Dear Edwin,
"There is no problem with the math of the Lorentz transformation;
the problem is in the ontology, i.e., the nature of physical reality."
I have written that there is a mathematical problem with the derivation of the Lorentz transformations. I can take correction. Here is why I have said the above: I wrote:
The Improper 'Derivation' of the Lorentz Transforms by Einstein
Excerpt:
In order to make the clearest case for how the mathematics is carried out, I will use the common simplified example of the light source and the two observers all being located at the same point before any activity occurs. At the start of the activity two things happen. One observer leaves the initial point with a relative velocity with respect to that original point called the origin and moving horizontally to its right.. At the same instant the the moving observer begins to move, a light pulse is sent out from the origin in all directions and equal distance.
There are a few more assumptions included in the derivation. It is assumed that these new transforms will not apply to length in the orthogonal directions. It is also assumed that time is not to be treated as unidirectional, which leaves it being omnidirectional. In other words, Equations for the vertical directions, if they include 't', for example velocity, may need transformation. I mention this because the derivation will be made for the horizontal direction to the right as if it is one dimensional. However, the handling of 't' gives it a simultaneous three dimensional treatment.
There is an origin where both observers and the light source wait for the action to begin. At the same time One observer leaves the origin with a velocity v and moves horizontally to the right. Simultaneously, a bubble of light is released from the origin and moves away from the origin at the speed of light.
Part of the wave front travels along the same path that the moving observer is on. It speeds on ahead of him in the same horizontal direction to the right. Both observers see that same wave front. Both observers measure the speed of that wave front with respect to themselves. They both measure the speed of it as C.
For the stationary observer, the speed of the light presents no surprise. However, for the moving observer, it is expected that the speed of light will measure less than C by the velocity of the observer. That does not happen. Regardless of the magnitude of v, the moving observer measures the speed of light moving ahead of him as traveling at the speed C.
There is a mathematical step coming up that I haven't seen in a long time, but, it bothered me as being mathematically unwarranted from the first time I saw it done. In the early part of the derivation of the Lorentz transforms, the observers write their equations for their measurement of the speed of light. The distance between themselves and the leading edge of the light is simply the application of the Pythagorean theorem. That measurement is divided by time t1 for the stationary observer and t2 for the moving observer.
This is not the problem that I mentioned and will be getting to next. However, this part assumes that time may be different for each observer. This assumption is leading the solution toward time-dilation. We don't have that solution yet, but by providing for its possibility in the mathematics makes the mathematics 'look' for it. In other words, whether it is correct or not, if there is a mathematical way of expressing it using what is in the setup of the equations, it can be expected to appear as part of the result.
Now for the problem I have been leading up to. There are two equations for the measured speed of light from the perspective of each observer. The expressions consist of the distance written in the form of the Pythagorean theorem divided by the time. Time t1 for the fixed observer, and, time t2 for the moving observer. They both measure the speed of light as C, so the distance divided by the time is set equal to C. Nothing wrong there yet. Measured velocity equals C. This can be written as Measured velocity -C equals zero. We have measured velocity 1 minus C equals zero. We also have measured velocity 2 minus C equals zero.
The problem is the very next step. We have two equations set equal to zero. Since zero equals zero, the two equations are set equal to each other. The problem has appeared. Any equation can be written with everything on the left side and zero on the right side of the equals sign. Any two equations can be set equal to one another in this manner. There is nothing about this practice that seems justified. I can write any number of nonsensical equations by writing them so that they are equal to zero and then setting them equal to any other equation prepared in the same way.
I mentioned earlier about putting the assumption that there could be two different rates of clock time into the equations gives direction to the mathematics that if it is possible for a solution to include two rates of time, to find that solution. Here in the problem just explained. setting two equations equal to one another just because they are written as both being equal to zero, appears to again give direction to the mathematics to find, if possible, an expected or anticipated solution. I see the mathematics being manipulated to head toward a particular kind of solution.
It could be that the solution will be the one that does best fit with physical reality. I don't see it that way because one of the solutions is that there are different rates of time simultaneously. I find no direct empirical evidence to support the solution called time-dilation. It may be that the speed of light varies, which seems far more physically reasonable to me. Many relativity type of effects can be accounted for by a remote perspective of a varying the speed of light instead a varying speed of time for each observer. There is a challenge in trying a new method of solution. This mainstream simple example used to achieve the Lorentz transforms is appears to physicists as a reasonable problem to solve.
It needs to be shown that an undirected solution, as opposed to the mainstream solution which gives some appearance of having been mathematically directed toward a solution, doesn't present us with a varying speed of light, then there is a conflict with the idea that the speed of light varies.
A derivation of the Lorentz transforms is at this link:
https://en.wikipedia.org/wiki/Lorentz_transformation
The equation that I spoke about where two separate equations were set to zero and then set equal to each other is the first equation on the equation line D3.
Getting back to the simple problem that is being solved. Going any further with the mainstream mathematics is not really worth it. There are several mathematical steps that need to be done. And, there is a lot of explaining the needs to be done. However, the Lorentz transforms are the solutions to that problem and, physicists have been using them for a hundred years, and like them. So the list of the Lorentz transforms is really all that most users need to know. Here is an introductory video about Special Relativity, using easy math:
https://www.youtube.com/watch?v=AjJwnsc4D40
With regard to maybe there being an alternative solution to the mainstream example problem; one can just use normal length and normal time keeping and the solution is that they will measure the speed of light to be different. Maybe the speed of light does vary, but it doesn't vary in that way. It is the case that all observers will measure the speed of light to be C when measured locally. Measured locally means that the equipment used to measure the speed of light experiences the same environment conditions as does the light who's speed is being measured. So, it looks like I need to consider a possible third solution.
I am looking to generate a better example problem that, after setting up the initial conditions, the mathematics will provide its own non-directed solution. I will include a few more paragraphs for the purpose of showing you what my thoughts are.
Here is a different problem: There are again two observers. They are in different environments. For observer #1 there is a beam of light coming toward him and beginning to pass him. Observer two is experiencing the same problem but is not located with observer #1. The environmental conditions are such that the light passing by observer #! is moving faster than the light beam that is passing by observer #2.
Each observer sees the beam passing by them and the beam passing by the other observer. The beams are traveling horizontally in the same direction. Observer #1 is located high above observer #2. Observer #1 measure the speed of the light passing by him and finds that it is traveling at the speed of C. Observer #2 does the same and finds that the speed of the light passing by him is C.
Observer #1 looks at the light passing by observer #2 and finds that its speed is substantially less than C. Observer #2 looks at the light .passing by observer #1 and finds that its speed is greater than C. Each observer used their own meter stick and clock located next to them. They don't see anything out of the ordinary when they measure the speed of the light passing by themselves. However, they measure each other's speed of light very differently. This finding does tell us something important. Their meter sticks for sure are not the same length. The difference in the lengths of their meter sticks should be enough to account for the differences in measurements of their own passing light and that of the other observer.
However, there is something about their two environments which changed the length of their meter sticks. Does it also affect the speed of light or perhaps the rates that their clocks tick at? The meter sticks don't answer that question. They may vary linearly or maybe not. What we do know is that the Lorentz transforms have been working and they are not linear. This example problem may have different solutions from the Lorentz transforms, but the results must be nonlinear and probably will look analogous to the Lorentz transforms.
Returning to the questionable mathematics of the 'derivation' of the Lorentz Transforms. I think that the two odd looking mathematical steps are unwarranted but they are both necessary because they are what lead to the time-dilation equation. It is a forced result. The inclusion, in the set up of the original math conditions, of a term that prepares the way for time to be transformed will not result in time dilation if the mathematics that follows is completed with normal mathematical steps.
Normal mathematical steps would not include the process of setting up the two equations where everything is moved to the left side of the equals sign resulting in each equation being set equal to zero. Then the illogical looking step of setting the two equations equal to one another simply because zero equals zero saves the derivation so that time dilation can appear in the results.
A normal mathematical step would have divided both equations by C with Ct in each denominator. Both equations would have been equal to one. Setting them equal to each other would have been a natural mathematical step. However. Ct would have been in the denominators of both equations. By moving all terms to one side of each equation and setting them equal to zero and then equal to one another, the product of speed of light C and t was not in the denominators by rather appears as a subtracted term in both numerators. This is obviously a cheat step that was necessary for the solution to include time dilation.
End of excerpt.
James
