Time in Special Relativity

Very good.

We have now almost completed the transformations of interest from the moving frame to the stationary frame (it's like getting blood from a stone). Would you like to complete the transformation of event c), or are you confident that I am competent? Note that we need the x values as well.

a) (x',t')=(0,r/c) (location of O')
\(x = \gamma vr/c\)
\(t = \gamma r/c\)

b) (x',t')=(-r,r/c) (location of light)
\(x = \gamma vr/c - \gamma r\)
\(t = \gamma r/c - \gamma vr/c^2\)

c) (x',t')=(r,r/c) (location of light)
\(x = ...\)
\(t = ...\)

d)(x',t')=(-vr/c,r/c) (location of O)
\(x = 0\)
\(t = r/\gamma c\)

Once that is complete, then we should make sure we agree on the transformations back to the moving frame.

Please, Jack, you need to make your questions more precise.
You've already asked and been answered what the time is at O':

So what are you asking this time?

 

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Be precise.
In the rest frame of O at t=r/γc, the moving clock shows a time of t'=r/c.

Again, precision is critical.
In the rest frame of O' at t'=r/c, the light flashes on the x-axis are at x'=r and x'=-r.


You seem to think that these two statements are contradictory?

 

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You are applying LT.

That is fine.

What you have not done is the following:
1) What is the elapsed time on the clock of O'?
2) If the clock at any emission point of light elapses r/c from light emission, what is the distance in all directions that light traveled?

Intellectually, you are only applying parts of SR and not the full theory.

Now, if I made any false statements, that should be easy to prove.

But, if you are not up to applying the full theory, you should not be operating in this context.

 

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If O elapses t=r/γc, the O' elapses t=r/c. I do not know how to be more precise.

I am not looking at the problem as this above.

I am applying time dilation are per Einstein.

Then, I apply the stationary light sphere as per Einstein.

I already know this contradicts LT or I would not have presented it.

You are digesting this fact.

So, apply time dilation and the light sphere as per Einstein.

 

Jack, this discussion is futile unless you can clearly demonstrate that you can do a Lorentz transform.
I've asked you several times to do some very simple transforms, but you haven't done so yet.
Why?

Humour me, Jack. Show me that you can do it.
Then we can continue:

c) (x',t')=(r,r/c) (location of light)
x = ...
t = ...

This is important, Jack.
The Lorentz Transform is a complete description of special relativity.
If you have a result that disagrees with a properly applied lorentz transform, then it is not a result of special relativity.

 

Well, you need to learn to be more precise, because in SR that statement is ambiguous.
It is true in the reference frame of O.
It is not true in the reference frame of O'.

This is indeed the heart of the problem, and I intend to fully explore it with you.
But before we begin, I need to be sure that you can do simple lorentz transforms.

Check that the following list is correct, and complete the missing transform:

a) (x',t')=(0,r/c) (location of O')
\(x = \gamma vr/c\)
\(t = \gamma r/c\)

b) (x',t')=(-r,r/c) (location of light)
\(x = \gamma vr/c - \gamma r\)
\(t = \gamma r/c - \gamma vr/c^2\)

c) (x',t')=(r,r/c) (location of light)
\(x = ...\)
\(t = ...\)

d)(x',t')=(-vr/c,r/c) (location of O)
\(x = 0\)
\(t = r/\gamma c\)

 

You keep claiming you're doing physics as per Einstein or SR, in an attempt to make it seem like you're following precisely what SR says. This is obviously false because you're saying you're doing the standard methodology employed by the physics and maths communities and yet no one gets the conclusion you do. And by this I don't mean that no one can do the algebra, I mean that people can and have done the algebra and not said "Its a contradiction". People have been through Einstein's work and all the work since many many times and no one reaches the conclusion you do, they disagree with your interpretation of the algebra.

You believe you're doing new algebra, new mathematics, but you aren't. And you'd know this if you bothered to read any books on SR, rather than immediately dismiss them because they don't tell you what you want to hear. If you correctly follow the methods of Einstein/SR you would reach the same conclusion as the mainstream community. These transformations have been examined for more than a century by millions of different people working for the most part independently (ie they'll do the algebra themselves) and none of them agree with you. Your entire position is an attempt to argue that millions of people doing these kinds of calculations billions of times made exactly the same mistake as one another, a mistake you supposedly haven't made. Even more so, you're claiming that all the times SR has been applied to a real world situation then somehow the calculations were all done incorrectly and once again all the different people made precisely the same mistake and it just so happened that this mistake didn't show up in experiments. So not only have billions of SR calculations all been done incorrectly in precisely the same way but that mistake actually provides the correct (to the limit of our ability to test) experimental predictions.

Am I wrong in this evaluation of your point of view? If so please explain to me why no one has ever spotted this supposed problem before, despite having tested it experimentally and it being related to algebra so simple its 1st year homework. Your ignorance of physics makes you believe you're treading new ground but you aren't. And the results you claim are a contradiction are well known (and indeed a physical requirement!) and not viewed as a contradiction.

You believe it contradicts LT but you haven't proven it. And you've already demonstrated you have no problem presenting things without justification or evidence.

All you need to do to see you're mistaken is consider the ramifications of the experimental fact that the motion of light is independent of the motion of the emitter and only dependent on the location of the emitter in space at the time of emission. Its therefore required by experiment that your thought experiment of two rigid spheres should result in there being frames where each sphere (separately) should be centred on the spacial location which forms the centre of the expanding light sphere. This is because the requirement to be independent of the emitter's motion means that it should be impossible to categorically state which sphere emitted the light. And since all frames should be equivalent if there's a frame such that the light sphere and a given rigid sphere share a common spacial centre then there should be (separately) frames for all the other spheres (since the thought experiment generalised to as many spheres whose centres all coincide at the same moment) where each sphere is individually centred on the light sphere.

A point p in space-time M uniquely determines a light cone (and vice versa). There's infinitely many vectors definable at p, ie [tec]v_{p} \in T_{p}M[/tex]. Given a point and a vector at that point you can solve the geodesic equation (which is easy in SR, unlike GR) and you get a particular worldline. This can be regarded as the worldline of the centre of one of the spheres. Different vectors, different worldlines, different spheres moving inertially but all in a bijective relationship (ie any one of them uniquely defines the other two). In a given set of coordinates the vector which is parallel to the unit time-like vector gives a worldline which is associated to a sphere which, in those coordinates, has the same centre as the light sphere. In a different set of coordinates its a different vector and thus a different sphere. This is not contradictory. The two sets of coordinates are Lorentz transform related and the transformation is defined at p so irrespective of which sphere is supposedly centred on the emission point at a later point in time all frames know the apex of the light cone. Further more the physics which follows from the spheres (if they were particles in a collider etc) is blind to which one of them appears not to be moving (ie centred on the same point in space as the light spheres at a point in time after the emission. Its irrelevant the position and motion of the spheres relative to some point in space, only their relsative position to one another matters. And the Lorentz transformations transform them into different configurations but such that the end results are the same. You can either let a bunch of objects interact and then transform to a new frame to get a result or you can transform to the new frame and let the objects interact. If LT are consistent then this would result in the same outcome. It does. This is tested every time a particle accelerator is used because precisely those methods are used in making predictions.

Different frames disagree on where the centre of the light sphere is, yes. But its not a contradiction as you claim. If you put an object into space to 'mark' that point you're doing nothing more than throwing in another rigid sphere. You can't 'mark' the point because you're effectively saying "There is a preferred frame", which is inconsistent with SR. If there's nothing to mark the point it has no physical properties or interactions or relevance so it doesn't lead to an inconsistency in physical predictions. If you mark it with a point you're trying to single out the marker's rest frame as preferred, which it isn't.

The physical FACT light's motion is independent of its emitter's motion means you can't use light to single out a particular frame and that means different frames with disagree on the light sphere centre. If they agreed that agreed upon point would single out a preferred frame. So not only is it not a contradiction its essential for any model of space-time and light to say if there is to be no preferred frame.

If you want the mathematics of this, go to the pseudo forum and look at my post on projections from a bundle to its base space. I've posted it twice without retort from you, maybe 3rd time's the charm.

 

Pete, I have done the transform.

I am simply showing LT does not agree with time dilation plus the light sphere logic.

What is so hard?

 

This statement is totally precise. It is a statement of time dilation.

If O elapses t=r/γc, the O' elapses t=r/c. I do not know how to be more precise.

I already said O and O' are clocks.

From the list above, you are going into the O' frame and using the light sphre.

That is fine but has nothing to do with the path I am takiing.

I use the path that when the clock at O elapses r/γc, the clock at O' elapses r/c.

When the clock at O' elapses r/c, light must be r in all directions.

This contradicts SR.

The only problem here is you are unwilling to accept this even though you do accept it.

Otherwise, refute this path becaue this is the point of my posts.

 

I am starting to get it.

Let's see light moves through space at one speed c.

Now, we know the earth is in orbit at 18.55 mps based on light aberration. So it is moving through space at 18.55 mps.

So, put a satellite at the equator on the horizon 10,000 from point A in the east at noon.

So, the satellite and A are in line with with the earth's 18.55 mps orbit.

Now shoot a signal at A.

The light moves through space at c.

Oh, the earth is also moving through space but at 18.55 mps toward the light.

So, A is cutting the distance light must travel to A.

So light does not measure c on the path and SR is false.

Got it.

 

c) (x',t')=(r,r/c) (location of light)
x = ...
t = ...

Can you fill in the blanks or not?

 

Sure

x = (x' + vt')γ = (r + vr/c)γ

And here is a trick from me to you.

t = x/c always.

so t = r/c(1 + v/c)γ


[Edit] You must not play chess. You cannot go anywhere with this vs my timer dilation and light sphere argument.

 

For events on the worldline of light emitted at or passing through the origin, of course.

And does that agree with the t you get from the lorentz transform?

We will see how time dilation follows from the lorentz transform. This will enlighten us about exactly what time dilation means and what it does not mean.

 

time dilation follows from setting x = vt in the stationary frame.

Why are you operating in the moving frame?

I get t' = t/γ by setting x = vt to explain the motion of O'.

 

No, you aren't.

.... relative to .....

So you've 'got it' by considering the motion of a non-inertial object held in orbit by gravity and proclaiming thus special relativity is wrong.

I'm still waiting for you to have the balls to submit your work to a journal. You responded to nothing I said. You demand people respond then you fail to reply to what they say. And you're stupid enough to think somehow no one will notice....

 

We will move on to time dilation and light spheres in good time.

First, I need to build a solid framework that we can both agree on about what the lorentz transform says about this scenario. OK? This discussion will be a lot shorter and less confusing if you just humour me and answer my stupid questions.

Look at this table of events. Take some time to figure out how it's structured. Let me know if it doesn't make sense.
It lists the positions of O, O', and both light flashes at various times (0, r/γc, r/c, and γr/c) in each reference frame, and transforms those events (ie those position/time combinations) to the other frame.
LL is the leftward light flash, LR is the rightward light flash.

To illustrate:
Near the top left of the table, you'll find:
(0, r/γc) in the O(x,t) column.
This is an event, showing that O is located at x=0 at time t=0 in the rest frame of O.

Next, you'll see (vr/c, r/c) in the O(x',t') column.
This indicates the transformation of the event to the moving frame:
That event is located at x'=vr/c at time t=r/c in the rest frame of O'.

When you figure out how I've structured it, you might like to take the time to check it through, and confirm that the transformations given are correct. Or at least check some cells. Yes, it's hard work, but I'm sure you won't mind expending some effort in the pursuit of expounding scientific truth.

If it is not clear what some part of the table is supposed to mean, please point it out. If I've made a mistake in any cell, please point it out.

Once we agree that this table is a correct application of lorentz transforms, we can explore what it tells us about time dilation.

\(\begin{array}{c|cc|cc|cc|cc} & O(x,\ \ t) & O(x',\ \ t') & O'(x,\ \ t) & O'(x',\ \ t') & LL(x,\ \ t) & LL(x',\ \ t') & LR(x,\ \ t) & LR(x',\ \ t') \\ \hline t=t'=0 & (0,\ \ 0) & (0,\ \ 0) & (0,\ \ 0) & (0,\ \ 0) & (0,\ \ 0) & (0,\ \ 0) & (0,\ \ 0) & (0,\ \ 0) \\ \hline t=\frac{r}{\gamma c} & (0,\ \ \frac{r}{\gamma c}) & (\frac{vr}{c},\ \ \frac{r}{c}) & (\frac{vr}{\gamma c},\ \ \frac{r}{\gamma c}) & (0,\ \ \frac{r}{\gamma^2c}) & (\frac{-r}{\gamma},\ \ \frac{r}{\gamma c}) & (-r(1-\frac{v}{c}),\ \ \frac{r}{c}(1-\frac{v}{c})) & (\frac{r}{\gamma},\ \ \frac{r}{\gamma c}) & (r(1+\frac{v}{c}),\ \ \frac{r}{c}(1 + \frac{v}{c}) \\ t=\frac{r}{c} & (0,\ \ \frac{r}{c}) & (\frac{-\gamma vr}{c},\ \ \frac{\gamma r}{c}) & (\frac{vr}{c},\ \ \frac{r}{c}) & (0,\ \ \frac{r}{\gamma c}) & (-r,\ \ \frac{r}{c}) & (-\gamma r(1+\frac{v}{c}),\ \ \frac{\gamma r}{c}(1+\frac{v}{c})) & (r,\ \ \frac{r}{c}) & (\gamma r(1-\frac{v}{c}),\ \ \frac{\gamma r}{c}(1-\frac{v}{c})) \\ t=\frac{\gamma r}{c} & (0,\ \ \frac{\gamma r}{c}) & (\frac{-\gamma^2vr}{c},\ \ \frac{\gamma^2r}{c}) & (\frac{\gamma vr}{c},\ \ \frac{\gamma r}{c}) & (0,\ \ \frac{r}{c}) & (-\gamma r,\ \ \frac{\gamma r}{c}) & (\frac{-r}{1-\frac{v}{c}},\ \ \frac{r}{c-v}) & (\gamma r,\ \ \frac{\gamma r}{c}) & (\frac{r}{1+\frac{v}{c}},\ \ \frac{r}{c+v}) \\ \hline t'=\frac{r}{\gamma c} & (0,\ \ \frac{r}{\gamma^2c}) & (\frac{-vr}{\gamma c},\ \ \frac{r}{\gamma c}) & (\frac{vr}{c},\ \ \frac{r}{c}) & (0,\ \ \frac{r}{\gamma c}) & (-r(1+\frac{v}{c}),\ \ \frac{r}{c}(1 + \frac{v}{c}) & (\frac{-r}{\gamma},\ \ \frac{r}{\gamma c}) & (r(1-\frac{v}{c}),\ \ \frac{r}{c}(1-\frac{v}{c})) & (\frac{r}{\gamma},\ \ \frac{r}{\gamma c}) \\ t'=\frac{r}{c} & (0,\ \ \frac{r}{\gamma c}) & (\frac{-vr}{c},\ \ \frac{r}{c}) & (\frac{\gamma vr}{c},\ \ \frac{\gamma r}{c}) & (0,\ \ \frac{r}{c}) & (-\gamma r(1-\frac{v}{c}),\ \ \frac{\gamma r}{c}(1-\frac{v}{c})) & (-r,\ \ \frac{r}{c}) & (\gamma r(1+\frac{v}{c}),\ \ \frac{\gamma r}{c}(1+\frac{v}{c})) & (r,\ \ \frac{r}{c}) \\ t'=\frac{\gamma r}{c} & (0,\ \ \frac{r}{c}) & (\frac{-\gamma vr}{c},\ \ \frac{\gamma r}{c}) & (\frac{\gamma^2vr}{c},\ \ \frac{\gamma^2r}{c}) & (0,\ \ \frac{\gamma r}{c}) & (\frac{-r}{1+\frac{v}{c}},\ \ \frac{r}{c+v}) & (-\gamma r,\ \ \frac{\gamma r}{c}) & (\frac{r}{1-\frac{v}{c}},\ \ \frac{r}{c-v}) & (\gamma r,\ \ \frac{\gamma r}{c}) \\ \hline \end{array}\)

 

It might be easier if we use real numbers.
If distance is measured in light-seconds, t in seconds, and v=0.6c, and r=1, then the above table above looks like this:

\(\begin{array}{c|cc|cc|cc|cc} & O(x,\ \ t) & O(x',\ \ t') & O'(x,\ \ t) & O'(x',\ \ t') & LL(x,\ \ t) & LL(x',\ \ t') & LR(x,\ \ t) & LR(x',\ \ t') \\ \hline t=t'=0 & (0,\ \ 0) & (0,\ \ 0) & (0,\ \ 0) & (0,\ \ 0) & (0,\ \ 0) & (0,\ \ 0) & (0,\ \ 0) & (0,\ \ 0) \\ \hline t=\frac{r}{\gamma c} = 0.8 & (0,\ 0.8) & (-0.6,\ 1) & (0.48,\ 0.8) & (0,\ 0.64) & (-0.8,\ 0.8) & (-0.4,\ 0.4) & (0.8,\ 0.8) & (1.6,\ 1.6) \\ t=\frac{r}{c} = 1 & (0,\ 1) & (-0.75,\ 1.25) & (0.6,\ 1) & (0,\ 0.8) & (-1,\ 1) & (-2,\ 2) & (1,\ 1) & (0.5,\ 0.5) \\ t=\frac{\gamma r}{c} = 1.25 & (0,\ 1.25) & (-0.9375,\ 1.5625) & (0.75,\ 1.25) & (0,\ 1) & (-1.25,\ 1.25) & (-2.5,\ 2.5) & (1.25,\ 1.25) & (0.625,\ 0.625) \\ \hline t'=\frac{r}{\gamma c} = 0.8 & (0,\ 0.64) & (-0.48,\ 0.8) & (0.6,\ 1) & (0,\ 0.8) & (-1.6,\ 1.6) & (-0.8,\ 0.8) & (0.4,\ 0.4) & (0.8,\ 0.8) \\ t'=\frac{r}{c} = 1 & (0,\ 0.8) & (-0.6,\ 1) & (0.75,\ 1.25) & (0,\ 1) & (-0.5,\ 0.5) & (-1,\ 1) & (2,\ 2) & (1,\ 1) \\ t'=\frac{\gamma r}{c} = 1.25 & (0,\ 1) & (-0.75,\ 1.25) & (0.9375,\ 1.5625) & (0,\ 1.25) & (-0.625,\ 0.625) & (-1.25,\ 1.25) & (2.5,\ 2.5) & (1.25,\ 1.25) \\ \hline \end{array}\)

Again, let me know if I've made mistakes.

 

Relative to a non rotating frame like all sagnac experiments.

 

Pete, I am not disputing waht LT calculates. I have no problem with the calculations.

I did them myself and corrected both you and James.

So I understand them.

But, what we need to do logically, is ao apply the full theory.

You are attempting to claim LT as the sole truth. You are claiming anything that refutes LT is wrong.

I am showing you SR refutes LT.

The logic I present is not mine but Einstein's. It is just noone put all of them together at once.

Anyway, I stipulate LT calculates a different result.