HYPER-FAST INTERSTELLAR TRAVEL VIA A MODIFICATION OF SPACE-TIME GEOMETRY
ABSTRACT:
We analyse difficulties with proposals for hyper-fast interstellar travel via modifying the space-time geometry,
using as illustrations the Alcubierre warp drive and the Krasnikov tube. As it is easy to see, no violations
of local causality or any other known physical principles are involved as far as motion of spacecrafts is
concerned. However, the generation and support of the appropriate space-time geometry configurations does
create problems, the most significant of which are a violation of the weak energy condition, a violation of
local causality, and a violation of the global causality protection. The violation of the chronology protection
is the most serious of them as it opens a possibility of time travel. We trace the origin of the difficulties
to the classical nature of the gravity field. This strongly indicates that hyper-fast interstellar travel should
be transferred to the realm of a fully quantized gravitational theory. We outline an approach to further the
research in this direction.
INTRODUCTION:
It is commonly accepted that for an interstellar travel to become of a practical interest and importance one
should acquire a capability to complete such a travel within a reasonable interval of time by the clocks of
both the traveller and the community remaining on the Earth. In most cases it is the total time of the round
trip that one is concerned but, under special circumstances, it might be also the one way time of arrival at
a destination.
The troubles with the time of an interstellar travel emerge as the result of an interplay between two major
contributing factors, (1) the necessity to cover very large distances to reach even the nearest stars, and (2)
the limitation on the maximal speed of a spacecraft imposed by relativity. This speed must be smaller than
the speed of light. Equivalently, the world line of the spacecraft must pass inside the local light cone in a
neighbourhood of each point of this world line.
In Minkowski space-time (the case of special relativity) the construction of the light cone is global and the
distance between the Earth and the star of destination in the frame of reference of the Earth’s observer
is fixed. Under these conditions the only means to reduce the travel time is to increase the speed of the
spacecraft within the limit determined by the null cone. The nature of the limitation on the minimal time
of travel in this case is best illustrated by the space-time diagram (Fig. 1). Points S and R on the world line
of the Earth’s observer are the events of the start and return of the interstellar expedition, while point A
on the world line of the star is the event of the expedition arrival at its destination. The world line of the
spacecraft SAR, and with it the the expedition duration for the traveller, can be made as short as desirable
by choosing pieces SA and AR sufficiently close to the null cones (moving at a speed sufficiently close to
the speed of light with respect to the Earth). Meanwhile, the distance between A and R along the world
line AR of the Earth (the duration of the expedition as perceived by Earth’s observers) obviously cannot be
made less than 24 where d is the distance between the Earth and the star of destination in the frame of the
Earth.
General relativity opens. at least seemingly, opportunities to circumvent this difficulty. As in the case of
special relativity, the speed of the spacecraft is limited by the speed of light, i. e. the world line of the
spacecraft should pass inside of the local light cone at each point of the world lime. However, the metric
and topology of space-time is not fixed, and presumably can be manipulated. The construction of the null
cone is not global in this new setting. The space-time geometry, and with it the tilt and the opening of the
local light cones can be manipulated in a controlled fashion. Such a manipulation allows in some cases to
reduce the distance to be covered by the spacecraft, which reduces the time of arrival at the destination as
well as the round trip time. In other cases it allows to transform the space-like separations into the time like
separations, which does not reduce the time of arrival but reduces the round trip time.
It is easy to write down an expression for a spacetimemetric that satisfies the basic chronometric requirements
of a feasible interstellar Bight (cf. the next section for some details and references). Whether such a metric
can be generated, supported and controlled in the real physical world without violating the most basic laws
of physics, is a different and, perhaps, the most troublesome issue. Whatever the final answer might be, any
attempt to analyse the emerging situation demands an expansion of the domain of application of the laws
of physics far beyond the conditions for which they were formulated originally.
1. A straightforward computation of the Einstein tensor shows that in all cases gravitational fields corresponding to desirable metrics demand “exotic” matter to be at least a part of their source. Such
”exotic” matter violates various commonly accepted energy conditions, including the weak energy condition, which means that it is supposed to have negative energy density. Although the negative energy
density is not ruled out by quantum field theory, understanding of its coupling to the gravitational field
is incomplete and it shows up sooner than one might expect.
2. In many cases (a good example is the Alcubierre warp drive space-time) the domain of modified geometry
is causally disconnected. This means that the appropriate configuration of the gravitational field cannot
be generated, sustained, or controlled as a result of any geometrodynamic evolution unless one has at
his disposal tachyonic matter, or some kind of a device capable of emulating tachyonic effects.
3. All known configurations of modified space-time geometry involved in a hyper-fast travel, when considered in conjunction with the principle of general co-variance, invariably lead to a capability of building
a time travel machine. Basic theorems characterising the causal structure of space-time [Geroch SC
Horowitz, 19791 tell one that, as a consequence, such space-time geometry cannot be a result of any
geometrodynamic evolution unless there are some means to contribute to the space-time structure externally (space-time singularities, appropriate boundary conditions, tachyonic effects, quantum effects,
topological effects, etc.).
The recent literature presents quite a few attempts to asses compatibility of the idea of modified space-time
geometry as a means of hyper-fast travel with basic concepts of physics (we will provide a more detailed
account and references in subsequent sections). Unfortunately, most of this literature has a tendency to rely
on standard theoretical constructions without any modifications or with minimal modifications, frequently
without clear understanding of their relations to observations. The results are all too predictable. This
approach leads one rather fast and in a relatively trivial fashion to ruling out virtually all particular geometry
modification proposals on the ground of their nonphysical nature via seemingly plausible arguments. The
arguments are relying heavily on rather involved computations and contain numerous assumptions necessary
for their tractability. Some of these assumptions are questionable, others leave one with the impression
that they might be circumvented via more sophisticated designs. In brief, the trouble with this direction of
thought is that it does not provide an understanding of the common mechanism of the failure.
The arguments of more general nature, less computationally demanding, and appealing to more basic and
dependable arguments are scattered in the literature, but have not been systematically used for more balanced
assessment of the problems and prospects of the field. In this paper we attempt to do so having as goals (1)
to determine the restrictions imposed on modifications of space-time geometry, (2) to determine the nature
and the origin of these restrictions, (3) to formulate the conditions on the mechanism that might circumvent
the restriction, and (4) to evaluate possible realisations of such mechanism.
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