How much Energy to create a 'warp field' according to White?

Question

An Alcubierre Drive is a hypothetical device that can move a place someplace else faster than the speed of light without violating known laws of physics. This paper provides some equations as to the energy density in such a field, as well as proposals about testing the theory experimentally and finding modes of applications.

One application is described as follows:

the spacecraft departs earth and establishes an initial sub-luminal velocity $v_i$, then initiates the field. When active, the field’s boost acts on the initial velocity as a scalar multiplier resulting in a much higher apparent speed, $v_{\text{eff}}= γ v_i$ as measured by either an Earth bound observer or an observer in the bubble. Within the shell thickness of the warp bubble region, the spacecraft never locally breaks the speed of light and the net effect as seen by Earth/ship observers is analogous to watching a film in fast forward. Consider the following to help illustrate the point – assume the spacecraft heads out towards $\alpha$ Centauri and has a conventional propulsion system capable of reaching $0.1c$. The spacecraft initiates a boost field with a value of 100 which acts on the initial velocity resulting in an apparent speed of $10c$. The spacecraft will make it to $\alpha$ Centauri in 0.43 years as measured by an Earth observer and an observer in the flat space-time volume encapsulated by the warp bubble.

A formula to arrive at $\gamma$ is given in the paper. Now ...

• I assume that the lower limit to create such a "warp bubble" is independent of the method or device or whatever used, there's a hard lower physical limit - is this even correct?

• Now, as an example, how much energy would be needed to create a boost factor $\gamma$ of 100 for a golf ball (or any other arbitrary sized volume)?

• Where does this energy go?

While White mentions an experimental setup to test the theory with only a ring of capacitors, others mention exotic matter to be neccessary. I don't want to go into details of implemementation in this question. Since my question is about the physical limits, it's here on Physics SE and not over at Space exploration.

Answer 1

It's very, very highly likely that none of these schemes are workable for the following reasons:

1. Superluminal travel violates causality. You can talk about "general relativity loopholes" all you want, but you can always envision that the "warp region" is going to be confined to a finite-sized subspace of the whole spacetime, and outside of this region, the superluminal observer will just be an ordinary superluminal traveller in special relativity. Have two of them travelling in opposite directions, and they can start communicating with each other out of temporal order
2. They require exotic matter-- all of these schemes require that you have some negative energy density matter in order to create the stretching behind/contracting ahead effect in spacetime that makes the scheme work. This nasa paper claims that you can do this with Casimir potentials, but there is no real indication that Casimir energies are even negative in the GR sense, or that you can get enough of this energy to:
3. These schemes require a LOT of exotic matter. It's difficult to bend spacetime on macroscopically significant scales because gravity is weak relative to the other forces. Futhermore, the bending of spaceitme has to be large enough to create an effect, but over a large enough region so that our travellers don't get torn apart by tidal effects. The "new" update of the design reduces the amount of exotic mass required from something like several stars to "just" a Jupiter's worth of exotic matter. Even if it was 1 mg, you still have the issue that exotic matter hasn't been observed, and many GR theorems break (cosmic censorship, causality [see 1.], the singularity theorems, etc.), and relativity ceases to be the strong, predictive theorem that we know.

Answer 2

White claimed that his modified Alcubierre Metric from 2011 for a vessel traveling at 10c and 10m wide only needs -700 kg of mass (or, -6.29 x 10^19 J of energy) in a NASA / DARPA Symposium in 2012. At first this seems to contradict Zo the Relativist's astute answer about a negative Jupiter mass (or, -1.898 x 10^27 kg). What's going on here? In short: White is ignoring a crucial constraint and it's baffling as to why. Zo is entirely correct in the end.

Later papers by White (2021) confirm that he is hoping to exploit the Casimir Effect to be the source of his negative energy densities. Sadly, this comes with severe mathematical constraints, which Alcubierre onward (1994+) abide by in making the warp wall thickness on the Planck scale. White reduces the energy needed to a mere -700 kg because he thickens the wall of the warp bubble, seemingly unaware of why everyone else is making them so immensely thin. One can only "borrow on credit" energy from the vacuum under severe constraints called "Quantum Inequalities." White ignores this for unstated reasons. Krasnikov is the only one I am aware of that tries to bypass QIs legitimately (2003), but White never cites Krasnikov.

I assume that the lower limit to create such a "warp bubble" is independent of the method or device or whatever used, there's a hard lower physical limit - is this even correct?

The method is currently unknown, besides vague talk of exploiting the Casimir Effect to quantities and densities of negative energy well beyond experimental observations. There's no lower limit, aside from the size of the warp bubble, the velocity it is going, and the wall thickness. The mathematical relation reiterated by Alcubierre in 2017 is:

$E ≈ - v² R² σ$

Where $v$ is the velocity (in the starship's frame of reference), $R$ is the radius of the warp bubble, and $σ$ is one over the wall thickness. You can see why White tampers with $σ$, but as I mentioned, he ignores the issue of Quantum Inequalities established in the literature.

Now, as an example, how much energy would be needed to create a boost factor γ of 100 for a golf ball (or any other arbitrary sized volume)?

Refer to the mathematical relation I gave above. Note: there's nowhere to input the "boost factor" (γ), because that is purely White's invention. White cites the 2000 paper by Chung and Freese to "modify" the Alcubierre metric, an extremely speculative paper about cosmology, not warp drives. It doesn't appear in the Alcubierre Metric from 1994, nor in any other literature aside from White's. Since White's math is suspect I hesitate to provide the answer, but I'll do so anyway since you asked (but using Alcubierre's math, since that is not suspect). Assuming his -700 kg figure is right, $v$ is kept the same, a golf ball has a diameter of 42.7 mm, and we keep White's wall thickness:

$E ≈ - (0.02135/5)²$

Your Golf Ball only needs -12.76 g of negative mass-energy.

Again, I can't stress how urgent it is to consult other literature in the field, because White is making a lot of assumptions.

Where does this energy go?

In both White's and Alcubierre's Metrics the negative mass-energy is in the distorted spacetime itself or the wall of the warp bubble. It is most concentrated in the regions perpendicular to the direction of motion, with virtually none directly in front of the starship and behind.