The article claims that a calculation of the Casimir energy (see this) for a system of two parallel metal plates using what is called world-line numerics predicts that the region between capacitor plates has a torus-like region inside which the vacuum energy density is negative (note that the vacuum energy depends on the shape and size of the cavity for a quantum field theory restricted inside the capacitor by posing suitable boundary conditions).

The observation is that the vacuum energy density resembles that for the so-called Alcubierre drive (see this) claimed to make possible space-time travel with superluminal speeds with respect to the time coordinate of an asymptotically flat space-time region. The idea is that if the space contracts in front of the space-ship and expands behind, super-liminality becomes possible. Inside the space-ship the space-time would be in a good approximation flat. Alcubierre himself suggests that the Casimir effect might produce the needed negative energy density.

It is easy for a skeptic to invent objections. Consider first the calculation of the vacuum energy behind Casimir force, which is a real effect and has been experimentally detected.

- The original calculation of Casimir was for van der Waals forces. It has been later show that Casimir effect could be interpreted as retarded van der Waals force (see this): the consideration of poorly defined vacuum energies was not needed in this approach.
Later emerged the proposal that one can forget the interpretation as an interaction between molecules and that the calculation applies by considering the system as idealized conducting capacitor plates. There are objections against this interpretation.

- Force is the negative gradient of energy. The predicted force is finite although the calculation of the vacuum energy gives an ultraviolet divergent infinite answer requiring a regularization. The regularization gives a finite result as an energy E per area A of the plates, which is negative and given by
E/A= -ℏπ

^{2}/720a^{3}.a is the distance between plates and force is proportional to 1/a

^{4}.Note that there is no dependence on the fine structure constant or any other fundamental coupling strengths. Casimir energy and force approach rapidly to zero when a increases so that practical applications to space-travel do not look feasible.

Also the notion of Alcubierre drive can be criticized.

- The basic problem of general relativity (GRT) is that the notions of energy and momentum and corresponding conservation laws are lost: this was the starting point of TGD. In weak gravitational fields in which space-time is a small metric deformation of empty Minkowski space-time, one can expect that these notions make approximately sense. However, Alcubierre drive represents a situation in which the deviation from a flat Minkowski space is large. Does it make sense to speak about (conserved) energy anymore?
- If one accepts GRT in this kind of situation one still has the problem that negative energy density violates the basic assumptions of GRT. Some kinds of exotic matter with negative energy would suggest itself if one believes that energy corresponds to some kind of particles
- One can also argue that the proposed effect is a kind of Munchausen trick. The situation must allow an approximate GRT based description by regarding space-time ship as a single unit whose energy is determined by the sum of the energy of the space-time ship and Casimir energy and is positive so that the space-ship moves in a good approximation along time-like geodesic of the background space-time. The corrections to this picture taking into account the detailed structure of the space-ship should not change the description in an essential manner and only add small scale motion superposed to the center of mass motion.

What about the situation in TGD?

- The notions of energy and momentum are well-defined and the classical conservation laws are not lost. The conserved classical energy assignable to space-time surface is actually analogous to Casimir energy although it is not assigned to vacuum fluctuations and consists of the contributions assignable to Kähler action and volume action. These contributions depend on Kähler coupling strength and cosmological constant which in the TGD framework is (p-adic) length scale dependent. Recall that for the parallel conductor plates at least, Casimir energy has no dependence on fundamental coupling strengths.
If the energy is positive definite in TGD as there are excellent reasons to believe, the basic condition for the Alcubierre drive is not satisfied in TGD.

- Here I must however counterargue myself. One can construct very simple space-time surfaces for which the metric is flat and Euclidean and they are extremals of the basic variational principle.
- Consider a surface representable as a graph of a map M
^{4}× CP_{2}given by Φ= ω t, where Φ is angle coordinate of the geodesic circle of CP_{2}. The time component g_{tt}= 1-R^{2}ω^{2}of the induce flat metric is negative for ω >1/R.The energy density associated with the volume part of the action is non-vanishing and proportional to (g

_{tt}^{1/2}g^{tt}and negative. The coefficient is analogous to cosmological constant. - Can these "tachyonic" surfaces correspond to preferred extremals of the action, which are physically analogous to Bohr orbits realizing holography. The 3-D intersections of this solution with two t= constant time slices are Euclidean 3-spaces E
^{3}or identical pieces of E^{3}. If the preferred extremal minimizes its volume action, then (g_{tt}^{1/2}=(1-R^{2}ω^{2})^{1/2}is maximum. This gives ω R=0 and a flat piece of M^{4}.Interestingly, the original formulation for what is it to be a preferred extremal (as a condition for holography required by the realization of general coordinate invariance), was that space-time surfaces are absolute minima for the action which at that time was assumed to be mere Kähler action. The twistor lift of TGD forced the inclusion of the volume term. It seems that Alcubierre drive is not possible in TGD.

It might be also possible to show this by demonstrating that the embedding of the Alcubierre metric as a 4-surface in M

^{4}× CP_{2}is not possible. - TGD also allows different kinds of Euclidean regions as preferred extremals. These correspond to what I call CP
_{2}type extremals. They have positive energy density and they have light-like geodesics as M^{4}projection and they serve as classical geometric models for fundamental particles.

- Consider a surface representable as a graph of a map M

Concerning superluminal teleportation, the problem is that in the standard quantum theory teleportation also requires sending of classical information. Maximal signal velocity makes superluminal teleportation impossible. This poses extremely stringent limits on the communications with distant civilizations.

In the zero energy ontology of TGD (see this and this), the situation changes.

- In the so-called "big" state function reductions (BSFRs), which are the TGD counterparts for ordinary SFRs, the arrow of time changes.
- For light-signal this means that the signal is reflected in time direction and returns back in time with a negative energy (this brings in mind the negative energy condition for Alcubierre drive). This is just like ordinary reflection but in time direction perhaps allowing seeing in time direction I have proposed conscious memory recall could correspond to this kind if seeing in time direction.
- This might also make practically instantaneous classical communications over space-like distances possible. This in turn would also make possible superluminal quantum teleportation.

## 2 comments:

I'm gonna have to publish this paper on minimal surfaces and show all y'all how the Riemann hyopothesis is related to Yang-Mills. The energy levels are surface areas of minimal surfaces and the 3 families of elementary particles is due to this: https://i.imgur.com/cLfjDC9.gif the non-convexity of the 3 island-like root structures rising above the background surface (correspond to trivial zeros 1 2 and 3 if the first one is labeled 0 at the origin). The non-trivial zeros correspond to particle-like excitations

Best wishes for the project.

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