- #141
matheinste
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cfrogue said:
But we already know the answer to the originally posed scenario.
Do I know what you are going to do next. Based on experience I would say you will prevaricate.
Matheinste.
cfrogue said:I assure you I am operating strickly from my logic and have not read any of this. I would not lie.
This method helps is see the answer.
Do you know what I am going to do next?
matheinste said:But we already know the answer to the originally posed scenario.
Do I know what you are going to do next. Based on experience I would say you will prevaricate.
Matheinste.
cfrogue said:Each frame agrees on a rest distance of d for two rods one for each frame.
We will label the endpoints of the rods as L, R, L' and R'.
A light source is centered on the rod of O'.
When the two rods happen to be centered and O' is moving in relative motion, the light is flashed from the light source of O'.
Now, in the frame of O, t(L) = t(R) by the light postulate, t(x) means light strikes the point.
In O', t'(L') = t'(R')
any disagreements?
atyy said:No disagreements.
cfrogue said:OK, do you agree with the equations?
I mean, that is the light postulate.
Further, this should not be a problem because I am only bounding the expanding light sphere.
cfrogue said:Well, Matheinste is walking around with protest signs.
atyy said:That's ok. You should take Matheinste seriously (he's very good), but we can have some fun anyway.
atyy said:That's ok. You should take Matheinste seriously (he's very good), but we can have some fun anyway.
cfrogue said:If I have a light source in a frame and emit the light, the light travels to equidistant points in the same time t = d/c.
Is this correct?
cfrogue said:In that case, with a rod length of d, O draws the following conclusions.
t'(L') = d/(2*λ(c+v))
and
t'(R') = d/(2*λ(c-v))
Yes,atyy said:I don't understand this step. Can you explain your reasoning in more detail?
cfrogue said:Yes,
given a rod in O', the right end point moves away from the expanding light sphere in O.
So, for example, let's do the right end point,
the light must travel the distance d/(2*λ) plus the distance O' moved in time t which is vt.
So, light travels ct in time t therefore,
ct = d/(2*λ) + vt.
ct - vt = d/(2*λ)
t( c - v ) = d/(2*λ)
t = d/(2*λ*(c - v))
Since this is the time in O, then it is t'(R').
You can also find this logic here.
http://www.fourmilab.ch/etexts/einstein/specrel/www/
At the end of section 2.
Einstein wrote it as tA - tB = rAB/(c-v)
cfrogue said:In that case, with a rod length of d, O draws the following conclusions.
t'(L') = d/(2*λ(c+v))
and
t'(R') = d/(2*λ(c-v))
t'(L') = d/(2*λ(c+v))
and
t'(R') = d/(2*λ(c-v))
atyy said:So this is incorrect (I think). t'(L') is the time that O' assigns to the photon crossing L'.
The time that O assigns to the photon crossing L' is something else, let's call that t(t'(L')).
cfrogue said:No, these are the standard equations for O.
cfrogue said:O' concludes the light strikes t'(L') and t'(R') at the same time.
atyy said:I think the LHS is wrong, but the RHS is correct.
Yes. But O does not conclude that the light strikes L' and R' at the same time. O concludes that the light strikes L' at t(t'(L')) and the light strikes R' at t(t'(R')).
I would try
t(t'(L')) = d/(2*λ(c+v))
and
t(t'(R')) = d/(2*λ(c-v))
OK, I'm not sure about that, let me do some algebra ...
atyy said:I think the LHS is wrong, but the RHS is correct.
Yes. But O does not conclude that the light strikes L' and R' at the same time. O concludes that the light strikes L' at t(t'(L')) and the light strikes R' at t(t'(R')).
I would try
t(t'(L')) = d/(2*λ(c+v))
and
t(t'(R')) = d/(2*λ(c-v))
OK, I'm not sure about that, let me do some algebra ...
cfrogue said:No...
In frame O it is centered around O origincfrogue said:In each frame the sphere is centered around that frame's origin, and NOT AROUND BOTH ORIGINS IN ONE FRAME. Therefore It doesn't matter if they remain coincident.
So, you are saying the light sphere is centered at the origin of each frame.
No it is a failed logic, because you confuse things from two different frames. In each frame there is only one origin that is the center of the light sphere. What you have to realize is this:cfrogue said:Yet, the frames' origins separate by vt after any time t. It is plain and simple logic that this implies there are two different light sphere origins.
Yes, that is correct the light sphere in O at any given t>0 is centered at x=0. Also, the light sphere in O' at any given t'>0 is centered at x'=0. Furthermore, the line formed by the centers of all of the light spheres in O' is given by the equation x'=0 or x=vt, and the line formed by the centers of all of the light spheres in O is given by the equation x=0 or x'=-vt'.cfrogue said:OK, now the light sphere remains centered at O and also at O' in your diagram and O and O' are also diverging at vt.
Is this all correct?
Yes, you can see this in the spacetime diagram. Note that L follows the line x=-1 and R follows the line x=1. You can easily see where each of those lines intersect the light cone and verify that both events occur at the line t=1.cfrogue said:Now, in the frame of O, t(L) = t(R) by the light postulate, t(x) means light strikes the point.
Yes, you can also see this in the spacetime diagram. L' follows the line x'=-1 and R' follows the line x'=1. Again, you can easily see where each of those lines intersect the light cone and verify that both events occur at the line t'=1.cfrogue said:In O', t'(L') = t'(R')
cfrogue said:Yes. But O does not conclude that the light strikes L' and R' at the same time.
In one sense, R of S it does not, but the light postulate in O' demands they are the same.
This would imply O' concludes the R' and L' are struck at the same time, but O cannot calculate this correctly with LT, so LT fails.
cfrogue said:This would imply O' concludes the R' and L' are struck at the same time, but O cannot calculate this correctly with LT, so LT fails.
I think you mean:cfrogue said:t'(L') = d/(2*?(c+v))
and
t'(R') = d/(2*?(c-v))
atyy said:I think the LHS is wrong, but the RHS is correct.
Yes. But O does not conclude that the light strikes L' and R' at the same time. O concludes that the light strikes L' at t(t'(L')) and the light strikes R' at t(t'(R')).
I would try
t(t'(L')) = d/(2*λ(c+v))
and
t(t'(R')) = d/(2*λ(c-v))
OK, I'm not sure about that, let me do some algebra ...
A.T. said:In frame O it is centered around O origin
In frame O' it is centered around O' origin
No it is a failed logic, because you confuse things from two different frames. In each frame there is only one origin that is the center of the light sphere. What you have to realize is this:
The frames don't agree which physical location coincides with the center of the light sphere.
A physical location is defined by a real object like the light source or the end of a rod. The center of the light sphere is not a physical location, but just a calculated coordiante. It is calculated from the coordiantes of those physical locations which are hit by the light simultaneously. But simultaneity is frame dependent and so is the center of the light sphere in respect to physical locations.
atyy said:Yup, that's wrong.
Editing ... it's almost right ... ok, it's right.
Calculation A
t(t'(R')) = d/(2*λ(c-v)
=d.sqrt(1-v^2/c^2)/[2(c-v)]
=d.sqrt((1-v/c).(1+v/c))/[2(c-v)]
=d.sqrt(((c-v)/c).((c+v)/c))/[2(c-v)]
=d.sqrt((c-v).(c+v))/[2c(c-v)]
=(d/2c).sqrt((c+v)/(c-v))
Calculation B
For O', t'(R')=d/2c and L'=d/2
The Lorentz transformation is "t' = ( t - vx/c^2 )λ"
In the above formula we put
" t' " as t(t'(R'))
" t " as t'(R')=d/2c
" x " as L'=d/2
" v " as -v
So
t(t'(R')) = [d/2c + vd/(2c^2)]λ
=(d/2c).(1+v/c).λ
=d/2c.(1+v/c)/sqrt(1-v^2/c^2)
=d/2c.(1+v/c)/sqrt((1-v/c)(1+v/c))
=d/2c.sqrt((1+v/c)/(1-v/c))
=d/2c.sqrt(((c+v)/c))/((c-v)/c)
=d/2c.sqrt((c+v)/(c-v)))
Calculation A from the viewpoint of O, and Calculation B as the Lorentz transformation of the viewpoint of O' into the viewpoint of O match up.
DaleSpam said:In the following reply I am going to make a strong distinction between the light cone and a light sphere; I will not use the two terms interchangeably. I am also going to make a strong distinction between the apex of the light cone and a center of a light sphere. The light cone is a single 4D geometric object in Minkowski spacetime which is the set of all events that are null separated (c²Δt²-Δx²-Δy²-Δz²=0) from the apex event. A light sphere is a 3D conic section of the light cone formed by the intersection of the light cone with any plane of simultaneity in any inertial reference frame, the center of a light sphere is the event which is equidistant from all of the events in the light sphere in the 3D plane of simultaneity used to form the light sphere.Yes, that is correct the light sphere in O at any given t>0 is centered at x=0. Also, the light sphere in O' at any given t'>0 is centered at x'=0. Furthermore, the line formed by the centers of all of the light spheres in O' is given by the equation x'=0 or x=vt, and the line formed by the centers of all of the light spheres in O is given by the equation x=0 or x'=-vt'.
I am certain that you will think this is a contradiction, but it is not. The key thing to remember is that there is not just a single light sphere, there are in fact an infinite number of light spheres one corresponding to every coordinate time in every inertial reference frame. Different frames disagree on the location of the center of the various light spheres because each light sphere refers to a completely different set of events.
In contrast, there is a single light cone, and the apex of the light cone does not change. All frames agree which event is the apex (although they may disagree on the coordinates assigned to the apex if it is not located at the mutual origin) and that event does not change in any reference frame.
DaleSpam said:Yes, you can see this in the spacetime diagram. Note that L follows the line x=-1 and R follows the line x=1. You can easily see where each of those lines intersect the light cone and verify that both events occur at the line t=1. Yes, you can also see this in the spacetime diagram. L' follows the line x'=-1 and R' follows the line x'=1. Again, you can easily see where each of those lines intersect the light cone and verify that both events occur at the line t'=1.
Jackadsa said:atyy has it correct. Let me call t'(R') and t'(L') the times that an observer in O' records the light pulse hitting R' and L', and t(R') and t(L') the times that an observer in O records these same events (note that my t(R') is atyy's t(t'(R'))). Then as atyy stated:
(I will use & for gamma, since I havn't yet figured out how to use equations)
t(R') = d/(2&(c-v))
t(L') = d/(2&(c+v))
which you get from considering the rear and leading ends of the rod moving towards and away from the light respectively.
also, if x(R') and x(L') are the spatial coordinates of the events as recorded by an observer in O, then:
x(R') = ct(R')
x(L') = -ct(L')
Now, the observer in O can use the lorentz transformation to calculate the coordinates in O'. In particular, we are interested in what he calculates t'(R') and t'(L') to be, and whether this is consistent with t'(R') = t'(L') = d/2c.
t'(R') = &[t(R') - vx(R')/c^2]
t'(R') = &[t(R') - t(R')v/c] (I have substituted x(R') = ct(R'))
t'(R') = &t(R')(1 - v/c)
t'(R') = &(1 - v/c)d/(2&(c-v)) (I have substituted t(R') = d/(2&(c-v)))
t'(R') = d/2c
A similar calculation shows t'(L') = d/2c
So O records both events as non simultaneous. But then when he uses the lorentz transformation to figure out what O' sees, he finds that the events are indeed simultaneous in O' as expected.
DaleSpam said:I think you mean:
t(L') = d/(2*?(c+v))
and
t(R') = d/(2*?(c-v))
Again, you can see each of these in the diagram. For t(R') look at R' (x'=1) and see where it intersects the light cone, and note that event occurs at t=2 which agrees with the equation above. For t(L') look at L' (x'=-1) and see where it intersects the light cone, and note that event is halfway between the lines for t=0 and t=1, so it occurs at t=0.5 which also agrees with the equation above.
Hopefully you are beginning to see the value of spacetime diagrams.
The light sphere doesn't "move" in either frame, each frame sees that the sphere expands out symmetrically from that frame's origin and remains centered around that frame's origin.cfrogue said:No, the equations i presented are just plain wrong.
No, I do not see the value of spacetime diagrams. They do not confess a diverging center of the light sphere and thus, they are incomplete.
They overlay the two origins of the frames on top of each other.
This does not show the behavior of the light sphere in O' moving with the origin at vt relative to the fixed origin in O at 0.
cfrogue said:I will look these over and thanks.
JesseM said:The light sphere doesn't "move" in either frame, each frame sees that the sphere expands out symmetrically from that frame's origin and remains centered around that frame's origin.
Do you understand in each frame, the "light sphere" at any given moment is really the intersection between the light cone and a surface of simultaneity in that frame? And that since the two frames have different surfaces of simultaneity, they are not referring to the same set of points in spacetime when they talk about a "light sphere" at a given moment? For example, pick an event E on the left side of the light cone. Then in frame A, the light sphere at the moment of E would contain some event E1 on the right side of the light cone which is simultaneous with E in A's frame. But in frame B, that same event E1 would not be part of the light sphere at the moment of E, instead frame B would say that the light sphere at the moment of E contains some different event E2 on the right side of the light cone which is simultaneous with E in B's frame. So they are each talking about a different set of events when they refer to the "light sphere at the moment of E".
Any disagreement/confusion here?
atyy said:No problem. I missed a couple of brackets, so feel free to ask about those. Jackadsa presented cleaner notation and a sleeker argument, so glad you agree with him too.
http://www.youtube.com/watch?v=NrkUYTCwJPc"
atyy said:No problem. I missed a couple of brackets, so feel free to ask about those. Jackadsa presented cleaner notation and a sleeker argument, so glad you agree with him too.
http://www.youtube.com/watch?v=NrkUYTCwJPc"