An Analysis of Alcubierre's Warp Drive

"By a purely local expansion of spacetime behind the spaceship and an opposite contraction in front of it, motion faster than the speed of light as seen by observers outside the disturbed region is possible."--Miguel Alcubierre

To read Alcubierre's original warp drive paper, click here. First, let's derive Alcubierre's warp-drive metric from Einstein's field equations. Here is a list of variables required:

We begin with equation 1, Einstein's famous general relativity equation, then we do a little algebra and contract the tensors to simplify the math:

Next, we put together the metric. Equation 12 below is the finished product:

Coordinates y and z are set to zero because the spaceship is moving along the x axis. Velocity v is further defined below. It is a function of the warp-bubble shaping function and the difference between the spaceship's position and the total distance along x.

At 15 notice the value inside the warp bubble is 1 and outside is zero. This allows time to dilate outside the warp bubble, while inside the bubble there is no time dilation. More on this later. For now, let's continue defining the other components:

Below is a crude diagram representing the warp bubble moving along the x axis:

"It is then easy to see that for the spaceship’s trajectory we will have: dτ = dt ."--Miguel Alcubierre

From Alcubierre's metric we can derive a Lorentz equation (see 19 below) that shows how time dilation is avoided. Since the value of the bubble shaping function is 1 inside the warp bubble the velocity v and velocity x/t cancel each other, so there's no time dilation inside the warp bubble. The function value is 0 on the outside, so there x/t is not cancelled and any time dilation would take place outside the warp bubble.

At 20 above, we see that proper time (t') equals coordinate time (t).

"The metric I have just described has one important drawback, however: it violates all three energy conditions (weak, dominant and strong). Both the weak and the dominant energy conditions require the energy density to be positive for all observers."--Miguel Alcubierre

It appears that negative energy density is required to achieve speeds greater than light speed. Negative energy density implies negative energy and negative mass. Check out the following equation system:

Notice that when v > c, the only way to get a value for m' that is real and not imaginary is to plug in -mv^2, a negative energy that has negative mass (-m). For photons, where v = c, mass m' is zero as expected. Particles that move faster than light (e.g. tachyons) have negative mass. Thus it seems such exotic matter is required to achieve superluminal speed. However, consider a satellite orbiting a black hole at radius r:

The black hole has positive mass, i.e., positive energy density. If we increase that positive energy density, we can imagine the satellite orbiting faster and faster. There seems to be no upper limit to velocity v in this gadanke experiment, since there is no apparent upper limit to positive mass m. Further, using Einstein's field equations we can show that unlimited positive energy density (pressure, heat, etc.) can yield unlimited velocity:

At equation 30 above it is plain to see that any increase in positive energy density on the right side will cause a corresponding increase in velocity on the left side. So it appears exotic matter is not really needed to achieve superluminal speed. But then there's this:

"[L]ight itself is also being pushed by the distortion of spacetime."--Miguel Alcubierre

This is an unfortunate choice of words. It gives the impression that the distortion of spacetime can push light faster than light. At least that's what needs to happen if the spaceship is going faster than light; otherwise, the ship's electromagnetic energy could be left behind! Let's see what happens to photons when spacetime is distorted:

On the left side of equation 36 we have a mass particle with velocity v. On the right we have a photon with frequency f. If velocity v increases due to spacetime distortion, the velocity of the photon does not increase; rather, its frequency increases. Now, just for fun, what would happen if we assume the photon's velocity could increase?

On the right side of equation 38, the denominator has c'--a special photon velocity that can rise above c. What is the consequence? A paradox! When energy density increases, so does the spacetime curvature on the left side. This causes c' to increase wich causes the spacetime curvature to reduce which causes c' to reduce and so on. Bottom line: you end up with some very screwed-up physics!

If we can't show that photons go faster than c, at least we can show that other particles can ... or can we? Imagine the spaceship moving toward a star system at velocity nc, which is greater than c. The star system sends photons toward the spaceship at velocity c. What is the combined velocity? It's not nc + c. The velocity addition formula reveals the answer:

As Einstein himself could have told you, the combined velocity is no faster than light! This means no matter how fast distorted spacetime moves the spaceship, if there are any photons along its path (and there will be!), it will move no faster than light.

So how should we interpret equations 30 and 33? Why does it appear that superluminal speed is possible? Equation 33 makes sense if we place ourselves inside the spaceship and allow time dilation to happen.

At 43 above, we multiply the real velocity by t to get the distance. The perceived velocity v is that distance over proper time t', the time experienced by the observers in the spaceship. Let's see what we can derive:

At 51 above, we see that as epsilon gets closer to 1, the perceived velocity gets closer to infinity. The real velocity (epsilon-c) never goes faster than light.

We can determine the value of alpha for equations 30 and 33 (see equation 55):

Now equations 30, 33 and 52 make sense when you consider that observers outside the spaceship will see it going no faster than c, but observers inside the spaceship will swear they covered distance vt in time t', which could be faster than light.

But then again, we have a time dilation problem, since the clock inside the spaceship won't agree with the clock of outside observers. This is definitely true if the spaceship moves and its departure point A and arrival point B are at rest. We illustrate this in the diagram below. The arrow represents the spaceship.

But what if the spaceship, points A and B move at the same rate?

If points A, B and the spaceship all move at the same rate as in the above diagram, their clocks will agree. All observers along that path will be under the impression that the spaceship traveled distance vt in time t', which could be faster than light, at least on paper.

Now, if the observers in the spaceship view the star system they're heading for, according to the velocity addition formula, the combined velocity of the ship and the photons coming from the star will not exceed light speed, so what do they see? We can speculate they see a shorter distance or distance vt' instead of distance vt. If they don't look where they are going, they might assume the distance is vt and conclude they covered it faster than light.

Given what we now know, we should be able to visit any star system in record time, and when we get back home, our clock will agree with clocks on earth. Albeit, Miguel Alcubierre's paper makes no mention of the impact the needed energy will have on the spaceship's clocks. Things like mass and energy also slow time. According to Varieshi and Burstein's paper (click here to read), the amount of energy (mass) needed to manipulate spacetime to achieve a seemingly superluminal speed is approximately 3.42 X 10^38 solar masses! Many orders of magnitude greater than our observable universe! So time dilation is a problem even if we can successfully get the spaceship and points A and B moving at the same rate.

Conclusion: Warp drive makes wonderful science fiction--and it will be quite some time before it becomes a scientific fact.

Update: So do we really need the spaceship and points A and B moving at the same rate to cure time dilation? Take the muon time dilation experiment. When muons are moving at velocity v, their time slows. But if an observer were riding the back of a muon and considered herself at rest, she would see the scientists moving at velocity v, so why doesn't their time dilate as well? If the muons and scientists that observe them are moving at v relative to each other, shouldn't their clocks agree?

But then a closer look at the Lorentz factor reveals what's going on. Time dilation really depends on how fast the muons and scientists are moving relative to the speed of light--and not to each other. Thus the scientists really are at rest (or slower than the muons)--a photon has no trouble catching them, but does have trouble catching a fast muon (according to a coordinate-time observer; a proper-time observer [the muon or scientist] will see the photon approaching at c). Thus, for their clocks to agree, points A, B and the spaceship must move at the same velocity relative to the speed of light, not to each other.

Do Electrons Leave Our Universe When They Move Inside the Atom?

Using a quantum microscope, it is possible to view the wave function of the hydrogen atom (see diagram above). Note the electron can only be found in the lit areas and not in the dark areas between. When the electron moves between energy levels, it seems to mysteriously vanish from our universe and then mysteriously reappears, hence the alternating pattern of dark and light circular bands.

Of course there's a string theory that explains this phenomenon: when the electron leaves an energy level it literally leaves our 3D space and enters a higher curled dimension. It loops around then re-enters our space:

At the diagram above you can imagine the electron (red dot ) moving left to right. The loop represents the extra hidden dimension.

Now notice the diagram below. It is the famous double-slit experiment. Notice the target screen has light and dark areas. We could postulate that the dark areas are due to the light disappearing into a higher dimension or an alternate universe--or perhaps it was abducted by extraterrestrials.

Of course this is all nonsense. We know why the dark sections are dark and why the light sections are light: constructive and destructive interference wave patterns. Where the red and black lines are parallel there's light (constructive interference), where they cross or move in opposite directions, there is darkness (destructive interference).

Since there is no empirical evidence of extra dimensions, it makes sense to propound an alternate hypothesis that uses established physics as its basis--the established physics being constructive and destructive interference. If you compare the two diagrams above, you will note a striking similarity: both have alternating light and dark areas on their respective target screens.

We know what causes the interference pattern of the double-slit experiment--it's the two slits. But what could possibly cause the interference pattern (discrete energy levels) of the hydrogen atom? That's what this post shall cover. First, let's define the variables needed:

At equation 1 below, energy (E) is charge (q) times voltage (V). Assuming energy E is an eigenvalue, it must have a probability (P(E)). That brings us to equation 2. Quantum mechanics tells us the probability is an amplitude squared (A^2, see equation 3). A pinch of algebra gives equations 5 and 6.

At equation 5 we see the positive charge from the hydrogen atom's proton corresponds to a positive amplitude squared. At 6 the negative charge from the electron corresponds to a negative amplitude squared. To get a negative amplitude squared we multiply the positive amplitude of a sine wave with its negative amplitude:

Notice there's no way to get a positive amplitude squared using this method. To get the positive squared amplitude, we need to do the following:

We use the absolute value of sine to create a positive squared amplitude. We now have what we need to model the hydrogen atom's constructive and destructive interference wave pattern. Check out equations 7 through 10 below:

When we plug in values for equation 7, we get the wave pattern below which maps beautifully to the light and dark areas of the hydrogen atom:

No extra dimensions needed, just old-school physics.

Why the Graviton Can't Be Found

Why hasn't the graviton been discovered yet? A thought experiment could shed some light on this question. Imagine a universe with only a Higgs field and nothing else. No strong, weak or electromagnetic interactions, no spacetime as we understand it. The basis of this universe is just the Higgs, so the only boson available is the Higgs boson. It's true the Higgs can decay into other particles, but let's focus on it while it is a Higgs.

Now, in such a universe, there should be no gravity, since there are no gravitons, right? (We performed a similar thought experiment in a previous post involving photons. Click here to read all about it.) Let's lay out the mathematics and see. First, we define the variables:

If we find gravity in our Higgs-only universe, that would explain why the graviton hasn't been found--it isn't necessary--so let's begin with the Higgs Lagrangian (L) at equation 1 below. At 2 we convert the Lagrangian to the Hamiltonian (H). To make the math less cumbersome we set the kinetic term equal to chi at 3.

We make a substitution at 4. Equation 5 is a Hamiltonian (H') with the same energy as H, but a different mass and kinetic energy. Equations 4 and 5 represent two adjacent fields whose centers of mass are r distance apart. At 6 we show the equality or conserved energy of the two fields. At 7 and 8 we equate the kinetic and potential energy differences.

Here is an overly simplified, crude diagram for illustrative purposes only:

As you can see the two adjacent fields are outlined with imaginary boxes and labeled blue (high kinetic energy/low mass) and red (low kinetic energy/high mass). The white dots represent the masses. Now, equation 8 fails to take into account distance r, so let's convert mass m as follows:

At equation 11 we have distance r where we want it. Equation 11 is the value of the kinetic-energy difference between the two fields. Classical kinetic energy is a function of velocity squared. What we want to know is the value of the velocity squared:

Now that we know the value of velocity squared, we can do one more step and determine the value of the gravitational constant for this Higgs universe (Gh):

We made a substitution at 14 above and end up with Newtonian gravity! And no gravitons! Equation 14 reveals that gravity is the net velocity squared of kinetic energy differences. If we divide both sides by another r, we get gravitational acceleration. Given these results, one could postulate that gravity is the net motion resulting from motion differences. And motion differences are caused by mass differences. Einstein suggested that matter curves spacetime. However, that assertion is very specific to our universe. A more general assertion is mass disturbs the status quo, whatever that may be, and causes kinetic energy variations. At the quantum scale, gravity does not seem to need its own boson. Information is passed using whatever boson is available. In this case, it's the Higgs.

Now, for extra credit, let's derive Einstein's field equations from equation 14:

If you are feeling ambitious, you can work backwards and derive the Higgs Langrangian from Einstein's field equations.

Update: Here is a couple of videos that falsify the graviton:

Why Gravitational Waves Fail to Confirm Extra Dimensions

According to the holographic principle, our four-dimensional universe, consisting of three space dimensions and one time dimension, is a surface area of a five-dimensional spacetime called "the bulk." The remaining dimensions of string theory or M-theory are allegedly compacted and rendered insignificant.

Gravity, compared to the other fundamental interactions, is weak due to the graviton's unique ability to move between the surface area (our spacetime) and the bulk. Other particles remain fully in our spacetime and thus have more intensity. At least that's how the story goes. Unfortunately, the gravitational-wave test described in the above video failed to confirm the existence of "the bulk" or any extra dimensions beyond our four-dimensional spacetime. This does not surprise me, given the problems extra dimensions can cause (click here to read all about it).

So why did the gravitational-wave test fail? Do we really need "the bulk" to explain the nature of gravity? We will explore these questions. First, let's define the variables we will use:

According to general relativity, gravity is a function of energy density, so let's begin with the energy density of an atom. An atom is mostly space, so let's only consider the volume of space taken up my the average nucleus and the electrons. That approximate volume can be found in the denominator of equation 1 below:

Of course if we put that volume in the numerator, we get the energy (E):

If we put a larger volume (V) in the denominator (equation 3), we get a reduced energy (E'). Reduced energy is consistent with weak gravity, so we are on the right track.

We don't want Energy units, so at 5 and 6 we use meters and Newtons to adjust the units:

Now, coincidentally, 10^-45/N is approximately equal to G/c^4, so we make the substitution:

We use distance D and the alpha scale factor to make more substitutions at equation 9. From there we derive equation 12.

Equation 12 is Newton's equation. We were able to derive this equation because we started with the premise that the intensity of gravity is determined by the actual amount of space a particle interacts with. For baryonic matter, that actual amount of space corresponds with the gravitational constant G. Note that no extra dimensions are needed to get equation 12. Our 4D spacetime is sufficient. So why should we be surprised that the gravitational-wave test failed to confirm "the bulk"?

Caveat: the above mathematics may work just fine for ordinary matter such as atoms and molecules, but what about singularities such as black holes? Theoretically, a singularity takes up no space, so there shouldn't be any interaction between the matter and space, but there is! To resolve this conundrum, we first need to establish that light speed is truly the top speed in our universe. Consider the familiar Lorentz equation:

The main problem with this equation is time (t) is arbitrary. Let's make it precise. Let's make time (t) equal to the age of the universe. When I say universe I mean everything including the megaverse if such a thing exists. What we want is the longest time ever lapsed--so we set t accordingly and define the other variables we need:

Now we derive 21 below:

Line 21 shows that no velocity (v) can exceed light speed (c). So what does this have to do with gravity and black holes? Given the fact that light speed is the top speed, we can derive the following:

Take a look at 25 and 26 above. At 25, G stays constant as long as the change in time (delta-t) is equal to or less than the age of the universe. Note that delta-t increases as radius r decreases, so G remains constant. But delta-t has an upper limit of t. If r continues to shrink, G must also shrink. Thus it appears the intensity of gravity is determined by how much space interacts with matter. The smaller the radius r, the smaller the space the matter occupies. Equation 27 shows that the intensity of gravity never exceeds the speed of light squared no matter how much radius r shrinks.

In conclusion, "the bulk" and extra dimensions are completely unnecessary to describe gravity.