Alcubierre warp drive in Bohmian Quantum Gravity

Abstract

Alcubierre warp drive metric is coupled to quantum mechanical scalar matter field. The requirement of the exotic matter for the warp drive is mapped into a conformal wave equation. This results in a fourth order partial differential equation in terms of the quantum mechanical density. Finding a proper quantum mechanical density obeying the proposed partial differential equation will be a resolution to the exotic matter problem of Alcubierre warp drive in Bohmian Quantum Gravity context.

Introduction

Generalization of Einstein’s theory of gravity is under intense theoretical exploration. The main motivation behind this research involves both mathematical and physical reasons. In a purely mathematical perspective, the Lagrangian density of Einstein’s gravity is simply linear in the Ricci scalar. One is always tempted to generalize this simple Einstein-Hilbert Lagrangian to more general ones, resulting in f(R) theories and f(R,T) theories. In addition, there are several physical motivations to study generalized Einstein theory; for example, the dark energy problem is one among them. Quantum gravity considerations are also there, which hint at the extension of Einstein’s theory for a concise physical picture.

Even though there are several mathematical routes for the generalization of Einstein’s theory, f(R) theory is mathematically more straightforward. Later it was shown that f(R) theory of gravity is equivalent to scalar tensor theory of gravity. Based on the nature of the covariant derivative of the metric tensor, Einstein’s gravity can also be generalized in a different manner. Metric compatibility, that the covariant derivative of the metric tensor vanishes, and zero torsion, that the connection is symmetric in its lower indices, are the two essential conditions in Einstein’s theory. Both of these conditions can be relaxed and different kinds of extensions of Einstein’s theory can be obtained. Relaxing the metric compatibility condition, one is naturally led to Weyl differential geometry. Due to the research in the last few decades, it is identified that the Weyl type of differential geometry is very useful to incorporate quantum mechanical scalar matter fields as a correction to Einstein’s theory. Based on the de Broglie-Bohm version of quantum theory, one is able to couple quantum theory with classical gravity. There are several research works along this interesting direction.

Usual quantum gravity approaches like Loop Quantum Gravity focus on the quantization of gravity, while Bohmian Quantum Gravity gives only a geometrical way to couple quantum mechanical scalar matter fields with Einstein’s gravity. Note that gravity is still treated as classical and we are only focusing on how classical field theories are coupled to each other. We will first focus on the essential equations in Bohmian Quantum Gravity and more specifically the stress-energy tensor is analyzed in connection with warp drive. Then we will explore Alcubierre warp drive metric, taking it as the background space-time coupled to quantum mechanical scalar matter field. Here we search for the properties of the quantum system in order to support Alcubierre metric as the background space-time.

Bohmian quantum gravity

In previous research works, an action is taken into account in which the quantum mechanical scalar matter field is coupled to gravity in a geometric, de Broglie-Bohm manner. Equation 1 in the source gives that action as a functional of the metric tensor, the conformal factor, the phase of the matter field, the matter density and a Lagrange multiplier. It has three parts, each an integral over four-dimensional spacetime of the square root of minus the metric determinant: a gravitational part, one over twice the gravitational coupling, containing the Ricci scalar multiplied by the squared conformal factor minus six times the squared gradient of the conformal factor; a matter part containing the density over the mass, multiplied by the squared conformal factor and the squared gradient of the phase, minus the mass times the density times the fourth power of the conformal factor; and a constraint part, the Lagrange multiplier multiplying the logarithm of the squared conformal factor minus the reduced Planck constant squared over the mass squared, times the d’Alembertian of the square root of the density divided by the square root of the density.

It is already shown that minimizing the action with respect to the density and to the phase leads to the real and the imaginary parts of the Generalized Klein-Gordon Equation. Equation 2 in the source is the real part: the squared gradient of the phase minus the mass squared times the squared conformal factor, plus a term in the reduced Planck constant squared divided by twice the mass, the squared conformal factor and the square root of the density, acting on the difference between the d’Alembertian of the Lagrange multiplier over the square root of the density and the multiplier times the d’Alembertian of the square root of the density over the density, all equal to zero. Equation 3 in the source gives the generalized continuity equation: the divergence of the density times the squared conformal factor times the gradient of the phase is zero.

Equation 4 in the source is the constraint equation: the squared conformal factor equals the exponential of the reduced Planck constant squared over the mass squared, multiplied by the d’Alembertian of the square root of the density divided by the square root of the density. The conformal factor is thereby identified as a quantum mechanical quantity, namely the exponential of the Bohmian quantum potential.

Varying the action with respect to the conformal factor gives an equation of motion for the scalar curvature, Equation 5 in the source: the Ricci scalar times the conformal factor, plus six times the d’Alembertian of the conformal factor, plus twice the gravitational coupling over the mass times the density times the conformal factor times the squared gradient of the phase less twice the mass squared times the squared conformal factor, plus twice the gravitational coupling times the Lagrange multiplier over the conformal factor, equal to zero.

Finally, varying the action with respect to the inverse metric tensor gives the conformally transformed Einstein equation along with its stress-energy tensors. Equation 6 in the source states that the Einstein tensor equals the sum of three stress-energy tensors: a matter term depending on the phase and the density, a quantum term depending on the conformal factor, and a mediating term depending on the Lagrange multiplier and the density.

Equation 7 in the source gives the matter stress-energy tensor: minus the gravitational coupling over the mass times the density times the symmetrised product of the gradients of the phase, plus twice the gravitational coupling over the mass times the density times the metric times the squared gradient of the phase, minus the gravitational coupling times the mass times the density times the squared conformal factor times the metric. Equation 8 in the source gives the quantum stress-energy tensor: the metric times the d’Alembertian of the squared conformal factor minus the second covariant derivative of the squared conformal factor, all divided by the squared conformal factor, plus six times the product of the gradients of the conformal factor divided by the squared conformal factor, minus three times the metric times the squared gradient of the conformal factor divided by the squared conformal factor. Equation 9 in the source gives the mediating stress-energy tensor, built from the gravitational coupling, the reduced Planck constant squared over the mass squared and the squared conformal factor, acting on the symmetrised gradients of the square root of the density and of the Lagrange multiplier divided by that square root, with the corresponding trace term subtracted.

The resulting Einstein equation is therefore composed of stress-energy tensors related to the matter, quantum and mediating field contributions. The vacuum density, the Lagrange multiplier, obeys a first order differential equation coupled to the square root of the density, given as Equation 10 in the source: it equals the reduced Planck constant squared over the mass squared times one minus the quantum potential, multiplied by the divergence of the multiplier times the gradient of the square root of the density divided by that square root. This equation is obtained by comparing the trace equation with the trace of the tensor equation. It is found that the Lagrange multiplier term can give additional corrections to standard quantum theory.

Alcubierre warp drive

Usually Alcubierre metric is studied using Einstein’s General Theory of Relativity and one is encountered with the problem of the exotic matter. Later, a considerable amount of research work was carried out in relation to the physical feasibility of the warp drive solution and its energy requirements. Advancing further, some modifications to the Alcubierre warp drive were also suggested. Recently various matter-energy sources have been explored in order to see whether they support the Alcubierre warp drive metric solution, and a relation to Burgers’ equation is found.

Here we focus on the more generalized theory of Einstein’s gravity, that is, Bohmian Quantum Gravity, which incorporates quantum mechanical matter field coupling to gravity using a conformal factor. In order to explore the warp drive in the context of Bohmian Quantum Gravity, we use the following assumption: that the background space-time is the Alcubierre warp-drive metric, coupled to a quantum mechanical scalar matter field whose amplitude is the square root of the density and whose phase carries the factor of the imaginary unit over the reduced Planck constant. In Bohmian Quantum Gravity this complex scalar matter field can be coupled to gravity, and the total metric of the system is the gravitational background metric multiplied by the squared quantum mechanical conformal factor.

Equation 11 in the source writes the Alcubierre metric in the plus-minus-minus-minus signature: the line element is one minus the squared bubble velocity times the squared shape function, times the squared time differential, plus twice the squared bubble velocity times the shape function times the product of the differentials in the direction of travel and in time, minus the squared differentials of the three spatial directions. The bubble velocity is the time derivative of the bubble centre position. Equations 12 and 13 in the source give the corresponding metric tensor and its inverse in matrix form.

The shape function is termed the shape function of the warp metric since it describes the shape of the warp bubble. Equation 14 in the source gives it as the difference between the hyperbolic tangent of the steepness parameter times the sum of the radial coordinate and the bubble radius, and the hyperbolic tangent of the steepness parameter times their difference, all divided by twice the hyperbolic tangent of the steepness parameter times the bubble radius. The steepness parameter is inversely related to the thickness of the warp bubble and the bubble radius is the radius of the bubble. The radial variable is defined as the distance from the centre of the bubble to an arbitrary point on the surface of the bubble. Alcubierre already had mentioned that this shape function can approach a step function as the steepness parameter goes to infinity: for radial distances inside the bubble radius the shape function is one, whereas for distances much greater than the bubble radius it goes to zero. The components of the Eulerian, that is normal, observers’ four-velocities are then fixed by the metric.

Equation 15 in the source gives the Einstein tensor contracted twice with the Eulerian four-velocity as minus the squared bubble velocity times the sum of the squares of the two transverse coordinates, divided by four times the squared radial variable, all multiplied by the square of the derivative of the shape function with respect to the radial variable. Using the field equation, Equation 16 in the source gives the same expression for the energy density measured by those observers — the negative quantity that is the usual statement of the exotic matter requirement.

Note that this expression is derived using geometric arguments only. Alcubierre had started with a warp drive metric and found a condition on the energy density in order to have the solution that he desired. Then one needs to look into the proper stress-energy tensor which can support the desired warp drive space-time structure. In usual general relativity one is compelled to accept the requirement of the exotic matter in order to get the desired warp drive solution. Warp drive metric is no longer a solution to Einstein’s General Relativity since it violates the energy conditions allowed in the theory. Hence one is instructed to look into a more generalized theory which can easily support the warp drive solution without much difficulty, and Bohmian Quantum Gravity is a good candidate for that.

Since we are taking into account the extended theory of Einstein General Relativity, that is a scalar-tensor theory which contains an additional scalar degree of freedom apart from Einstein’s gravitational tensor field, note that the scalar field is quantum mechanical in origin. There are many different types of scalar tensor theories, but Bohmian Quantum Gravity imposes a special physical meaning on the scalar field, and it is intimately related to the conformal factor in the theory, which is quantum mechanical in nature. The quantum mechanical conformal factor is getting coupled to gravity. Once the quantum mechanical matter field couples with gravity, one can find the exact expression for the stress-energy tensor, and it contains three contributions. Substituting that expression into the equation, one can find the condition which needs to be satisfied in Bohmian Quantum Gravity.

Even though the space-time is rigid in the Riemannian sense, it is not so in the Weyl sense. In Riemannian geometry, the length of a vector remains constant and only the orientation changes during parallel transport, while in Weyl geometry the length of a vector and its orientation change during parallel transport. Equations 17 and 18 in the source state the distinction: in Riemannian geometry the covariant derivative of the metric is zero, and in Weyl geometry it equals a vector field times the metric.

In Riemannian geometry the covariant derivative of the metric is zero while it is not true in Weyl geometry. Since the length changes during the parallel transport, the covariant derivative of the metric gives a nonzero contribution. In Weyl geometry, one can think about different conformal frames where the metric is different by a conformal scaling but the physics remains the same. In the Bohmian Quantum Gravity framework, one can see that the quantum potential appears as the Weyl scalar field, since the covariant derivative of the total metric equals the gradient of the quantum potential times that metric, and it is purely a quantum mechanical effect. Parallel transporting a vector along a Weyl manifold will make a change in the length of the vector, and such a freedom is not there in a Riemannian manifold. In other words, space-time is rigid in the Riemannian sense but it is not so rigid in the Weyl sense, hence using quantum effects one is able to manipulate space-time structure easily. This interesting fact is reflected while analyzing the contribution coming from the doubly contracted stress-energy tensor, and the major contribution is from the quantum term. Note that the inverse gravitational constant appearing as a term in front of the usual expression for the exotic matter is not present when considering that quantum term. Since we adopt the Weyl geometry, it can be seen that the Weyl length change is associated with the squared conformal factor which is purely quantum mechanical in origin. Hence quantum mechanical contribution can make space-time manipulation much easier than the usual gravitational bending of space-time.

From our previous studies, it is found that there is a dynamical cosmological term which appears along with the metric term. The dynamical cosmological term is written in terms of the quantum potential and the Lagrange multiplier. Hence, focusing on the metric part of the quantum stress-energy contribution and ignoring the other contributions, which are smaller due to the presence of the gravitational coupling, one is led to Equations 19 and 20 in the source: the doubly contracted stress-energy tensor is approximately the d’Alembertian of the squared conformal factor minus three times the squared gradient of the conformal factor, all divided by the squared conformal factor.

Combining this result from Bohmian Quantum Gravity with the previous expression for the exotic matter gives Equation 21 in the source, in which that same quantity is set equal to the negative Alcubierre energy density. Rearranging the terms, one is led to a Klein-Gordon type wave equation in terms of the squared conformal factor, Equation 22 in the source: the d’Alembertian of the squared conformal factor, minus three times the squared gradient of the conformal factor, plus the Alcubierre energy density expression multiplied by the squared conformal factor, equal to zero.

Recalling the expression of exotic matter as the Alcubierre exotic function of the three spatial coordinates, Equation 23 in the source becomes the same wave equation with that function in place of the explicit expression. Here the Alcubierre exotic function appears like the positive mass term of the conformal wave equation. Note that the conformal wave equation is purely a quantum mechanical feature, since the squared conformal factor is intimately related to the quantum potential, which in turn is related to the square root of the quantum mechanical matter wave density. Equation 24 in the source restates the constraint: the squared conformal factor is the exponential of the reduced Planck constant squared over the mass squared times the d’Alembertian of the square root of the density divided by that square root, which is the exponential of the quantum potential.

Note that the squared conformal factor has a double property: on one hand it can be written as a purely quantum mechanical quantity, and on the other hand it has a definite geometrical meaning as a conformal factor which can be written in terms of a Weyl scalar field. Substituting its expression into the wave equation, one can write the equation in terms of the quantum potential. Equation 25 in the source is that form: the d’Alembertian of the quantum potential, plus one quarter of the squared gradient of the quantum potential, plus the Alcubierre exotic function, equal to zero.

Substituting the expression for the quantum potential in terms of the quantum mechanical matter wave density, given as Equation 26 in the source — the reduced Planck constant squared over the mass squared, times the d’Alembertian of the square root of the density divided by that square root — one obtains Equation 27 in the source, which carries a term of second order and a term of fourth order in the reduced Planck constant together with the Alcubierre exotic function. Ignoring the higher order terms, that is the terms of fourth order in the reduced Planck constant, one obtains Equation 28 in the source: the d’Alembertian of the d’Alembertian of the square root of the density divided by the square root of the density, plus the mass squared over the reduced Planck constant squared times the Alcubierre exotic function, equal to zero.

Equation 29 in the source gives the Laplace-Beltrami operator on a curved space-time as one over the square root of minus the metric determinant, times the divergence of the square root of minus that determinant times the inverse metric times the gradient. Since the square root of minus the determinant is one for the Alcubierre metric, the fourth order contribution takes the reduced form of Equation 30 in the source, and Equation 31 in the source becomes the full fourth order equation: the divergence of the inverse metric acting on the divergence of the inverse metric acting on the gradient of the square root of the density, divided by that square root, plus the mass squared over the reduced Planck constant squared times the Alcubierre exotic function, equal to zero. Expanding this equation in terms of the components of the metric tensor, one is led to an expression containing the fourth order space-time derivative of the square root of the density with other terms involving the Alcubierre shape function and the velocity of the warp bubble.

It is difficult to find the analytical solutions of these kinds of fourth order differential equations, but one can take very simplified toy versions of the wave equation in order to have an idea of the new physics emerging out of the equation presented here. The main achievement we have made is mapping the problem of the exotic matter into finding a suitable quantum density for the matter field as a solution to the fourth order partial differential equation on curved space-time. If the square root of the quantum density satisfies that wave equation, it is equivalent to having exotic matter in usual general relativity. Thus one needs to expand the idea of having exotic matter into the determination of a proper quantum mechanical density. Hence the gravitational problem is mapped into a purely quantum mechanical problem. In the Bohmian Quantum Gravity sense, these quantum mechanical densities will result in a conformal wave equation.

Now let us assume a simple fourth order wave equation where the space-time is flat and the quantum density is confined along one direction, and instead of the Alcubierre exotic function we have just a constant raised to the fourth power. Equation 32 in the source is that toy equation: the fourth time derivative of the square root of the density, minus its fourth spatial derivative, plus the constant to the fourth power times the square root of the density, equal to zero.

By the method of separation of variables, it is possible to find interesting solutions for the quantum density. Apart from the usual trigonometric functions, hyperbolic functions also appear as solutions, which is not allowed in usual quantum theory since there the wave equation is only second order. But for this toy equation one can see that the temporal part of the square root of the density is a sum of cosine, sine, hyperbolic cosine and hyperbolic sine terms with constant coefficients and a constant rate. Similarly one can solve the spatial part of the quantum density as a sum of cosine, sine, hyperbolic cosine and hyperbolic sine terms in the spatial coordinate, whose rate is the fourth root of the sum of the fourth powers of the constant and of the temporal rate. The total square root of the quantum density is the product of the temporal and spatial parts. Hence the final solution consists of the product of trigonometric and hyperbolic functions. The presence of the hyperbolic terms arises from the fourth order derivative of the toy equation, and these are extra solutions which only appear from the conformal waves appearing in Bohmian quantum gravity. Trigonometric solutions are already there in usual quantum theory, but the hyperbolic solutions might be bringing the new physics here, due to the fact that they are the extra solutions appearing in the theory. Exotic matter that we perceive in usual General Relativity is a simple effect arising from the complicated conformal wave equations presented here.

Conclusions

We have analyzed the Alcubierre warp drive in the Bohmian Quantum Gravity context. The exotic matter problem is mapped into a fourth order quantum mechanical density equation involving the Alcubierre shape function. It is found that the fourth order wave equation in space-time can contribute extra solutions of quantum density which are usually not present in quantum theory, since usual quantum theory is second order in nature. We have taken a simple toy model of the partial differential equation and found that hyperbolic functions of quantum density can appear, which may contribute to the new physics appearing in the theory. If one can find a viable distribution of quantum mechanical density which obeys the wave equation given here, we will resolve our problem of exotic matter for the Alcubierre warp drive in the Bohmian Quantum Gravity framework. The exact solution of the quantum density can be evaluated in further research works.

Sijo K. Joseph, GITAM Deemed to be University, Hyderabad, Telangana, India. Published as Physica Scripta 98 (2023) 095012; author’s version arXiv:2205.02780, licensed CC BY 4.0.

(Reference-number markers have been dropped and display equations rendered in words with their source equation numbers; the thirty-four references and the equations in their original form are at the source.)