Seems like someone's already got their Nobel acceptance speech all written up and proofread, and is reluctant to have to shred it. I see nothing more here than yet another version of the typical "giant truck in tiny garage" problem, which obviously will look like a paradox when you choose to throw the concept of time out the window. While we're at it, let's neglect mass too and see what contradictions we might find in that case. Anyone game?
The burn mark problem
- Thread starter Jack_
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Well, no... I don't think that geometry is a property of a transform at all. I think that geometry is a property of a manifold (like space, or spacetime, or a surface). A transform might be a property of a geometry, in a sense, but we're getting into mathematical semantics.
But regardless, your diagram is not "that of LT". That makes no sense.
Perhaps you meant that your diagram is one that can be transformed by a Lorentz Transform?
Thanks for that, Jack, I appreciate your patience. Working through communication barriers is difficult.
You can link to images using tags, like this:
[img] (image url here)
OK, so we have two snapshots, one at time t=zero, and one at time t=d/c.
Great!
Now, work with me for a second while I make sure that I properly understand what you mean.
- It looks like O is moving to the left toward the burn mark, while O' is stationary.
Is that right, or should O and O' be swapped around?- In the second snapshot, there are two green lines.
Are they light beams?
Does that mean there are there two light sources, one carried by O and one carried by O'?
Thanks again, Pete.
Thanks Pete for the IMG thing.
If O' is stationary, then yes O moves left.
However, the basis is to assume O is stationary and O' is moving right.
Those "green" lines are supposed to look yellowish.
Those are the light beam views of the frames.
There is one light source when O an O' are co-located. By the light postulate, it does not matter which has the light source.
You may join in.
The objective of this thread is to prove the light path length is not logically decidable within the frame.
To me, this is obvious.
You would not believe the results after this.
Nope , we do not need the GR manifold in SR.
Liken it to the light cone. After any time t, there exists a 3-d picture.
That is what I presented.
And, yes, there are two snapshots one at time t=t'=0 and one at the burn mark O co-location.
I assume you can think through the other t's.
Well, look at the diagrams. As t increases (ie as we look toward the right of the diagram), O and O' separate. In the rest frame of O, O stays at x=0 (that's why the black line is horizontal), while O' moves backward (which is why the red line has a negative gradient).
A manifold is any space. It's not limited to GR.
That's not a light cone, that's a simultaneous slice of spacetime.
Yes, you did. And no, it's not "that of LT", or a Minkowski diagram. It's a simultaneous slice of spacetime at a given instant in a given reference frame.
Thanks!
Now, those yellow lines...
So this is a single flash of light? It's not a maintained beam?
The lines represent the historical path of the light, not a current beam at this moment?
Sorry if I'm being dense, but I really don't want to misinterpret anything.
Well, look at the diagrams. As t increases (ie as we look toward the right of the diagram), O and O' separate. In the rest frame of O, O stays at x=0 (that's why the black line is horizontal), while O' moves backward (which is why the red line has a negative gradient).
You make such excellent drawings.
Let's do the math now.
OK, agreed.
Well, I can combine the frames on the unifying logic of the co-location of the burn mark and O.
The tools you have do not describe this. Hence, they are inferior.
Yes, you are not being dense, I was not clear. You are correct.
Jack_:
From this diagram, it looks like in the O frame the light beam travels distance d and in the O' frame it travels distance $$\lambda d$$.
Is that correct?
If so, what is the problem you are trying to raise?
The O' frame contends light moved a distance
(d + (v/c)d)λ from the burn mark.
I thought I made this clear.
The drawings are generated in Excel. The data comes from applying the Lorentz transorm.
You can find the spreadsheet here: http://cid-0cada106989e1f68.skydrive.live.com/self.aspx/.Public/BurnMarkProblem.xls
But just going back a step...
This doesn't match your diagrams or your original description.
In your diagrams, O' is in the same place at two different times (ie it's stationary), while O has moved.
And in your description, you had O' moving toward the burn mark, not O.
Are you sure you haven't mixed up O and O' in your diagrams?
Your diagrams show two snapshots of space at different times in a single rest frame. They don't accurately describe the other rest frame (you are still just using length contraction instead of a Lorentz transform), and they don't describe what happens in between the two times.
My diagrams show all information you have presented, plus more.
I'm not sure you understand what my diagrams represent... perhaps it would help if I invert them to make the x-axis horizontal like your diagrams, highlight where your diagrams fit on mine, and add some length labels.
Will post them later - I have a clas..
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What do the green lines in your diagram represent?
:LOL:
Green lines.
They represent the Irish sun.
No, I did not say any of the lines represented (d + (v/c)d)λ.
This was not the point.
When O is with BM, it concludes the light beam is d from BT.
However, O' contends the light beam is (d + (v/c)d)λ when O is with BM.
That is the point.
Yea, I change it to match the standard configuration of LT.
I am sure you can handle it, but you are right.
No, let's face facts. This is a real world exercise and you do not want to place this experiment in the real world.
The fact is, O and O' are co-located and a pulse is shot down the x-axis.
We need to describe this with real world pictures and I have.
Ok. I'll ask again. What are the green lines?
They represent each frames view of the light emission.
I'm not sure what you're thinking here.
In your original description, O is never colocated with the burn mark and O' is, but your diagrams show the reverse.
There's no difficulty with diagramming the original scenario, so why change it?
Yes, O and O' are co-located and a pulse is shot down the x-axis. That's what my diagrams represent. I'm puzzling over how to make them easier to understand. This might help...
On this diagram, the horizontal shaded sections indicate the times of your snapshots.
The lower section is at t=0. It shows that O, O', and the light flash are all at x=0, and that the burn mark is at x=-vd/c.
The upper section is at t=vd/c. It shows that O' is at the burn mark at x=-vd/c, the light flash is at x=d.
Look at the shaded sections more closely. Do you see how they show they represent the scenario according to O at two moments in time?
If you can follow how that diagram represents your scenario in the rest frame of O, then we can move on to figuring out what SR says about the rest frame of O'.
If you still have trouble, we might have to try animations.
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This is not a coherent sentence, irrespective of whether LT means 'Lorentz transformations' or 'a Lorentz transformation'. A Lorentz transformation is a map from one set of coordinates to another. Each set of coordinates is defined on a tangent space (or rather there's a natural link between the coordinates of M and those of TM), which can sometimes be (and in relativity is) given an inner product in the form of a metric. The Lorentz transformations are defined by the requirement they leave this metric invariant in SR. A Lorentz transformation is representable by a matrix. It doesn't have 'geometry', the underlying metric space (M,g) has a geometry (where M is the space-time manifold and g is the metric). You can represent a Lorentz transformation by the difference in how a particular system is represented in two different coordinate systems but such diagrams are space-time diagrams (of the kind used in this thread) and the Lorentz transformation is deduced from the world lines of each frame. You can talk about the structure (or loosely, the 'geometry') of the group of Lorentz transformations, such as the fact its not simply connected (or even connected) but this is something entirely more abstract and talks about how you embed all matrices which represent Lorentz transformations within a particular matrix set, such as $$SO(n,1) \subset GL(n+1,\mathbb{R})$$.
Your misuse of such terminology, your earlier throwing around of Godel's work and your mention of 'the GR manifold' makes it seem like you don't understand the terminology you're using and you're simply trying to make it appear you grasp more than you actually do.
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