What Einstein Didn't Tell You About Time

According to Einstein, if you are sitting in your chair, you are moving more through time and less through space, so time is moving faster for you than it is for a speeding jet. Unlike you, the jet is moving more through space and less through time. If a particle is completely at rest, it moves exclusively through time. If a particle is propagating at the speed of light, it is moving exclusively through space. Below is a Minkowski diagram illustrating the point. The vertical arrow represents the particle at rest and the horizontal line represents a photon. The diagonal arrow could be anything going less than light speed but not at rest.

Now, think about how we humans measure time. We see repeating patterns like day and night, the four seasons, the moon's phases. We then assign time units to these patterns. But imagine a universe where there are no patterns or events, where everything is at rest. How could time be measured? Who would do the measuring?

According to Einstein, everything in that universe is moving through time only, but how can we verify that? There are no clocks to track the alleged passing time. Consider his thought experiment where he has a clock consisting of a light beam oscillating in a boxcar. In the diagram below we mimic that thought experiment with an imaginary photon oscillating vertically inside a box:

The photon is moving at light speed, so special relativity says it experiences no time, but we do. Why? Because we see something happening. We see something that is not at rest. We decided to make that moving photon our clock. Ironically, in order to have time or know time exists, we need something to move through space.

Here's another irony: If all the particles in our imaginary universe were to change to a new overall state, we would say time has moved forward. If all the particles returned to a previous state, we'd say time has gone backwards. If you grow older, time is progressing; if you grow younger, time is regressing. Now, suppose you didn't age or grow younger? Suppose all particles stopped changing states? We would no doubt say time has stopped.

But how can that be? Didn't Einstein say when particles are at rest they move through time?

What gives a sense of time is ongoing change and/or repeated patterns. Time feels real when stuff is happening. When nothing happens, it is as if time has stopped. When things happen faster, it's as if time has sped up, when things slow down, time seems slower. However, if the photon box moves through space, the photon does not appear to complete the cycle as fast, so from our point of view, time has slowed:

But here's the thing: do all clocks slow down when they move faster through space? We will demonstrate the answer to this question is definitely NOT a "yes." First, let's define the variables we will be using:

Here's a new thought experiment: We take a photon clock like the one illustrated above. We have it oscillate in a larger box at velocity u. The larger box is moving at velocity v. Let's assume both velocities are well below the speed of light. We use the Pythagorean theorem to calculate the combined velocity. Below is a diagram for visual reference:

Of course the photon in the smaller box is still moving at velocity c, but this is no longer the clock we want to track time with. We decide to use the larger clock with the small box ticking off the time at a steady rate of velocity u. The question is, what happens to this new rate of time when velocity v is increased? To figure this out requires a little math. We start with equation 1 below which gives the relative mass of the entire system. From there we derive equation 9:

Equation 9 is just Einstein's full energy equation. Let's assume energy is conserved and the combined velocity of v and u is held constant. If we increase velocity v, u must decrease accordingly. So far, our new clock is consistent with relativity's prediction that time slows when velocity v is increased.

Now suppose we add energy to the system. Since both v ad u are well below light speed, we can use the additional energy to increase v without decreasing u! The system mass will increase, but time stays the same!

And what of the photon clock? It slows down as expected. What we now have are three time rates: t', t, and u. We can relate them as follows:

If we only change velocity v, velocity u can remain steady while time t' changes. However, if the combined velocity of u and v is light speed, when v increases, u must decrease.

So to keep our new clock ticking at the same rate, we must keep our combined velocity below light speed. That shouldn't be too hard.

Below we derive equations 16 and 17. These equations express u in terms of time rather than velocity:

Thus when when something moves through space it doesn't necessarily move less through time, and vice versa.

Why Einstein's Time Theory Works So Well

Experiments involving mu-mesons confirm Einstein's special relativity theory and the value of the Lorentz factor. When a particle is moving fast along a straight line, its time slows down. But is this the only circumstance where time slows? This post will address that question and more. First we define the variables we will use:

Einstein's classic thought experiment involved a speeding train passing by an observer who sees the inside of a boxcar as the train whisks by. The rectangular diagram below represents the inside of the moving boxcar. A person inside the boxcar shines a light from the floor to the ceiling (see red line in diagram). If the train is at rest, the light beam goes straight up to the ceiling. When the train moves, the beam goes up at an angle as indicated in the diagram. Using the Pythagorean theorem, we can derive the Lorentz equation (see equation 7 below):

The train and the vertical light beam can be considered a clock. We can say that one unit of time passes when the beam goes from the floor to the ceiling and vice versa. The Lorentz factor is perfect for this clock. As the train moves faster, it takes longer for the light to reach the ceiling--thus our time unit takes longer, i.e., time slows down.

Now, suppose the light beam went from side to side instead of up and down? Would the Lorentz factor still work or does time behave differently? In the diagram below, we have the beam going from right to left. From that, we derive equation 12 below. Note that equation 8 is the velocity addition formula where the total velocity never exceeds light speed.

In the next diagram, the beam is moving left to right. From that we derive equation 18:

Let's take the average of the right sides of equations 12 and 18 at 19. Doing the math we see that the result at 20 is equal to the Lorentz equation at 21. So when the beam goes horizontally, time behaves the same way as when the beam is vertical. But what if the beam goes at an angle? It would have a horizontal and vertical component. At 22 we use sine and cosine for the vertical and horizontal components.

Equation 23 shows it doesn't matter whether the light beam is horizontal, vertical or some arbitrary angle. In any case, we get the Lorentz-equation value of time (t'). It appears the Lorentz factor works splendidly in flat spacetime and where particles, trains, light all move in straight lines. What about curves?

At 24 and 25 below we start with flat spacetime. At 26 we include the metric tensor. From there we derive equation 31:

Equation 31 confirms the Lorentz factor works in curved spacetime. Now what about crazy, oddball geometries (see diagrams below)? We can still derive the Lorentz equation (see equation 41).

Equation 41 will accommodate horizontal light beams, and beams going at any arbitrary angle or curve. At 42 below we calculate the horizontal component which is equal to the Lorentz equation at 41:

To cap things off, we use basic calculus for curves and squiggly lines to derive the Lorentz equation yet again (see 50 below):

We know that the velocity along a curve varies from point to point, so why is velocity (v) in the above equations unvaried? Because velocity v is the average velocity along the whole curve distance. We take the velocity at each point, add them all together, but that would give us infinity, so we need to divide by n, the number of velocities:

OK, so we've established the Lorentz factor works in all situations where a particle, train or whatever is moving at velocity v along any line. We know that mass can slow time as well as velocity. We also know that protons have their mass due to quark oscillations. So we can think of mass as oscillating velocity. The equations below show the relationships between time, mass, angular velocity and the Lorentz factor:

Update: Below are some diagrams (and calculus) that help one visualize how a light beam (or photon) moves along a curved horizontal component. The first diagram below shows the photon (red dot) moving from left to right along epsilon-x. Epsilon-x is also moving from left to right along x. The total distance covered is x + epsilon-x. The second diagram shows the same except the photon moves right to left. The total distance is x - epsilon-x.

In the next two diagrams, everything is the same except x and epsilon-x are now warped, distorted, curved, etc.

Below is a mathematical proof showing that the magnitudes of x and epsilon-x are invariant no matter how they are distorted:

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.

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?