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Warp Drive Mathematics and Physics

"Scotty!" barked Captain Kirk, "we need more power!" "I don' know, Cap'n!" replied Scotty, "w...

Wednesday, May 16, 2018

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.

Thursday, May 10, 2018

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.

Monday, April 30, 2018

How to Beat the Light-Speed Barrier

Is faster-than-light speed or warp drive possible? If Scotty were here to explain how warp drive works, he'd probably say, "Sorry, Capt'n, we don't 'ave enough power!" But assuming we have enough power, warp drive shrinks the spacetime in front of the Enterprise while it stretches the spacetime behind it. That way the Enterprise is closer to its destination and further from its starting coordinates without having moved through space. According to Scotty, this prevents time dilation, so the time on the Enterprise stays in sync with the time on earth.

You may be familiar with the twin paradox where one twin stays on earth and the other travels through space at high speed. According to special relativity, time on the spacecraft passes more slowly than the time on earth. When the space-twin returns home, he finds his brother has aged considerably while he has aged hardly at all. Warp drive allegedly solves this problem by not requiring the Enterprise to move through space. However, notice spacetime must be altered for warp drive to happen. This implies time as well as space is being altered, since, according Einstein, space and time are intrinsically connected.

Einstein also insisted that nothing goes faster than light, but as we shall see, the light-speed barrier can be beaten under the right circumstances. First, let's define the variables we are going to use:

Equation 1 below contains the famous Lorentz factor. At equation 2 we can see why Einstein believed nothing goes faster than light. Any particle with mass would be infinitely massive at light speed, which implies that an infinite amount of energy is needed to get the particle up to that speed. Since energy is limited, the logical conclusion is light speed for anything with mass is out of the question.

What Einstein insisted, is true if velocity (v) is not a function of mass. But suppose velocity is a function of mass, like in a gravitational field, for instance. What happens then? We know our earth orbits our sun. We know each is affected by the other's gravity. The equations below assume the sun's mass (M) and the earth's mass (m) increase as the Lorentz factor says they should.

So now the velocities are a function of mass, but this creates a runaway positive feedback loop!

The red arrows above show that an increase in the earth's mass causes an increase in the sun's mass which in turn causes another increase in the earth's mass and so on. According to NASA's database, the masses of the earth and sun do not need to be continually updated. Those masses stay reasonably constant. The same is true for other celestial bodies within our solar system. Also, at equation 8, notice how the sun and earth start out with less energy than they end up with. This violates energy conservation. That can only mean one thing: equations 6 and 7 are wrong.

We know equation 1 is right, however. So let's sort out this conundrum. We can derive equation 1 if we assume momentum is conserved. At equation 9, mc is constant and m' increases when velocity u decreases. Velocity u is defined at equation 10:

Using a little elementary algebra we derive equation 1 and restate it at 13 below:

So why doesn't equation 13 work for a gravitational field? Gravity is truly something special. Momentum is not conserved (see inequality at 13b). Different masses fall at the same rate. When a falling mass accelerates, the instant velocity (v) at a given point along the path is the same no matter how big or small the mass (m). Using this fact, we derive equations 20 and 21 below:

At equation 20 and 21, notice how the masses of the earth and sun don't change. No more positive- feedback loop and energy is now conserved.

Now, let's see what happens when a mass (m) moves at light speed in a gravitational field:

Mass m does not change! It remains the same no matter how fast it moves! This implies that infinite energy is no longer needed to achieve light speed. Higher speeds are possible when you consider that mass M has no upper limit and radius r has a zero-limit ( see equation 20). The only energy needed is GMm/r.

We know that time is slower where gravity is stronger and faster where gravity is weaker. If we replace mass (M) with time (t) at equation 21 we can derive equation 26:

Equation 26 confirms that time dilates in a gravitational field. This makes total sense when you consider gravity is a function of warped spacetime. This also explains why the Enterprise's warp drive must somehow place the ship in a protective spacetime bubble where the spacetime within is not warped, so time can be the same on the ship as it is on earth. Only the spacetime surrounding the bubble is warped.

Now, if mass m goes no faster than light speed, energy conservation is not a problem, but faster than light speed suggests a problem. If there are speeds faster than light then E does not necessarily equal mc^2. You would start out with that amount of energy but end up with more energy. Faster-than-light velocity might yield E = m(greater than c)^2. But if we begin with Einstein's complete energy equation (equation 27), we can derive equation 34 below:

At equation 34 we see that energy E is conserved no matter how fast mass m moves. At equation 37 we see why. Any increase in gravitational velocity is offset by a decrease in time velocity (u). When you add the square of those velocities you get c^2. NICE!

OK, another question: does time run backwards during faster-than-light spacetime travel? We work out the math below:

The variable u is time velocity. If it is negative, then proper time (t') is running backwards (see equation 41). However it appears earth time (t) is still running forward. At equation 44 we see that negative u cancels itself.

Imagine a spaceship (not protected by a warp-drive bubble) moving faster than light. Let's plug in some specific numbers and see what happens:

At equation 47 above, we see that two solutions are possible: positive and negative proper time (t') and imaginary numbers to boot. We factored out i to make the number real, but even then we have two possible solutions: proper time is running forward or backward. Forward time is more probable. To see why, click here. In either case, equation 53 confirms that earth time is not altered. It is still moving forward at a rate of t:

Apparently Hollywood and sci-fi novels failed to consider that a person going back in time is not exempt. If you went back in time, you would grow younger and younger. You would only be able to go back as far as your conception. Before that, you didn't exist. This might explain why we don't get tourists from the future.

Additionally, you are only going back in time in your reference frame. The rest of the universe is moving forward in time. So, if you are a space-twin, when you return to earth, your brother will still be years older than when you left, and, you will be much younger than when you left--perhaps just an embryo! The math below confirms this:

Thus the warp-drive bubble that protects the star ship Enterprise is an absolute necessity! We need to control time. Once we have this technology, we need the finite energy of a black hole to hurl across the universe faster than light!