How to Beat the Light-Speed Barrier--or Not

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!

Update: If a spaceship can go faster than light in a gravitational field, does light also go faster than light? Let's see what the math says. Here are the variables we will need:

Equation 58 is velocity according to special relativity (see derivation below). As the velocity v grows, so does the mass, creating an upper speed limit of c or light speed.

Equation 59 is velocity in a gravitational field. As we demonstrated above, mass m does not grow as velocity increases. There doesn't seem to be any upper limit to this type of velocity.

Equation 60 is light speed. If energy is added, the frequency increases, but the wavelength shortens. No matter how much energy is added, the increase in frequency is offset by the decrease in wavelength. As a result, a photon can't go faster than c.

At 61 below, we compare the momentum of a falling body to a photon's. The falling body's velocity will increase in a gravitational field with no apparent upper limit. The photon, on the other hand, will experience a frequency increase, but no apparent velocity increase.

"Captain," Scotty might say, "we have a problem!" If fermions in a system go faster than light, then the system's photons must also go faster than light or be left behind. If they are left behind, there's an energy loss that might reduce the fermions' speed, keeping it within the light-speed limit, and/or, the fermions could lose mass. No doubt Einstein is smiling in his grave. Anyway, as promised, here is the derivation of equation 58:

Update: What about galaxies beyond the cosmological horizon? Aren't they hurling away from us faster than light? Perhaps, but our galaxy's photons can't reach them and their's can't reach us, but said photons can reach galaxies that are closer. Locally, each galaxy is moving well under light speed. The relative velocity (v) of a galaxy is determined by Hubble's constant (H) and distance (D). Where v = HD. If HD < c, a system's photons can come, go, or stay. Within any galaxy, HD is less than c.

Update: Below is further proof that c is the top speed in our universe:

The Probability of Backward Time, Forward Time and No Time

Is backward time possible? Yes it is, but what is the likelihood? What is the probability of going back in time? Imagine you have a three-sided coin. The sides are labeled -1, 0, and +1. Suppose we define time as follows: when the coin goes back to its previous state, it has gone back in time. When it goes to a new state, it goes forward in time. If it stays the same, there is no time.

Side 0 is the coin's current state, i.e., the present. Side -1 is the previous state--the past. Side +1 is the future. Below we calculate the probabilities:

At equations 2 through 4 we see the probability of going to the past is the same as going to the future, and staying in the present is just as likely. The coin above could be analogous to a simple quantum system; perhaps a single particle. Where the number of particles and states are few, backward time and "no time" are highly probable.

Now, let's make our simple system above more complex. Let's place the coin in a lattice with four cells. We decide that a change in state includes a change in position. If the coin moves to a new cell, it has moved forward in time. To move back in time, it must go back to its previous state which includes -1 and the cell it previously occupied. To have "no time" means no changes at all. Below is the relevant math. At equation 7 we normalize the total number of possibilities so the total probability is 1.

Here are the probabilities for our more complex system:

At equation 10, notice how the probability of forward time has increased to 10/12. At equations 8 and 9 we can see that backward time and "no time" have lost some ground--they are now less probable. Their likelihood decreases as we add more and more particles, states, and cells, and, the likelihood of forward time increases. Below are some general equations that determine the probability of past, present, and future.

But take note of equation 14. There's a question mark. Our model of time is incomplete. So far, we have only included what happens when the coin is tossed, i.e., when there is an interaction between, say, you and the coin. Assuming you toss the coin at a steady rate, there are three basic states: you toss the coin and get -1, you toss the coin and get +1, you toss it and get 0. But what happens if you don't toss it, i.e., if there isn't any interaction? Nothing changes and time stands still.

Thus there are two ways time can be zero: no interactions or an interaction where you get back the same state. To get the true probabilities of time, we need to take relativity into account. (The variables we will be using are defined below.) We know that time can slow down at high velocities and where there are large masses. The slowing of time implies that there are more instances where time is "no time" and fewer instances where time is moving forward or backward.

Equations 15 through 19 give us the relationships between mass, energy, velocity and time:

Equations 15 and 16 show how increased linear velocity (v) reduces the time rate (t') and increases the mass (m'). It follows that there is a correlation between reduced time rate and increased mass. It is understood that increased mass reduces the time rate, but how? Especially if it is at rest. Oscillators are the key. Equations 17 through 19, which involve Hooke's law and Einstein's mass-energy equivalence, show how oscillators increase mass.

If all variables, except angular velocity, are held constant at equation 19, It becomes apparent that increased angular velocity increases mass, or, mass is a function of angular velocity. Looking carefully at equations 15 through 19, it follows that increased linear velocity (v) causes increased angular velocity. In the case of both mass and linear velocity, there is increased angular velocity or oscillations. Could increased oscillations cause slower time? If so, that would explain why both linear velocity and mass slow time. Let's see if we can prove it:

Equation 34 clinches it! Increased oscillations cause a reduced rate of time. Anytime we add energy to a system, the oscillators oscillate more. Why does this reduce the time rate? Take a look at the left side of equation 34. It has the variable "u"--the 'relative' emission and absorption rate of gauge bosons within a system of harmonic oscillators. Bosons move no faster than light speed. They can't speed up when fermions speed up. When fermions are at rest, gauge bosons are relatively faster and carry force faster than when fermions are in motion (oscillating), so time is faster when fermions are at rest and slower when fermions are in motion.

Using our coin-toss analogy, if you can't increase your speed and have to chase the coin and catch it before you can change its state, you can change its state more often if the coin is at rest than if it is moving at high speed. So time, as we defined it earlier, has more instances of "no time" if the coin is hard to catch (i.e. time is slower). Also, at 34, notice the plus and minus sign in front of the radical. The square root can be negative as well as positive. This allows for backward time. The only question that remains is, "What are the odds?"

If slower time increases the instances of "no time" due to no interaction, we must reduce the probabilities of the other three possibilities: forward time, backward time, "no time" with interactions. To do this we use the Lorentz factor from equation 34.

Equations 36 through 38 show that forward time still dominates within complex systems due to more degrees of freedom. To get the probability of "no time" due to no interactions, we subtract the above probabilities from the total of 1:

This probability is zero when matter is at rest, and it grows when matter is in motion. We now have a complete probability distribution for backward time, forward time, and no time.

Update: What about the oscillations? Couldn't they count as changes of state? Sure, why not? But the net value of time would still be "no time." Take for example a pendulum. We decide if it swings right, time is going forward, but when it swings left, time goes backward and washes out the forward time, so time makes no progress until something more happens than mere oscillations.

Caveat: One could ask what is the duration of a no-time state or any time state for that matter. How long must each state last in order to count as a state? The obvious answer is more than zero arbitrary time units. So do changing states really make up time or does time make up changing states? Perhaps we've gone as far down the philosophical rabbit hole as we can go.

Would Chance Exist If We Knew All There is to Know?

This post is a continuation of the previous post "What We Didn't Know About Time Travel Could Surprise Us."

It is commonly believed that if we went back in time, history would repeat itself exactly as before if we started with the same initial conditions. It is also commonly believed that chance is a function of our ignorance. If we knew everything that goes into a coin toss, we could predict the outcome 100% of the time. Today we put these beliefs to a rigorous test.

For illustrative purposes, and to keep our first test simple, let's cut our 3D universe down to 2D. Imagine a pencil balanced on its tip. It can fall either right or left. A hand grabs the eraser end and tilts the pencil to the right at some unknown angle that's less than 90 degrees. Which way will it fall? Notice we are not completely informed. We don't know the exact angle of the tilt. Any estimate we make can be way off; yet, we can predict the outcome with 100% accuracy.

The pencil will always fall to the right. This demonstration shows that our ability to make 100% accurate predictions is not dependent on perfect knowledge. There are many things we don't know, like the tilt's angle, but our ignorance does not deter us. More importantly, it does not create a chance environment. Thus the so-called correlation between ignorance and chance breaks down here.

Since it is possible to be ignorant and make 100% accurate predictions, is it also possible to know all there is to know and be uncertain about an outcome?

Imagine a simple universe that contains a particle called B. As time (t) passes, B can move either right or left. Which way will it move? We don't know--so B's movements seem random to us. If we knew what causes B to move right or left, we could predict B's moves and the randomness would disappear, right?

We get all the scientists together and give them a big fat grant. Their mission (if they choose to except it) is to find the initial condition(s) that cause B to move as it does. They discover particle A. When particle A moves right, B moves right. When particle A moves left, B moves left. Perfect! We simply watch A to see how it moves, then we can predict how B moves.

The randomness is gone ... or is it? What causes A to move right or left? Well, nothing. "A" is the "initial condition." Nothing causes it. Nothing precedes it--or it would not be initial. So when A moves, it is completely random.

The above example demonstrates that you can have complete knowledge; i.e., know the initial condition(s) and still have randomness. But suppose the initial condition A moved left the first time history unfolded. We go back in time and A moves left again. Will history unfold the same way?

Let's see. If A has gone left, then B will move left. So far, so good. But which way will B move next? We can't look at A; it has done its job of initiating the universe's evolution. B moves on its own--either right or left. We don't know which direction now that A is out of the picture. So we can't predict with certainty that history will repeat itself because it doesn't have to--even when we know the initial condition with absolute certainty!

Since we no longer can predict B's movements, does that mean we are ignorant? Not if our thought-experiment universe consists only of A, B, time, space, and the rule that particles can go right or left. If that's all there is to know then we know it all--except what B will do next.

We don't know what B will do next because chance is not a function of ignorance in this case--it is a function of more than one option. If, for example, B could only move left, we could be as ignorant as a retarded cockroach and still predict B's movements with 100% precision. On the other hand, we could be geniuses with access to supercomputers and have no clue what B will do next if B can move in a gazillion number of ways (or exist in a gazillion number of varied states).

Finally, history need not repeat itself even if the initial conditions are the same as before and we have perfect knowledge of them.

What We Didn't Know About Time Travel Could Surprise Us

According to Hollywood and even some scientists, going back in time means seeing history repeat itself. We worry that if we interfere with the past, the future might unfold differently, unexpectedly and even disastrously.

Imagine Hitler winning World War II or your parents never getting together, so you are never born--all because you spit on the sidewalk in the year 1933, which set off a chain of events that changed the future.

What I described above is rather nonsensical if we take entropy into account (s=change in entropy; k=Boltzmann's constant; ln is the natural log; omega=number of ways particles can arrange themselves):

If our most common belief about time travel is correct, then entropy would have to shut down, so history could repeat itself exactly as before. But then we might ask, why should history repeat itself exactly as before (assuming we don't interfere)?

To satisfy our curiosity, what would happen if entropy remains intact during our time-travel excursion? Imagine a universe with one particle in it. It can go either right or left as it passes through time. The probability (p) is 0.5 for each move just like a coin toss:

The particle makes three moves, following a path that is one out of eight possible paths. The path it takes is outlined in red:

The red path represents its history. The entropy starts with a single possibility and evolves into eight possible paths.

Now let's find a wormhole and travel back in time to the particle's starting point and watch its history unfold again:

Oh wow! History failed to repeat itself. We didn't interfere. We were good. We watched from afar ensconced in our time-travel VW bug. Relax, it's not our fault. Entropy was just doing its job. There was only a one-in-eight chance history would repeat itself and a seven-out-of-eight chance it would not. I mentioned earlier that we didn't interfere, but we did go back in time and that in itself is a kind of interference that could possibly alter history.

OK, so maybe going through a wormhole is not the best way to go back in time. Sometimes going back in time is described as a movie going in reverse. If we flip a cosmic projector switch and make the particle go back in time, it should follow the red path back to its starting point:

Oh no! What happened? The particle didn't behave like a backward-running film. It took a different path back. And why not? It can go left or right at any node along the path. To get the equivalent of a backward-running flick, the particle's probability distribution would have to exit the theater.

Backward time travel, as we imagine it, requires an entropy of zero: one path the particle can take and not the usual eight. If we treat the particle's present as a single starting point, the change in entropy would evolve as follows:

Just for fun, let's assume the particle's probabilities of going left or right stay intact when time is reversed. Its past would most likely be different than expected. The past it remembers is only one of eight possible paths to its present time.

Now apply what we've learned to your past. Imagine the implications! You go back in time, but you see a different past than you remember. You think to yourself, "This is not my life!" Your past self looks different, like a brother or sister--because you may not be the same person or sex! A different sperm cell beat all the others to your mother's egg. Your parents still gave you the same name (assuming your sex is the same) but you're watching the life of a different person who took your place because you just had to try out that bloody time machine!

Since your past is messed up anyway, you might as well interfere with it, i.e., take advantage. You read in a history book that the racehorse, Fools-errand, won the Kentucky Derby, so you decide to place a bet, but don't place that bet--it's a fool's errand.

For more on this topic, click here or here or here.

How to Travel Back in Time

You might be aware that relativity makes it possible to travel forward in time, but going back in time is thought to be impossible due to the light-speed limitation. It has been suggested that to go back in time, we must go faster than the speed of light.

‘t = (1-(v^2/c^2))^.5 * t (1

Equation 1 assumes velocity squared (v^2) is less than or equal to the speed of light squared (c^2). The best we can do here is slow our time (‘t) by flying in our spaceship at a very fast sub-light speed. When we arrive back on earth, earth time (t) will be years ahead of our time (t’). Our grandchildren will be older than we are. The price of a stick of gum will be $1,000.98. We’ve seen the future, but now we want to go back. But not back to the present. A little before that, so we can bet on winning horses and stock investments. How do we do it? We know that gravitational acceleration (g) = Gm/r^2 (G=gravitational constant, m=mass, r=radius), and that v^2 = Gm/r. We can make the following substitution:

‘t = (1-(Gm/rc^2))^.5 * t (2

Equation 2 shows that mass (m) can also slow time (‘t) and take us into the future. But as far as we know, mass (m) has no upper limit like velocity. that means the mass equivalent of velocity (Gm/r) can exceed light speed. We can now travel back in time. All we have to do is fly our spaceship near a massive object that exceeds C^2. You might ask, “If that really works, then where are all the visitors from the future?” Good question! There are a few theories:

1. The visitors from the future are the UFOs. They keep a low profile and avoid contact for fear they could seriously alter the future. If you visit a time before your birth, your mother might fall in love with you instead of your father. You would then vanish because you would not be born. So you don’t make contact. You stay in your flying saucer and do a couple of photo-ops, but no more than that.

2. The visitors from the future disguise themselves as homeless war veterans. No one pays much attention to them, so they don’t alter the future significantly. They are watching us, though. That’s why they stand or sit at crowded street corners and watch us go by. They don’t need to take notes regarding our activities. They have a chip implanted in their brains that record everything they see. When they go back to the future, they upload the data to Facebook.

3. They flew too close to a black hole and were crushed before they could travel back in time.

4. There are no visitors from the future.

Theory number 4 is true if Gm/r has an upper limit of C^2 like V^2 does. That means a black hole’s mass per radius can never be infinite, and, if someone claims it can be, ask, “Where are the visitors from the future?”