Discussion in 'Alien Hub' started by waitedavid137, Jul 31, 2020.

1. ### waitedavid137Honorable

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193
So lets say an observer outside of any warp uses coordinates [cT, X, Y, Z], define
$image=https://latex.codecogs.com/gif.latex?%5Csmall%20r_%7Bs%7D%3D%5Csqrt%7BX%5E%7B2%7D+Y%5E%7B2%7D+%5Cleft%20%28%20Z-%5Cint%20%5Cbeta%20dcT%20%5Cright%20%29%5E%7B2%7D%7D&hash=c93dfd81cfd0887c63c6243f594034c2$
where $image=https://latex.codecogs.com/gif.latex?%5Csmall%20%5Cbeta&hash=75e23280445a45e8de4d2058f805e534$ will be a function of T for this will be a standard for the distance from somewhere to the center of a warp ship.
The spacetime given by
$image=https://latex.codecogs.com/gif.latex?%5Clarge%20ds%5E%7B2%7D%3D%5Cleft%20%5B%20A%5Cleft%20%28%20cT%2Cr_%7Bs%7D%20%5Cright%20%29%20%5Cright%20%5D%5E%7B2%7DdcT%5E%7B2%7D-%5Cleft%20%5C%7B%20dX%5E%7B2%7D+dY%5E%7B2%7D+%5Cleft%20%5B%20dZ-%5Cbeta%20%5Cleft%20%28%20cT%20%5Cright%20%29f%5Cleft%20%28%20r_%7Bs%7D%20%5Cright%20%29dcT%20%5Cright%20%5D%5E%7B2%7D%20%5Cright%20%5C%7D&hash=9fda066beaf1646c0e3fa1c4bf97f241$
will be an Alcubierre warp drive spacetime given boundary conditions
$image=https://latex.codecogs.com/gif.latex?%5Clarge%20%5Cbegin%7Bmatrix%7D%20f%20%5Crightarrow%201%20%26%20as%20%26%20r_%7Bs%7D%5Crightarrow%200%20%5Cend%7Bmatrix%7D&hash=2d72f71cf1157b49958768bda7b0eab6$
$image=https://latex.codecogs.com/gif.latex?%5Clarge%20%5Cbegin%7Bmatrix%7D%20f%20%5Crightarrow%200%20%26%20as%20%26%20r_%7Bs%7D%5Crightarrow%20%5Cinfty%20%5Cend%7Bmatrix%7D&hash=d6d624abc019870bac98606385b8e685$
$image=https://latex.codecogs.com/gif.latex?%5Clarge%20%5Cbegin%7Bmatrix%7D%20A%5Crightarrow%201%20%26%20as%20%26%20r_%7Bs%7D%5Crightarrow%20%5Cinfty%20%5Cend%7Bmatrix%7D&hash=918441f0e5f166f8976483aec382fa2f$
A is called the lapse function because its value at the ship's location determines time dilation for the ship during the trip. The ship will have a speed of
$image=https://latex.codecogs.com/gif.latex?%5Clarge%20v_%7Bs%7D%3D%5Cbeta%20c&hash=52cd6df0fbfd7eaa4b0b377674d91d4b$
For the case that
A=1 everywhere, time does not lapse for the ship with respect to the frame for which it was initially at rest when
$image=https://latex.codecogs.com/gif.latex?%5Clarge%20%5Cbeta%20%3D0&hash=1c5c54be85b2114cf08aaa573613001a$
So what I'm interested in is the energy density according to coordinates that are actually appropriate for the ship. After all it is the ship that we want to generate the warp. So what we start out doing is Let A=1 everywhere and then do the transformation
$image=https://latex.codecogs.com/gif.latex?%5Clarge%20%5Cbegin%7Bmatrix%7D%20ct%3DcT%5C%5C%20x%3DX%20%5C%5C%20y%3DY%20%5C%5C%20z%3DZ-%5Cint%20%5Cbeta%20dcT%20%5Cend%7Bmatrix%7D&hash=7ab64c5111b9f0ae468af5c0f98d6cae$
The expression for the spacetime becomes
$image=https://latex.codecogs.com/gif.latex?%5Clarge%20ds%5E%7B2%7D%3Ddct%5E%7B2%7D-%5Cleft%20%5C%7B%20dx%5E%7B2%7D+dy%5E%7B2%7D+%5Cleft%20%5B%20dz+%5Cbeta%20%5Cleft%20%28%201-f%20%5Cright%20%29dct%20%5Cright%20%5D%5E%7B2%7D%20%5Cright%20%5C%7D&hash=acc86a796f6a71572d5385006f17f5dd$
And so for simplicity of calculation we define a new function g
g=1-f
So it becomes
$image=https://latex.codecogs.com/gif.latex?%5Clarge%20ds%5E%7B2%7D%3Ddct%5E%7B2%7D-%5Cleft%20%5C%7B%20dx%5E%7B2%7D+dy%5E%7B2%7D+%5Cleft%20%5B%20dz+%5Cbeta%20%5Cleft%20%28%20ct%20%5Cright%20%29%20g%20%5Cleft%20%28%20x%2Cy%2Cz%20%5Cright%29dct%20%5Cright%20%5D%5E%7B2%7D%20%5Cright%20%5C%7D&hash=10fb3fda08ff7ce619fa2d3f9df5d4fb$
defining
$image=https://latex.codecogs.com/gif.latex?%5Clarge%20r%3D%5Csqrt%7Bx%5E%7B2%7D+y%5E%7B2%7D+z%5E%7B2%7D%7D&hash=d4a47314b5b61b7f8d5b5e58582a14d2$
Yields the boundary conditions to be
$image=https://latex.codecogs.com/gif.latex?%5Clarge%20%5Cbegin%7Bmatrix%7D%20g%5Crightarrow%200%20%26%20as%20%26%20r%5Crightarrow%200%5C%5C%20g%5Crightarrow%201%20%26%20as%20%26%20r%5Crightarrow%20%5Cinfty%20%5Cend%7Bmatrix%7D&hash=19611ebdfd538e8e0810085eaa6d638b$
From the perspective of the ship observer, he floats at zero velocity with no pushes or pulls in a locally inertial frame while it is the stars that are at warp in the opposite direction.
Now here's the first trick. I am going to reintroduce what was called the lapse function term into this standard of coordinates with the modifying exception that I am going to change its boundary conditions.
The modified warp drive spacetime will be
$image=https://latex.codecogs.com/gif.latex?%5Clarge%20ds%5E%7B2%7D%3D%5Cleft%20%5B%20A%5Cleft%20%28%20ct%2Cx%2Cy%2Cx%20%5Cright%20%29%20%5Cright%20%5D%5E%7B2%7Ddct%5E%7B2%7D-%5Cleft%20%5C%7B%20dx%5E%7B2%7D+dy%5E%7B2%7D+%5Cleft%20%5B%20dz+%5Cbeta%20%5Cleft%20%28%20ct%20%5Cright%20%29g%5Cleft%20%28%20x%2Cy%2Cz%20%5Cright%20%29dct%20%5Cright%20%5D%5E%7B2%7D%20%5Cright%20%5C%7D&hash=9de2351a27d77953322c0cc09a92bd9c$
with
$image=https://latex.codecogs.com/gif.latex?%5Clarge%20%5Cbegin%7Bmatrix%7D%20A%5Crightarrow%201%20%26%20as%20%26%20r%5Crightarrow%200%2C%5Cinfty%20%5Cend%7Bmatrix%7D&hash=d1a9a6f80b4bef297b8b1acdb914a6ad$
You calculate or compute from the metric $image=https://latex.codecogs.com/gif.latex?%5Clarge%20G%5E%7B00%7D&hash=3e6e71a5770cfa8d65f9383c788bc3ae$, and use $image=https://latex.codecogs.com/gif.latex?%5Clarge%20T%5E%7B00%7D%3D%5Cfrac%7Bc%5E%7B4%7D%7D%7B8%5Cpi%20G%7DG%5E%7B00%7D&hash=4f84cb562ba80379a8434e0ff222d342$to arrive at the ship frame energy density term to be
$image=https://latex.codecogs.com/gif.latex?%5Clarge%20T%5E%7B00%7D%3D-%5Cfrac%7Bc%5E%7B4%7D%7D%7B32%5Cpi%20G%7D%5Cfrac%7B%5Cbeta%20%5E%7B2%7D%7D%7BA%5E%7B4%7D%7D%5Cleft%20%5B%20%5Cleft%20%28%20%5Cfrac%7B%5Cpartial%20g%7D%7B%5Cpartial%20x%7D%20%5Cright%20%29%5E%7B2%7D%20+%5Cleft%20%28%20%5Cfrac%7B%5Cpartial%20g%7D%7B%5Cpartial%20y%7D%20%5Cright%20%29%5E%7B2%7D%20%5Cright%20%5D&hash=b5b1331e24e56481aae6ccb959a171ac$
So from the perspective of the ship, which is what we want to produce the stress-energy tensor for the warp, the energy density can be arbitrarily reduced by allowing A to be large in the region of the warp matter where the negative energy density would have otherwise been large. Yet with its boundary conditions there need be no lapse in time between the ship observer, and the inertial frame observers he was initially at rest with respect to before going into warp.
More later.

Last edited: Jul 31, 2020
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2. ### waitedavid137Honorable

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The warp itself tends to deflect geodesics it goes against away from it, so an observer off to the side would see light coming from anything in the direction it is going as coming from it.
I think star trek took that idea from Chris Van Den Broeck that showed that doing such a thing could reduce the negative energy density's magnitude.
I usually describe it from the perspective of the craft when in warp with respect to some remote observer who is at rest with respect to the ship when beta=0.
This version doesn't radiate, but when we get a fully practical version, I'm sure it would radiate both electromagnetic and gravitational radiation.
My version does, yes.

Last edited: Jul 31, 2020
3. ### Dejan CorovicCelestial

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Five more to settle UFO lore, and I am done:

1) Does the infamous lenticular shape, aka flying saucer shape, have any privileged status in your equations? Or even somewhat asymmetrical lenticular shape, with the top dome being more convex than the bottom one.

2) Under your equations will stationary hovering spacecraft, near Earth's surface, need to spontaneously rotate around its own major axis of symmetry at about 0.3Hz?

3) Would the spacecraft's hull be able to have any openings, aka windows? This one might be going too far into the engineering side.

4) I guess this is given: these spacecraft would be able to instantaneously accelerate to great speeds, from standing still.

5) What would be the stronger propulsive direction? A) along the major axis of symmetry? or B) perpendicular to the axis of symmetry?

4. ### waitedavid137Honorable

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193
Why self impose limits?
I don't think scout/shuttle craft small ones work on this kind of warp drive. However the more you keep the warp matter azymuthally symmetric while flattening it, the less you will need in total.
No, such a thing is more likely a reaction to internal gyroscopic stabalization.
Again, I don't think shuttle types use this, but as for interstellar craft, a much sturdier design would be to have an essentially solid shell with outer placed camera and scanners relaying information to the interior via something like radio communication.
Yes and with no g-force felt by the pilots.
Like I've mentioned the negative energy is around the sides. The direction is along the axis of symmetry, but its not the energy density term that tells you which way along it.

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5. ### waitedavid137Honorable

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So the next thing I want to point out is that this energy density does not determine which direction along z the ship goes into warp. Its distributed around the sides.

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7. ### Dejan CorovicCelestial

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Maybe I should rephrase the Q.

Is it easier to move the craft sideways, in XY plane, or to move it along the Z axys of symetry?

OK, so broadly speaking, flatter symetrical shape, like lenticular, is beneficial for UFO.

Last edited: Aug 3, 2020
8. ### waitedavid137Honorable

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193
The geometry makes sense for a warp drive, and its warp velocity is along the Z axis of symmetry.

9. ### Dejan CorovicCelestial

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Wa-Wa-BOOM! *

I have a whole bunch of electro-magnetic observations by witnesses that 100% UNQUESTIONABLY confirmed by physics. But this is the first time General Relativity and UFO witness testimonials connected.

This is my personal collection of UFOs that flipped 90º just before the departure, that I collected since I noticed pattern in witness testimonials. I knew that at some point I shall stumble onto a proper scientific explanation for that behavior. Now we know why UFOs flip to the side. Because of the shape of the warp field, it's easier for UFOs to move along the axis of symmetry, in spite of flying in direction of maximum air resistance not having any aerodynamic sense.

Indiana, 2005, drawing

As far as I remember, that chemical photo was taken somewhere in Australia.

Photo by car mechanic Paul Villa, 1953, who was considered fraud. Photo was taken in US, in desert states like New Mexico or Nevada.

Another photo by Paul Villa of a different craft. Circa 1953.

Lake Isabella, US, 1957

off coast of San Diego, CA, US Navy, 2015

video by a construction worker in Acosta, Costa Rica, on November 22, 2007

Poor quality video shot on an old-style mobile phone, but close to 02:20, in a single frame UFO can be seen doing 90º flip, before leaving the scene.

jump to 03:04, Jeff Willes, 2003, Phenix, AZ, US. Same footage as previous, shows UFO flipped to a side against blue sky, moving out of a sight behind eves of a roof. Warning, music is horrible.

The last two are from Arizona. I guess that's your neighborhood.

Shout to @Thomas R. Morrison, check this thread out.

In this way General Relativity vindicates all the witnesses whose credibility was questioned, like Paul Villa's.

* That was jingle used in relatively famous car ad for French Renault.

Last edited: Aug 3, 2020
10. ### waitedavid137Honorable

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193
Now how the metric elements vary between the origin where the ship is, and infinity actually don't effect that you have a warp drive as long as long as the solution reduces to this both at the origin and infinity. Also what standard for coordinates the ship based observer are not restricted to this choice remote from him since there exist no globally rectilinear coordinates when you have spacetime curvature. So lets look at the equation at infinity from the perspective of the ship based observer. Given the stated boundary conditions it reduces to
$image=https://latex.codecogs.com/gif.latex?%5Clarge%20ds%5E%7B2%7D%3Ddct%5E%7B2%7D-%5Cleft%20%5B%20dx%5E%7B2%7D+dy%5E%7B2%7D+%5Cleft%20%28%20dz+%5Cbeta%20dct%20%5Cright%20%29%5E%7B2%7D%20%5Cright%20%5D&hash=1d7e563305d1b06cb66f9fbef61bb16f$ at infinity.
Now as long as the solution reduces to this at infinity and was the metric of special relativity at the origin, it doesn't matter how it transitions in between as to whether it is a warp drive. You still have one. Also again the ship observer is not restricted to this particular choice of coordinates at infinity. Consider a constant warp speed. Also consider at infinity the ship observer standard of time warps to
$image=https://latex.codecogs.com/gif.latex?%5Clarge%20dct%3Ddct%27+%5Cfrac%7B%5Cbeta%20%7D%7B1+%5Cbeta%20%7Ddz&hash=3f9c804a5f13cf25c0f9d96fdfb25ec5$
The solution is then expressed
$image=https://latex.codecogs.com/gif.latex?%5Clarge%20ds%5E%7B2%7D%3Ddct%27%5E%7B2%7D-%5Cleft%20%28%20dx%5E%7B2%7D+dy%5E%7B2%7D+dz%5E%7B2%7D%20%5Cright%20%29-%5Cbeta%20%5E%7B2%7D%5Cleft%20%28%20dz+dct%27%20%5Cright%20%29%5E%7B2%7D&hash=5146696dd3254be8aed3a8efea025f2d$
So as long as the solution is that of special relativity at the origin and becomes this at infinity, no matter how it varied in between, you would still have a warp drive. Now I'm going to drop the primes in further expressions. Consider
$image=https://latex.codecogs.com/gif.latex?%5Clarge%20ds%5E%7B2%7D%3Ddct%5E%7B2%7D-%5Cleft%20%28%20dx%5E%7B2%7D+dy%5E%7B2%7D+dz%5E%7B2%7D%20%5Cright%20%29-%5Cbeta%20%5E%7B2%7Dh%5Cleft%20%28%20x%2Cy%2Cz%20%5Cright%20%29%5Cleft%20%28%20dz+dct%20%5Cright%20%29%5E%7B2%7D&hash=1295dde9e9dd19499947dd05b7526f7d$
$image=https://latex.codecogs.com/gif.latex?%5Clarge%20%5Cbegin%7Bmatrix%7D%20h%5Crightarrow%200%20%26%20as%20%26%20r%5Crightarrow%200%5C%5C%20h%5Crightarrow%201%20%26%20as%20%26%20r%5Crightarrow%20%5Cinfty%20%5Cend%7Bmatrix%7D&hash=48f23446a793762dfc608f87669079d8$
This is a warp drive, but differs from Alcubierre's in that the metric transitions in between differently. So now calculate or compute the Einstein tensor in order to get the energy density. You will find that is now is
$image=https://latex.codecogs.com/gif.latex?%5Clarge%20T%5E%7B00%7D%3D-%5Cfrac%7Bc%5E%7B4%7D%5Cbeta%20%5E%7B2%7D%7D%7B16%5Cpi%20G%7D%5Cleft%20%28%20%5Cfrac%7B%5Cpartial%20%5E%7B2%7Dh%7D%7B%5Cpartial%20x%5E%7B2%7D%7D+%5Cfrac%7B%5Cpartial%20%5E%7B2%7Dh%7D%7B%5Cpartial%20y%5E%7B2%7D%7D%20%5Cright%20%29&hash=753388ca9e3fde87e4974b7907e1fc5e$
Now the significance of what this does is that it allows one to localize the negative energy density somewhere inside the ship rather than having it spread all throughout the warp like it is in Alcubierre's warp drive. You might for example choose
$image=https://latex.codecogs.com/gif.latex?%5Clarge%20%5Cbegin%7Bmatrix%7D%20h%20%5Capprox%201-%5Cprod%20%5Cleft%20%28%20%5Cfrac%7Bz%7D%7BZ%7D%20%5Cright%20%29%5Cleft%20%5B%201-%5CTheta%20%5Cleft%20%28%20%5Crho%20-R%20%5Cright%20%29%20%5Cright%20%5D%5C%5C%20%5Crho%20%3D%5Csqrt%7Bx%5E%7B2%7D+y%5E%7B2%7D%7D%20%5Cend%7Bmatrix%7D&hash=6e811f9b2415a82bcb743ee3944716eb$
This constrains the negative energy to a thin cylindrical shell of finite length.

Last edited: Aug 3, 2020
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11. ### Dejan CorovicCelestial

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David, do you appreciate the fact that we explained the observations of UFO witnesses over the last 70 years with fundamental physics of General Relativity. Your equations proved that historically observed UFOs are unambiguously real!

This is the biggest scientific discovery of the 21st century and in this dedicated UFO forum, nobody except wwkrik, is even bothering to read about this .
So absurd and funny . YOUR PAPER IS A DAWN OF NEW INDUSTRIAL REVOLUTION, and nobody is realising it's just here at arm's lenght

It's very similar to when Wright brothers flew the first powered aeroplane at Kity Hawk beach. Idea that heavier than air machine was so unnatural that not a single local or national newspaper wanted to publish that story for 5 years. Yes, 5 years it took to break the mould. It's the same moment here.

Man, what do you say about the photographs that I compiled to support your GR solution? Do you see anything interesting there? I have a lot more UFO propulsion related observations. By the end of this thread, we'll completely reverse engineer them

One more video that supports the notion that UFOs have to move in a non-aerodynamic direction (along the axis of symmetry, aka "vertical" Z-axis). During the well known 1989 Belgium UFO flap, whenever two F-16 pilots locked targeting radars onto UFO, UFO escaped by flying vertically several thousand feet in one second. Each time UFO did that radar lock was broken. Here, nobody less than Col. Wilfried Brouwer, a commander of Belgian Air Force, gives journalists a briefing and shows F-16's HUD and vertical UFO maneuver. UFO chase lasted 75 min. and object was seen on total of 5 radars, two in the air and 3 on the ground.

Comment is in French, but HUD display is just numbers, so one can still follow.

Here is the most accurate breakdown of Belgium F-16 HUD, showing how fast was UFO moving in a vertical, non-aerodynamic, direction. Vertical move of 2,500 ft in 7 seconds:

ufo - UFOS at close sight: the Belgium UFO wave 1989-1993, March 31, 1990, about 00:30 a.m. at Thorembais-les-Beguines

Last edited: Aug 4, 2020
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12. ### waitedavid137Honorable

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I put the approximation in because in the real world Heiviside step functions and top hat functions are physically unrealistic limits meant only to be approximation. This limit description really only serves to show that the warp matter is around the sides of it and that its total can be reduced by compressing it to a ring.
Perhaps this is a better way to explain what's going on. With this warp drive the energy density depends on what the concavity of h is doing with respect to x and y. When h is concave up with respect to them the energy density can not be positive. When h is concave down with respect to them the energy density can no longer be that negative energy of interest. h has to go to zero at the origin and 1 at infinity, which means that on the inside it is concave up and outside concave down with respect to those variables. So the negative energy which is the part we are concerned with becomes constrained to the inner region where h is concave up with respect to x and y. The negative energy is inside the warp bubble or ship. Not extended all throughout the warp like in Alcubierre's. This also means I solved the causality problem with dropping out of warp. Take the plates apart or whatever you are using to produce the negative energy density state which is now inside the warp drive and completely causally connected.

Last edited: Aug 4, 2020
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13. ### waitedavid137Honorable

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Btw this is basically the Laplacian of h with respect to a radial coordinate so a constant plus or minus a log function of a radial coordinate also yields zero for the energy density in a transition region between the inside and outside as vacuum.

Last edited: Aug 4, 2020
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14. ### Dejan CorovicCelestial

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Do you have any comments regarding all these photos of UFOs that flip 90º in order to start moving?

Aerodynamically speaking, one would expect UFOs to fly edge-on so that their sharp edge can cut through the air. But as it turns out, UFOs fly in the direction of normal to their flat face. That means they fly in the direction of the largest aerodynamic resistance.

So, if your warp drive solution suggests preferred movement direction is in a direction of normal onto UFO's flat face, then this is a huge opportunity of scientifically validating UFO witness observations.

And, really, reaffirm the path for all the future research.

"O brother, where art thou?" @Thomas R. Morrison, check this thread.

Last edited: Aug 5, 2020
15. ### Dejan CorovicCelestial

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In one UFO case, aliens invited witnesses to enter the cabin of the spacecraft. Witness duly asked how the ship operates and was told that there was an incandescent ring, running around the circumference of the craft and rotating faster than a speed of light.

I am not saying that this account is true, but I am just curious would this fit in in the context of your equations. Maybe ring was incandescent because it was running high alternating current and then rotating such ring at a high speed one would get a mind-boggling magnetic field. Of course, classical centrifugal forces on such ring would be out of this world as well

16. ### Dejan CorovicCelestial

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Just on offchance, can these concepts be visualized with MathLab or WolframAlpha?

17. ### Thomas R. MorrisonMeh

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Sorry pal - this is fascinating stuff but I have two new employees to train and I need to get a break from work before I can study this thread as closely as it merits.

An interesting aside here: Oliver Heaviside (after which the Heaviside step function is named) discovered the principles of gravitoelectromagnetism (the linearized weak-field approximation of gravitational laws) in 1893...21 years before Einstein began publishing his papers on the general theory of relativity (GR) which convey the correct nonlinear equations that fully elucidate the inductive features of GR.

I'm seeing some interesting points being raised here that remind me of the papers by Dr. Harold White (with contributions by Dr. Eric Davis) when they were laying the foundations for NASA's fairly recent experiments with gravitational field propulsion using a toroidal electromagnetic device - if you're looking for some interesting reading Dejan, you might enjoy reading through those papers.

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18. ### Dejan CorovicCelestial

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Thanks, I read Dr. White's paper a few years back. And Heaviside was really the first relativist, although Einstein stole the whole limelight. As well, David said that Emmy Noether's work is even more important for warp drives than Einstein's.

Yeah, real UFOs prefer moving in a non-aerodynamic direction of symetery and it seems David's solution suggest the same. Maybe things became clearer with all these historical photos.

19. ### waitedavid137Honorable

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Lets say you've engineered or even back-engineered a disc whose hover isn't entirely stable. Ideally what you do to correct that is in torus vacuum suspend rings that you can spin up electromagnetically for gyro stabilizers. When you give one a kick up in angular velocity it would have a reaction on the craft to rotate it the other way which you can counter by having two you spin in opposite directions. If you want to deliberately rotate the craft some amount you can kick up one and then the other. Lets say you want communication to be more secure then radio. You can use lasers mounted at some places on it to constrain the data to go only only toward an intended target. If its your policy to send a communicae prior to shooting off into orbit then you rotate it before takeoff so that a laser is directed toward the receiver.

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20. ### waitedavid137Honorable

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193
So lets go back to the 1st modified warp drive according to the ship frame
$image=https://latex.codecogs.com/gif.latex?%5Clarge%20ds%5E%7B2%7D%3D%5Cleft%20%5B%20A%5Cleft%20%28%20ct%2Cx%2Cy%2Cx%20%5Cright%20%29%20%5Cright%20%5D%5E%7B2%7Ddct%5E%7B2%7D-%5Cleft%20%5C%7B%20dx%5E%7B2%7D+dy%5E%7B2%7D+%5Cleft%20%5B%20dz+%5Cbeta%20%5Cleft%20%28%20ct%20%5Cright%20%29g%5Cleft%20%28%20x%2Cy%2Cz%20%5Cright%20%29dct%20%5Cright%20%5D%5E%7B2%7D%20%5Cright%20%5C%7D&hash=9de2351a27d77953322c0cc09a92bd9c$
As mentioned the ship frame energy density
$image=https://latex.codecogs.com/gif.latex?%5Clarge%20T%5E%7B00%7D%3D-%5Cfrac%7Bc%5E%7B4%7D%7D%7B32%5Cpi%20G%7D%5Cfrac%7B%5Cbeta%20%5E%7B2%7D%7D%7BA%5E%7B4%7D%7D%5Cleft%20%5B%20%5Cleft%20%28%20%5Cfrac%7B%5Cpartial%20g%7D%7B%5Cpartial%20x%7D%20%5Cright%20%29%5E%7B2%7D%20+%5Cleft%20%28%20%5Cfrac%7B%5Cpartial%20g%7D%7B%5Cpartial%20y%7D%20%5Cright%20%29%5E%7B2%7D%20%5Cright%20%5D&hash=b5b1331e24e56481aae6ccb959a171ac$
is around the sides and does not tell you which direction along z the ship goes into warp. What is directional and tells you this is the time-z terms of the stress-energy tensor which according to the ship frame are
$image=https://latex.codecogs.com/gif.latex?%5Csmall%20T%5E%7B0z%7D%3DT%5E%7Bz0%7D%3D%5Cfrac%7Bc%5E%7B4%7D%7D%7B32%5Cpi%20GA%5E%7B4%7D%7D%5Cleft%20%5C%7B%20%5Cbeta%20%5E%7B3%7Dg%5Cleft%20%5B%20%5Cleft%20%28%20%5Cfrac%7B%5Cpartial%20g%7D%7B%5Cpartial%20x%7D%20%5Cright%20%29%5E%7B2%7D+%5Cleft%20%28%20%5Cfrac%7B%5Cpartial%20g%7D%7B%5Cpartial%20y%7D%20%5Cright%20%29%5E%7B2%7D%20%5Cright%20%5D-2A%5Cbeta%20%5Cleft%20%5B%20%5Cleft%20%28%20%5Cfrac%7B%5Cpartial%20A%7D%7B%5Cpartial%20x%7D%20%5Cright%20%29%5Cleft%20%28%20%5Cfrac%7B%5Cpartial%20g%7D%7B%5Cpartial%20x%7D%20%5Cright%20%29+%5Cleft%20%28%20%5Cfrac%7B%5Cpartial%20A%7D%7B%5Cpartial%20y%7D%20%5Cright%20%29%5Cleft%20%28%20%5Cfrac%7B%5Cpartial%20g%7D%7B%5Cpartial%20y%7D%20%5Cright%20%29%20%5Cright%20%5D+2A%5E%7B2%7D%5Cbeta%20%5Cleft%20%5B%20%5Cfrac%7B%5Cpartial%20%5E%7B2%7Dg%7D%7B%5Cpartial%20x%5E%7B2%7D%7D+%5Cfrac%7B%5Cpartial%20%5E%7B2%7Dg%7D%7B%5Cpartial%20y%5E%7B2%7D%7D%20%5Cright%20%5D%20%5Cright%20%5C%7D&hash=bacec8ea9d84fd3da048b96aa8529eaa$
Whose magnitude is reduced by having A large where g significantly varies with respect to x and y. This is often interpreted as a momentum term, but it can be produced by a static field. For example if one has a charged parallel plate capacitor and you fix magnets just outside the gap, the stress-energy tensor of the static electromagnetic field will have time space cross terms like this.

Last edited: Aug 6, 2020
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