B A thought experiment of the relativity of light

[CClyde]

@CClyde I reread this thread and confess I am confused as to your goial.

Is is:
1. To clear up your misconceptions?
2. To convince us that re;ativity is internally inconsistent and you are the first person in more tnan a century who notic4d?

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

 

[Dale]

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

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

Several problems with your reasoning have been pointed out as have several points where your communication is unclear. So the goal has been resolved as the first one:

you have misconceptions (some already identified and some remaining to be clarified).

 

[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).

What do you mean by this?

 

[Nugatory]

If there were some reference, some coordinate you could actually prove cannot move, not just state it as you are, but prove the existence of, and by law that such coordinates cannot move, well, then you could measure your position over time relative to such coordinates

There is such a reference. If you are moving inertially (you determine this by the absence of proper acceleration - an accelerometer you are holding reads zero) you can choose your position in three-dimensional space to be at rest with coordinates (0,0,0) and your position in four-dimensional spacetime to have coordinates (t,0,0,0) where t is your wristwatch time. Once you’ve done that, you can determine the coordinates of any other point in space, and it is a simple calculation to convert these values to other coordinate systems.

However, you may be hung up on an earlier misunderstanding:

and know if it is you or the object that is moving.

There is no such distinction. When we say “THIS is moving and THAT is not” we are actually saying “We have chosen to use a coordinate system in which THIS is moving and THAT is not”. Thus we are free to consider either to be at rest, just by choosing our coordinates.

In fact, we choose coordinates for our convenience and change them so routinely that we often aren’t aware of it. When driving a car I nearly always consider the surface of the earth to be at rest - but I instantly and effortlessly switch to coordinates in which I and the car are at rest when I reach for the cup of coffee in the armrest cup holder.

 

[Frabjous]

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

Here’s a rule of thumb: If you do not do the math, you have not found an inconsistency. Words are not precise enough to perform physics with.

 

[Nugatory]

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).

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.

 

[Vanadium 50]

Great, we can work with that.

It's probably worth pointing out that if you options are "I misunderstand something" and "I discovered something major that Einstein and everybody who followed him missed" it'd a good bet to go with the former.

As @Dale points out, relativity is known to be mathematically consistent. So that's a non-starter.

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.

 

[vanhees71]

So, where do you go from here? I would claim you are new to the concept of having the same events described in two different reference frames and, in particular, cannot reconcile the invariance of $c$ with your mental picture of the motion of light. What do you do next? Accept you are wrong? Persist in your error? The choice is yours. No one can force you to understand SR.

Let's see, whether we can make the entire trouble concrete. Since it's only about kinematics, let's just consider a massless scalar field. Wo so we can simply look at the spherically symmetric solution of the wave equation,
$$\Box \Phi(x)=0, \quad \Box=1/c^2 \partial_t^2-\vec{\nabla}^2.$$
Then to have a nice model for a "light source", just consider a spherical harmonic wave, i.e.,
$$\Phi(x)=\phi(r) \exp(-\mathrm{i} \omega t), \quad r=\vec{x}.$$
The solution "outgoing" solution reads
$$\phi(r)=\frac{1}{r} \exp(\mathrm{i} k r), \quad k=\omega/c.$$
This is the solution for a light source sitting at rest at $r=0$ (the singularity of the wave).

Now it's convenient to write this in a manifestly covariant way, using the four-velocity of the light source $(u^{\mu})=(1,0,0,0)$. Then you get
$$\Phi(x)=\frac{1}{\sqrt{(u \cdot x)^2-x \cdot x}} \exp[-\mathrm{i} k (u \cdot x-\sqrt{(u \cdot x)-x \cdot x}].$$
In a frame of reference $\Sigma'$, where the light-source moves with velocity $v= c \beta$ in the positive 1-direction you have
$$u'=\gamma (1,\beta,0,0).$$
Then in this frame the wave is represented by
$$\Phi'(x')=\frac{1}{\sqrt{\gamma^2(x'-v t')^2+y^{\prime 2}+ z^{\prime 2}}} \exp[\mathrm{i} k [\gamma (ct'-\beta x')-\sqrt{\gamma^2(x'-v t')^2+y^{\prime 2}+ z^{\prime 2}}],$$
where I made use of $\gamma=1/\sqrt{1-\beta^2}$ to get
$$r=\sqrt{(u' \cdot x')-x' \cdot x'}=\sqrt{\gamma^2(x'-v t')^2+y^{\prime 2}+ z^{\prime 2}}.$$
As you see for an observer in this frame the singularity is at
$$\gamma^2 (x'-v t')^2+y^{\prime 2}+z^{\prime 2}=0 \; \Rightarrow \; x'=y'=0, \quad x'=v t'$$
i.e., the singularity moves with velocity $v$ in the positive $x'$-direction along the $x'$ axis, as constructed.

At the same time the motion of the surfaces of constant phase $0$ is given by
$$\gamma (ct'-\beta x')-\sqrt{\gamma^2(x'-v t')^2+y^{\prime 2}+ z^{\prime 2}}=0,$$
and indeed after some algebra you find that it's given by
$$c^2 t^{\prime 2}=\vec{x}^{\prime 2},$$
i.e., the surface of constant phase is moving with the speed of light out from the origin.

That must be so because of the properties of the Lorentz transformation, i.e., because $x \cdot x=x' \cdot x'$.

 

[Nugatory]

Let's see, whether we can make the entire trouble concrete….

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

 

[hutchphd]

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).

If there are two emission events, different observers will not agree as to either their distance apart or their simultaneity. You cannot intuit the details (nor can I)

 

[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.