Thread: A new way to calculate kinematic time dilation on an atomic clock ?

I think I'm pretty close, but I could use some help with the math.

I would like to go through a thought experiment to explain what I call, 'the speed limit law of light theory' , in which all matter must obey the speed limit of light.

First, I'll leed with a macroscopic analogy. Lets say that a car is traveling, and must obey the speed limit or the police (the law) will slow him down. The car is already traveling at the speed limit and experiences a huge side wind. The side wind combined with the car's original speed, would send the car on a new path in which it would be traveling slightly faster than the speed limit. In order to obey the speed limit, the driver must slow down his car.

Secondly, microscopically now consider a electron, traveling around an atom, at the speed limit of light. Whenever the atom is shifted in any direction, the electron must slow down. Of course, the electron has no brakes, but still must obey the speed limit of light. So, if an atom were traveling at the speed of light, the electron would have to stop moving in any other direction in order to obey the speed limit of light.
But before that, at speeds before the speed of light, in the same direction of travel, an electron will be allowed to move the remainder of motion allowable ((the speed of light) minus (the velocity of the traveling atom)). Relatively, however, an electron would be allowed travel at the speed of light in the direction opposite the direction of travel of the atom.

I understand how to calculate the kinematic time dilation for bodies moving perpendicular to the direction of travel, with the application of the pythagorean theorem as seen here,
Time dilation - Wikipedia, the free encyclopedia.

I need help in coming up with an equation for describing the kinematic time dilation for a body moving in a circular motion and determining the average electron slowing that occurs as the atom travels in the x-positive direction, with a velocity between 0 and the speed of light.

I figured an easy way to start solving this problem might be to model the electron as it orbits in the xz-plane, xy-plane, and yz-planes . And then maybe find some way 'average' the results.

I've attached my sketches of the planes.
x-y plane.jpgy-z plane.jpgx-z plane.jpg

There are a couple of givens that I've figured out already. .

The electron travels the speed of light in the x-negative direction at point D in the x-y plane.

The velocity at points C and E are equal and can be found using pythagorean theorem.

For the y-z plane, the velocity is equal to C and E at all points.

For the x-z plane, the values mirror that of the x-y plane.

I'm confused about whether I can simply average the velocity of the points or not. If I do, I get (C minus (C minus the velocity of the atom)) / 2 for all planes. This way of thinking would suggest that the maximum time dilation that can occur is between 0 and 50%. II think it differs from the established 0 to 100%, which only considers the motion perpendicular to the direction of travel.

From wikipedia, at Time dilation - Wikipedia, the free encyclopedia

" Time dilation by the Lorentz factor was predicted by Joseph Larmor (1897), at least for electrons orbiting a nucleus. Thus "... individual electrons describe corresponding parts of their orbits in times shorter for the [rest] system in the ratio :" (Larmor 1897). Time dilation of magnitude corresponding to this (Lorentz) factor has been experimentally confirmed, as described below. "

I am not sure what you are trying to do, but first of all, the time dialation of an object moving in a circular orbit, as seen from an observer outside, is, I believe, simply the same as that of an object moving at the same speed in a straight line. Keeping in mind that "time dialation" refers to what the moving clock looks like to an outside observer, and not to any deviation of the angular speed from v/(2*pi*r), where v and r are the externally observed speeds and radii, which is unaffected by how the moving clock reads.

However, an orbiting electron is not a classical particle. It is a quantum object. The frequency of its emissions is not determind by its "orbital speed" but rather by the difference in frequencies of oscillation between two energy states, each corresponding to a different "orbit". The possible speeds of an orbiting electron is actually a more complex question that it first seems, because it depends upon what you mean by that. In a stable energy state, the electron has a fixed angular velocity. However, being a wave, it is not confined to a single radius. Therefore, its linear speed cannot be specified unambiguously. Interestingly, in some states its linear "speed" about the nucleus is infinite. These are the states having zero angular momentum, and therefore an infinite angular phase velocity. In unstable states, i.e., radiative transitions between one state and another, it is the group velocity, and not the phase velocity, that is of primary interest. This may be the velocity of greatest interest to you, because it is the one related to emission frequency. (But not directly; it also depends upon the number of nodes, which equals the difference in angular momenta of the 2 states). All in all, a correct description of the behavior of electrons in atoms requires a quantum analysis. I have doubts that your methods will be successful.