space.”
The expansion of space between two bodies is antigravity, aka negative gravity, by definition. That's why the dark energy field is regarded as an antigravity field and is associated with a gravitational blueshift (time contraction), whereas the contraction of space between two bodies describes a positive gravitational field between them and is associated with a gravitational redshift (time dilation).
Sure, the nomenclature isn’t steeped in the rigorous formal vernacular of the tensor calculus equations of GR – but it was good enough for Alcubierre, and Dr. White the Advanced Propulsion Team Lead for the NASA Engineering Directorate, and Dr. Hal Puthoff.
So yes – this is clearly a gravitational dipole, and it makes zero sense to argue that it’s not: all of these diagrams depict the spacetime distortions that represent positive and negative gravitational fields. The same diagram can be seen in this paper published by Dr. Puthoff with the warp metric clearly depicted as positive and negative regions of the gravitational field, alongside his discussion of Alcubierre's concept in terms of "gravity/antigravity propulsion":
So although Robert Forward’s concept of a self-accelerating gravitational dipole is produced in a much simpler manner than the Alcubierre metric - Forward's notion simply places a body with gravitational field in front of a body with negative gravitational (antigravity) field and off they go:
...whereas with the Alcubierre metric the negative mass-energy region surrounds the spacecraft in a toroidal ring perpendicular to the direction of acceleration...the core motive principle at work appears to be identical in nature: a positive and a negative gravitational pole self-accelerating in the direction of the positive pole, dragging everything in-between right along with them.
Unfortunately none of this pertains to the topic of this thread: your gravitational field propulsion concept that’s based on the electric field. If you actually want anyone to understand your proposal, then you’ve failed to do that. I would be thrilled if you explained it in a manner that the rest of us here could understand, but I’m not convinced that you have any interest in doing that because you've made no effort to do so.
And that’s a shame, because I’m only aware of three viable theories of gravitational field propulsion: the proposals of Robert L. Forward (negative matter propulsion), Miguel Alcubierre (the Alcubierre metric), and Jack Wisdom ("swimming in spacetime," which depends on a gravitational field gradient in the vicinity of the device but not produced by the device...which limits its utility to regions immediately surrounding highly massive astronomical bodies).
I've always wanted to add a fourth viable gravitational field propulsion concept to that very short list, but none of us here can do that because you haven't bothered to explain it in a manner that the rest of us here can comprehend. You've obviously spent years performing post-grad-level derivations of general relativity; nobody else here has that level of experience. And you know that. So I'm unconvinced that you're correct, because the discovery of a gravitational field propulsion concept that doesn't require either very substantial negative mass-energies, or the context of an intense external gravitational field, would make headlines all around the world. Maybe even win you the Nobel Prize in physics. If it was actually viable.
I ran into similar obfuscation and intransigence last year when I asked Dr. Jack Sarfatti about his unpublished (and therefore not peer-reviewed) low-power warp drive concept, so I'm inclined to disbelieve both claims of a reasonably accessible approach to gravitational field propulsion. Because if I were to someday come up with a potentially world-changing breakthrough in the theoretical physics of gravitational field propulsion, I'd be delighted to get the gist of the idea across to anyone who was interested.
But aside from posting a couple of links to well-trodden electrical field equations, you still haven't made any effort to explain your revolutionary breakthrough gravitational field propulsion concept at all. Why is that?