sine

The simple harmonic oscillator is in yellow, its velocity in blue, and acceleration in red. The scalar value for the a low velocity system is equal to one while the scalar acceleration is zero. The differences between both frequencies and amplitudes change the relative lengths of the blue and red lines, velocity and acceleration respectively. In this example, the frequency decreases while the amplitude increase going from x to y to z.

The oscillator is in yellow, its first time derivative in blue, and second time derivative in red. I understand why the velocity has a fixed scalar value equal to 1, that is what gamma is for low velocities. Acceleration in special relativity must be handled with care. The math is easy: tack the derivative of a constant, and zero will result. The implications of that math are not clear to me.

When one take the time derivative of the oscillator, . I puzzled over that obvious result for days, since its meaning was not clear. Then I recalled in relativistic physics, one takes the derivative with respect to the interval tau, so . For tiny values of velocity, gamma will equal 1. What people often graph is the velocity parameterized by time. They don't wish to treat the 4-velocity like a 4-velocity. If you keep the books consistent, you can get fun surprises.

Part of the great mystery of quantum mechanics is that all particles are identical. There is no adding a tag or painting one red while the rest are yellow. As a programmer, we can cheat, do something not allowed in Nature, and mark all those where t = 0 in red. The shift is the same as before, but now we can spot its trail.

What is so tricky in quantum mechanics is not the vectors - those we can always point at with our fingers. Instead it is the scalars that provide the challenge, the unpointables. Each scalar is connected to three vectors to make 3 complex numbers, but the scalar can be shared by other events. The scalar become the thread within a pattern of events, and between separate patterns of events. It is wonderfully ironic that the simplest core component can be so confusing by playing many roles.

The wave function of a wave equation in spacetime does not move in a bumpy wave through space. It moves along military straight lines (unless you choose different coordinates, which is perfectly valid). The movement starts with pair creation, an agreed apon parting of ways. Movement is slowest at the maximal separation. There is a rush to collide and destroy. Because the animation going in looks like the one going out, there is time reflection. Because there are always two points dancing toward or away from each other, there is a reflection in space. Much of the mystery in interference experiments of quantum mechanics centers around this symmetric spacetime function.

Each of these sets of events starts out pointing in a different direction. Yet the x, y, and z values of the input is never altered. This is why you can spot the spatial origin, the point in the center of all the moving points. It would be simple enough to shift these arbitrary oscillators around precisely the origin to arbitrary oscillators around arbitrary points - just add in an arbitrary value as a last step.

In classical physics, the amount of change in time measured in the same units as space vastly exceeds changes in space. In other words, relativistic velocities are low. In these animations, time changes by 100 while changes in space are only 20.

The linear input is in yellow, the odd sine function is in red, the even cosine in blue. The cosine has only one apparent line diving into the origin because that line is a doublet. I thought I knew what cosines and sines looked like, but it feels like there is a frightening large amount of unknown diversity possible for these formerly familiar friends. The cause of the oddness maybe due to varying t, x, y, and z at the same time.

Sines and cosines have to do with circles. By fixing x, y, and z, the circle stays fixed. What direction the line in space points to is arbitrary.

The line in yellow is the input for the The length of the line in space is the amplitude.