In this article, we consider the Alcubierre spacetime, such a spacetime describes a ``bubble'' that propagates with arbitrary global velocity. This setting allows movement at a speed greater than that of light. There are some known problems with this metric, e.g., the source's negative energy and the violation of the source's energy conservation when the bubble accelerates. We address these two issues within the realm of the Teleparallel Equivalent of General Relativity (TEGR). The energy conservation problem can be solved when considering the energy of the gravitational field itself. The total energy of the spacetime, gravitational plus source, is conserved even in accelerated motion. We explicitly show the dependence of energy and gravitational energy flux on the frame of reference, one adapted to a static observer and the other to a free-falling one in the same coordinate system. Addressing the problem of energy negativity of the source, we find that a static observer measures positive energy of the source, while an Eulerian observer measures a negative one. Thus, we surmise that negative energy may be a reference problem.
The expectation value of the stress-energy tensor of a free conformally invariant scalar field is computed in a two-dimensional reduction of the Alcubierre ``warp drive'' spacetime. The stress-energy is found to diverge if the apparent velocity of the spaceship exceeds the speed of light. If such behavior occurs in four dimensions, then it appears implausible that ``warp drive'' behavior in a spacetime could be engineered, even by an arbitrarily advanced civilization.
It can be shown that negative energy requirements within the Alcubierre spacetime can be greatly reduced when one introduces a lapse function into the Einstein tensor. Thereby reducing the negative energy requirements of the warp drive spacetime arbitrarily as a function of A(ct,r_s). With this function new quantum inequality restrictions are investigated in a general form. Finally a pseudo method for controlling a warp bubble at a velocity greater than that of light is presented.
The Alcubierre metric is a spacetime geometry where a massive particle inside a spacetime distortion, called warp bubble, is able to travel at velocities arbitrarily higher than the velocity of light, a feature known as the warp drive. This is a consequence of general relativity, which allows global superluminal velocities but restricts local speeds to subluminal ones as required by special relativity. In this work we solved the Einstein equations for the Alcubierre warp drive spacetime geometry considering the dust matter distribution as source, since the Alcubierre metric was not originally …
To gain more insights about what could be the source of the Alcubierre warp drive, a technique designed to find an orthogonal basis from the metric expression is used. It is deduced that exotic fluids can produce the same stress-energy tensor as the Alcubierre warp drive. For different configurations of warp acceleration and velocity, constraints on density, pressure, total positive mass, total negative mass, absolute value of the equation of state as well as null-energy condition violation are provided. The Alcubierre warp drive Einstein tensor expression and the optimization of fluid propert…
In a seminal paper, Alcubierre showed that Einstein's theory of general relativity appears to allow a super-luminal motion. In the present study, we use a recent eternal-warp-drive solution found by Alcubierre to study the effect of Hawking radiation upon an observer located within the warp drive in the framework of the quantum tunneling method. We find the same expression for the Hawking temperatures associated with the tunneling of both massive vector and scalar particles, and show this expression to be proportional to the velocity of the warp drive. On the other hand, since the discovery of…
The null geodesic equations in the Alcubierre warp drive spacetime are numerically integrated to determine the angular deflection and redshift of photons which propagate through the distortion of the ``warp drive'' bubble to reach an observer at the origin of the warp effect. We find that for a starship with an effective warp speed exceeding the speed of light, stars in the forward hemisphere will appear closer to the direction of motion than they would to an observer at rest. This aberration is qualitatively similar to that caused by special relativity. Behind the starship, a conical region f…
The Alcubierre warp spacetime yields a fascinating chance for comfortable interstellar travel between arbitrary distant places without the time dilation effect as in special relativistic flights. Even though the warp spacetime needs exotic matter for its construction and is thus far from being physically feasible, it offers a rich playground for studying geodesics in the general theory of relativity. This paper is addressed to graduate students who have finished a first course in general relativity to give them a deeper inside in the calculation of non-affinely parametrized null and time-like …
The Alcubierre warp drive is an exotic solution in general relativity. It allows for superluminal travel at the cost of enormous amounts of matter with negative mass density. For this reason, the Alcubierre warp drive has been widely considered unphysical. In this study, we develop a model of a general warp drive spacetime in classical relativity that encloses all existing warp drive definitions and allows for new metrics without the most serious issues present in the Alcubierre solution. We present the first general model for subluminal positive-energy, spherically symmetric warp drives; cons…
A process for using curvature invariants is applied to evaluate the metrics for the Alcubierre and the Natario warp drives at a constant velocity.Curvature invariants are independent of coordinate bases, so plotting these invariants will be free of coordinate mapping distortions. As a consequence, they provide a novel perspective into complex spacetimes such as warp drives. Warp drives are the theoretical solutions to Einstein's field equations that allow the possibility for faster-than-light (FTL) travel. While their mathematics is well established, the visualisation of such spacetimes is une…
It is commonly believed that Alcubierre's warp drive works by contracting space in front of the warp bubble and expanding space behind it. We show that this expansion/contraction is but a marginal consequence of the choice made by Alcubierre, and explicitly construct a similar spacetime where no contraction/expansion occurs. Global and optical properties of warp drive spacetimes are also discussed.
I show how a minor modification of the Alcubierre geometry can dramatically improve the total energy requirements for a `warp bubble' that can be used to transport macroscopic objects. A spacetime is presented for which the total negative mass needed is of the order of a few solar masses, accompanied by a comparable amount of positive energy. This puts the warp drive in the mass scale of large traversable wormholes. The new geometry satisfies the quantum inequality concerning WEC violations and has the same advantages as the original Alcubierre spacetime.
The Alcubierre metric describes a spacetime geometry that allows a massive particle inside a spacetime distortion, called warp bubble, to travel with superluminal global velocities. In this work we advance solutions of the Einstein equations with the cosmological constant for the Alcubierre warp drive metric having the perfect fluid as source. We also consider the particular dust case with the cosmological constant, which generalizes our previous dust solution (arXiv:2008.06560) and led to vacuum solutions connecting the warp drive with shock waves via the Burgers equation, as well as our perf…
"Warp drive" spacetimes are useful as "gedanken-experiments" that force us to confront the foundations of general relativity, and among other things, to precisely formulate the notion of "superluminal" communication. We verify the non-perturbative violation of the classical energy conditions of the Alcubierre and Natario warp drive spacetimes and apply linearized gravity to the weak-field warp drive, testing the energy conditions to first and second order of the non-relativistic warp-bubble velocity. We are primarily interested in a secondary feature of the warp drive that has not previously b…
A class of spacetimes (comprising the Alcubierre bubble, Krasnikov tube, and a certain type of wormholes) is considered that admits `superluminal travel' in a strictly defined sense. Such spacetimes (they are called `shortcuts' in this paper) were suspected to be impossible because calculations based on `quantum inequalities' suggest that their existence would involve Planck-scale energy densities and hence unphysically large values of the `total amount of negative energy' E_tot. I argue that the spacetimes of this type may not be unphysical at all. By explicit examples I prove that: 1) the re…
We introduce analog quantum simulations of 1+1 dimensional sections of exotic 3+1 dimensional spacetimes, such as Alcubierre warp-drive spacetime, Gödel rotating universe and Kerr highly-rotating black hole metric. Suitable magnetic flux profiles along a SQUID array embedded in a superconducting transmission line allow to generate an effective spatiotemporal dependence in the speed of light, which is able to mimic the corresponding light propagation in a dimensionally-reduced exotic spacetime. In each case, we discuss the technical constraints and find the optimal region of parameters for the …
We study Rotating Boson Star initial data for Numerical Relativity as previously considered by Yoshida and Eriguchi, Lai (arXiv:gr-qc/0410040v2), and Grandclement, Somé and Gourgoulhon (arXiv:1405.4837v3). We use a 3 + 1 decomposition as presented by Gourgoulhon (arXiv:1003.5015v2) and Alcubierre, adapted to an axisymmetric quasi-isotropic spacetime with added regularization at the axis following work by Ruíz, Alcubierre and Núñez (arXiv:0706.0923v2) and Torres. The Einstein-Klein-Gordon equations result in a system of six-coupled, elliptic, nonlinear equations with an added unknown for the sc…
It is pointed out that in curved spacetime, one cannot define the sum of energy-momemtum 4-vector over a space-like hypersurface. The difficulty in finding satisfactory gravitational energy-momentum complex stems from misunderstanding of this question. The law of conservation of energy-momentum holds valid only approximately when spacetime is not seriously curved.
The Alcubierre warp drive metric is a spacetime construction where a massive particle located inside a spacetime distortion, called warp bubble, travels at velocities arbitrarily higher than the velocity of light. This theoretically constructed spacetime geometry is a consequence of general relativity where global superluminal velocities, also known as warp speeds, are possible, whereas local speeds are limited to subluminal ones as required by special relativity. In this work we analyze the solutions of the Einstein equations having charged dust energy-momentum tensor as source for warp veloc…
Freely falling point particles in the vicinity of Kerr black holes are subject to a conservation law, that of their Carter constant. We consider the conjecture that this conservation law is a special case of a more general conservation law, valid for arbitrary processes obeying local energy momentum conservation. Under some fairly general assumptions we prove that the conjecture is false: there is no conservation law for conserved stress-energy tensors on the Kerr background that reduces to conservation of Carter for a single point particle.
Simplicial lattices provide an elegant framework for discrete spacetimes. The inherent orthogonality between a simplicial lattice and its circumcentric dual yields an austere representation of spacetime which provides a conceptually simple form of Einstein's geometric theory of gravitation. A sufficient understanding of simplicial spacetimes has been demonstrated in the literature for spacetimes devoid of all non-gravitational sources. However, this understanding has not been adequately extended to non-vacuum spacetime models. Consequently, a deep understanding of the diffeomorphic structure o…
In quantum field theory, there exist states in which the expectation value of the energy density for a quantized field is negative. These negative energy densities lead to many problems. Although quantum field theory introduces negative energies, it also provides constraints in the form of quantum inequalities (QI's). These uncertainty principle-type relations limit the magnitude and duration of any negative energy. We derive a general form of the QI on the energy density for both the quantized scalar and electromagnetic fields in static curved spacetimes. In the case of the scalar field, the …
A notion of geometric symmetry is introduced that generalizes the classical concepts of Killing fields and other affine collineations. There is a sense in which flows under these new vector fields minimize deformations of the connection near a specified observer. Any exact affine collineations that may exist are special cases. The remaining vector fields can all be interpreted as analogs of Poincare and other well-known symmetries near timelike worldlines. Approximate conservation laws generated by these objects are discussed for both geodesics and extended matter distributions. One example is…
A set of exact quasi-local conservation equations is derived from the Einstein's equations using the first-order Kaluza-Klein formalism of general relativity in the (2,2)-splitting of 4-dimensional spacetime. These equations are interpreted as quasi-local energy, momentum, and angular momentum conservation equations. In the asymptotic region of asymptotically flat spacetimes, it is shown that the quasi-local energy and energy-flux integral reduce to the Bondi energy and energy-flux, respectively. In spherically symmetric spacetimes, the quasi-local energy becomes the Misner-Sharp energy. Moreo…
In this paper we will review some facts, both classical and recent, concerning the geometry and analysis of the Kerr and related black hole spacetimes. This includes the analysis of test fields on these spacetimes. Central to our analysis is the existence of a valence $(2,0)$ Killing spinor, which we use to construct symmetry operators and conserved currents as well as a new energy momentum tensor for the Maxwell test fields on a class of spacetimes containing the Kerr spacetime. We then outline how this new energy momentum tensor can be used to obtain decay estimated for Maxwell test fields. …
A new quasigroup approach to conservation laws in general relativity is applied to study asymptotically flat at future null infinity spacetime. The infinite-parametric Newman-Unti group of asymptotic symmetries is reduced to the Poincaré quasigroup and the Noether charge associated with any element of the Poincaré quasialgebra is defined. The integral conserved quantities of energy-momentum and angular momentum are linear on generators of Poincaré quasigroup, free of the supertranslation ambiguity, posess the flux and identically equal to zero in Minkowski spacetime.
New electromagnetic conservation laws have recently been proposed: in the absence of electromagnetic currents, the trace of the Chevreton superenergy tensor, $H_{ab}$ is divergence-free in four-dimensional (a) Einstein spacetimes for test fields, (b) Einstein-Maxwell spacetimes. Subsequently it has been pointed out, in analogy with flat spaces, that for Einstein spacetimes the trace of the Chevreton superenergy tensor $H_{ab}$ can be rearranged in the form of a generalised wave operator $\square_L$ acting on the energy momentum tensor $T_{ab}$ of the test fields, i.e., $H_{ab}=\square_LT_{ab}/…
The lecture explains the geometric basis for the recently-discovered nonholonomic mapping principle which specifies certain laws of nature in spacetimes with curvature and torsion from those in flat spacetime, thus replacing and extending Einstein's equivalence principle. An important consequence is a new action principle for determining the equation of motion of a free spinless point particle in such spacetimes. Surprisingly, this equation contains a torsion force, although the action involves only the metric. This force changes geodesic into autoparallel trajectories, which are a direct mani…
In the present work, the spin-coefficient formalism of Newman and Penrose is used to formulate geometric constraints for the existence of Kerr-Schild groups, i.e. continuous groups of generalized Kerr-Schild transformations. In addition, by characterizing the geometric structure of the deformed Einstein tensor of the generalized Kerr-Schild class, restrictions are imposed on the existence of apparent conservation laws in generic spacetimes, which are defined via considering special Kerr-Schild currents whose associated Kerr-Schild vector fields coincide with timelike Killing vector fields of p…
We review the treatment of conservation laws in spacetimes that are glued together in various ways, thus adding a boundary term to the usual conservation laws. Several examples of such spacetimes will be described, including the joining of Schwarzschild spacetimes of different masses, and the possibility of joining regions of different signatures. The opportunity will also be taken to explore some of the less obvious properties of Lorentzian vector calculus.
The quasi-local energy conservation law is derived from the vacuum Einstein's equations on the timelike boundary surface in the canonical (2,2)-formalism of general relativity. The quasi-local energy and energy flux integral agree with the standard results in the asymptotically flat limit and in spherically symmetric spacetimes.
From the constructions of the quantum spacetime, a four dimensional quantized spacetime can be embedded in a five dimensional continuous spacetime. Thus to observe from the five dimensional continuous spacetime where the four dimensional quantized spacetime is embedded, there exist the energy-momentum flows between the five dimensional continuous spacetime and the four dimensional quantized spacetime. This makes the energy-momentum not locally conserved generally on the four dimensional quantized spacetime. We propose that energy-momentum tensors of noncommutative field theories constructed fr…
As Harvey Brown emphasizes in his book Physical Relativity, inertial motion in general relativity is best understood as a theorem, and not a postulate. Here I discuss the status of the "conservation condition", which states that the energy-momentum tensor associated with non-interacting matter is covariantly divergence-free, in connection with such theorems. I argue that the conservation condition is best understood as a consequence of the differential equations governing the evolution of matter in general relativity and many other theories. I conclude by discussing what it means to posit a ce…
We present a practical framework for ideal hyperelasticity in numerical relativity. For this purpose, we recast the formalism of Carter and Quintana as a set of Eulerian conservation laws in an arbitrary 3+1 split of spacetime. The resulting equations are presented as an extension of the standard Valencia formalism for a perfect fluid, with additional terms in the stress-energy tensor, plus a set of kinematic conservation laws that evolve a configuration gradient. We prove that the equations can be made symmetric hyperbolic by suitable constraint additions, at least in a neighbourhood of the u…
We construct a general relativistic conservation law for linear and angular momentum for matter and gravitational fields in a finite volume of space that does not rely on any spacetime symmetries. This work builds on our previous construction of a general relativistic energy conservation law with the same features. Our approach uses the Brown and York quasilocal stress-energy-momentum tensor for matter and gravitational fields, plus the concept of a rigid quasilocal frame (RQF) introduced in previous work. The RQF approach allows us to construct, in a generic spacetime, frames of reference who…
We explain the necessity of application of semi-metric in general relativity. A detailed discussion on the energy-momentum conservation in the general relativity is presented using the mathematical tool of semi-metric. By means of the general covariant spacetime translation transformation, the most general covariant conservation law of energy-momentum is obtained, which is valid for any coordinates and overcomes the flaws of the expressions of Einstain, Landau and Moller.
We examine the basic conservation laws for diffeomorphism symmetry in the context of spontaneous diffeomorphism and local Lorentz-symmetry breaking. The conservation laws are used as constraints on a generic series of terms in an expansion around a flat background. We find all such terms for a two-tensor coupling to cubic order in the metric and tensor field fluctuations. The results are presented in a form that can be used for phenomenological calculations. One key result is that if we preserve the underlying diffeomorphism symmetry in a spontaneous-symmetry breaking scenario, one cannot deco…
The renormalized expectation value of the stress energy tensor of the conformally invariant massless fields in the Unruh state in the Schwarzschild spacetime is constructed. It is achieved through solving the conservation equation in conformal space and utilizing the regularity conditions in the physical metric. The relations of obtained results to the existing approximations are analysed.
The approximate stress-energy tensor of the conformally invariant massless spin-1/2 field in the Hartle-Hawking state in the Schwarzschild spacetime is constructed. It is shown that by solving the conservation equation in conformal space and utilizing the regularity conditions in a physical metric one obtains the stress-energy tensor that is in a good agreement with the numerical calculations. The back reaction of the quantized field upon the spacetime metric is briefly discussed.
A new, conserved, symmetric tensor field for a source-free Maxwell test field on a four-dimensional spacetime with a conformal Killing-Yano tensor, satisfying a certain compatibility condition, is introduced. In particular, this construction works for the Kerr spacetime.
We briefly recall the problem of defining conserved quantities such as energy in general relativity, and the solution by introducing a symmetric background. We apply the general formalism to perturbed Robertson-Walker spacetimes with de Sitter geometry as background. We relate the obtained conserved quantities to Traschen's integral constraints and mention a few applications in cosmology.
We study the Hawking radiation as charged particles' tunneling across the horizons of the Hot-NUT-Kerr-Newman-Kasuya spacetime by considering the spacetime background as dynamical and incorporating the self-gravitation effect of the emitted particles when the energy conservation, the angular momentum conservation, and the electric charge conservation are taken into account. Our result shows that the tunneling rate is related to the change of Bekenstein-Hawking entropy and the radiant spectrum is not pure thermal, but is consistent with an underlying unitary theory. The emission process is a re…
Minkowski spacetime is transitive under ordinary translations, a transformation that do not have matrix representations. The de Sitter spacetime, on the other hand, is transitive under a combination of translations and proper conformal transformations, which do have a matrix representation. Such matrix, however, is not by itself a de Sitter generator: it gives rise to a conformal re-scaling of the metric, a transformation not belonging to the de Sitter group, and in general not associated with diffeomorphisms in spacetime. When dealing with variational principles and Noether's theorem in de Si…
We argue that the conventional quantum field theory in curved spacetime has a grave drawback: The canonical commutation relations for quantum fields and conjugate momenta do not hold. Thus the conventional theory should be denounced and the related results revised. A Hamiltonian version of the canonical formalism for a free scalar quantum field is advanced, and the fundamentals of an appropriate theory are constructed. The principal characteristic feature of the theory is quantum-gravitational nonlocality: The Schroedinger field operator at time t depends on the metric at t in the whole 3-spac…
The ``warp drive'' metric recently presented by Alcubierre has the problem that an observer at the center of the warp bubble is causally separated from the outer edge of the bubble wall. Hence such an observer can neither create a warp bubble on demand nor control one once it has been created. In addition, such a bubble requires negative energy densities. One might hope that elimination of the first problem might ameliorate the second as well. We analyze and generalize a metric, originally proposed by Krasnikov for two spacetime dimensions, which does not suffer from the first difficulty. As a…
The "conformal mass prescriptions" were used recently to calculate the mass of spacetimes in higher dimensional and higher curvature theories of gravity. These definitions are closely related to Komar integrals for spacetimes that are conformally flat at great distances from the sources. We derive these relations without using the conformal infinity formalism.
