A thought experiment of the relativity of light

[PeroK]

Ahhhh…. That would be #2 of this thread.
But it would be a good exercise to actually calculate the position of the spreading flash of light twice, using the two coordinates systems in which each of two observers moving relative to one another are at rest.

I suspect calculations are anathema to the OP.

 

[vanhees71]

Although there is nothing wrong here, there are easier ways of approaching the math and OP may want to start with those.

If you have a more simple model for a "light source", let's see it. Perhaps it's my limited imagination that I couldn't find a simpler one ;-)).

 

[vanhees71]

I suspect calculations are anathema to the OP.

Then, s/he has no chance to ever talk about physics in such a way that s/he can understand it.

 

[Vanadium 50]

Well, B-level probably does preclude a D'Alembartian (and there should be a squared there! Just like the Oxford Comma!")

But the general point is valid - one cannot prove a mathematical inconsistency withoutm you know, mathematics,

 

[vanhees71]

Where should be a "squared"?

If you are not allowed to use the wave equation to discuss waves, then the question cannot be answered adequately at any level.

 

[Vanadium 50]

\Box^2 over \Box

 

[vanhees71]

That's a very strange notation. Does anybody use this?

 

[Vanadium 50]

Everybody.

It emphasizes that it;s a second derivative.,

 

[Dale]

Please explain how two light spheres from two emission events (0,0,0,0) and (0,1, 1, 1) move away from each other such that at a time > 0 the centers of these spheres, are no longer at (0,0,0,0) and (0,1, 1, 1).

So, instead of talking about light spheres, for a moment let's talk about light cones. In this case in some given (unprimed) inertial frame with coordinates $(t,x,y,z)$ we have two light cones: $$\mathrm{A}=(t,x,y,z) : \ t^2=x^2+y^2+z^2$$$$\mathrm{B}=(t,x,y,z) : \ t^2= (x-1)^2+(y-1)^2+(z-1)^2$$ So here we have two light cones, each of which is a full 4D object. These 4D objects represent right circular cones with the axis along the $t$ direction and with the apex at $(0,0,0,0)$ and $(0,1,1,1)$ respectively.

Now for any $t=t_1>0$ we can take the full 4D objects $\mathrm{A}$ and $\mathrm{B}$ we can obtain the 3D objects that are 3D "slices" of the light cones $\mathrm{A}$ and $\mathrm{B}$ at the specific time $t_1$. These 3D objects are $$\left. \mathrm{A} \right|_{t=t_1} = \mathrm{A}(t_1) = (x,y,z) : \ t_1^2=x^2+y^2+z^2$$$$\left. \mathrm{B} \right|_{t=t_1} = \mathrm{B}(t_1) = (x,y,z) : \ t_1^2=(x-1)^2+(y-1)^2+(z-1)^2$$ These are 3D objects at the time $t=t_1$, and each represents a sphere of radius $t_1$ with the center at $(0,0,0)$ and $(1,1,1)$ respectively.

Now, with that clarification and notation. Please write what you mean about light spheres moving away from each other.

 

[PeroK]

My guess is that the confusion stems from implicity using the Galilean transformation with universal simultaneity and intuitively noting that this is incompatible with an invariant speed. And, since the Lorentz transformation is "mathematics", it's not an intuitive alternative.

 

[CClyde]

If you wanted to go and prove something mathematically inconsistent, the way you are going about it is not the way to do tt. You would provide two calculations of the same thing and show they get different results: i.e. you would use 10x as many numbers as you are using and one-tenth as many words.

You want "if I calculate x thsi way [numbers numbers numbers[ I get 7 and when i calculate x thsi oter way [numbers numbers numbers[ I get 11," You don't want increasingly complex scenarios with lots of words,

But as mentioned, this will be futile. as it is known that SR is internally consistent.

I am not claiming a mathematical inconsistency in SR. That would be a futile ambition for the most competent mathematician. And in case it is not yet obvious, mathematics is not my day job.

Once you accept the principle of equivalence of rest and uniform motion extends beyond inertial bodies to electrodynamics, the rest is just a matter of making the math as pretty as possible.

This is why I stated earlier that the principles need to be understood before the math can make it a concise, axiomatic story.

 

[CClyde]

Relativity is self-consistent. If any chain of reasoning leads you to believe otherwise then that chain of reasoning is incorrect.

You may be right, relativity may be self-consistent.
But the jury is still out, not on the math, on the principles.
Time is dilated in frames in motion and all motion is a relative measure.
As far as I know, as of today there is no solution to this internal inconsistency.

There are solutions to the anthropomorphic analogy, but they do not address the conflict. They require the a priori knowledge of frame change described in the analogy. They do not resolve which of two frames in uniform motion experience time dilation.

 

[vanhees71]

But math is the language needed to formulate the principles to begin with.

 

[vanhees71]

You may be right, relativity may be self-consistent.
But the jury is still out, not on the math, on the principles.
Time is dilated in frames in motion and all motion is a relative measure.
As far as I know, as of today there is no solution to this internal inconsistency.

There is no internal inconsistency whatsoever. What do you think is inconsistent? Of course, you cannot explain this without math.

There are solutions to the anthropomorphic analogy, but they do not address the conflict. They require the a priori knowledge of frame change described in the analogy. They do not resolve which of two frames in uniform motion experience time dilation.

???

 

[CClyde]

But math is the language needed to formulate the principles to begin with.

No, math is a very concise and sometimes beautiful language, but it is a language.You can, as did Dirac, find previously unknown, even unbelievable, physically real entities drop out of mathematical statements, but that does not mean all mathematics describe something physically real. Nor does it mean only mathematics can.

 

[Sagittarius A-Star]

My goal is to find out if I have misconceptions, or if I have discovered an internal inconsistency in the principle of relativity.

You have the following misconception.
You wrongly think, that the center of the expanding light-sphere must move together with the object "light-source":

All coordinates move relative to an appropriately chosen frame of reference.

The correct conception would be the following.

If an object's spatial coordinate is translated i.e. according to
$x' = 0 \Rightarrow x = vt$
then $x$ is a function of time. The object is moving in the unprimed frame.

If an event's spatial coordinate is translated i.e. according to
${x'}_0 = 0 \Rightarrow x = vt_0$
then $x = {x}_0$ is constant.

 

[Dale]

I am not claiming a mathematical inconsistency in SR.

Good, then we can close this thread.

If you wish to open up a separate thread on any inconsistency with experimental results you may do so. Since SR is internally self consistent the only possible inconsistency remaining is with experimental results. If you wish to disprove SR, it can only be done experimentally.

But the jury is still out, not on the math, on the principles.

You have a misunderstanding. The principles are the math. Since there are no inconsistencies in the mathematics of SR and since the principles are the mathematics there is therefore no inconsistency in the principles either.

The jury is in, the verdict is rendered.