Positive energy warp drive from hidden geometric structures

Abstract

Warp drives in Einstein's general theory of relativity provide a unique mechanism for manned interstellar travel. It is well-known that the classical superluminal soliton spacetimes require negative energy densities, likely sourced by quantum processes of the uncertainty principle. It has even been claimed by few that negative energy densities are a requirement of superluminal motion. However, recent studies suggest this may not be the case. A general decomposition of the defining variables and the corresponding decomposition of the Eulerian energy are studied. A geometrical interpretation of the Eulerian energy is found, shedding new light on superluminal solitons generated by realistic energy distributions. With this new interpretation, it becomes a relatively simple matter to generate solitonic configurations, within a certain subclass, that respect the positive energy constraint. Using this newfound interpretation, a superluminal solitonic spacetime is presented that possesses positive semi-definite energy. A modest numerical analysis is carried out on a set of example configurations, finding total energy requirements four orders of magnitude smaller than the solar mass. Extraordinarily, the example configurations are generated by purely positive energy densities, a tremendous improvement on the classical configurations. The geometrical interpretation of the Eulerian energy thus opens new doors to generating realistic warp fields for laboratory study and potential future manned interstellar travel.

1. Introduction

The prospect of travelling to other far away astronomical systems has been a tantalizing thought for many millenia, however its bleak reality has only been realized in the past couple centuries. The vastness of space is the greatest barrier to interstellar travel. The closest planetary system to Earth is Alpha Centauri at a distance of 4.3 light-years [1]. Using the fastest man-made object, the Parker Space Probe, it would take approximately 65 centuries to reach Alpha Centauri [2]. This is many times longer than the average human life span. Fortunately, there may exist other methods to manned intersystem travel using more fundamental theoretical processes.

It is well-known that an empty, flat, and non-dynamical spacetime does not permit massive particles from exceeding the speed of light. However, there have been cases in more complex systems of particles exceeding the local speed of light, such as highly energetic massive radiation in a medium giving rise to Cherenkov radiation [3]. In some instances, even spacetime itself can travel faster than the vacuum speed of light. This is the case during the inflationary or dark energy phases of the universe. Such distortions of spacetime manifest as objects travelling faster than light relative to distant observers. This effect can be observed today by measuring the recession speed of distant galaxies and noting they are receeding faster than light away from the earth [4, 5].

The dynamical nature of spacetime giving rise to superluminal relative speeds may be able to be exploited for interstellar travel. The field equations of general relativity (GR) permit inertial observers to be transported at superluminal rates relative to distance observers. The first solution possessing such features was the Alcubierre spacetime [6]. This solution describes a region of spacetime in which an observer inside a "spacetime bubble" could be transported superluminally relative to observers outside the region. This is due to the contraction of spacetime in front of the central observer and a corresponding expansion behind. In theory, this solution permits interstellar travel of a payload on arbitrarily short timescales.

Unfortunately, this solution possesses numerous pathologies that would prevent its construction using realistic matter fields. Invoking quantum inequality restrictions enforced on the stress-energy tensor, as derived by [7], one finds this solution requires an enormous amount of negative energy, several orders of magnitude more than all the energy content in the observable universe [8]. Even if one appealed to quantum processes to generate the negative energies, the shear quantity requirement yields this configuration highly unphysical. To generate the energy densities required by the Alcubierre configuration using the Casimir force, the length scales would have to be millions of times smaller than the diameter of a proton. Additionally, the spacetime hosts horizons that would render the soliton uncontrollable [9]. Other pathologies include subluminal weak energy violations and Planck-scale dimensions [8, 10].

A number of other configurations have been constructed, each with varying goals and properties. The Natario spacetime is another example of a superluminal spacetime [11]. This solution, by construction, has trivial expansion. The advantage of this configuration is that it preserves volumes, as opposed to the Alcubierre spacetime which dramatically alters them. However, this configuration still comes with many of the same problems that the Alcubierre spacetime hosts. The Broeck configuration aims to reduce the massive amount of negative energy by altering the geometry of the warping region [12]. They reduce the surface area of the "bubble" while maintaining the internal volume. This drastically reduces the total energy requirements from universal-scale to solar mass-scale. Nonetheless, negative energies are still required in order to generate all previously mentioned spacetimes. In recent years, theoretical research has hinted at possible superluminal configurations generated by purely positive energy densities [13]. This is accomplished by analyzing constraint equations enforced by GR and finding relationships that permit superluminal motion generated by positive energies. Other studies have claimed to generate superluminal motion from positive energies by first choosing a particular form of the stress-energy tensor and solving the Einstein equations in the Alcubierre spacetime [14–16]. These studies appear to be in contrast to the claimed proof that superluminal motion requires negative energy densities [17].

Modified theories of gravity provide alternate descriptions of the gravitational force that could yield more physically realistic superluminal solitons [18]. One relevant example is the Debenedictis et al. configuration developed in the Einstein-Cartan theory [19–21]. In this theory of gravity, the torsion tensor no longer vanishes, resulting in new couplings between the spacetime geometry and matter fields (a similar setup can be constructed in terms of non-metricity [22–24]). An interesting feature of the Einstein-Cartan theory is that deviations from GR only manifest in the presence of matter. In this theory, the spin of the matter fields appears in the field equations. Debenedictis et al. exploited this coupling to generate the Alcubierre configuration using positive energy densities only. This, however, solves only one pathology of the Alcubierre configuration. It doesn't solve the horizon problems, causality problems, etc. Nonetheless, it illustrates the possible advantages of utilizing alternate theories of gravity to construct more physically realistic superluminal solitonic spacetimes.

With the presence of negative energy densities being one of the main pitfalls of the classical solitonic configurations, the aim of this work is to analyze the energy equation at the classical level in the context of superluminal spacetimes. The main question of this work is could a compact superluminal soliton described by the classical equations of general relativity be generated only using positive energy densities? Moreover, the geometric nature of the Eulerian energy within a certain subclass of spacetime configurations is investigated, providing insight into how and why the classical configurations necessitate negative energies. A few spacetimes hosting superluminal motion generated by positive energies will also be presented, along with some of the subtleties of the Eulerian energy.

This paper is then divided into three main sections. Section 2 will introduce the relevant theoretical tools and their relevance to the superluminal solitonic spacetimes. Section 3 will present the results of this study and its significance. Finally, section 4 will conclude with a summary and discussion on potential future work.

(Section 2, Theory — the 3+1 decomposition, the lapse, shift and extrinsic curvature, the Hamiltonian and momentum constraints, and the statement of the weak energy condition — is omitted for length; the complete text is at the source.)

3. Results

Presented in this section is a class of solitonic spacetimes generated by positive energy densities, alongside a new geometric interpretation of the Eulerian energy distribution. The considered class of spacetimes permit superluminal transportation of an inertial observer. Several comparisons of the presented configurations can be drawn to previous constructions in the literature. The most widely used ansatz will also be employed here, namely the shift vector field will be used to fully describe the spacetime. Moreover, potential fields will be introduced to further analyze the constraining equations, similar to the analysis in [13], albeit without enforcing extra constraints on the metric variables. The presented work appears to be in good agreement with the study performed in [27], namely the non-truncation of the gravitational field leads to positive semi-definite energies, axisymmetric configurations can lead to interesting solitonic configurations, and modifying the geometry of the soliton can lead to a reduction of the energy requirements.

There are several advantages to the presented configurations compared to those previously presented in the literature. The foremost advantage is the absence of negative energies, in contrast to the Alcubierre, Natario, and Van Den Broeck configurations among others. The presented solitonic configurations are also tuneable, which is a first of its kind. This means the central observer can have a variable speed, relative to distant observers, by simply altering the geometry of the Eulerian energy distribution while maintaining the total energy. The speed is based upon the geometry of the energy distribution and not the total energy content, making it highly desirable for interstellar transportation. The configurations are also highly modifiable due to the geometric interpretation offered by the Hamiltonian constraint. This will greatly aid any work seeking to minimize the energy requirements.

3.1 Generalities

In order to explore the feasibility of constructing a positive-energy spacetime soliton, the Eulerian energy is investigated in a suitably general ansatz. The shift vector field is taken to be time-independent with no other constraints placed on it. Moreover, the lapse function is taken to be unity, and the induced metric is the identity. The Hamiltonian constraint then reduces to equation (5): sixteen pi times the energy density equals twice the sum of the three products of like-index derivatives of the shift components — the x-derivative of Nx times the y-derivative of Ny, plus the x-derivative of Nx times the z-derivative of Nz, plus the y-derivative of Ny times the z-derivative of Nz — minus one half of the sum of the squares of the three mixed derivative pairs.

Some observations are already in order. The first term in the above equation has an indeterminate sign and is quadratic in derivatives of different components. The second term is negative semi-definite and involves the square of linear combinations of derivatives. This explains why some of the previously proposed solitonic configurations exhibit negative energies. Their configurations depend only on a single component of the shift vector. For example, the Alcubierre ansatz reduces the above equation to sixteen pi times the density equalling minus one half of the sum of the squares of the y- and z-derivatives of Nx. In order to satisfy the WEC, any ansatz must contain at least two non-zero components of the shift vector field.

Equation (5) does not provide any further information regarding its positivity. One must dive deeper into the internal structure of the equation to investigate its properties. This can be accomplished by considering a general decomposition of the shift vector field. There are a number of decompositions of vector fields, but the most commonly employed, and the one utilized here, is the Helmholtz decomposition, equation (6): the shift vector is the gradient of a scalar field phi plus a solenoidal field omega whose divergence vanishes. The metric tensor in line element form, equation (7), is then minus the squared time interval plus the sum, over the three spatial directions, of the squared quantity formed by the coordinate differential plus the corresponding component of that gradient-plus-solenoidal shift times the time differential.

After tedious algebra, the Hamiltonian constraint can be shown to reduce to equation (8): sixteen pi times the density equals twice the quantity formed by the sum of the three second-order principal minors of the Hessian matrix of phi, minus twice the Frobenius product of the Jacobian matrix of omega with the Hessian, minus the Frobenius product of the Jacobian with itself, plus one half of the squared magnitude of the curl of omega. Importantly, the Frobenius product is a positive-definite form, implying the third term in equation (8) is negative definite. The momentum constraint admits a simple decomposition: sixteen pi times the momentum density is the Laplacian of omega, which is minus the curl of the curl of omega. Notably, this implies irrotational fields will always enforce a vanishing momentum density.

The key result of this decomposition is the geometrical interpretation of the energy density in the purely irrotational sector. For shift vector fields described by a single scalar field, that is, the shift equal to the gradient of phi, the constraint equations in the 3+1 decomposition of the Einstein equations reduce to equation (9): eight pi times the density equals the sum of the three second-order principal minors of the Hessian of phi, and the momentum density vanishes. The WEC then enforces the constraint that the sum of those three minors is greater than or equal to zero.

The scalar component of the shift vector field contributes a geometric term to the Eulerian energy density. The second-order principal minors describe the curvature of the scalar field in the two-dimensional subspaces. Although it is not directly related to the convexity of the entire function, convexity of the two-dimensional subspaces of the scalar field phi determines the positivity of the Eulerian energy density. Convexity (concavity) of the entire function is indeed a sufficient condition, but it is not necessary. As long as one of the principal minors is sufficiently large, the other two may be negative while still satisfying the positive energy constraint.

Although not evident from equation (8), its a simple task to show that purely solenoidal shift vector fields have negative semi-definite energy. This follows directly from the Hamiltonian constraint in equation (4), noting that the trace of the extrinsic curvature is the divergence of the shift. If the shift vector field were purely solenoidal, then that divergence vanishes, implying the density equals minus one over sixteen pi times the fully contracted square of the extrinsic curvature, which is less than or equal to zero, as is the case for the Natario spacetime.

3.2 Positive Energy Spacetime Soliton

Using the geometrical interpretation offered by equation (9), it is a simple matter to construct, using geometric intuition alone, a positive energy solitonic spacetime. Assume the shift vector field is purely irrotational and let phi be the piecewise field of equation (10). The shift vector field is then calculated as the gradient of phi. The scalar field phi fully describes the spacetime, including the energy density and expansion. See Figure 1. The direction of travel for this configuration is in the negative z-direction. The total integrated expansion is zero and the magnitude of the shift vector in the central region around the origin is 1.3, which is superluminal. Moreover, the energy density approaches zero in the central region, a perfect location for a vehicle. Additionally, the shift vector field portrays the same directionality as the Alcubierre configuration, namely the shift vector field points away from expanding regions and towards contracting regions. It is important to note that the scalar field is not continuously differentiable.

One could construct several variations of the above configuration. The key trait is that its energy is purely positive. One could seek to minimize the energy density while maintaining superluminality of the central region by altering the geometry of the scalar field. However, the discontinuities present in the configuration could pose a large problem when attempting to construct such a configuration.

Figure 1. Configuration described by a piecewise-defined shift vector potential. Figure 1b shows the energy density of the configuration in the y equals zero plane. Note the large discontinuities. Figure 1c shows a 3D density plot of the energy density. Opacity and color are a proxy for magnitude. The magnitude is maximal on the outer edge of the toroids and minimal inside. Figure 1d shows a density plot of the volume expansion. The magnitude of the expansion is maximal on the outer face of the toroid and minimal on the inside. Positive expansion is occurring in the positive z region, while contraction is occurring in the negative z region. Figure 1e is the y equals zero slice of the two shift vector components. In a neighborhood of the origin, the shift vector field is almost purely in the negative z-direction (towards the negative expansion).

One must be careful when dealing with piecewise functions in spacetime. Cusps, corners, and other discontinuities aren't physical. As suggested by the geometrical nature of equation (9), negative energies will manifest if the scalar field phi has strongly-curved saddle points in any of the three two-dimensional subspaces. One could "hide" the saddle points in the discontinuities of the function, such as a corner. Any physically realistic field should be sufficiently smooth, since any physical construction will inevitably possess a certain fall-off in the otherwise geometrical corner. Additionally, the length scale of such a discontinuity is infinitesimally small, way beyond the regime of classical gravity. One would need a description of the small-scale nature of spacetime to even begin to describe configurations containing scales of this order.

In the interest of constructing a physically interesting superluminal configuration, an additional smoothness condition is enforced on the spacetime. The shift vector potential field phi is required to be at least twice continuously differentiable. This forces the resulting energy density distribution to be continuous, largely avoiding issues associated to many piecewise configurations. One such example of a scalar field satisfying this condition is equation (11), built from Gaussian factors, error functions and two spatial functions m and n that set the central gradient, with parameters for the radius, the depth and the Gaussian weight. For certain choices of those spatial functions and of the exponent, this is twice continuously differentiable everywhere except the origin and approaches zero at the origin. For the parameter choice given in the text, together with suitable spherically symmetric choices for m and n, the energy density is everywhere positive and highly localized, and the total energy, the volume integral of the density, is finite.

The central region inside the radius r has asymptotically vanishing shift vector. The only superluminal shift vector is in the exterior region beyond r, which approaches Minkowski space at asymptotic infinity. A simple modification to the spatial functions m and n results in a configuration possessing superluminal shift in the interior region, Figure 2. By letting m exceed n in the positive z region and n exceed m in the negative z region, the gradient of the phi-field increases inside the radius r, generating a high and level shift. The direction of travel is now in the positive z-direction. Compare this to the spherically symmetric case for m and n. The phi-field becomes spherically symmetric, resulting in the energy density being uniformly distributed in a spherical shell around the central region. Modifying the central gradient causes the energy density distribution to be highly peaked in the negative z region. For the specific choice of parameters used in Figure 2, the central shift magnitude is 1.26, i.e. superluminal.

This could serve as a method of generating acceleration without modifying the total energy on the hypersurface. One could start with a finite energy distribution highly concentrated and uniformly distributed in a spherical shell around a spacecraft. The energy density is then increased behind the spacecraft and decreased in front, without destroying or creating any additional energy. This increases the shift vector magnitude in the region where the spacecraft is located, transporting it to non-zero speeds relative to faraway observers. If the energy density is sufficiently concentrated behind the central observer, the relative speed becomes superluminal. This then operates as a tuneable solitonic configuration capable of superluminal speeds. Whether the central vehicle can actually manipulate the energy density in the luminal or superluminal regimes remains in question. The spacetime might still possess some of the other pathologies associated to the classical solitonic configurations, such as the formation of horizons. Although, there are two important deviations from the classical configurations, namely the absence of negative Eulerian energies and the exterior region no longer behaving like Minkowski space. The exterior region of the warp bubble is now Schwarzschild-like and satisfies the prerequisites of Birkoff's theorem only in the case when m and n are spherically symmetric, which the spacetime in Figure 2 does not. This non-truncation of the gravitational field outside the warping region is likely the reason why negative energies are absent from this configuration, in agreement with [27]. Interestingly, by tuning the Gaussian weight, the energy density could be reduced without modifying the central gradient of the phi-field. This could be used to avoid the formation of a black hole due to high energy densities. This configuration is then highly modifiable, thanks to the geometric interpretation of the Eulerian energy. Compared to the configuration presented in [13], the shift vector potential in equation (11) is more general as it does not assume any relations between the shift vector components. Moreover, as there are no relations between the derivatives of the shift vector potential, the scalar field can be easily adjusted in many ways in order to optimize the physical construction of the configuration.

Figure 2. Configuration defined from an at least twice-differentiable shift vector potential, shown in Figure 2a. The energy density, Figure 2b (the y equals zero slice) and Figure 2c (3D density plot), is highly localized. It has a finite integral over the three-dimensional leaves of the foliation. The expansion, Figure 2d (3D density plot), is everywhere non-negative, with greater expansion in the negative z region. Opacity and color in the density plots serve as a proxy for magnitude. The shift vector field, Figure 2e, is highly radial and asymptotically Minkowskian in the exterior region and highly linear in the interior with a central magnitude of about 1.26, i.e. superluminal.

With the Eulerian energy distribution in hand, the full WEC can now easily be investigated to study the nature of the energy distribution in other frames of reference. By analyzing the Lorentz-invariant eigenvalues of the stress-energy tensor calculated using equation (7), it turns out the WEC is violated in compact regions within the energy distribution. It follows from the Lorentz-invariant eigenvalue equation that the Eulerian energy, equation (9), is the first eigenvalue (whose eigenvector is the normal vector) and thus satisfies the first constraint enforced by the WEC for a type-I stress-energy tensor, which the configuration in Figure 2 appears to be [28, 29]. However, the principle momenta constraints is where the violations occur, namely, the density plus one of the principal momenta is negative for at least one of the three directions in compact regions in the distribution. No amount of modification to the configuration could get rid of these WEC-violating regions. However, the WEC is not violated everywhere and if the configuration in [13] indeed satisfies the WEC, as claimed, then it may still be possible to satisfy the WEC in the presented configurations too, given sufficient modifications.

The ansatz of equation (5) contains an interesting property related to an O(3) operation. The dynamical equations of equation (4) are not invariant under the total spacial inversion of the shift vector. Additionally, the expansion is not invariant either, since under this operation the trace of the extrinsic curvature changes sign. However, since the induced metric is the identity, the inversion is equivalent, according to the dynamical equations, to the temporal inversion. As expected, the shift vector inversion is equivalent to an inversion of the foliation label, which can be interpreted as "time running backwards." This means that the same static energy distribution can produce two different configurations, one in which spacetime possesses positive expansion and one with negative expansion. It makes sense, then, to consider the strong energy condition (SEC) in this case. The geometric SEC is the constraint that the Ricci tensor contracted twice with any timelike vector is greater than or equal to zero. In the case of a purely irrotational shift vector field, this condition in the Eulerian frame takes the simple form of minus the Frobenius product of the Hessian with itself, minus the gradient of phi dotted into the gradient of the Laplacian of phi.

It is interesting that this equation is invariant under the inversion of the sign of phi. This means that if the spacetime possessing positive expansion everywhere violates the SEC, then so too will the spacetime obtained by flipping that sign, which possesses negative expansion (contraction) everywhere. In fact, it follows by simple calculations that if the time derivative of the divergence of the shift — that is, the time derivative of the expansion — vanishes, then the Eulerian contraction of the Ricci tensor is invariant under the total inversion of the shift. This means that if the expansion is independent of co-ordinate time, then the geometric side of the SEC constraint in the Eulerian frame is invariant under that inversion. Evidently, the configurations in Figure 1 and Figure 2 violate the SEC.

Even though the Eulerian momenta vanish, and thus satisfies the constraint enforced by the dominant energy condition in the Eulerian frame, the full dominant energy condition is also violated due to the fact that the magnitude of the density is smaller than the magnitude of one of the principal momenta in compact regions in the distribution, where those principal momenta are the Lorentz-invariant eigenvalues.

One of the main issues with warp drive spacetimes is trying to find possible stress-energy sources that generate the proposed spacetimes. Warp drive research typically starts with a given geometry and attempts to find what stress-energy distribution sources the geometry via the Einstein equations, as the study presented here does. This doesn't guarantee the resulting stress-energy source to be physical. Starting with a stress-energy source in a given geometry and finding what conditions must be respected for the source to be physical is still on-going research. An explicit study of this sort is out of the scope of this paper; however, some points can still be mentioned. The configuration proposed in [13] is based on the same ansatz studied in this paper. In principle, a similar model could be applied to the configurations presented here. Moreover, the sources discussed in [14–16] could be applied to the configurations here as well. More complete statements on possible stress-energy sources would require a deeper analysis, which is reserved for future studies.

3.3 Chronological Protection

When studying superluminal motion, one inevitably runs into the problem of generating closed timelike curves (CTCs) and the associated chronological protection conjecture presented in [30]. As discussed, the configurations presented in Sec. 3.2 admit superluminal inertial observers. This naturally leads to the question of whether these configurations can be used to generate CTCs or not. It may be that these configurations can be used to violate causality in the superluminal regime, however there are two possible arguments for reconciliation with the chronological protection conjecture. The first is that one of the preconditions of the presented configurations is that the spacetime is globally hyperbolic. This means that the leaves of the foliation are Cauchy surfaces and hence CTCs cannot form. The second is that as the energy distribution is being manipulated from the spherically symmetric distribution into the axi-symmetric distribution (Figure 2), horizons form preventing manipulation of the energy distribution past the speed of light barrier. In other words, horizon formations prevent the configurations from transporting an inertial observer from the subluminal regime to the superluminal regime. This agrees with the discussion in [27]. More work will need to be done to analyze the formation of horizons and the evolution from the subluminal to superluminal regimes in the presented configurations.

3.4 Numerics

To get a sense of the magnitude of the energy densities in the previous section, Newton's constant and the speed of light are reinserted into the Hamiltonian constraint. With this definition of the units, the energy density becomes the fourth power of the speed of light, divided by sixteen pi times Newton's constant, multiplying the same combination as before: twice the sum of the three principal minors, minus twice the Frobenius product of the Jacobian with the Hessian, minus the Frobenius product of the Jacobian with itself, plus one half of the squared magnitude of the curl of omega. For the case of an irrotational field, this simplifies, as before, to the fourth power of the speed of light divided by eight pi times Newton's constant, multiplying the sum of the three principal minors.

The maximum value of the energy density for the configuration shown in Figure 2, as measured by the Eulerian observer, is found to be about 3.2 times ten to the twenty-sixth kilograms per cubic metre. This is an astronomical energy density concentrated in a very small region. More than likely, any attempt to construct this specific configuration will form a black hole. The total energy present on any initial hypersurface can be found to be approximately 9.25 times ten to the forty-third joules, which is still enormous, but quite small in astronomical terms. For comparison, the rest mass energy of the sun is approximately 1.78 times ten to the forty-seventh joules. Hence, the total energy required to form the configuration of Figure 2 is four orders of magnitude smaller than the rest mass energy of the sun. This is a drastic improvement on the energy scales of the classical superluminal solitons. Moreover, thanks to the geometric interpretation of the Eulerian energy in the considered class of spacetimes, it would be a simple matter to drastically reduce the energy requirements, as discussed in Sec. 3.2, and thus avoid the formation of a black hole.

4. Conclusion

The results discussed in the previous sections provide great insight into the properties of the weak energy condition for superluminal solitonic spacetimes. The geometric interpretation of the Eulerian energy density opens new doors for constructing realistic superluminal spacetimes from feasible energy densities. Numerical techniques could be employed to investigate the nature of the irrotational sector of the Hamiltonian constraint, such as a Finite Element Method for solving the second order partial differential equation. The techniques available during this study could not solve the PDE, owing to its high non-linearity. More than likely, a new PDE solver would need to be constructed specifically for this differential equation. One could also perform other decompositions of the shift vector field to investigate other hidden properties. This may illuminate hidden interpretations that would allow easier construction of a shift vector field composed of irrotational and solenoidal components.

The energy densities discussed in Sec. 3.4 could easily be reduced to more manageable scales by minimizing the curvature of the shift vector potential. One could even utilize computational power to search for configurations that minimize the energy density while maximizing the central shift vector magnitude.

These results are extremely exciting. They shed new light on physically-realistic superluminal warp drives and brightens the future of manned interstellar travel.

Acknowledgements

LH is supported by funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme grant agreement No 801781 and by the Swiss National Science Foundation grant 179740.