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.

How Entropy, Temperature and Chemical Reactions Impact Time Dilation

How do we measure time? Normally we take some event that happens over and over again, and, we assign it a time unit. For example, the complete rotation of the earth we call a day. We can take that time unit (or any fraction thereof) and assign it to other events--that may or may not be periodic--such as a chemical reaction, for instance. We say the chemical reaction happened in time t.

What if that chemical reaction were to slow down? Can we say its rate of time has slowed? If we define time as the rate of change, then the change in the rate of chemical reactions, entropy, or the earth's rotation would indeed impact the rate of time.

Consider the famous twin paradox, where one twin boards a rocket that hurls him into outer space close to light speed. According to Einstein, his time slows, he ages more slowly than his twin back on earth. His body's biochemical reactions are slower, entropy is reduced--at least that is the implication.

When he arrives back on earth, he will be younger than his twin brother. But his twin has a plan: while he's out in space, his twin cryogenically freezes himself. His twin will be thawed out by the time he gets back to earth. If his twin's plan works, they will both be the same age. Now, did the twin on earth reduce his time rate? Can the rate of time be reduced by means other than high velocities and mass density? This post shall address these questions, but first, let's define the variables we will use:

Let's start with entropy. If we somehow reduce the rate of entropy, will the time rate also be reduced?

If we take the Boltzmann constant (which has entropy units) and multiply it by the unit-less Lorentz factor's squared reciprocal, we derive equation 2 below:

Equation 2 shows that entropy is reduced when velocity (v) is increased. The time rate is also reduced. So far it appears we have a correlation between entropy and time. If equation 2 is valid, we should be able to use it to derive a more standard entropy equation:

Equation 6 above confirms the validity of equation 2. So we can say the twin in outer space, traveling near light speed, has reduced his entropy. Now the twin on earth wants to freeze himself, i.e., lower his body's temperature. Will this reduce his entropy? Equation 8 below confirms that it will. From 8 we derive equation 12 which shows lowering the temperature (T) reduces the time rate.

According to equation 12, the twin on earth will age more slowly. Does this imply the rate of his body's biochemical reactions will slow down?

Equation 13 is the Arrhenius equation, where k is the rate of a chemical reaction. When temperature T is reduced, so is the rate of the chemical reaction. From 13 we derive a new entropy equation at 19:

Equation 19 tells us that when the rate of a chemical reaction is reduced, so is entropy. And the reduced chemical-reaction rate reduces the rate of time:

So the age difference between the twins could be nil when the space twin arrives back on earth. If the earth twin cryogenically freezes himself, he may be younger than the space twin. Both twins have found a way to reduce their respective time rates, they have found ways to time travel into the future. Here are four ways to reduce the time rate:

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:

Why a Discrete Minimum Distance Fails

In the previous post entitled "Why Strings Don't Exist", we showed it is possible to have a length shorter than the famed Planck length. The question becomes, what is the minimum discrete distance, assuming there is such a thing? Normally, we use infinitesimal points to build geometrical objects like lines, planes, rectangles, etc. What happens if we use a line with the smallest magnitude possible that is greater than zero? Before we delve into these questions, let's define the variables:

OK, let's assume the fundamental building block is a line somewhere between zero and the Planck length. We'll call it d:

Let's try building a square with d:

So far, so good. All the distances appear to be no less than d. But what is the distance along the diagonal (or hypotenuse)?

The diagonal distance is a little bit more than d. The additional distance is marked in red. Notice this distance is not an integer multiple of d. To make this distance, we need a distance d plus a distance less than d. There can be no distance less than d, so we can't draw the above square. Here's an idea: draw a rectangle with 3X4 d-units. Thanks to Pythagoras, the diagonal will be 5 d-units:

Because all distances must be integer units of d, our geometry does not include squares, rectangles or triangles that have diagonals and sides that fail to have magnitudes that are integer multiples of d. But at least we found one rectangle that works--or maybe not:

If we draw lines from each d to every other d, we once again have distances that are not integer multiples of d. In the example above, we have a distance (c) of 3.6d. To have that distance, we need 3d plus a 0.6d. In our geometry, there's no such thing as 0.6d. Thus we can't draw the 3X4 rectangle. In fact, every rectangle and triangle we draw will have some distance that includes a fraction of d.

Perhaps we'll have better luck with circles? Check this out:

If we take distance d and shape it into a circle circumference (C), the diameter (D) will be less than d! D = d/pi (where C=d). If we try to draw any circle with d-units, we hit a brick wall. You see, you calculate the circumference with pi * D. If D is an integer (n) multiple of d, pi * n won't equal a circumference with an integer multiple of d. Pi is an irrational number.

OK, so perfect circles are out. How about imperfect circles? Perhaps we can replace pi with something more rational. Even if we do, circles have the same problem we encountered earlier:

There's always a line between two d's, that is not an integer multiple of d. This is also true with any shape imaginable:

So far, things look pretty hopeless for our geometry based on d. But unfortunately, there's more pain. Let's go back to the beginning and reexamine d:

Distance d is a line along the x-axis. But what is it along the y and z axis, i.e., what is its cross section?

Line d has a cross section of zero magnitude! That zero magnitude is just a single point in space with zero distance! Zero distance is not allowed, so a distance-d line is not allowed. Perhaps we can convert the line into a cylinder, so the cross-section will have a d-magnitude. Let's look at our new cross section:

Oops! The cross section is shaped like a circle. We can easily draw a red line from one d to the other that isn't an integer multiple of d. Thus it is now abundantly clear that a geometry based on distance d (instead of a point) is a dismal failure.

Update: Here is a formal proof that shows that, at any two non-parallel adjacent unit lengths (in this case the Planck length), it is possible to draw lines (lines a, b) shorter than the unit length. The following diagram is an example of an arbitrary shape:

Note that the shape is made up of connected unit lengths. Here are two examples of adjacent-unit lengths blown up to a convenient size for inspection:

Now let's take one arbitrary pair of connected unit lengths and label the lines and angles:

Now we are ready to do the proof. Here it is:

Update: The following proof demonstrates why a curved Planck-length string creates distances shorter than the so-called shortest distance. Below are some examples of curved strings:

The examples above clearly make the point, but is the point generally true? Here's the proof: