3
The previous example shows how one can use an expansion of spacetime to move away
from some object at an arbitrarily large speed. In the same way, one can use a contraction
of spacetime to approa ch an object at any speed. This is the basis of the model for hyper-
fast space travel that I wish to present here: create a local distortion of spacetime that
will produce an expansion behind the spaceship, and an opposite contraction ahead of
it. In this way, the spaceship will be pushed away from the Earth and pulled towards a
distant star by spacetime itself. One can then invert the process to come back to Earth,
taking an arbitrarily small time to complete the round trip.
I will now introduce a simple metric that has precisely the characteristics mentioned
above. I will do this using the language of the 3+1 formalism of general relativity [1, 2],
because it will permit a clear interpretation of the results. In this formalism, spacetime
is described by a foliation of spacelike hypersurfaces of constant coordinate time t . The
geometry of spacetime is then given in terms of the following quantities: the 3-metric γ
ij
of the hypersurfaces, the lapse function α that gives the interval of proper time between
nearby hypersurfaces as measured by the “Eulerian” observers (those whose four-velocity
is normal to the hypersurfaces), and the shift vector β
i
that relates the spatial coordinate
systems on different hypersurfaces. Using these quantities, the metric of spacetime can
be written as:
2
ds
2
= − dτ
2
= g
αβ
dx
α
dx
β
= −
α
2
− β
i
β
i
dt
2
+ 2 β
i
dx
i
dt + γ
ij
dx
i
dx
j
. (1)
Notice that as long as the metric γ
ij
is positive definite for all values of t (as it
should in order for it to b e a spatial metric), the spacetime is guaranteed to be globally
2
In the following greek indices will take the values (0,1,2,3) and latin indices the values (1,2,3).
