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: