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Faster Than Light Travel in Warp Drives and PHYS 471: Introduction to Relativity and Cosmology

One of the most tantalizing ideas in is that we might someday be able to break the “light barrier” and travel at speeds exceeding that of, well, light! One of the most sobering tenets in physics is that nothing can travel faster than light (nice going, Einstein!). The idea of “superluminal propulsion” hasn’t escaped the minds of gravitational physicists, though, so in the following few pages, we’ll review several different proposals for circumventing the universal speed limit.

But first, as always, we’ll need a little bit of general relativity foundation to guide our exploration...

1 The Stress Tensor and Energy Condi- tions

As we know from Einstein’s equations, the stress energy tensor Tµν defines the source of curvature, i.e. and energy. For a , we have T µν = 0, while for a point mass we have T µν = diag(ρ, 0, 0, 0). A distribution of “dust” can be described µν by the tensor T = diag(ρ, px, py, pz), where ρ is the energy density and pi the pres- sure.

We can make a few more statements about how matter “behaves” in by introducing what are known as energy conditions. These are Lorentz invariant statements about how the matter behaves over time and through space. In a sense, they are statements about across different frames of ref- erence. Thinking back to , we know there are three types of four vectors:

• Timelike vectors:

µ 2 2 2 2 u uµ > 0 =⇒ u0 > u1 + u2 + u3 These are vectors describing points within the light cone, i.e. for any observer traveling at v < 1. An example is uµ = (1, 0, 0, 0).

• Null vectors: µ 2 2 2 2 k kµ = 0 =⇒ k0 = k1 + k2 + k3 These are vectors describing points on the light cone, i.e. for things traveling at v = c = 1. An example is kµ = (1, 0, 0, 1). • Spacelike vectors:

µ 2 2 2 2 w wµ < 0 =⇒ w0 < w1 + w2 + w3

These are vectors describing points outside the light cone, i.e. between two causally-disconnected points. You thought I was going to say “for things trav- eling faster than light”, but there are no such things!

µ We evaluate the inner product (“dot product!”) of Tµν with a four vector x , on which certain constraints are imposed to define each specific condition:

• Weak Energy Condition (WEC):

µ ν Tµνu u ≥ 0

The WEC guarantees that, for an observer in any frame of reference moving with v < c, the energy density measured by will be positive (or rather, non-negative). Since the expression is Lorentz invariant, this means if it’s true in one frame, it’s true in every frame. So, picking uµ = (1, 0, 0, 0) (the rest frame) and Tµν given above, it’s easy to show that

µ ν Tµνu u = ρ ≥ 0

That is, the total energy density of a distribution is constant (and positive!) in every reference frame.

• Null Energy Condition (NEC):

µ ν Tµνk k ≥ 0

This conditijon looks like the WEC, but it holds specifically for null vectors. It basically says that matter will interact with (and gravitions... if they exist!) in such a way as to guarantee a positive mass density. For the null vector kµ = (1, 0, 0, 1), we again see that

µ ν Tµνk k = ρ ≥ 0

• Dominant Energy Condition (DEC):

µ ν Tµνn n ≥ 0

The DEC is basically a combination of the WEC and NEC, because nµ is defined to be a non-spacelike vector. We generally want this to hold, ’cause it’s a twofer... one! • Strong Energy Condition (SEC):

 1  T − g T xµxν ≥ 0 µν 2 µν

In the above condition, the constraint is strengthened to include a term involving µ gµνT , where T = T µ is the contraction of Tµν. This looks a lot like Einstein’s equations rearranged in the way we saw in Problem Set 5, so we can interpret this condition to say that the curvature of spacetime must be induced by a non- density.

In summary, the energy conditions tell us that energy density must be positive unless there is nothing there!

As you might have guessed by now, the key to superluminal travel probably involves some type of violation of the energy conditions. That is, it requires the energy density somewhere to somehow become negative! This conceptually makes sense, since the energy conditions constrain the energy density to the light cone, because that’s how we guarantee that physics makes sense (causality and all that). To travel between points outside the lightcone, one must travel faster than c... Aha!

Whenever the Enterprise is in peril, Kirk asks Scotty for a solution, to which the response is usually “Captain, I canna change the laws of Physics!”. Dissatisfied with that answer, Kirk inevitably turns to Spock and reiterates the question. Spock’s retort is always much more optimistic: “There may be a way to do it, Captain. In theory...”.

2 The Alcubierre

In theory, the Alcubierre Warp Drive [1] is a means of achieving faster-than-light travel without the pesky side effects of special relativity (time dilation and all that). It was proposed by Mexican physicist in 1994, which for us geeky / sci-fi nerds in graduate school at the time was the most awesome paper ever (surpassing the one about traversable wormholes by and Mike Morris...). The stipulations imposed by Alcubierre were as follows:

• The ship must be able to travel at arbitrary speeds v > 1

• The ship itself must remain in an inertial frame – i.e. flat spacetime.

• Special relativitstic effects should be confined to the acceleration phases only.

• The metric should be asymptotically flat. The warp drive is predicated on a matter distribution described by some Tµν that is able to accomplish the above tasks. But Einstein’s equations have two sides, so we can also look to the other to try and “engineer” the appropriate energy density. That is, we consider how the curvature around the ship must behave, and that will lead to a value of Tµν.

As the general relativity game goes: find your metric! Alcubierre’s proposed the following:

2 i 2 i i j ds = α − βiβ dt − 2βidx dt − γijdx dx This doesn’t look quite like the metrics we’ve seen for black holes. The most striking difference is that it has off-diagonal terms, but that’s OK! Many metrics have them, we just haven’t had the need to use them to date. In fact, through a suitable coordinate change, you can always make a metric diagonal if it isn’t (and vice versa). Anyway, the terms in the metric are:

• α =⇒ the lapse function. This tells you how the proper time interval changes with respect to coordinate time of observers, i.e. α = dτ . In special √ dt relativity, the equivalent is α = 1 − v2. For the Schwarzschild metric, it’s 2GM  α = 1 − r . ds • βi =⇒ the shift vector. This term is βi = dxi , and shows how the spatial components are affected by the metric.

• γij =⇒ the usual spatial part of the metric, γij = δij.

The forms of these functions are set by the stipulations placed on the warp drive. We suppose the ship is traveling along a path xs(t) in thex ˆ direction, and so its spatial dxs(t) velocity is vs(t) = dt . The shift vector is then

y z βx = −vs(t)f(rs(t)) ; β = β = 0

To minimize time dilation effects, we set α = 1.

tanh (σ(r + R)) − tanh (σ(r − R)) f(r (t)) = s s , R > 0, σ > 0 s 2 tanh(σR)

2 2 2 2 2 ds = dt − (dx − vsf(rs) dt) − dy − dz 0 Note that if we re-define the coordinate dx = dx − vsf(rs)dt, this looks like a flat spacetime metric! Indeed, the spacetime is completely flat within the bubble. Figure 1: The Alcubierre “warp bubble” function f(rs), for σ = 8, xs = 0.5,R = 1. The spacetime is flat inside r < R, and the width of the boundary of f(rs) is determined by σ.

We can study how the spacetime is warped by the function f(rs) by calculating the extrinsic curvature of the spatial components. In terms of the metric parameters α and βi, this is 1  ∂g  K = ∂ β − ∂ β − ij ij 2α i j j i ∂t In this formalism (called the “ADM” formalism), we can understand how the metric stretches or compresses spacetime by calculating what’s called the volume expan- sion element, i θ = −αK i i where K i is the trace of the extrinsic curvature. Applying this to the Alcubierre metric, we find xs df θ = vs · rs drs A spatial plot of θ is shown in Figure 2.

The energy density of this spacetime ρ = T00 can be calculated from the metric through Einstein’s equations. Plugging this in, we find

p 2 2  2 1 vs y + z df T00 = − 2 < 0 (1) 8π 4rs drs Figure 2: The Alcubierre “warp bubble” volume expansion element θ, for σ = 8, xs = 1,R = 1. The ship is traveling at speed vs in thex ˆ direction. In front of the ship, the volume expansion is negative, while behind the ship it is positive. This implies spacetime is contracted in front and expanded behind the warp bubble. The magnitude of the expansion scales linearly with the ship’s velocity vs, which can theoretically take on any value.

Since all the values are positive, the energy density is strictly negative! Using the energy conditions outlined earlier, it is clear that this violates all three. The upshot is that no form of matter known presently (i.e. baryonic matter) can be used to generate the warp field. This type of matter is rather exotic, so we call it... uh... um... oh! We call it .

The shape of the energy density distribution (1) is shown in Figure 3. It is a ring whose perpendicular axis lies along the direction of motion. This informs the shape of the engines required to generate the warp bubble. This type of “ring ship” is shown in Figure ??, and unfortunately doesn’t look quite like the we’ve seen in Star Trek!

3 Travel Time at Warp Speed

Let’s do a quick analysis of a ship traveling at warp speed from planet A to planet B. Since the speed of the warp bubble vs is unlimited, the travel times can be made arbitrarily small. The original paper provides a simple analysis to show this fact. The warp drive involves significant curvature deformations of spacetime, which can produce incredibly large (albeit local) tidal effects. As such, the ship must first move Figure 3: Left: The density distribution for the Alcubierre warp drive is toroidal about the x-axis, creating a ring perpendicular to the direction of motion. The density is most negative at the center. Right: A hypothetical warp drive would require a similarly toroidal housing for the exotic matter. Pictured is the IXS Enterprise, designed by Dutch Mark Rademaker in collaboration with NASA. very far from its point of origin (A) before engaging the bubble. Let’s call this dis- tance d. It must also drop out of warp an equivalent distance from its destination B. This leaves a distance D − 2d to travel by warp.

The ship first travels the distance d from planet A by conventional propulsion (let’s call it impulse power!). This is affected by special relativistic time dilation, and so the proper time as observed on the planets is d 1 τimpulse = , γ = 2 γvimpulse 1 − vimpulse Following this, the ship stops and engages the warp drive. Since the proper time in the bubble is the same as coordinate time on the planets, the calculation is free of time dilation. If we assume the ship undergoes positive acceleration for half the trip D ( 2 ), and then slows down symmetrically for the latter half of the journey, the travel time at warp is calculated by Newtonian means as r D − 2d τ = 2 warp a The total travel time including the last impulse leg is therefore d √ τtot = 2 + 2 D − 2da γvimpulse Assuming D  d, we can ignore the impulse time dilation legs to find √ τtot ∼ 2 Da

You can see that this time can be made arbitrarily small by making a large. The limitations are simply the amount of available energy, and hence exotic matter.

4 Wormholes

Another way of achieving faster than light travel – while never actually travelling faster than light – is to pass through a in spacetime. You’ve no doubt encountered this idea in popular media and/or fiction, but based on your un- derstanding of curvature, energy conditions, and tidal , you can now understand them on a new level!

Strictly speaking, a wormhole is a solution to Einstein’s equations that resembles a Schwarzschild , except that the “throat” interior to the does not squeeze off to a singularity. Instead, the throat contracts to a minimum radius Rthroat and then starts expanding back open until the spacetime becomes flat again. This tunnel in spacetime can connect two regions of spacetime that are separated by an arbitrarily vast distance in space that would take ages to cross. Depending on its characteristics, however, travel time could be arbitrarily small!

5 Basic Wormhole

The most basic set-up of a wormhole is as follows, and came about long before Morris and Thorne’s in the last 1980s. We ascribe1 to a region of spacetime the metric

ds2 = c2dt2 − dr2 − (r2 + b2)(dθ2 − sin2 θ dφ2) (2)

This unassuming looking thing has some very peculiar properties that we can analyze using basic geometry. First, we note that it is nowhere singular, which is always good. Second, when r  b, the spacetime is flat. Third, when r = 0, the parameter b seems to play the role of some minimum distance, namely Rthroat = b.

We can evaluate the curvature tensors (and scalars!) using Maple, and we find the non-zero ones to be b2 b2 sin2 θ R = ,R = ,R = −b2 sin2 θ rθrθ b2 + r2 rφrφ b2 + r2 θφθφ 1Who uses “ascribe” anymore? Figure 4: A twormhole is a connected pair of Schwarzschild solutions with mini- mum throat radius Rmin = b, linking two otherwise causally-disconnected regions of spacetime. The throat is stabilized by the presence of exotic matter. as well as the permutations from Bianchi identities. Likewise, the non-zero Ricci tensor is 2b2 R = − rr (b2 + r2)2 Only one! Furthermore, the Ricci scalar is the negative value of that, since

2b2 R = gµνR = grrR = µν rr (b2 + r2)2

In fact, this is very interesting, because it tells us something very interesting about the stress-energy of the wormhole. The LHS of Einsteins’s equations are

1 1 2b2 b2 R − g R = 0 − = − tt 2 tt 2 (b2 + r2)2 (b2 + r2)2

1 2b2 1 2b2 b2 R − g R = − + = − rr 2 rr (b2 + r2)2 2 (b2 + r2)2 (b2 + r2)2 1 1 2b2 b2 R − g R = 0 + (r2 + b2) = + θθ 2 θθ 2 (b2 + r2)2 b2 + r2 1 1 2b2 b2 sin2 θ R − g R = 0 + (r2 + b2) sin2 θ = + φφ 2 φφ 2 (b2 + r2)2 b2 + r2 π The first two are negative, and the second two are positive and equal if we set θ = 2 . Remember that Einstein’s equations tell us how matter curves spacetime: 1 8πG R − Rg = T µν 2 µν c4 µν where the stress-energy tensor will have the form

 ρ(r) 0 0 0   0 −τ(r) 0 0  Tµν =    0 0 p(r) 0  0 0 0 p(r)

The quantities in this tensor are:

• ρ(r) =⇒ the energy density in the wormhole throat

• τ(r) =⇒ the tension of the throat

• p(r) =⇒ the in the throat which we can write as c6 b2 ρ(r) = − 8πG (b2 + r2)2 c4 b2 τ(r) = 8πG (b2 + r2)2 c4 b2 p(r) = 8πG b2 + r2 Note that ρ(r) < 0, so whatever it is that is curving spacetime into this wormhole isn’t normal matter, but rather is matter with negative energy density! Exotic matter strikes again! This is one of the major problems with the wormhole solution. A secondary problem comes from the fact that the tension τ(r) is incredibly large. From our results, we see that for a wormhole of solar mass M = M with a throat size of b = 10 km, the energy density and tension are

c6 J ρ(r)c2 = − ∼ −1049 8πG m3 c4 τ(r) = − ∼ 1032 Pa 8πG Ulp! These numbers are huge, and indicated that they energy requirements to main- tain a wormhole are not only exotic, they are HUGE!! Nevertheless, note that an observed moving through this wormhole feels no tidal forces t GM whatsoever, since with four-velocity u = 1 1 − r (remember your assignment!), we find t t t ar,tidal = −R rtrtu u = 0 because that component of the Riemann tensor vanishes. So, perhaps there is hope in the !...

6 Traversable Wormhole

A more robust attempt at studying the characteristics of a traversable wormhole was made in 1988 by Mike Morris and Kip Thorne (or Interstellar fame!... and also of LIGO frame!.... and also of Nobel Prize fame!...... and also of traversable wormhole fame). The took the original metric described in the previous section and added a bunch of new constraints, particularly on the magnitude of the energy requirements, as well as the subsequent conditions experienced by an observer passing through the wormhole.

In brief, their metric was of the form

2 2 −2Φ(r) 2 dr 2 2 2 2 2 ds = e dt − b − r dθ − r sin θ dφ 1 − r This looks suspiciously like a Schwarzschild solution (in fact, it’s very similar), except the gtt coefficient is slightly different, and the Schwarzschild radius has been replaced with the minimum throat width b.

Without going into too much detail, Morris and Thorne imposed the following con- ditions on their wormhole:

• The energy requirements are reasonable (even if exotic), i.e. and tensions not like those in the center of a star!

• The time to traverse the wormhole must be reasonable short, in terms of typical human lifespan (i.e. passage of up to a year).

• Time dilations effects must not significantly affect the traveller and his family he left behind (i.e. no noticeable effects).

• The traveller must survive the trip through the throat, i.e. no tidal accelerations exceeding 1 g. The Morris-Thorne wormhole has become quite popular over the years, and ranks up in the “what if?” file with the Alcuiberre warp drive. In fact, it was the theoret- ical basis for the wormhole featured in Interstellar [3], and marked one of the first times gravitational physics research was funded by a special effects company! The ref- erences (and a few extras) are given below for those who are interested in following up!

Until time: Live Long and Prosper!

References

[1] M. Alcubierre, “The Warp Drive: Hyperfast travel within general relativity”, Classical and Quantum 11, L73-L77 (1994).

[2] M. S. Morris and K. S. Thorne, “Wormholes in spacetime and their use for : A tool for teaching general relativity”, Am. J. Phys. 56, 395 (1988).

[3] O. James, E. von Tunzelmann, P. Franklin, K. S. Thorne, “Visualizing the Interstellar Wormhole”, Am. J. Phys. 83, 486 (2015).