Worldline numerics applied to custom Casimir geometry generates unanticipated intersection with Alcubierre warp metric

Abstract

While conducting analysis related to a DARPA-funded project to evaluate possible structure of the energy density present in a Casimir cavity as predicted by the dynamic vacuum model, a micro/nano-scale structure has been discovered that predicts negative energy density distribution that closely matches requirements for the Alcubierre metric. The simplest notional geometry being analyzed as part of the DARPA-funded work consists of a standard parallel plate Casimir cavity equipped with pillars arrayed along the cavity mid-plane with the purpose of detecting a transient electric field arising from vacuum polarization conjectured to occur along the midplane of the cavity. An analytic technique called worldline numerics was adapted to numerically assess vacuum response to the custom Casimir cavity, and these numerical analysis results were observed to be qualitatively quite similar to a two-dimensional representation of energy density requirements for the Alcubierre warp metric. Subsequently, a toy model consisting of a 1 µm diameter sphere centrally located in a 4 µm diameter cylinder was analyzed to show a three-dimensional Casimir energy density that correlates well with the Alcubierre warp metric requirements. This qualitative correlation would suggest that chip-scale experiments might be explored to attempt to measure tiny signatures illustrative of the presence of the conjectured phenomenon: a real, albeit humble, warp bubble.

1 Background

Work being conducted under a DARPA Defense Sciences Office grant is investigating the implications of the dynamic vacuum model for the possibility of structure to the Casimir energy distribution manifested in a parallel plate cavity. The dynamic vacuum model predicts that the negative vacuum energy density present in the parallel plate cavity is not isotropic, rather there is a varying energy density field present in the cavity with an average value that corresponds with that predicted by the traditional equation for parallel-plate Casimir energy density. The structure predicted to be manifest in the cavity takes the form of a larger magnitude negative vacuum energy density concentrated along the cavity mid-plane that relaxes non-linearly to the unperturbed state at the cavity boundaries. Based on detailed studies of the atomic orbitals of the hydrogen atom, and deriving the acoustic wave equation from the Schrödinger equation, it is speculated that the energy density structure in a Casimir cavity is coupled to a small polarization field in the vacuum fluctuations resulting in a small but non-zero electrostatic field originating along the cavity mid-plane and terminating at the grounded cavity walls. It has been further reasoned in the literature that it may be possible to construct a customized Casimir cavity equipped with small pillars placed at the mid-plane as depicted in Fig. 1 such that when the pillar channel is sampled by a high impedance oscilloscope, the scope would detect a transient non-zero voltage signal that would rapidly go to zero as the stored energy in the polarization field is depleted from the measurement process.

(Footnote in the original: a recently published paper details an experimental campaign using an asymmetric Casimir cavity arrangement where one cavity has a small separation and the other cavity has a much larger effective separation. This experimental campaign observed a current flow from the larger cavity electrode to the smaller cavity electrode. Our cavity is analogous in that the two parallel plates define a large cavity and the plate-pillar system plays the role of the smaller cavity.)

If these cavities can be made small enough and arrayed together in large enough numbers to increase the stored energy, the magnitude and duration of this predicted transient voltage signal may be detectable in a laboratory setting. The equation that predicts the magnitude of this small but non-zero electrostatic field was derived in the earlier work and depends on the inverse fourth power of the cavity gap. A quick calculation for a Casimir cavity with a 4 µm gap predicts a magnitude of the electrostatic potential arising from the polarization of the vacuum fluctuations along the mid-plane of the cavity to be about 0.7 mV.

Figure 2 shows some examples of the current nanofabrication trial runs that are ongoing as part of the effort. The left panel in the figure depicts a recent result obtained utilizing a Nanoscribe 3d printer to evaluate a range of cavity gaps and pillar sizes. The material being used in the manufacture is SU-8 2025 permanent epoxy negative photoresist. After printing and curing, the cavities then undergo electroless plating to add a layer of Ag to the outer surfaces. The right panel shows recent etching results using Deep Reactive Ion Etching (DRIE) equipment to etch the high-aspect ratio cavity planes into a silicon wafer substrate. This approach uses standard SiO2 wafer materials with the expectation that the final concepts will have a metallic layer added by means of evaporation. The results depicted were focused on achieving the high-aspect ratio etch, and future work will incorporate the masking necessary to produce pillars.

A technical concern with the proposed design implementation is how the presence of pillars might affect the predicted Casimir energy density distribution within the cavity — would there be self-screening that occurs within the pillar that minimizes the magnitude of the negative vacuum energy density present inside the pillar, and hence the magnitude of the detectable signal? It was during the analysis process seeking to address this screening question that an unanticipated intersection with the Alcubierre metric was found.

The anticipated geometry of each cavity is approximately 4 µm wide with each plate measuring 40 µm by 40 µm. The pillars are expected to be approximately 1 µm in diameter.

2 Introduction

The literature search phase of this DARPA project discovered a numerical methods approach known as worldline numerics that can be used to study and quantify the Casimir energy density and force. The curious aspect of this modeling approach that makes it of high interest to the DARPA project is that it predicts that there is structure (spatial variation) to the negative vacuum energy density in a Casimir cavity analogous to the predictions of the dynamic vacuum model. The primary value of considering an implementation of this model technique is that it provides a high fidelity prediction of the perturbed vacuum state inside model geometry (e.g. the evanescent fields in structure) along with predictions for the perturbed vacuum state within the cavity gaps. An additional benefit of the worldline numerics method — also referred to as the loop cloud method — for studying the Casimir effect is that it can be used to address any type of geometry with effectively no restrictions on curvature or lack of smoothness.

Due to the similarities with the dynamic vacuum model and its computational flexibility coupled with maturity, this worldline numerics technique has been implemented to consider the custom Casimir cavities and determine the predicted negative vacuum energy density distribution in the cavities and within the pillars. Figure 3 depicts the numerical analysis results from our implementation of the worldline approach considering a 3 dimensional sphere with a radius of 4 µm separated from an infinite flat plate by a separation of 4 µm. The approach was implemented using Open MPI-enabled c-code and the analysis was run on 100 2.40 GHz Intel Skylake CPUs. The model grid was a 50 × 50 × 50 grid with a 2000 unit-loop ensemble. The top panel shows a 2 dimensional section cut of the predicted distribution to the negative vacuum energy density between the two bodies where it should also be noted that the field gradients extend into both the body of the sphere and the flat plate in the form of evanescent fields. The bottom panel shows the distribution of forces across the surfaces of the model and was generated using COMSOL. The following section provides a brief summary of the details behind the worldline numerics analytic approach.

3 Synopsis of Casimir worldline numerics

The string theory inspired worldline numerics approach to determine the Casimir effect is developed in detail in the work of Gies, Langfeld and Moyaerts, and the critical aspects of the analysis technique are briefly summarized here for convenience. With the objective of evaluating the Casimir interaction energy (normalized) arising from the coupling of a real scalar quantum field of finite mass with a background potential that represents the Casimir geometry, the key equation is the effective action.

(Equations 1 to 9 of the published article are dense mathematical typesetting that did not survive text extraction cleanly. They are given here as named results; the exact expressions are in the source.)

• Equation 1 is the effective action: an integral over proper time of the expectation value of a Wilson loop functional, minus one, taken over four-dimensional space, with a mass-dependent exponential weighting.
• Equation 2 defines that expectation value as the average of the loop ensemble over all closed loops with Gaussian walks — a ratio of two path integrals, one weighted by the Wilson loop functional and one not.
• Equation 3 is the Wilson loop identity itself: the exponential of minus the proper time multiplied by the integral, along the unit loop, of the background potential evaluated at the shifted and proper-time-scaled loop position.

Here the loop path is a unit closed loop, the position shift places the unit loop in model space, and the proper time serves to scale the unit loops.

Equipped with this information, one can calculate the unrenormalized Casimir energy as the effective action divided by the "volume" in the time direction. When considering the Casimir force, the portion of the Casimir energy that has a dependency on the relative positions of the bounding geometries can be obtained by subtracting the energies of the single objects from the total Casimir energy — Equation 4, the Casimir interaction energy.

The Casimir force can be obtained by taking the negative spatial derivative of this interaction energy, and further, this process has removed any UV divergences. In the Dirichlet limit and for a massless scalar field with Dirichlet boundaries in three space and one time dimension, the worldline representation of the Casimir interaction energy reduces to a proper-time integral over the model volume of a worldline functional — Equation 5. The worldline functional is zero if the re-scaled unit loop does not intersect any Casimir geometry, and is one minus n if the re-scaled loop intersects n bodies of the Casimir geometry, where n is one or more.

The numerical evaluation process requires two discretization steps. The first is the discretization of the path integral into an ensemble of random paths, with each path forming a closed spacetime loop. The second is the discretization of the proper time interval into N steps such that an individual closed loop consists of N points per loop. Transporting and rescaling the ensemble of unit loops to a point in the model amounts to adding the square root of the proper time multiplied by the unit loop coordinate to the centre-of-mass point. Applying these two discretizations to the Casimir interaction energy yields Equation 6, the ensemble-averaged numerical form actually evaluated.

As the worldline numeric approach for the Casimir phenomenon is based on (massless) scalar fields, the technique can currently only assess idealized behaviour for bounding geometry and cannot assess any frequency dependence of materials. Additionally, the approach developed to date in the literature does not account for the impacts of temperature. However, it is still a very capable and appealing technique in that it can provide quick and fairly accurate assessments for very complicated geometries where analytic techniques are not practical.

3.1 Generating unit loops, computational approach, and implementation validation

The developers of the worldline numerics for the Casimir phenomenon explored numerous ways to generate ensembles of unit loops with Gaussian distribution ranging from a heat bath kernel to random walks, and finally landing on a technique denoted as the "v-loop" algorithm. The curious reader is encouraged to review the referenced manuscript for a thorough discussion of the benefits and shortfalls of the different techniques explored. The "v-loop" technique was selected as it can computationally generate an ensemble of loops each having N points per loop without having to perform multiple iterations on each loop to realize a closed random walk/worldline with the required statistical characteristics. Figure 4 shows several examples of unit-loops generated by the v-loop methodology ranging from a 100 point unit loop to a 5000 point unit loop.

A summary of the computational procedure steps are provided here to facilitate the reader's understanding of the "v-loop" approach:

1. generate N minus 1 numbers with a Gaussian distribution (e.g. using the Box–Müller method);
2. calculate N minus 1 normalized numbers from them — Equation 7, a normalization that scales the first entry by the square root of two over N and each subsequent entry by a ratio involving N plus one minus the index over N plus two minus the index;
3. calculate the increment values with a recursive correction that subtracts a running partial sum divided by N plus two minus the index — Equation 8;
4. a unit loop can now be created by accumulating those increments, with the first point set by a weighted sum and the last point set so that the loop closes — Equation 9;
5. this procedure is repeated to create the full unit loop ensemble.

The benefit of this numeric worldline approach is that it can be used to address any type of geometry while other approaches such as Proximity-Force Approximation (PFA) are not as flexible. Additionally, the approach has no dependency on the choice of model grid spacing or grid choice. The answer for a single point of interest in space does not have any interdependency on any other model grid points and may be calculated in total isolation if that is all that is needed. Figure 5 provides a pictorial representation of the analysis process for a parallel plate Casimir cavity. As indicated in the figure, once the loop ensemble has been generated the computational process to calculate the Casimir interaction energy follows the below enumerated steps:

1. The loop ensemble is moved to each model grid point of interest and scaled using proper time until two or more bodies in the model are pierced;
2. the scale at which an individual loop pierces two or more bodies defines the integral limits for the Casimir interaction energy integral;
3. the energy at the geometric point of interest in the model is increased based on wavelength (loop scale) and loop weight factor;
4. this scaling process is repeated for each loop in the ensemble at a geometric point of interest in the model;
5. the above steps are repeated for each geometric point of interest in the model.

(Footnote in the original: while the figure depicts a regularized model grid for communication purposes, the computational result at an individual model point is not dependent on adjacent points making the technique independent of grid choice.)

Validation of our implementation of the numeric worldline approach was done on a plate-plate case and a corresponding plate-sphere case and was compared to documented results in the literature. For the reader's awareness, the referenced work conducts extensive analysis to compare analytic results to the numeric results produced by the worldline technique for the simple plate-plate scenario and plate-sphere scenario. The referenced study explored the impact of number of points per unit loop, number of unit loops in an ensemble, separation distance of geometries, coupling, and mass. It is not the intention of this paper to duplicate the viability of the overall worldline numerics approach as this has already been done in the literature as noted, rather the intention of this paper is to apply this very powerful and flexible technique to fairly complicated geometries where only numerical methods can effectively be used. In our validation effort, we confirmed that our model predict the correct Casimir force for a given plate-plate or sphere plate scenario, and subsequently compared their Casimir interaction energy density results from their numeric worldline algorithm to our interaction energy density results from our numeric worldline algorithm. The subsequent more complicated geometries we consider forthwith as part of this work do not have trivial analytic solutions which is why the numeric worldline technique is employed.

A plot of the results from our implementation is provided in Fig. 6 for the two cases and the plot also includes a plate-blade case. The geometry of all three cases is such that the closest point of separation between all three cases is identical allowing for comparison of the results to evaluate the effects of curvature. The plots reflect the energy density as measured along a line normal to the plate-plate geometry and these geometric points of interest are the same for the plate-sphere and plate-blade cases. The colors of the Casimir energy density plots correspond to the colors of the toy geometry also overlayed on the plot facilitating comparison of the results and to clearly see the impact of curvature. The magnitude of the energy density distribution clearly shows a decrease in magnitude as the cases go from plate-plate to plate-blade. Additionally, the plots show that there is a shifting of the peaks to the right due to the curvature effects. Comparing our results to those in literature indicates that our algorithms are functioning properly.

4 Analysis results and unanticipated findings

As discussed in the opening of the manuscript, the critical concern for this project is if the presence of a pillar in the Casimir cavity would serve to screen itself in such a way that it would be unable to see the negative vacuum energy density gradient predicted to be present in the cavity if the pillars were not present. A model was built to assess a 4 µm cavity with a 1 µm diameter pillar placed in the middle of the cavity. The model discretization was a 35 × 35 grid running plus and minus 4 µm in both the x and y axis. The x-axis is the vector normal to the parallel cavity plates, and the y-axis is orthogonal to the x-axis and defines the 2D surface for the energy density plot. The origin of the coordinate system is at the center of the pillar.

The analysis results are shown in Fig. 7 with a two dimensional representation of the energy density depicted in the left panel and a log of the energy density levels in the right panel. Inspecting the log-plot on the right shows that while the presence of the pillar in the cavity does perturb the field, it actually serves to slightly increase the effective negative vacuum energy density seen in the pillar by a factor of 3–5 compared to the density level present without the pillar in the cavity. These analysis results would suggest that the pillar does not adversely self-screen itself in a manner that prevents it from seeing the field magnitude in the cavity with no pillar present. Rather, due to the slightly elevated state, it could be reasoned that the pillar seems to focus the gradient in a manner that would at most allow the pillar to drain the stored energy in the cavity at a quicker rate once it is connected to a high impedance oscilloscope. This effect might result in a need for more cavities to provide enough stored energy such that the duration of the transient voltage signal will last long enough for detection.

While the analysis results discussed above are encouraging for the project objective of attempting to measure the presence of structure in the negative vacuum energy density within a customized Casimir cavity, the implications of this particular predicted negative vacuum energy density distribution is quite intriguing for an altogether different reason. As it so happens, the structure of the field around the pillar in the two dimensional plot is qualitatively very similar to a plot of the negative vacuum energy density necessary for the Alcubierre warp metric. Figure 9 shows a zoomed view on the numeric worldline analysis of the plate-pillar case on the top, and the energy density field for the Alcubierre metric on the bottom.

(Footnote in the original: the Casimir phenomenon was first discussed as an alternative source to exotic matter for the idea of a wormhole by Morris and Thorne, and later expanded on in Visser's book on wormholes. It was also identified by Alcubierre in his seminal paper as an alternative source to exotic matter for the manifestation of a warp bubble. The Casimir phenomenon has more recently been explored by Garattini as a sourcing material for "benign" wormholes.)

Before fully exploring the implications of this unanticipated intersection between these two models, the critical elements of the Alcubierre model will be identified and discussed that lead to the energy density that allows the "trick" to work. The motivation for the Alcubierre metric was to develop a model within the context of general relativity that would mathematically encapsulate the idea of a space warp that would allow for hyper fast travel between arbitrarily distant stellar objects. The metric and shaping function are provided in Equation 10, with G and c set to one, where vs is the speed of the craft, f(rs) is the shaping function, σ is the shell thickness parameter that controls the thickness of the warp bubble wall, and R is the radius of the warp bubble:

ds² = −dt² + (dx − vs f(rs) dt)² + dy² + dz²

f(rs) = [tanh(σ(rs + R)) − tanh(σ(rs − R))] / (2 tanh(σR))

The critical element of the model that enables stellar hyper fast transit is conjectured to be the York Time which is a measure of the expansion and contraction of space associated with the metric. A plot of the York Time is provided in Fig. 8. The York Time field is depicted as a grid that has a wave-like appearance with a simple representation of a notional craft overlayed on top of the field to show the connection between the spacetime disturbance and the source of the negative vacuum energy density. The York Time plot indicates that space is expanding behind the spacecraft and contracting in front of the spacecraft. The craft depicted in the plot has a central part located in the center of the warp bubble in the region where the spacetime is flat, the proper acceleration is zero, and local clocks are synchronized with external clocks on earth. The craft is equipped with a ring structure that represents an encapsulation of exotic matter or negative vacuum energy density distributed throughout.

(Footnote in the original: it was shown by putting the metric into canonical form that the catalytic mechanism was not the York Time, rather it was the boost field serving as a multiplier of the ship's initial velocity, akin to watching a movie in fast-forward. With the canonical form of the metric, the expansion and contraction of space is viewed as a response of spacetime as the hyperfast craft transits through space — space piles up in front of the craft and stretches out behind the craft.)

The York Time and the energy density distribution are shown in Equations 11 and 12 respectively. The York Time is proportional to the craft velocity multiplied by the ratio of the along-axis position to the radial coordinate and by the derivative of the shaping function. The energy density is proportional to minus the square of the craft velocity, multiplied by the square of the off-axis radial distance divided by the square of the radial coordinate, multiplied by the square of the derivative of the shaping function. The velocity term again represents the velocity of the craft, the position term represents the position of the center of the craft (and hence the fields), and the transverse coordinates just define a radial distance from the central x-axis.

Anecdotally, a sensitivity analysis of the field equations conducted in earlier work showed that by varying the shell thickness parameter, one could reduce the magnitude of the York Time, and as a result the total energy required to make the concept work. The analysis effort created two animations available online that show the response of the York Time field and the energy density field to variation of the shell thickness parameter. The animations show that as the warp bubble wall thickness increases the peak energy density decreases significantly, and as the warp bubble wall thickness decreases the peak energy density increases. The reasoning behind this response is that the York Time can be viewed as a sort of 3-dimensional strain of space, and as the shell thickness increases, the amount of 3-dimensional strain needed to manifest a target speed decreases which is accompanied by a decrease of energy density, and thus a reduction in total energy. This is not without a cost — as can be seen in the online animation of the York Time field, the region of flat spacetime available for the critical portions of the craft, say for a crew or science instruments, is decreased as the bubble wall thickness is increased. So the energy optimization process has a competing constraint in the form of the required size for the warp bubble to adequately encapsulate sensitive cargo within the flat spacetime region inside the bubble wall.

Now that the Alcubierre metric has been introduced and the critical elements have been identified and discussed, a comparison between the exotic matter requirements of the warp concept and the numeric worldline analysis results for the custom Casimir cavity may now be made. The top panel of Fig. 9 shows a close up view of the predicted response of the quantum vacuum within the custom Casimir cavity, and the bottom panel shows a 2-dimensional representation of the energy density necessary for the Alcubierre model. The concentrations in the negative vacuum energy density due to the presence of the pillar in the Casimir cavity are qualitatively very similar to the 2D representation of the energy density for the Alcubierre model. It should be noted that the 2-dimensional plot for the Casimir cavity is in effect a linear extrusion extending up from the surface of the paper meaning the lenticular shaped concentration is rod-like, while the Alcubierre plot is a revolution which yields a toroidal distribution.

Based on the custom Casimir cavity results for the parallel plate cavity with a cylindrical pillar at the mid-plane, a toy model comprised of a 1 µm diameter sphere suspended in the middle of a 4 µm diameter cylinder was implemented and the numeric worldline analysis technique was used to find the predicted Casimir energy density. Figure 10a shows a section cut of the toroidal Casimir energy density for the sphere-cylinder system which correlates well with the toroidal Alcubierre energy density requirements. The approach was implemented using Open MPI-enabled c-code and the analysis was run on 660 2.40 GHz Intel Skylake CPUs; the model grid was a 100 × 100 grid with a 2000 unit-loop ensemble.

If one could manufacture a chip with these types of nano structures (nano-spheres suspended in nano-tubes), an experiment might be designed and attempted to conduct a test to measure transit time of say a current (alternately a photon or electron) through a tiny conductor (alternately open bore) routed through the center of the sphere or spheres. This transit time could be compared to the time it takes for a current (photon or electron) to run through a mirror system that has no external tube (control test). If need be, many of these nano structures could be arranged in parallel to increase the time resolution of the notional experiment (see Fig. 10b). If a difference in transit time were observed, this would be an empirical confirmation of the generation of a real nano scale warp bubble on a chip. To be clear, this would not be some simple analogue or proxy representation of a space warp phenomenon, rather it would be a genuine implementation of the idea in physical fact with observable consequences in the laboratory — just not in the dramatic form of a craft bound for a distant stellar destination.

(Footnote in the original: it could be speculated that a nano sphere might be made to translate through a nano cylinder as a more direct implementation of the Alcubierre model with the provision that it may be viewed as a space warp/wormhole hybrid with the cylinder serving as the connecting pathway between two points and also enabling the formation of the necessary negative vacuum energy density around the sphere to boost the effective velocity.)

5 Conclusions

The impetus for the work discussed in this manuscript was to explore the implications of the dynamic vacuum model as applied to a custom Casimir cavity geometry. The dynamic vacuum model suggests that the state of the negative vacuum energy density in the cavity is not just an isotropic value that is constant throughout the enclosing geometry, rather it has spatial variation and can manifest complicated structure. In the process of the team exploring the literature, a technique called worldline numerics was discovered that also predicts that there is structure to the perturbed vacuum state that is predicted to exist in a notional Casimir cavity. This technique was used to evaluate the predicted state of the vacuum in response to the presence of small pillars placed at the mid-plane of a Casimir cavity. The analysis showed that the pillar would not adversely screen itself from the predicted background field that exists in response to just the presence of the plates. The analysis also showed a possible intersection with a model developed in the context of general relativity to understand how hyperfast stellar travel might be manifested mathematically. The qualitative correlation would suggest that a chip-scale experiment might be explored to attempt to measure a tiny signature illustrative of the presence of the conjectured phenomenon.

(Acknowledgements — the work was supported by the DARPA Defense Sciences Office Quest for Undiscovered Energy Storage and Thrust programme under funded agreement HR00112090082 — the data availability statement and the reference list are omitted here; the complete text is at the source. Reference numerals have been removed from the body for readability.)