Time Travel Research Center © 2005 Cetin BAL - GSM:+90  05366063183 - Turkey / Denizli 

In special relativity, travel faster than the vacuum speed of light must be forbidden in order to preserve causality(effect never precedes cause). And in Special relativity the vacuum speed of light is globally c (Eqn. 1.1.7 c = 2.99792458m/s exact by definition). Therefor in special relativity there is no faster than c travel. However, in general relativity the vacuum speed of light is only locally invariant and so the remote coordinate speed of light is not always c, and in some cases it is greater than c. There are metrics as shown below for which the coordinate speed of light c' can be increased above c. Because of this there are situations in general relativity where objects travel faster than c.

Lets consider for a moment the invariant interval for the world line of a particle moving in the x¹ direction, and dx¹ is a coordinate distance displacement.

ds² = g₀₀dct² + 2g₀₁dctdx¹ + g₁₁dx¹dx¹

(13.1.1)

If this is a photon, then ds² = 0.

0 = g₀₀dct² + 2g₀₁dctdx¹ + g₁₁dx¹dx¹

0 = g₀₀ + 2g₀₁(dx¹/dct) + g₁₁(dx¹/dct)²

0 = (1/2)g₀₀ + g₀₁b + (1/2)g₁₁b²

b = {-g₀₁ ± [(g₀₁)² - g₀₀g₁₁]^1/2}/g₁₁

So the remote coordinate speed of light is given by,

c' = {-g₀₁ ± [(g₀₁)² - g₀₀g₁₁]^1/2}c/g₁₁

(13.1.2)

 163

164 Chapter 13 Warp Drives

So we see that objects can travel faster than c in the + x¹ direction if

{-g₀₁ + [(g₀₁)² - g₀₀g₁₁]^1/2}/g₁₁ > 1

(13.1.3)

Problem 13.1.1

Consider light instantaneously undergoing a coordinate angular displacement in a spacetime described by

ds² = (A² - B²b²f²)dct² + 2B²bfrdctdq - B²r²dq²

Show that c' = - B²bf ± AB

______________________________________________________________________________

To the extent that we restrict our physics to special relativity and demand that the traditional order of causation is always correct, no information can travel faster than the speed c. However, special relativity is only a special case of physics within a more general theory known as general relativity.

In general relativity the vacuum speed of light is locally invariant, but not globally invariant. The vacuum speed of light remote from a given observer need not be c. It can be greater than c and can vary with direction depending on the gravitational field involved. Because this is allowed starships that travel between the stars faster than c are not ruled out a-priori within the physics of general relativity.

In 1994 physicist Miguel Alcubierre of the University of Wales published how to use general relativity to consider a space-time geometry, actual warp drives, that would undo the special relativistic time dilation effects and allow for faster than c travel in the May issue of Classical and Quantum Gravity.  His space-time geometry would also allow for acceleration of the ship without any internal forces crushing the crew. They would feel weightless as the ship accelerated. Unfortunately in the arbitrary manner that the metric was developed it was shown by Pfenning & Ford to require a lot of negative energy that would violate a few quantum mechanics energy conditions and a greater magnitude of it than all the mass of the known universe.  This section will demonstrate a way to arbitrarily lower the magnitude of the negative energy requirement.

13.2 Alcubierre-Broeck Warp Drives 165

A case of the space-time geometry that he had derived can be represented in the following equation

ds² = dct² - (dx - bfdct)² - dy² - dz²

(13.2.1)

f can be any function of the coordinates that is one at the location of the starship and zero far from it. Transforming coordinates to a particular choice of the starship coordinate frame and allowing it to travel faster than c (b > 1), coordinate singularities crop up in the equation corresponding to event horizons in front of and behind the ship enclosing it in a mathematical bubble. This is called the warp bubble. Even though the event horizons remain for any choice of ship frame, there is a choice for star ship coordinates for which the singular nature is transformed away.

June 1999 Chris Van Den Broeck of the Institute for Theoretical Physics at the Catholic University of Leuven, Belgium, came up with an alteration for this space-time geometry that would retain all of the desired warp drive qualities but reduced the negative energy requirements down to the order of a transversable wormhole.  His space-time geometry can be represented by this equation

ds² = dct² - B²[(dx - b fdct)² + dy² + dz²]

(13.2.2)

B can be any function that is large near the starship and one far from it though he used a specific top hat function for his example calculations. That not only brought the negative energy requirements down to a hopefully one day reachable goal but it also solved one of the quantum mechanics energy condition violations.

The above metrics were designed to study linear motion and Alcubierre himself wrongly believed that the effect of the acceleration beyond c speeds of the Starship was a result of the space behind the ship happening to be in a state of expansion while the space in from of the ship is in a state of contraction. This isn't really the cause. There are other Warp drive spacetimes besides the one above. Consider the following warp drive loop.

ds² = dct² - dz² - dr² - (rdf - bfdct)²

(13.2.3)

Let b be any function of time. Let f be a function that is zero far from the warp loop and is at a maxima of one at (z = 0, r = R). This spacetime has a geodesic at the loop such that an object initially following the loop at its initial warp speed b will continue on accelerating according to however b is changed.

Notice that in this spacetime there is no diametric nature to the space immediately behind and in front of the ship.

166 Chapter 13 Warp Drives

The Van Den Broeck Warp Drive was actually a case of a more general type of Alcubierre-Broeck Warp Drive SpaceTime. A more general case of an Alcubierre-Broeck Type Warp Drive SpaceTime can be written

(13.2.4)

Where the Metric tensor reduces to that of special relativity far from the ship, and dx¹ represents a coordinate distance displacement.

The transformation of coordinates to the ship frame and the interval in the ship frame were given by Hiscock for the case of constant velocity in the consideration of only two dimensions.  We here show how his transformation extends to four dimensional spacetime with arbitrarily time dependent acceleration. We also present the ship frame energy density T⁰⁰ from a four dimensional calculation and note that the 4d classical calculation is everywhere finite.

Consider an Alcubierre interval given according to a remote frame's cylindrical coordinates by

ds² = (1 - b²f²)dct² + 2bfdctdz - dz² - dr² - r²df²

(13.2.5)

Where f is a function that is 1 at the location of the ship and zero far from it.

We start out with the first transformation Eqn 13.2.7

z' = z - ò^ctbdct

Where b is first expressed here as a function of time ct.

With some algebra for simplification this results in

ds² = [1 - b²(1 - f)²]dct² - 2b(1 - f)dctdz' - dz'² - dr² - r²df²

Let g = 1 - f and this becomes

ds² = (1 - b²g²)dct² - 2bgdctdz' - dz'² - dr² - r²df²

(13.2.6)

Notice that this returned the original intervals form with a reversal on the sign of b and a reversal of the boundary conditions for g.

Now we notice that at r = 0, this interval becomes the interval for special relativity transformed to cylindrical coordinates. Thus, we have found a transformation to a frame based local to the ship. One can also verify that in these coordinates the relevant affine connections vanish at r = 0.

13.2 Alcubierre-Broeck Warp Drives 167

Therefor this interval works not only for a frame local to the ship, but also for ship frame itself. So the global coordinate transformation between the remote frame and the ship frame is

ct' = ct

z' = z - ò^ctbdct

r' = r

f' = f

 (13.2.7)

Using r² = z ' ² + r² and cosq = z '/r , based on this interval, the ship frame energy density works out to be

T⁰⁰ = - (b²/4)(c⁴/8pG)(dg/dr)²sin²q

(13.2.8)

The total ship frame energy is

E = òT⁰⁰dV

(13.2.9)

E = -ò₀^¥ò₀^p 2p (b²/4)(c⁴/8pG)(dg/dr)²r²sin³qdqdr

E = -ò₀^¥(b²/12)(c⁴/G)(dg/dr)²r²dr

(13.2.10)

Other terms in the stress-energy tensor involve d²g/dr² and so that this doesn't diverge at the boundaries leaving a delta function matter distribution there we can also require dg/dr to be continuous at the boundaries. If the warped region is modeled as a spherical shell of inner radius R and outer radius R + D then we might choose for this region

dg/dr = {-6(r - R)[r - (R + D)]/[(R + D)³ - R³]}{(3R²+3DR+D²)/D²}.

and dg/dr = 0 outside of this warped region.

This results in a total ship frame energy requirement of

  E/c² = -{(1/35)(b²c²/G)[21R² + 35RD + 15D²]D⁵/[(R + D)³ - R³]²}{(3R²+3DR+D²)/D²}².

In the case that we choose D << R this approximately simplifies to

  E/c² = -(3/5)(b²c²/G)R²/D.

168 Chapter 13 Warp Drives

And if we choose D >> R then we have

E/c² = -(3/7)(b²c²/G)D

These results are within an order of magnitude of the calculation of Pfenning and Ford Class. Quantum Grav. 14 (1997) pg 1750 equation 28. However, dimensions between the extremes do NOT result in more energy magnitude requirements than contained in the universe. The calculation that led to that result was an extreme limit where D << R. This limit was required by the assertion that a quantum inequality restriction due to the presence of negative energy matter limited the thickness of the warp shell (the matter responsible for the warp).

For an example calculation take b = 1, traveling right at the speed c. Take R = 50m, D = 60m. At this point, these dimensions are not really allowed by a quantum inequality due to the presence of the negative energy, but we later show how to even further reduce the negative energy result which then puts these dimensions within the limit allowed by the quantum inequality. These inputs result in a total energy requirement of

E/c² = -0.068 solar masses

The reasons for the difference are

1.) They have restricted their warped region's thickness by a quantum inequality where we have not. We do not as we later show how this negative energy can be arbitrarily reduced until any particular dimensions are allowed by the inequality.

2.) We have chosen a much different behavior for the function f between the boundaries than Alcubierre had originally chosen, and we looked at an intermediate case for the extremes of R and D . We could consider speeds greater than c in which we must still address other issues such as the locally tachyonic motion of the matter at the outer edge of the warp, but at speeds just under c the only real showstopper is the weak energy condition violation. Fortunately there are ways in which the weak energy condition can be violated in nature such as production of Casimir effect energy, or otherwise lowering the zero point energy of the vacuum.

For the next result it is convenient to make a definition

H(ct, r) º 1 - b²(1 - f)²

(13.2.11)

Using this the interval can be expressed in the form

ds² = H[dct - b(1 - f)dz'/H]² - dz'²/H - dr² - r²df ²

(13.2.12)

13.2 Alcubierre-Broeck Warp Drives 169

Instead of using, t' = t, IF b is a constant, say b₀, and IF we consider only two dimensions (ct,z), then the spacetime may be diagnalized by doing the next transformation

ct" = ct - b₀ò{[1 - f(z')]/H(z')}dz'

(13.2.13)

Then we arrive at Hiscock's interval

ds² = H(z')dct"² - dz'²/H(z')

(13.2.14)

Now we see that singularities associated with the event horizons occur in the invariant interval where H(z') = 0, but these singularities only exist with such a particular choice of time for ship frame coordinates. With our previous choice of time coordinates these singularities were transformed away and using that frame we don't have problems with divergence of the energy density.

Next we will go back to the ship frame Alcubierre metric Eqn.13.3.7

ds² = (1 - b²g²)dct² - 2bgdctdz' - dz'² - dr² - r²df²

Alcubierre's original more general metric had a time dilation term in the remote observer's frame

A(ct,r). I call it a time dilation term because of how it is related to U⁰ in Eqn.13.2.19 below. Alcubierre prefers to call it by the lapse function. We will reintroduce such a term into the ship frame's interval. Only we will use different boundary conditions for it. We will keep A = 1 both at the location of the ship, and far from it, but allow it to become large in the warped region. The interval in the ship frame becomes

ds² = (A² - b²g²)dct² - 2bgdctdz' - dz'² - dr² - r²df²

(13.2.15)

The solution for T⁰⁰ is

T⁰⁰ = - (b²/4)(c⁴/8pG)(dg/dr)²sin²q/A⁴

(13.2.16)

Now we see that T⁰⁰ can be arbitrarily lowered by making A arbitrarily large in the warped region. The exact solution for the

170 Chapter 13 Warp Drives

entire Einstein tensor developed in spherical coordinates according to the ship's frame can be found at http://www.geocities.com/einswarp. Notice that with the use of the trig identity sin²q + cos²q = 1 all the denominators can be greatly simplified. Also, the covariant Einstein tensor can be viewed at http://www.geocities.com/alcwarp.

Another concern over the Alcubierre warp drive is that it once the ship goes superluminal, there may be a problem controlling the warp to turn it off and slow the ship. At superluminal speeds, there are event horizons for this spacetime where

g₀₀ = 0, even in the case that we have transformed away the singularities there. These two horizons can be constructed as two half spheres enclosing the ship in a warp bubble. Information can not be sent from behind the ship outside the bubble to the inside. Also, information can not be sent from inside the ship to the region to outside, in front. For the warp drive, part of the matter region producing the warp, or the warp shell extends across the horizons. A signal sent from inside the ship can not reach the matter extending in front of the horizon that is in front of the ship. This piece of the warp shell can not then be turned off. This lead to the concern over the ability to control the warp at superluminal speeds. We can work around the problem as follows. The matter controlling the behavior of A(ct,xⁱ ) can be arranged prior to the matter controlling the behavior of

b(ct)g(r).

It is the behavior of b(ct)g(r) and not A(ct,xⁱ ) that controls the speed of the ship. The horizon occurs where

A(ct,xⁱ ) = b(ct)g(r).

For a conceptual view of the problem from the perspective of an external observer's inertial frame

The function g(r) is then manipulated so that it goes to 1 at a smaller distance from the ship then where

A(ct,xⁱ ) = b(ct).

In other words, function A(ct,xⁱ ) is larger than b(ct) for an interval extending beyond the place where g becomes 1. Now the portion of the matter shell that was formed to control the behavior of A(ct,xⁱ ) that is in front of the horizon will not be accessible by a signal from the ship once superluminal speeds have been reached, but as far as controlling the speed of the ship goes, this does not matter. What matters is that the portion of the matter shell that controls the behavior of

b(ct)g(r) is totally contained inside the horizons ensuring that the ship speed is controllable even after the ship has gone superluminal.

 Finally in investigation of the new quantum inequalities, we return to the Alcubierre spacetime according to the ship frame modified by inserting A(ct,r) into the ship frame's metric

ds² = (A² - b²g²)dct² - 2bgdctdz' - dz'² - dr² - r²df²

13.2 Alcubierre-Broeck Warp Drives 171

The solution for T⁰⁰ from 13.2.16 can be written

T⁰⁰ = - (b²/4)(c⁴/8pG)(dg/dr)²(r/r)²/A⁴.

Pfenning applied a quantum inequality for a free, massless scalar field to the alcubierre warp even though he did not include the possibility of an A(ct,r) other than A = 1, and even though the Alcubierre spacetime is not really the result of a free, massless scalar field. Even so, we can go back and redo the calculation along the same lines with a variable A(ct,r) included. The quantum inequality he applied which we recalculate is

(13.2.17)

In order to apply it we must find T^mnU_mU_n and t₀ for a Euclerian observer. A Euclerian observer is an observer who starts out just inside the warp shell with zero initial velocity according to the ship frame who "samples" the time it takes t₀ for some flux of negative energy to pass him.

The Euclerian observers for this modified spacetime interval will be those with a velocity along the z' axis given by

U = -bgcA^-1

(13.2.18)

From the interval we have

ds² = (A² - b²g²)dct² - 2bgdctdz' - dz'²

c² = c²(A² - b²g²)(dt/dt)² - 2bgc(dt/dt)U - U²

Inserting U we wind up with

dt/dt = A^-1

Again, this result is why I term the lapse function a time dilation term.

This results in 

(13.2.19)

172 Chapter 13 Warp Drives

Inserting T^mnU_mU_n = A²T⁰⁰c² into the inequality we have

we can make the replacement

r² = z ' ² + r²

Pfenning

then makes an order of magnitude aproximation of the geodesic motion of the Euclerian observer. Equation (5.11). Since we are using the ship frame with A(ct,r) inserted, we must include its effect here as well.

z' ~ -vg(r)A^-1(r)t

r ² = r² + v²g²A^-2t²

We make a definition corresponding to Pfenning's equation 5.15

y =r/[vg(r)]

(13.2.20)

And choose dg/dr = 1/D corresponding to his choice for 5.4

Consider A kept constant with respect to z, and therefor t , through the negative energy region. According to Pfening's 5.16 the integral is approximately

(13.2.21)

13.2 Alcubierre-Broeck Warp Drives 173

So the inequality becomes

(13.2.22)

This result is the same as Pfenning eq 5.17 with the exception that there is an A(r) is not necessarily 1. Pfenning then asserts that this result must hold for sample times that are small compared to the square root inverse of the largest magnitude of the Riemann tensor components as calculated for the local observers frame.

If for simplicity A is kept approximately constant through the negative energy region, then the largest Riemann tensor component for this spacetime is

|R^r_00r| » b²(dg/dr)² = (v/D)²

(No sum on r which is only the index for the ship frame radial distance, not a variable index.)

So in this case, the sampling time must be restricted to

t₀ = aD/v

where 0 < a < 1.

Inserting this into the inequality and looking at the case of large A approximately constant A₀ through the negative energy region leads to

Choosing a = 0.10, and making the approximation 3/p ~ 1, this becomes

D £ 10²(v/c)L_PlanckA₀

(13.2.23)

Now we see that letting A become arbitrarily large also arbitrarily thickens the minimum warp shell thickness. Therefor the -0.068 solar mass calculation had a reachable shell thickness. All that remains then is to divide this -0.068 solar mass result by an A₀⁴ which would allow the thickness chosen, and which will lower the energy magnitude even farther by several orders of magnitude.

Problem 13.2.1

Let A be approximately constant throughout the region that g varies. What does A₀ have to be to allow the chosen shell thickness in the example energy calculation of -0.068 solar masses? What does this reduce the energy calculation to?

___________________________________________________________________________________________

174 Chapter 13 Warp Drives

We have seen that the remote coordinate speed of light for light moving in the ± x¹ direction is given by Eqn.13.1.2

c' = {-g₀₁ ± [(g₀₁)² - g₀₀g₁₁]^1/2}c/g₁₁

when dx¹ is a coordinate distance displacement. We can use this to make faster than c travel possible in two ways. The first way we will demonstrate results in a general Alcubierre-Broeck type of warp drive space-time. The second results in a new type of warp drive.

First, consider a starship moving along a geodesic in the x¹ direction with a speed v given by

b = v/c = - g₀₁/g₁₁ . Since this speed is never outside of the boundaries of the speed of light given above, we can allow any speed, even speeds greater than c. So consider a spacetime where

g₀₁ = g₁₀ = - g₁₁b at the location of the ship and g₀₁ = g₁₀ = 0 far from it. Let f be any function that is 1 at the location of the ship and 0 far from it. We can then do this by requiring

g₀₁ = g₁₀ = - g₁₁bf.

Then g₀₀ can be chosen to be A² + b²f²g₁₁ which ensures that the remote coordinate speed of light is globally real.

This leads to a general Alcubierre-Broeck type of warp drive space-time given by Eqn.13.3.4

where dx¹ is a coordinate distance displacement. Here, f is any function that is 1 at the location of the ship and 0 far from it. A is a function of the coordinates that is 1 far from the ship. The other metric components must also reduce to those of special relativity far from the ship.

For the interval between events along the world line of the ship dt/dt = A₀, where A₀ is A evaluated at the location of the ship.

The second way to go about achieving faster than c travel along these lines goes as follows. Consider the case of a diagonal metric.

The coordinate speed of light is then given by

c' = ± [- g₀₀/g₁₁]^1/2c

This speed of light is greater than c when [- g₀₀/g₁₁]^1/2 > 1. This leads us to a new type of warp drive space-time.

13.3 Faster Than c Travel By A NEW Warp Drive 175

Consider the following interval expressed in a remote observer's coordinates

(13.3.1)

Far from the ship

T = Z = Y = X = 1

The coordinate speed of light moving along the z direction is

(13.3.2)

This is the new maximum speed for objects in motion in the z direction. Recall how in section 12.4 we patched together different vacuum field solutions through a region of a spherical shell matter. Here we will do the same thing. We will use two concentric matter shells to patch together three vacuum field solutions. Out side of the outer shell the interval in the external observer's coordinates will be a vacuum field solution that reduces to far from the ship.

For the vacuum region between the outer and inner shells the interval will be given by

(13.3.3)

Also to avoid local tachyonic motion of the matter in creating the disturbance in the metric, we will let the the metric tensor at the outer edge of the outer shell equal the metric evaluated at the outer edge of the inner shell. Let single primes represent the ship frame coordinates. The transformation between the external observer's coordinates and the ship frame coordinates is a composition of the two transformations that results in

176 Chapter 13 Warp Drives

(13.3.4)

From these we find the velocity transformation equation for motion in the z direction to be

(13.3.5)

From this we find that the remote coordinate speed of the ship is given by

(13.3.6)

The only speed limitation is -1 < b < 1

In the ship frame the stress energy tensor of the inner matter shell is given by the solution in section 12.4. Using this to achieve any b, we see the ship does achieve faster than c speeds in the remote observer's coordinates.

Problem 13.3.1

Verify that Eqn. 13.3.3 is a vacuum field solution by showing that it is a transformation of the metric of special relativity.

Problem 13.3.2

Lets say a ship can reach b = 0.01. If T(z = 0) = 25, what does Z(z = 0) have to be for a remote coordinate ship speed of 10c?

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