14.1 The Krasnikov Tube
The Krasnikov Tube of general relativity was presented [Krasnikov, S. V., 1998] as a two dimensional metric. It is represented by the following interval.
ds2 = (dct - dz)(dct + k(ct,z)dz)
ds2 = dct2 + [k(ct,z) - 1]dctdz - [k(ct,z)]2dz2
k(ct,z) goes to 1 as x goes to if the tube is not to be infinitely long. In the case that
k = 1 the metric reduces to ds2 = dct2 - dz2 which is the two dimensional case for the metric of special relativity. A photon traveling in the + z direction is described by ds = 0 and the coordinate velocity that results from this interval is dz/dt = c.
Using (ct',z') for the accelerated ship frame coordinates, the invariance of the interval results in
(dct - dz)(dct + k(ct,z)dx) = dct'2 - dz'2
for the interval between events local to the ship.
Events describing the location of the ship are displaced by dz' = 0.
(dct - dz)(dct + k(ct,z)dz) = dct'2
The interval between the events described is of course timelike. This is described by the
178 Chapter 14 Other Faster Than c Methods
. Consider the case of a faster than c ship . If k = 1 as in special relativity, then this results in which is spacelike and a contradiction to . This is an arguement that faster than c is not possible in special relativity. Next, consider the case of a faster than c ship where k is negative at the location of the ship. This results in because each term in the product is positive. This eliminates the contradiction which disallowed the faster than c travel in the special relativity case. The interesting thing about this spacetime is that since we have shown that there are photons moving at c, when the ship speed is greater than c, according to the remote coordinate frame the ship speed is also greater than this light. This is similar to spacetime folding which occurs in black holes.
It is easy to extend this interval to 4 dimensional examples for general relativity. For instance,
Unfortunately there are problems with constructing such a spacetime. The spacetime itself requires negative energy matter. Also, as with other metric engineering attempts, it also results in large stress energy tensor terms.
Work out, or have a computer work out the Einstein tensor and there from the stress-energy tensor for Eqn. 14.1.3. Verify that none of the stress-energy tensor terms diverge. GRTensorII is a free downloadable tensor calculus package that is good for this kind of work.
14.2 Faster than c that does occur naturally
Consider the interval given for general relativity by
A will go to 1 as r goes to . A will be -1 at r = 0 for a finite length along z of (a,b) and again A will go to 1 as z goes to . A coordinate transformation that reduces this interval to the form of special relativity at the location of an observer between (a,b) at r = 0 is given by
ct = z'
z = ct'
14.2 Faster than c that does occur naturally 179
Time ct and space z are folded over in this region. This results in a velocity transformation
u = c2/u'
Thus an object still with respect to this local frame moves from point a to point b instantaneously.
Now a boost can be done from the local inertial S' frame to another local inertial frame S" and the transformation will be Lorentz. This leads to the general transformation from a local frame to the remote frame given by
Yes these are inverted compared to the Lorentz transformation equations. is the velocity of S" with respect to S'. This leads to the velocity transformation
Now the fastest S" can move with respect to S' is v = c and this results in
u = c
Thus we see that for this case of space-time folding c is the speed minimum instead of the speed maximum. If this all sounds far out, closely compare this situation to matter falling into a Schwarzschild black hole. The same thing happens to r and ct there as happens to z and ct here. Thus , according to a remote observer using Schwarzschild coordinates, under the event horizon the coordinate speed of light falling radialy into a Schwartzschild hole is a speed minimum for in falling objects rather than a speed maximum.
Finally, consider the Space-time Interval given by
A will vanish outside of the interval from z = 0 to z = a. and it will also vanish at r = infinity. An object in this interval and at r = 0 has a ruler position of
180 Chapter 14 Other Faster Than c Methods
Even if it has a zero coordinate velocity, it will still have a nonzero rigid ruler velocity of
This velocity can be anything even greater than c, even though the coordinate velocity dz/dt is zero. This is the same phenomenon as the ruler velocity of galaxies becomes greater than c far enough away. v = HL (See Hubble's "constant".)
This is in essence because we are expanding space itself instead of trying to move objects next to each other within the space at faster than c velocities.
This Method of faster than light travel is closely related to type of warp drive of section 13.3. In order to use this for method for travel to another star, it is not enough to manipulate the rigid ruler velocity of the ship. Since the star sits at some fixed coordinate position, one can use this method, both expanding space between here and the ship as well as contracting space between the ship and the star in order move the rigid ruler distance of the ship arbitrarily close to the star, but at some point the ship will have to cross the remaining coordinate displacement to the star. Contracted distance between the ship and the star is described by
This results in the remote observer coordinate speed of light between the ship and the star being greater than c, which in accordance with section 13.3 allows the ship to cross the final coordinate displacement to the star with a greater coordinate velocity than c.
With this type of travel, one can in essence contract the distance between the ship and the destination, in a way, bringing the destination to the ship. The ship then transverses the remaining distance, and then spacetime is allowed to return to normal.
Work out, or have a computer work out the Einstein tensor and there from the stress-energy tensor for Eqn, 14.2.1. grtensorII is a free downloadable tensor calculus package easily found on Google, that is good for this kind of work. Notice that if this interval is used without modifications then there will be divergence problems at the event horizon.
14.2 Faster than c that does occur naturally 181
Show, or have a computer work out that the Einstein tensor for Eqn. 14.2.6 is.
GRTensorII is a free downloadable tensor calculus package that is good for this kind of work. This gives the stress-energy tensor by . Notice that none of the stress-energy tensor terms diverge so long as A is never -1. Show that the matter field's invariant mass density is given by
Krasnikov, S. V., 1998, "Hyperfast Interstellar Travel in General Relativity," Phys. Rev. D 57, 4760-4766