# Light sphere question

by cfrogue
Tags: light, sphere
P: 687
 Quote by JesseM Why do you say it's false? If we set y and z to zero (so we're only looking at where the light is on the x-axis), then x'² + y'² + z'² = t'²c² reduces to t'²c² = x'² which is the same as ct' = ±x'.
Because we proved there exists two different t's in O such that ct' = ±x' is satisified and this violates logical consistancy.

So, I am saying that is not the way to approach the SR light sphere.

If it is approached as t increases and each point is transformed to satisfy,
x'² + y'² + z'² = τ²c²
then we have logical consistancy.

On the other hand, if we include ±x', we get different length radii for each directiion due to the simultaneity difference in each direction of the light ray in O.

It can only be approached as taking each spacetime coord in O and LT'ing it, then applying
x'² + y'² + z'² = τ²c²

That is why I say the normal logic of ct' = ±x' does not work.

The following points in O satisify that condition

t1=r/(λ(c-v)) for the ray point right and
t2=r/(λ(c+v)) for the ray point left.

If you run these through LT with x=+ct and x=-ct as appropriate, you will find two different satisfactions of the equation ct' = ±x' at two different times in O which creates two different light spheres which is a logical inconsistancy.
 P: 687 I have concluded LT will not give me the center of the light sphere after and time t in the frame of O. I realize Einstein assumed it is at vt. Does anyone have anything else on this that I am not seeing?
 P: 687 Also, if you run this point through LT, x = λvt/(1+λ) t' = t( t - vx/c² )λ You will find t' = t with collinear relative motion. Thus, we must be very careful not to conclude LT decides time dilation between frames.
P: 8,430
 Quote by cfrogue Because we proved there exists two different t's in O such that ct' = ±x' is satisified and this violates logical consistancy.
What violates logical consistency? I don't see anything illogical about the fact that there can be two events with values of (x,t) that satisfy ct = ±x, and with t being different for each event, such that when you apply the Lorentz transformation to the (x,t) coordinates of each event to get the (x',t') coordinates of the same event, then both events can have (x',t') coordinates that satisfy ct' = ±x' but with the new feature that the t' coordinate is the same for each event. Are you saying this is illogical or impossible?
 Quote by cfrogue That is why I say the normal logic of ct' = ±x' does not work. The following points in O satisify that condition t1=r/(λ(c-v)) for the ray point right and t2=r/(λ(c+v)) for the ray point left. If you run these through LT with x=+ct and x=-ct as appropriate, you will find two different satisfactions of the equation ct' = ±x' at two different times in O which creates two different light spheres which is a logical inconsistancy.
Again, what's a logical inconsistency? These events occur at different times in O, so naturally they are part of two different light spheres at different times in O. The equation ct = ±x was never supposed to define a single light sphere, rather it is an equation defining all points that lie on the light cone in 1D, it can include events at different times in O. Exactly the same is true of the equation x'² + y'² + z'² = τ²c², τ is a variable so this too is the 3D version of a light cone which includes events that lie on the path of the light at different times in O.
P: 8,430
 Quote by cfrogue I have concluded LT will not give me the center of the light sphere after and time t in the frame of O. I realize Einstein assumed it is at vt.
Why have you concluded that? The center of the light sphere in O' is always at x'=0, and if you apply Lorentz transformation to some event which has an x' coordinate of 0 you'll always get an event whose x and t coordinates satisfy x=vt. Do you disagree? If not, what's the problem?
P: 687
 Quote by JesseM Why have you concluded that? The center of the light sphere in O' is always at x'=0, and if you apply Lorentz transformation to some event which has an x' coordinate of 0 you'll always get an event whose x and t coordinates satisfy x=vt. Do you disagree? If not, what's the problem?
I get x=(vt)λ
P: 8,430
 Quote by cfrogue I get x=(vt)λ
Suppose the event has coordinates x'=0, t'=T' in O'. Then the coordinates in O are:

x = gamma*(x' + vt') = gamma*vT'
t = gamma*(t' + vx'/c^2) = gamma*T'

So, x/t = (gamma*vT')/(gamma*T') = v, and if x/t=v then x=vt.
P: 687
 Originally Posted by cfrogue Because we proved there exists two different t's in O such that ct' = ±x' is satisified and this violates logical consistancy.
 Quote by JesseM What violates logical consistency? I don't see anything illogical about the fact that there can be two events with values of (x,t) that satisfy ct = ±x, and with t being different for each event, such that when you apply the Lorentz transformation to the (x,t) coordinates of each event to get the (x',t') coordinates of the same event, then both events can have (x',t') coordinates that satisfy ct' = ±x' but with the new feature that the t' coordinate is the same for each event. Are you saying this is illogical or impossible?
No, I am saying uisng that method creates two different light spheres. I gave you the points. Use them and apply the radius in both directions.

 Quote by JesseM [ Again, what's a logical inconsistency? These events occur at different times in O, so naturally they are part of two different light spheres at different times in O. The equation ct = ±x was never supposed to define a single light sphere, rather it is an equation defining all points that lie on the light cone in 1D, it can include events at different times in O. Exactly the same is true of the equation x'² + y'² + z'² = τ²c², τ is a variable so this too is the 3D version of a light cone which includes events that lie on the path of the light at different times in O.
I have been down this road.

It does not achieve anything. I created two different light spheres in O' because of the simultaneity differential of the direction of the light rays in O.

Yet, if I do strictly point by point transforms as suggested by Einstein, I have a consistant resolution.
P: 687
 Quote by JesseM Suppose the event has coordinates x'=0, t'=T' in O'. Then the coordinates in O are: x = gamma*(x' + vt') = gamma*vT' t = gamma*(t' + vx'/c^2) = gamma*T' So, x/t = (gamma*vT')/(gamma*T') = v, and if x/t=v then x=vt.
Good one.

Thanks. That works.
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P: 15,552
 Quote by cfrogue I clearly see a difference of when an event will occur concerning light and time dilation.
Obviously. Time dilation requires Δx=0 and Δx never equals 0 for light, so there will always be a difference.
 Quote by cfrogue It is false that ct' = ±x'
You already agreed it was true, and it was proved conclusively several different ways. If you want to suddenly go back and ignore the previous 570 posts then I am done with this. What is the point of continuing the conversation if you will just ignore that amount of proof?
P: 7,398
 Quote by cfrogue Yet, if I do strictly point by point transforms as suggested by Einstein, I have a consistant resolution.
Yes, that is the only way to do it, and that is what I've been doing all along, although that was not apparent to you. Regardless, this is correct, and fundamental.
P: 687
 Quote by DaleSpam Obviously. Time dilation requires Δx=0 and Δx never equals 0 for light, so there will always be a difference. You already agreed it was true, and it was proved conclusively several different ways. If you want to suddenly go back and ignore the previous 570 posts then I am done with this. What is the point of continuing the conversation if you will just ignore that amount of proof?
I am the one that introduced ct' = +-x'.

That was my argument.

Then, I discovered it created two light spheres. You agreed.

The question for you is how do you create logical consistancy with ct' = +-x'?

I would like to see this.
P: 687
 Quote by atyy Yes, that is the only way to do it, and that is what I've been doing all along, although that was not apparent to you. Regardless, this is correct, and fundamental.
Agreed.
P: 687
 Quote by atyy Yes, that is the only way to do it, and that is what I've been doing all along, although that was not apparent to you. Regardless, this is correct, and fundamental.
Well, you could be more specific.

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P: 15,552
 Quote by cfrogue The question for you is how do you create logical consistancy with ct' = +-x'? I would like to see this.
We already demonstrated this more than 500 posts ago. If you would like to see it just use your browser to review the thread.