Warp Drive Mathematics and Physics

"Scotty!" barked Captain Kirk, "we need more power!"

"I don' know, Cap'n!" replied Scotty, "we're on impulse engines alone!"

This classic exchange comes from the Star Trek series. It takes place in the 23'rd century, a time when there is warp-drive technology. In this post we work out the mathematics and describe the physics behind warp drive.

What exactly is warp drive? According to the series, it is powered by dilithium crystals. Warp drive pulls the starship's destination closer and pushes the ship's starting coordinates further back. Essentially, the spacetime shrinks in front of the starship and stretches out behind it. This implies shorter spacetime wavelengths in front and longer spacetime wavelengths in back. It is possible to derive an equation that models this. Let's begin with the classic Hamiltonian:

Why the Hamiltonian? It is the sum of kinetic and potential energy. We can think of kinetic energy as energy needed to move a particle through space. Potential energy is, of course, stored energy, or, time energy, since a particle moves through time when it is at rest.

Energy conservation suggests that when there is more kinetic energy (more movement through space), there is less potential energy (less movement through time), and vice versa. At equations 4 and 5 below, we show the equivalency of time and potential energy; and, space and kinetic energy:

We can also create a Minkowski diagram:

From the Minkowski diagram we can derive the Lorentz factor (see equation 11 below):

If we start with the Planck mass squared, we can derive and define the spacetime wavelength (lambda) as well as proper time (lambda/c). (See equations 16 and 17):

Using a scale factor (alpha) we can build a second energy equation equal to the one we derived from the Minkowski diagram.

At equation 20 we set the kinetic energy equal to the gravitational energy. Gravitational energy is the warped spacetime that allows the starship to stay at rest, yet, seemingly move through space. It actually moves with space rather than through it. This enables the starship to reach destinations at super-light speeds.

At 21 and 22 we equate the classical Hamiltonian with the energy's quantum representation. The alpha scale factor makes this possible. Also, notice energy would not be conserved without it. When gravitational energy increases, the wavelength (lambda) decreases. This conserves energy on the right side of equation 22, but the left side can become infinite. Dividing the left side by alpha fixes this problem.

Using a bit of algebra we derive equation 29 below:

Equation 29 is the warp-drive equation. We know that massive galaxies move away from us faster than light if they are far enough away. Equation 29's first term contains Hubble's constant and bar-lambda. This is a velocity with long wavelengths or vast distance. The second term contains a velocity with short wavelengths or distance. The greater the difference, the faster the starship moves with space. It's like dark energy pushing from behind and gravity pulling in front. We can use an integral to sum every point in space along the path between the longest wavelength to the shortest:

At 30 and 31 we show how energy is conserved in spite of the fact that gravitational energy seems to have no upper limit. Shorter wavelengths (lambda) offset the longer wavelengths (bar-lambda):

Below we restate equation 26 at 32. From there we show how Einstein's field equations can be derived.

The fact we can derive the field equations confirms that the warp-drive equation is a solution. Caveat: Unfortunately there is still that pesky second postulate of special relativity and the apparent fact that the photons within any system can't ever be observed going faster than light.

Update: "Alcubierre drive shifts space around an object so that the object would arrive at its destination faster than light would in normal space without breaking any physical laws."--Wikipedia.

OK, so how long does it take the Alcubierre drive to shift space around? Let's say the goal is to bring point B closer to point A. If no physical laws are broken, then the minimum time (t) needed is t = (B-A)/c, where c is light speed. Distance B-A = ct, the shortest distance possible. So if a spaceship goes light speed, it will cover the distance just as fast or faster than if you take the time to shift point B closer to point A and then pretend you covered the distance faster than light.

Gravity is Weak Because of Light Speed

According to the above video, gravity lives outside our universe in a higher dimension or parallel universe. It then leaks into our universe. By the time it gets to us, it has a weak signal. However, using extra dimensions and universes to explain what we don't understand only increases the overall mystery. It also increases the burden of proof of the scientists who postulate such a hypothesis. Surely higher dimensions and parallel universes are more poorly understood than the gravity we can actually observe and measure. For these reasons, they make a poor hypothesis.

Today we are going to explain the weakness of gravity using what we already know. First we define the variables we will be using:

Time for a thought experiment. Imagine a cubic section of our universe. Divide this cube into two halves: A and B. A and B contain a vacuum with a few photons whizzing around with momentum p. A and B each have fairly equal numbers of these photons.

From another part of our universe we take mass m and add it to section B:

At equation 1 above we add section B's momentum and mass energies. It's obvious that A's total energy is less than section B's (see inequality 2 below):

Now, look what happens when we divide both sides by p^2:

Number 3 above is an absurdity. How can light speed (c^2) be less than another speed? According to Einstein, light speed is the maximum speed limit in our universe. Maxwell showed mathematically that light is an electromagnetic wave. The speed of light is a function of the permittivity and permeability of free space. Since these values are constant, light speed is also constant in a vacuum.

To date, no particles have been discovered that can exceed the speed of light. But is light speed really the top speed? The inequality at 3 suggests otherwise. Here's why we currently believe nothing goes faster than light in a vacuum. Below is a Minkowski-ish spacetime diagram.

Velocity vector A is at rest; it is moving through time (t'), so its velocity is zero (see equation 3b). B is moving through both time and space, so its velocity is greater than zero (see 3c). C is moving through space only. It's proper time (t') is zero and its velocity ... are you ready for this? ... is infinite (see 3d).

Of course this infinite speed is from the photon's point of view. We calculate what an observer sees at 3e above. At 3f, we assert that light speed is infinite speed if proper time is used. Can't beat infinity, therefore light speed is the top speed.

Since light speed is the top speed, the inequality at 3 is wrong. Equation 4 below makes better sense:

Notice we simply included a reduction factor of epsilon^2. What can we learn from this? We learn that the full energy of section B does not come into play. It is weakened by the fact that nothing can go faster than light. To restore equilibrium completely, let's multiply the left side by a time-dilation ratio. (To see how this ratio was derived, click here.)

From equation 5 we can derive equation 7 below (Newtonian gravity). Equation 9 reveals the nature of the reduction factor.

Equation 9 tells us that gravity is weak due to time dilation and the ratio of section A momentum to section B momentum. In particular, gravity is weak because momentum p is weak. Why is p so weak? It's the momentum you find in the vacuum of space (with a few photons thrown in). Where there is low vacuum energy there is weak gravity.