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.

How to Whip Time Dilation During High-Speed Interstellar Space Flight.

In our previous post we showed that infinite energy is not necessary to accelerate a mass to light speed. Click here and read all about it. From the Einstein Field equations we were able to derive equation 10 below. Equation 10 shows that faster-than-light speed is possible in a gravitational field. (Equations 11 and 12 show how energy is conserved.)

Of course the problem with going at, near, or above light speed is the time dilation problem. If you travel at light speed, theoretically time does not pass for you. It's as though you reach your destination instantaneously. The only problem is, the exoplanet you were planning to visit and colonize is long gone, its sun was a supernova eons ago. That's because time passed normally for the rest of the universe. If we are to explore the cosmos, we need to solve this time dilation problem.

To find a solution we need to take a closer look at the Lorentz factor and its limitations. If we create a time unit using Planck's constant, we can derive equation 18:

With another step we derive equation 19. At equation 20 we assume a particle is traveling at light speed. We get eye-opening results at 23 and 24:

Equation 23 shows the particle has infinite energy! This is expected if the particle has mass. But what if the particle is a photon? Most photons don't have or require infinite energy to go light speed. Assuming the particle is a photon and said photon has finite energy, then equation 24 shows that proper time for said photon is greater than zero!

We can spot another problem if we assume the particle is going faster than light. Equations 25 through 29 demonstrate that faster-than-light speed requires less energy than light speed!

To make matters worse, the energy for faster-than-light speed is an imaginary number. Surely we want a real number.

Below are some more problems. First the value of proper time (t') depends on the value of t. The Lorentz equation fails to give us an exact proper time. It just gives a relative proper time. Further, the energy needed to go velocity v is the same for all masses! The Lorentz equation fails to take into account how much mass a particle has.

But that's not all! If you change your energy units from electron volts to Joules, the energy is increased! Just compare equations 30 and 31 above. And, more significantly, the value for proper time (t') changes! Yet, we are talking about the same particle, going the same velocity, with the same energy.

What we want is better precision. We want a true value of proper time (t'). Perhaps we can get that by dividing Plank's constant by the energy (E):

At 33 and 34 we realize we can convert the non-specific proper time into a specific number by using a unit-less conversion factor alpha. Using a Minkowski diagram we can graphically show the conversion factor works:

We derived the Lorentz equation at 36. Equation 37 demonstrates that time (t) does not know how big or small it should be, so why not set it to t/a? That way the precise proper time we get from equation 34 agrees with the Lorentz equation.

OK, let's take what we discovered and derive a precise way of finding proper time in a gravitational field with energy GMm/r. We derive equation 51 below:

At equations 52 and 53 we steal an idea from quantum physics and apply it to conserving the energy of a star:

At 53, even if the gravitational energy is infinite (a black hole?), time (t') is zero and the finite energy of the the original star is conserved. So far, so good. But notice the proper time will never go to zero unless energy is infinite. Is this really true? Photons allegedly experience zero proper time with their finite energy. To resolve this contradiction, we need to understand the nature of time better. We can do this by building a quantum clock. First, here are the variables:

At equation 54 we put the gravitational energy into one mass variable. At equation 55, momentum (rho) is conserved--an increase in mass (m) causes a decrease in velocity (v). At 56, momentum (p) is not conserved--an increase in mass does not change velocity (c). At 57 we give time (t) and frequency (omega) definite values by dividing Planck's constant by ground-state energy. At 58 we take the ratio of conserved momentum to non-conserved momentum. (Note when mass increases, velocity v decreases and proper time decreases.) At 59 we cancel the masses. At 60 and 61, we convert the velocities into oscillators. Because oscillators are cyclical, they make excellent little clocks.

At equation 62 we see that when velocity v slows, the wavelength lambda shortens. So we discover a correlation between more mass, shorter wavelengths, and slower time. At 63 through 65 we prove that the wavelength is proper time t' multiplied by light speed c. At 66 through 68, we set the final parameters for our quantum clock.

Below is a diagram of the quantum clock. The clock's imaginary hand is radius mu. When there is more mass or energy, it goes around the clock slower. To stop the clock requires infinite energy. Thus, it appears to be true! Photons with finite energy experience non-zero proper time.

To be sure we are right about proper time, let's take another look at the Lorentz factor equation for mass. Photons have zero mass and have velocity c (see equations 69 and 70). At 73 we get a relative mass (m') that is between zero and infinity. At 74 we convert the relative mass into photon energy divided by c^2. Thus we confirm the photon's energy can be less than infinite.

At 75 we convert the photon's frequency into the reciprocal of its proper time (t'). If we convert the photon's zero mass into photon energy, variable t would need to be infinite (see equations 76 and 77). At 78 we make some substitutions and with a few more steps we get 81 and 82.

Looking at 81 and 82 we see that a photon's proper time does not have to be zero. And, equations 83 to 88 confirm that infinite energy is required to have zero proper time.

Thus our formula for calculating a precise proper time is correct. Our final formula for a gravitational field is at equation 90:

What does equation 90 tell us. It tells us that proper time never goes below zero even if velocity squared Gm/r goes to infinity! This means if you have a twin, if he/she stays on earth, and you traveled to the nearest star at, say, 24 times the speed of light (using a gravitational warp drive), your round trip would take about four months (instead of years at light speed). Your twin will only be four months older than you. The age gap decreases if you travel at even faster speeds. Plus, you can reach that exoplanet mentioned earlier in a timely fashion. Of course this is more science fiction than science. It is not clear how one can travel faster than light without violating the second postulate of relativity which states that all observers must see the photons in your spaceship going light speed in a vacuum.

The Essence of Time

What exactly is time? Is it just an abstract idea? Or does it exist independently of human imagination and perception? Atomic clocks reveal that the rate of time appears to run slower on the earth's surface than way out in space. Assuming time is a real entity, what is it made of? What is its essence? We start our investigation by defining some variables:

We know that nothing goes faster than light in a vacuum. If we add velocity (v) to velocity (c) we still get the speed of light (c).

Equation 1) above seems absurd. When we combine velocities, we should get a higher velocity than c ... unless ... the rate of time (t') shrinks. Equation 2) below works:

And from equation 2) we can derive the famous Lorentz equation:

When velocity (v) increases, time (t') shrinks, but something else happens that's also strange: mass (m') increases. We can verify this if we start with Einstein's energy equation below.

From equation 6) we can derive the relative mass equation:

Equation 13) confirms that when velocity (v) is increased, mass (m') increases. Now, let's take equation 10) and derive equation 14) below:

Equation 14) shows why increased velocity increases mass. When the velocity of a system increases, velocity (u) decreases. To conserve momentum, mass (m') must increase. We start with momentum (mc) and end up with (m'u). Of course m'u must always equal mc. But what exactly is this velocity u? I call it the velocity of time.

The rate of time is the relative speed (u) of a photon or (c^2-v^2)^.5. If a system is moving at velocity (v) and we assume that system is at rest, then the photons in that system may still appear to be moving at c, but relative to v they have slowed to velocity u. Since photons are bosons, their relative speed (how fast they carry force) will determine how fast or how slow the system evolves. If the system is your watch, your watch will noticeably slow down if it moves at a significant fraction of light speed. This suggests that time is real and not just a concept. After all, the original concept of time was that the rate of time is fixed.

Below is a Feynman diagram where velocity v is zero, so velocity u equals velocity c. At the beginning of time (t), two electrons colide. Next, a photon is emitted, then the electrons fly apart.

In the next diagram imagine that the entire diagram is moving through space at velocity v. If we assume the diagram is at rest, the emitted photon will be relatively slower and the measure of time will also be slower:

The electrons in the above diagram are moving faster, but photons can't increase their speed, so, relative to the electrons, the photon is slower. This seemingly slow moving photon is the time we measure or proportionate to the time we measure. Now, just for fun, what happens if the electrons go faster than light?

Time reverses! Your watch is now running backwards. In the diagram above, the particles fly apart, then comes the photon, and finally, the particles collide.

Before we conclude that time is (or is proportionate to) the speed of photons relative to the other particles in a system, let's look at how gravity impacts time, and take a closer look at light, i.e., electromagnetic waves. Once again we define some variables:

We include the variables for permittivity and permeability. Taken together, they determine the speed of an electromagnetic wave. A photon's relative speed (u) has its own corresponding permittivity and permeability. Free space is a vacuum. That is where light has a velocity of c. If the permittivity of free space, for example, had a lower value, light would go faster. Increasing velocity v causes permittivity and permeability to increase (so does additional mass/energy). As a result, EM waves (light) slow down or are relatively slower. The mathematical proof below provides further insight into the essence of time:

Equations 23) and 24) show that time is the reciprocal of frequency. Equations 24) and 26) define the rate of time (with variable t[sub o] set to 1) in one of two ways: The ratio of the relative speed (u) of a photon to light speed (c); or, the ratio of permittivities and permeabilities. Both ways are equivalent. To put it more simply, time (at the quantum level) is the measure of the rate bosons can carry force between particles. If that process is disrupted by high speeds or increased mass/energy, that process will slow down. Anything that is a function of that process will also slow down--including your watch.

Update: Quantum particle-waves are transverse waves: their oscillations are perpendicular to their propagation direction. The following is a mathematical proof that shows that the Lorentz equations above work for transverse waves.

Equation 37) is the formula used to calculate the velocity of a transverse wave. We were able to derive it from the Lorentz equation for relative mass. This shows that the two are intrinsically connected. Equation 39) predicts what we expect: increased mass (m') reduces the time rate (t').