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.