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:

Does Gravity Really Exist?--Dark Energy's Equivalence Principle

According to the equivalence principle, when you drop a pen to the floor, the pen can be considered at rest while the floor moves to the pen. Because the pen is allegedly at rest, no force is needed to act on it and it can be any mass. Now here's the tricky part, because the pen has mass, it has a tiny bit of gravity of its own. It attracts the earth, or rather, the earth is at rest and the pen moves to the earth, so the earth can be any mass and no force needs to act on it. So one has to wonder: Is the earth moving to the pen or vice versa? How can they both be at rest and be moving? Is gravity really a force? It has been argued that it is not. As we shall see, one could also argue that gravity does not exist, especially when one takes a closer look at dark energy.

Galaxies are accelerating apart as the universe expands. Allegedly they are not moving through space, but rather, they are moving with space--and because they are moving with space, they can be any mass. No force is needed to push them apart? Kind of reminds you of gravity and the equivalence principle, doesn't it? It seems that dark energy is a kind of negative gravity, or, maybe gravity is positive dark energy? So many questions! But these questions will be addressed in this post. First, we define the variables we need:

Both gravity and dark energy have something in common: spacetime. To understand why galaxies move apart and why pens fall to the floor, we need to examine spacetime. What is it exactly? On the surface, it's a vacuum with dimensions of space and time. If we were put in charge of building a universe, how would we go about creating this thing called spacetime?

We could start by taking all forces (strong, weak, Higgs, EM, and unkown), add them together to get an energy field (see equation 1 below). We use that energy to derive equation 5: the Planck length.

At equation 4, notice how the energy field cancels itself. It could be any magnitude and any type of energy or combination of energies. It doesn't matter. In any case we get the Planck length unit. We can use this unit to create an arbitrary length (r) at equation 6.

Normally when we think of dimensions of space we think of fixed Cartesian coordinates (x,y,z), but our universe is not static; it is expanding--so how fast does our coordinate system move? We derive the answer to this question below (see equation 12).

According to equation 12, our universe's coordinate system is expanding at light speed, but this seems to contradict equation 13 below, which shows the expansion velocity (vd) depends on the distance r. Then again, the following equations show the velocity depends on which clock you use. If you use Hubble's constant, you get a velocity of Hr. If you use time t, you get light speed.

At equation 15, note that velocity Hct can go to infinity. This does not violate the light-speed limit if we assume galaxies are moving with space instead of through space. Moving with space means they are at rest. If they are at rest then no force or pressure is moving them. The math below demonstrates that the galaxies move the way they do regardless of how much or how little pressure vacuum energy has. The implication is "dark energy" isn't energy or vacuum pressure.

At equations 20 and 21 we see that mass (m) cancels itself no matter how big or small it is. Whether spacetime has a little or a lot of energy (mass), it moves at the rate of Hr or light speed (c), depending on the clock used.

To prove that dark energy and gravity are the same phenomenon we set up a series of Minkowski diagrams. The first one below breaks velocity Hr into a time component (Hrt) and a space component (Hrs).

At 23 we integrate over all the varying velocities at each point in spacetime to get the Pythagorean relation between Hr and it's components.

In the next diagram we express the distances (r's) in terms of spacetime (ct) and make some substitutions:

In the next diagram we measure velocity using time (1/t) instead of Hubble's constant:

Time t cancels itself and that brings us to the familiar Minkowski diagram that can be used to calculate gravitational velocity (vo):

At 28 we once again integrate over all velocities at all points in spacetime and get the Pythagorean relation between time velocity (u), space velocity (vo) and light speed (c).

We extract dark-energy equation 29 below from equation 23. From 29 we derive Newton's gravity (see equation 32).

Taking a few more steps and making a few more substitutions we derive Einstein's gravity (see equation 39):

We can also take equation 31 (restated at 40 below) and derive the Lorentz factor (equation 45):

Bottom line? We can start with dark energy and derive gravity! Or vice versa!

Now, the following 1D diagrams show how the equivalence principle works in terms of dark-energy. We use time t rather than Hubble's constant. That means all masses are coupled with c^2--hence the famous equation: E = mc^2. However, our first diagram is a universe with pure spacetime, no masses, so we just have c^2 in all directions:

Notice points A and B. They are static, i.e., going nowhere. At equation 46 we see why. The c^2's are equal and opposite and sum to zero.

Suppose we add a mass at point B. Doing so causes time to slow at that point. As a result, velocity c is reduced to velocity u. Equation 47 confirms this causes point A to move to point B.

Let's add a mass (m') at point A. Time is reduced, reducing c to w. We can see from the equations and diagram below that point B moves to point A and vice versa. The masses, no matter how large or small, located at those points move at the same rates as those points. When we add masses m' and m to points A and B, c^2 becomes m'c^2 and mc^2 respectively.

Now, it is commonly believed that gravity and dark energy are two opposing forces. One pulls everything together and the other pushes them apart. The following diagrams show this is not the case. The universe will expand at the rate of c or Hr no matter how much mass there is. For comparison, let's start with a universe with no mass and measure its expansion rate:

In the above diagram we interpret one direction as positive and the opposite direction as negative, so the universe expands at +/- c or +/- Hr (taking the square root of equation 54). (If we used a 3D diagram, each velocity would have an equal and opposite velocity; however, we only need 1D to illustrate the point.)

In the next diagram we introduce a mass and this appears to reduce the overall expansion rate, but the following math suggests this is relative, it depends on where the observer is located. An observer some distance from the mass will sum up a total value of c^2. According to the above Minkowski diagram, if space expands less, time expands more or vice versa. The total expansion rate is always Hr.

Note that equation 57 agrees with the corresponding Minkowski diagram.

The final diagram shows how the thing we call "dark energy" causes the thing we call "gravity" when matter is present. The red arrows are pens falling on opposite sides (pens moving with spacetime). the blue and red curves are satellites or photons moving along geodesic paths. Objects not moving through space move with space at the same rate, since they are technically at rest (mc^2).

Now you might wonder why mc^2 is rest mass energy and not light-speed energy. Well, you might think you are at rest, but to an observer a cosmological-horizon distance away, you are moving away at light speed! In fact, every point in the universe is moving at light speed relative to such an observer. So "at rest" really means you are moving with space and not through it. Oddly enough, if time is measured in units of r/c instead of Hubble's constant's reciprocal (1/H), points near and far (and their respective masses) are always moving at the speed of light! So rest-mass energy truly is E = mc^2.

Below we derive the dark-energy Lagrangian and equations of motion:

At 64 we recognize that the Ricci tensor and scalar have the same units as quantum wave numbers (k). As we shall see, it doesn't seem to matter what kind of wave numbers we use.

At 72 we have the Lagrangian (L). At 73 we have the speed of mass-less particles in a gravitational field (light speed c or Hr at the cosmological horizon). At 74 we have the speed of mass particles in a gravitational field--or is it a dark-energy field?

Final thoughts: Assuming objects can move with space and need no force to act on them, does this imply that no force-carrying particle is needed? The above math and diagrams show that there is no attractive force, per se. Gravity appears to be a by-product of expanding spacetime in different reference frames. Could this be the reason the "graviton" has eluded particle physicists?

How to Build a Dark Matter Galaxy and a Baryonic Matter Galaxy

According to the above video, an ultra-diffuse galaxy (far fewer stars than ours) contains little or no dark matter. Light passes through it in a seemingly straight line (little or no gravitational lensing). Then there's the dark matter galaxy, with far fewer stars than our galaxy. Most galaxies appear to have approximately five to ten times more dark matter than baryonic matter.

Using Einstein's field equations, we shall build these different types of galaxies and derive an equation that models each of them. However, to accomplish this task, we make the following assumptions: A dark matter particle is an excitation of a field, possibly dark energy or the vacuum. Like the vacuum of space, dark matter particles are diffuse. They have no electromagnetic force. They don't interact significantly with ordinary-matter particles or with themselves. There is, however, gravitational attraction. Finally, dark matter has something similar to the Pauli exclusion principle which prevents DM particles from gravitationally collapsing into a single point in space.

Because dark matter is diffuse, its gravity is insignificant near, say, a black hole, but becomes more significant as we increase the distance (r). Below is a crude illustration of a dark-matter galaxy. It has a black hole in the center. Inside the smaller circle, there's the big black dot (black hole) surrounded by only a few small dots (dark matter). By contrast, inside the large circle the small dots dominate (make up most of the mass).

Now that we've defined "dark matter," let's define the variables we will need:

We are ready to begin working with Einstein's field equations:

Let's change the cosmological constant term and make a substitution:

We make the substitution at equation 3. At equation 4 we put the term on the other side of the equals sign:

To make the math less clunky, let's change the terms into some simple variables:

After making more substitutions we get equation 12 which has a surprising implication!

Spacetime can be flat no matter how much mass (energy density, pressure, etc.) you have! Variables C and D are the energy-stress or mass terms. As long as they are equal, spacetime is flat--which implies no gravity! How can this be?

If there is no gravity between C and D, they will not move (or they will be pushed apart by dark energy). Imagine a red bowling ball and a blue one sitting apart from each other on a trampoline. Imagine the trampoline is expanding. It might look something like this:

Each bowling ball makes a dent in the trampoline, but they don't roll to each other. But around the region of each dent, there are some marbles that roll toward each bowling ball. The analogy demonstrates the following: Gravity is local and not global. If two or more masses have equal energy densities, there is no attraction. Where they are not equal, there is attraction.

The next diagram shows why galaxies move away from each other rather than towards each other:

The above diagram is a toy universe consisting of four multi-colored galaxies. They have equal masses and are the same distance apart. The arrows assume there is gravity between them, but the red galaxy is equally attracted to the green and blue galaxies. The other galaxies are in a similar predicament. The gravity of one galaxy is cancelled by the gravity of another galaxy. So there's no movement. This universe is static. Even though there is plenty of mass, spacetime may as well be flat and gravity may as well be zero. Add some dark energy and this universe will expand indefinitely.

So how do we get some gravity going in our toy universe? We break the symmetry. One way is to alter the distances between the galaxies (see first diagram below). Another way is to vary the masses (see second diagram below).

Now, you may be thinking, this is all fine and good, but what does this have to do with dark matter? Well, to understand dark matter, we have to have a better understanding of how gravity works and how it doesn't work. The main point you want to hang on to is equal masses that are equidistant aren't attracted to each other or that attraction is cancelled. Thus galaxies stay apart and move apart except when the symmetry is broken:

Now, what is true for massive galaxies must also be true for smaller stuff like dark matter/energy particles:

So all we have to do to make a dark matter galaxy or any galaxy containing dark matter is break the symmetry by introducing, say, a huge mass:

Let's assume the big mass is baryonic matter. Why baryonic matter? Baryonic matter is electromagnetic. It can form into molecules, planets and stars. It can come in large clumps. Whereas dark matter is non-electromagnetic and does not form clumps. It remains diffuse much like dark energy or the vacuum of space. When its particles are equally spaced, they are not gravitationally attracted to each other, so they need a catalyst, a symmetry breaker. Baryonic matter comes in different-sized clumps and is a perfect symmetry breaker.

In our toy universe we can imagine little dark matter particles falling into the spacetime dent created by the huge clump of baryonic matter. The dent, of course, gets deeper, allowing even more DM particles to enter:

So when does this process end? It looks like it could go on indefinitely! Dark matter pours into the dent, the dent gets deeper, more particles pour in, and so on. There must be an equilibrium where the mass of the dark matter is around five to ten times greater than the baryonic matter. One way to create such an equilibrium is to use the inverse-square law and Hooke's law. Below we use these laws to limit how deep the spacetime dent gets. We treat it like an elastic surface that springs back into place when mass is removed.

At equation 21 above, the two main variables are M (total galactic mass) and x (how deep is the dent). As dark matter is added, M and x grow proportionately (see 19), but as x adds to the dent's magnitude (equation 21's second term), it has an inverse-square effect (equation 21's first term) and the gravity is reduced until equation 21 equals zero--equilibrium is achieved (see 23 below). Using a pinch of algebra, we calculate the final mass of the galaxy (see equation 28).

We see the total mass is the dark-matter constant (we set to 10) plus 1 times the original mass (Mo) we used to break the symmetry. If we assume that baryonic matter is a catalyst for the build up of dark matter within a galaxy, then we can predict that galaxies with few stars, i.e., a small amount of baryonic matter will have little or no dark matter. And we can predict that galaxies that are nearly exclusively dark matter, have or have something equivalent to a massive black hole at their center. Finally, we predict that normal galaxies with lots of dark matter also have far more stars than galaxies lacking dark matter. These predictions lead to the final equation:

Where the radius (r) is small, equation 30 reduces to Newton's equation, but when r is large, dark matter rules!

Update and caveat: Click here to read a new article that shows there are galaxies that have no dark matter.