Alcubierre Metric Energy Equation

[McC]

I know the Alcubierre metric has its problems, and is generally disliked. However, I like the concept behind it and want to use it in a universe I'm working on. However, I'd also like to be consistent with the actual math behind it, rather than just hand-waving and making shit up (a pet-peeve of mine in sci-fi, when a mathematical solution really can exist).

I see it said frequently that not only does Alcubierre's metric require negative energy densities, but also a staggering amount of it. I've been wading through the paper, but I feel like a lot of this math is so far beyond me. He ultimately gives the "energy density" equation (Equation 19), as this (curly brackets denote a subscript):

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-1/(8pi) * (v{s}^2*p^2)/(4r{c}^2) * (df/dr{s})^2

I haven't the first clue what any of those variables are. A few of them he seems to define earlier, as functions of t (v{s}(t), f(r{s}), and r{s}(t)). However, he introduces a new variable here, r{c}, which I haven't seen anywhere else in the paper. He does define p, which is

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p = (y^2 + z^2)^(1/2)

but when it comes to the functions, he seems to drop the actual function aspect of them. For example, he earlier defines v{s} as a function of t (after equation 5), in the form:

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v{s}(t) = dx{s}(t) / dt

and also defines r{s}(t) (which I assume is the integral of dr{s}) as:

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r{s}(t) = [(x-x{s}(t))^2 + y^2 + z^2]^(1/2)

and finally, defines f(r{s}), which I assume to be the integral of df, as:

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f(r{s}) = (tanh(σ(r{s} + R)) - tanh(σ(r{s} - R)))/(2tanh(σR))

Here, again, r{s} loses the (t) function indicator, which he does a lot, and confuses me greatly. He also introduces σ and R. R is, I think, the radius of the 'bubble' created (page 5, first paragraph), but I have no idea what units it's in, nor do I have any idea how σ, which I assume is an angle, comes into play. He also has x, y, z, and t coordinates, which obviously factor in, but I don't know how.

Ultimately, I'm just looking for a way to calculate how much energy input is required to produce a given "speed" for a given size region. If someone can show me how to understand the formulae to come to this answer, I'd very much appreciate it. A formula I can use to "plug and play" would be fantastic.

Thanks!

[Ariphaos]

All I remember was something like a two-hundred-foot long vessel taking some ten times the mass-energy conversion of the known Universe to travel at warp 2. The article didn't say for how long, and most of it seems like it was because of the 'negative energy interest'. I have to wonder, given how thin the bubble has to be anyway, if a more logical drive would in fact cheat, by creating trillions of this little warp zones to mitigate the effect.

[McC]

According to this paper, the metric can be refined such that only -10kg of exotic matter are required to make the whole thing work. However, I'm still at a loss for really understanding any of this. Anyone who could give me a "this is what all this shit means" primer would be my hero.

[Kuroneko]

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-1/(8pi) * (v{s}^2*p^2)/(4r{c}^2) * (df/dr{s})^2

I haven't the first clue what any of those variables are. A few of them he seems to define earlier, as functions of t (v{s}(t), f(r{s}), and r{s}(t)). However, he introduces a new variable here, r{c}, which I haven't seen anywhere else in the paper.

It's a typo. Alcubierre means r{s}. Otherwise, it doesn't make much sense.

Here, again, r{s} loses the (t) function indicator, which he does a lot, and confuses me greatly. He also introduces σ and R. R is, I think, the radius of the 'bubble' created (page 5, first paragraph), but I have no idea what units it's in, nor do I have any idea how σ, which I assume is an angle, comes into play. He also has x, y, z, and t coordinates, which obviously factor in, but I don't know how.

They're arbitrary parameters. Yes, R is essentially the radius of the bubble. The role of σ is desribed in equation (7)--it's essentially a free pameter whose role is nothing but to approximate a step function analytically. As for units, they're noted in the same page--geometric units, G = c = 1. In other words, it doesn't matter as long as they're consistent. For example if your time units are seconds, then length units should be [c] (light-seconds), and mass units should be [G/c²]. The point of these units is that all quantities are dimensionless.

I haven't taken a look at the second paper.

[Gil Hamilton]

I'd stay away from the math of such things. If you are writing science fiction for your star drive, even if it is based on a real theory, to not be described in depth and just assumed to work.

Just a suggestion, as it actually works to make your sci-fi harder. Describe effects, not mathematics.

[McC]

The mathematics won't make it into the actual story, but I need them for myself. I want to keep things consistent, and I'd rather not just "make up" equations when they already exist.

[Enola Straight]

Perhaps Albicurrie's formulas and equasions can be compared to Planck units.

http://en.wikipedia.org/wiki/Planck_units

[Ariphaos]

According to this paper, the metric can be refined such that only -10kg of exotic matter are required to make the whole thing work. However, I'm still at a loss for really understanding any of this. Anyone who could give me a "this is what all this shit means" primer would be my hero.

Oooh nifty.

Well, it begins with the Broeck Metric, which (so I read) is a slightly modified version of the Alcubierre, that cheats by creating an initially extremely tiny (nucleus-sized or so) warp bubble instead, and then warping space so that it can encompass something macroscopic. Even the Broeck version requires several solar masses between positive and negative energy. to function.

Anyway, the first five sections of the paper seem to focus on the problem whereby the warp bubble becomes causally disconnected from the ship, and claiming that this does not need to be the case. It seems to claim that photons will actually be able to exit the front of the bubble, which I don't quite get.

Section 6 discusses hitting something at warp, fairly straightforward in the text, at least.

Section 7 references this paper and others, which seems to go over a number of loopholes. In one, a negative milligram of negative energy is used to create and maintain a kind of dimensional pocket, of arbitrary size, and building the shortcut for the plank-sized 'hole' or transporting it, is far more feasible than trying to move a more massive vessel. I'm not sure where the -10kg reference comes from, though.

[McC]

Anyway, the first five sections of the paper seem to focus on the problem whereby the warp bubble becomes causally disconnected from the ship, and claiming that this does not need to be the case. It seems to claim that photons will actually be able to exit the front of the bubble, which I don't quite get.

When it comes to what I'm doing with it, this little hiccup (the part where the negative energy bubble would still be in non-warped space) is the portion that I would "ignore" in favor of having the FTL mechanism work. But I'm still interested in accurately representing the energy requirements, radii, and resultant velocities those two factors allow for.

Section 7 references this paper and others, which seems to go over a number of loopholes. In one, a negative milligram of negative energy is used to create and maintain a kind of dimensional pocket, of arbitrary size, and building the shortcut for the plank-sized 'hole' or transporting it, is far more feasible than trying to move a more massive vessel. I'm not sure where the -10kg reference comes from, though.

Neither am I, but it's there plain as day on page 7 and 8.

[Ariphaos]

The mathematics won't make it into the actual story, but I need them for myself. I want to keep things consistent, and I'd rather not just "make up" equations when they already exist.

The thing is, as far as energy requirements are concerned, they don't seem to be set in stone:

In the search for realistic shortcuts an obvious line of attack would be to look for situations in which the quantum inequality (4) does not hold. Note in this connection that for non-Minkowskian spacetimes (4) has never been proved4, though some arguments in its substantiation were brought forward in [14]. Moreover, in their recent paper [17] Olum and Graham showed that a system of two interacting scalar fields may violate (4). (It is especially interesting that the violation takes place near a domain wall. One might speculate on this ground that perhaps the similarity of the shortcuts considered in the introduction with the domains of false vacuum has far-reaching consequences.) Thus the inequality (4) is, at least, non-universal.

Earlier, assuming I'm reading it right, quantum interference is basically shown to be the reason for the plank-scale energy density, thus giving values on the order of 10^32 solar masses for energy requirements. If we're not in Minkowski spacetime, then these massive energies may not be needed.

...

Honestly though, in the end, many people still dig steampunk genre, for example. If you research every last detail to death, you won't stop researching, and never start writing. Also, it can be all too easy to do so much research in one subject that you forget about a bunch of others.

[McC]

Here's yet another paper by yet another person on the topic.

[McC]

I've been trying to teach myself the math on this stuff with mixed results. However, in the process, I found that a large chunk of my problem was not being able to really 'see' the equations in question. As such, I wrote up a little PHP script that would define each variable (such as v_s or r_s) and then sub it into the next equation called for. It can be viewed here. I also learned, after writing the part of the PHP script represented there, that v_s was a way of writing dv/ds (with d in this case representing the partial derivative). I didn't know that before, so that significantly improves my understanding of this.

Still haven't "cracked" it yet, though.