Explaining C: How Space Changes with Speed

[zoobyshoe]

Once again, I deny Lorentz' point of view that the effect of Length contraction has anything to do with electrodynamic equilibrium.

Yes, I understand where you stand on this. I meant, you agree with Lorentz answer to the question "Is length contraction real or apparent?"

I was answering your question regarding the "stability" of kilometers and seconds. If you look at the LT and perform such a velocity transformation for light from a moving source, you will see that the variances in "kilometers" and "seconds" completely vanish when transforming the speed of light from one frame to another. This is because your kilometers and mine, and your seconds and mine, are different by precisely the same multiplicative factor. So when we divide our respective kilometers and seconds to determine the speed of light, those differences cancel out exactly.

Yes, that is what bothers me. The speed of light is all things to all kilometers and all seconds, however contracted or dilated.

The origin of this, according to selfAdjoint, is that the Maxwell equations "leave no room for" the speed of the emitter or the reciever. I wonder what he was thinking. Then there's this Faraday thing Einstein was trying to clear up, but I don't think he understood the circumstance Faraday was referring to when he said the motion between conductor and magnet wasn't relative. I have to check on that.

 

[quantumdude]

Yes, I understand where you stand on this. I meant, you agree with Lorentz answer to the question "Is length contraction real or apparent?"

My position is in fact much more in line with Eddigton's. The length of a rod is really shorter to a moving observer, and it makes no sense to speak of the length of the rod without specifying an inertial frame from which the length measurement is made.

Yes, that is what bothers me. The speed of light is all things to all kilometers and all seconds, however contracted or dilated.

The length contraction and time dilation does not occur in some arbitrary way though. It occurs precisely in such a way as to preserve the invariance of the speed of light.

Consider a light source that is stationary with respect to observer S. The source is a distance D away from S, and at time t=0 a light pulse is fired. Observer S detects this pulse in a time equal to T=D/c, and of course he measures a velocity of -c (negative because it is moving towards him).

Now let an observer S' be moving with speed v towards the source along the line joining the source and observer S. At time t=t'=0, the origin of S' is coincident with that of S. What speed will S' measure for the light pulse?

Event 1: Pulse Emitted
Spacetime coordinates in S: x₁=D, t₁=0
Spacetime coordinates in S': Use Lorentz transformation.
x₁'=γ(x₁-vt₁)=γ(D-0)=γD
t₁'=γ(t₁-vx₁/c²)(0-vD/c²)

Event 2: Pulse Detected
Spacetime coordinates in S: x₂=0, t₂=D/c
Spacetime coordinates in S': Use Lorentz transformation.
x₂'=γ(x₂-vt₂)=γ(0-vD/c)=-γvD/c
t₂'=γ(t₂-vx₂/c²)=γ(D/c+vD/c²)

Now let the speed of the light pulse in S be u and let the speed of the pulse in S' be u'.

u=Δx/Δt=(0-D)/(D/c-0)=-c

u'=Δx'/Δt'=[γ(-vD/c-D)]/[γ(D/c+vD/c²)]

Now factor the numerator and denominator as follows.

u'=Δx'/Δt'=[γ(v/c+1)[/color](-D)]/[γ(v/c+1)[/color](D/c)]

See how the factors in blue[/color] are exactly the same? They cancel out every time.

So...

u'=(-D)/(D/c)

u'=-c

So to answer your earlier question, yes the speed of light will always come out the same in every frame, and there really isn't any need to do a Lorentz transformation. But that doesn't mean it's not instructive to do the Lorentz transformation.


edit: fixed a subscript bracket

 

[zoobyshoe]

My position is in fact much more in line with Eddigton's. The length of a rod is really shorter to a moving observer, and it makes no sense to speak of the length of the rod without specifying an inertial frame from which the length measurement is made.

I think when you say the length of the rod is "really" shorter you miss the fine point Eddington puts on the matter when he said 1). nothing whatever happens to the length of the rod, because 2.) length is not a property of the rod.

It occurs precisely in such a way as to preserve the invariance of the speed of light.

Yes, this illuminates my point. The LT assumes c is always c and is designed to make it so in all cases.
(Incidently, what do the question marks in your equations represent?)

The time dilation and length contraction can only be figured from c and c, as you have just shown, can be figured from the time dilation and length contraction. The LT is "rigged" so to speak. I find this circular.

There is only one way to make the postulate good, rob peter (time and space) to pay paul (light), and Einstein decided to do that rather than to wonder if the postulate had anything wrong with it. This is why I am wondering what Maxwell was thinking when he "left no room" for addition and subtraction of velocity for light. The postulate must have been primarily intended to conform to Maxwell's thinking because Einstein frequently said he concieved of the postulate prior to hearing about Michelson-Morley, so that MM didn't have much bearing on the matter, except to confirm the postulate was on the right track.
--------
I realize that raising the kinds of questions I do about relativity is regarded as a sign of being a crank or crackpot but the claims SR makes are just too extraordinary for me to accept without completely understanding every relationship of all parts to the whole and the whys of everything. It isn't enough, in my mind, to say that if we just shift our conception of time and space then this postulate about the properties of light can be accommodated with no further problems. Nor is the fact that Lorentz and Einstein found a convenient way to do this anything like a proof, in my mind, that this is what should be done. Before time and length are rendered flexible to accommodate the postulate it seems a much better case has to be made that the postulate needs accomodating. This is why I wonder what Maxwell was thinking when he "left no room" for addition and subtraction of velocities. In other words, I am more prone to look to the work of a human mathametician for something that needs adjustment, than to enormities like time and space.

 

[quantumdude]

I think when you say the length of the rod is "really" shorter you miss the fine point Eddington puts on the matter when he said 1). nothing whatever happens to the length of the rod, because 2.) length is not a property of the rod.

I am not missing any point of Eddington.

The simple fact of the matter is that one is free to measure the length of a rod in the rest frame of the rod. Furthermore, one is free to measure the length of the rod as the rod moves by at a speed v. And when one does measure the length of the rod as it speeds by, one measures its length to be something less than the "proper length".

By any measure, that implies that the rod is "really" shorter, and it does not contradict Eddington in any way, shape, or form.

Yes, this illuminates my point. The LT assumes c is always c and is designed to make it so in all cases.

How does this illuminate your point of "instability" of kilometers and seconds?

(Incidently, what do the question marks in your equations represent?)

There are no question marks in my equations. Perhaps your browser is not interpreting some symbols correctly.

The time dilation and length contraction can only be figured from c and c, as you have just shown, can be figured from the time dilation and length contraction. The LT is "rigged" so to speak. I find this circular.

It is not circular, it is simply a matter of being consistent.

There is only one way to make the postulate good, rob peter (time and space) to pay paul (light),

What are you talking about? The speed of light postulate has been experimentally verified (some decades ago, in fact). The postulate doesn't need anyone to "make it good". The postulate is "good" all by itself.

I realize that raising the kinds of questions I do about relativity is regarded as a sign of being a crank or crackpot

No, your questions do not mark you as a crackpot.

They mark you as someone who is curious about physics, and who needs to spend more time studying it.

A lot more.

 

[Janus]

This is why I am wondering what Maxwell was thinking when he "left no room" for addition and subtraction of velocity for light.

You make it sound as if you think it was a conscious decision on his part to do so; this is not the case. What he did was uncover rules of electromagnetic behavior that already existed.

 

[zoobyshoe]

You make it sound as if you think it was a conscious decision on his part to do so; this is not the case. What he did was uncover rules of electromagnetic behavior that already existed.

Thanks for addressing that. SelfAdjoint's wording suggested the slight possibility to me that "a conscious decision" was behind it, that Maxwell may, somehow, have had the option of including or excluding the addition and subtraction of velocities, and chose not to. You are clearly saying that he had no such option, which is what I was wondering about.

 

[selfAdjoint]

"Divergences" and "curls" are what, terms from calculus?

Scalar meaning having magnitude but no direction, no?

Two four vector equations? I am familiar with the four equations that comprise the Lorentz transformation as presented by Einstein in the above quoted book on relativity. I thought there was just one for each vector: x,y,z,t. What am I misconstruing here?

If you have a three dimensional vector field, say U=(u,v,w), with the components, u, v, and w as functions of x, y, and z, then the divergence of U is
\nabla \cdot U = \frac {\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z}. the curly d's are partial derivatives from multivariable calculus. A divergence is a scalar, a number field.

The curl of U is a vector \nabla \times U = ((\frac{\partial u}{\partial y} - \frac{\partial v}{\partial x}), (\frac{\partial v}{\partial z} - \frac{\partial w}{\partial y}), (\frac{\partial w}{\partial x} - \frac{\partial u}{\partial z})). The three expressions in parentheses are the components of the vector.

So a vector equation has three components, which are scalar equations, but a scalar equation has only one component. Thus the Maxwell equations break down to 2(3) + 2 = 8 scalar equations.

 

[zoobyshoe]

So a vector equation has three components, which are scalar equations, but a scalar equation has only one component. Thus the Maxwell equations break down to 2(3) + 2 = 8 scalar equations.

OK, I see: all the terminology I didn't follow is from calculus.

 

[zoobyshoe]

By any measure, that implies that the rod is "really" shorter, and it does not contradict Eddington in any way, shape, or form.

Eddington does not consider the question "Is the rod really shorter?" a proper question under the circumstances. If he did he could just have said "Yes, it is really shorter." Instead, he phrases the whole situation such that the questioner is diverted from asking about it in those terms, which, he feels, are not enlightening.

How does this illuminate your point of "instability" of kilometers and seconds?

It doesn't. It illuminates my complaint that light doesn't seem to have a property to which the concept of speed can accurately be attached.

There are no question marks in my equations. Perhaps your browser is not interpreting some symbols correctly.

Yes, it is probably my browser. The same thing happened to some equations someone else gave me in another thread.

It is not circular, it is simply a matter of being consistent.

I will mull this over.

What are you talking about? The speed of light postulate has been experimentally verified (some decades ago, in fact). The postulate doesn't need anyone to "make it good". The postulate is "good" all by itself.

What I'm talking about, obviously, is not proving the speed of light postulate, but explaining it in terms of everything else. If the speed of light is the same to all observers in all inertial frames it is doing something it shouldn't be able to do. The speed of light postulate bothered Einstein for something like ten years:

"In short, let us assume that the simple law of the constancy of the velocity of light c (in vacuum) is justifiably believed by the child at school. Who would imagine that this simple law has plunged the conscientiously thoughtful physicist into the greatest intellectual difficulties? Let us consider how these difficulties arise."

Then he goes on to explain how light doesn't comply with the addition and subtraction of velocities. That being the case, he says, we find ourselves faced with an impossibility.

"In view of this dilemma there appears to be nothing else for it than to abandon either the principle of relativity or the simple law of the propagation of light in vacuo."

That's from a chapter entitled "The Apparent Incompatibility of the Law of Propagation of Light with the Principle of Relativity", which is chapter VII of his book Relativity, The Special and the General Theory.

Instead of abandoning either, Einstein has arrived at a brilliant solution that makes both good, an extremely creative and counter-intuitive theory:

"This theory has been called the special theory of relativity to distinguish it from the extended theory, of which we shall deal later. In the following pages we shall present the fundamental ideas of the special theory of relativity."

(Quotes from pages 17, 19, and 20, respectively, of the 1961 edition of that book.)

The speed of light postulate is not "good" by itself. At least, Einstein didn't think so. He felt it needed some intense explaining. So much so, that in order to accommodate it, he felt justified in theorizing that time and space were not the absolute things we think them to be.

Chapter 7. The Apparent Incompatibility of the Law of Propagation of Light with the Principle of Relativity. Einstein, Albert. 1920. Relativity: The Special and General Theory
Address:http://www.bartleby.com/173/7.html

 

[selfAdjoint]

Eddington does not consider the question "Is the rod really shorter?" a proper question under the circumstances. If he did he could just have said "Yes, it is really shorter."

No that is not a proper question. What observer corresponds to "really"? The correct statement is that the observation of someone in another inertial frame, who measures the rod as shorter, is just as valid physics as the observation of someone in the rest frame of the rod, who sees it unshortened. There is no "reality" higher or deeper than this.

 

[quantumdude]

Eddington does not consider the question "Is the rod really shorter?" a proper question under the circumstances. If he did he could just have said "Yes, it is really shorter." Instead, he phrases the whole situation such that the questioner is diverted from asking about it in those terms, which, he feels, are not enlightening.

What I mean is that the length of the rod when it is measured in two different states of motion is really different. This another way of saying that the length of the rod is "indeterminate" (to use Eddington's term) until the state of motion of the rod is specified. Note that, due to the reciprocity of relative velocities, specifying the state of motion of the rod is the same as specifiying the observer.

It doesn't. It illuminates my complaint that light doesn't seem to have a property to which the concept of speed can accurately be attached.

But it does. If light covers a distance of 3x10⁸ m in 1 second, then that's the speed of light, and it's perfectly well-defined.

What I'm talking about, obviously, is not proving the speed of light postulate, but explaining it in terms of everything else. If the speed of light is the same to all observers in all inertial frames it is doing something it shouldn't be able to do.

Says who?

Physics is not an a priori discipline, because the universe is not known a priori. What light should and should not be able to do is determined by experimental results and nothing else.

Then he goes on to explain how light doesn't comply with the addition and subtraction of velocities. That being the case, he says, we find ourselves faced with an impossibility.

It's not an impossibility unless one insists on holding to the notion that velocities should be combined by simple addition and subtraction.

The speed of light postulate is not "good" by itself.

When I say "good", I don't mean "a comprehensive theory of motion", I mean "true".

In that sense, the speed of light postulate is "good" all by itself. It is a feature of the universe we inhabit, and no sleight of hand is required to justify it.

At least, Einstein didn't think so. He felt it needed some intense explaining. So much so, that in order to accommodate it, he felt justified in theorizing that time and space were not the absolute things we think them to be.

You keep posting all this history of the development of SR, but it is really not necessary. We all know that SR in its finished form is necessary to understand the whole picture, and we all know that neither postulate of relativity is consistent with the Galilean transforms.

 

[quartodeciman]

Lewis Epstein in the book "Relativity Visualized" does a trick I have not seen exploited anywhere else.

Instead of explaining the Minkowski 4-space idea with its new ct dimension, he postulates that all motion is in 4 spatial dimensions (x, y, z, s). 's' is, of course, the same as the space-time metric of an event from the zero event in a given frame of reference. But Epstein exploits our more current tolerance of extra spatial dimensions to postulate s as a 4th added to the classical 3. Then he postulates that natural motions of bodies in a framework always have total speed of c through all 4 dimensions. A lightlike object travels only through the first 3; the rate of motion along s is 0. Nothing can travel through the classical 3 dimensions at a faster rate without violating the total speed c assumption. A sub-lightlike object travels through s, meaning the speed through x, y and z must be less than c. In the case that the rate is 0 for x, y and z, the object is stationary in the reference frame and the full speed of c is carried in the s dimension.

Of course, this just transfers the open question from why light travels only at speed c to why things travel at exactly speed c in the 4 dimensions. Also, I don't know just how this helps with handling spacelike intervals of Minkowski theory or General Relativity, and I don't recall that Epstein exploits it there. I decided for myself that Epstein's 4th dimension s is a quadratic slack quantity. Standard linear slack (time expended) is something like one task team of a project having like 3 hours of inactivity before they can continue their contribution to the whole project because some concurrent activity is critical for completion of the current milestone and requires more time.

In this application, the total motion is

x² + y² + z² + s² = (ct)²

. s is just as quadratic slack quantity of incompleted classical spatial motion.

Again, rearrange and get

(ct)² - x² - y² - z² = s²

 

[zoobyshoe]

What I mean is that the length of the rod when it is measured in two different states of motion is really different. This another way of saying that the length of the rod is "indeterminate" (to use Eddington's term) until the state of motion of the rod is specified. Note that, due to the reciprocity of relative velocities, specifying the state of motion of the rod is the same as specifiying the observer.

The way you have just put it here is a good paraphrase of Eddington.

But it does. If light covers a distance of 3x10⁸ m in 1 second, then that's the speed of light, and it's perfectly well-defined.

I haven't managed to express what is confusing me well enought to convey it to you.

Says who?

I think you understand what I'm talking about: the fact the speed of light doesn't behave according to the addition and subtraction of velocities was a great surprise to everyone.

Physics is not an a priori discipline, because the universe is not known a priori. What light should and should not be able to do is determined by experimental results and nothing else.

People have expectations and are surprised if a measurement doesn't support what they expected.

It's not an impossibility unless one insists on holding to the notion that velocities should be combined by simple addition and subtraction.

That is right, but you are trivializing what an enormous step it was for Einstein to decide if that is what was called for or if the postulate or the principle of relativity should be abandoned.

When I say "good", I don't mean "a comprehensive theory of motion", I mean "true".

Yes I know, but I used it first, so I have dibs on the meaning.

In that sense, the speed of light postulate is "good" all by itself. It is a feature of the universe we inhabit, and no sleight of hand is required to justify it.

Again, you are trivializing the fact that to accommodate it Einstein had to theorize that time and space were not absolute. That's interesting. That's exiting. That's controversial. It's not peanuts.

You keep posting all this history of the development of SR, but it is really not necessary.

I posted it to answer your question "What are you talking about?"

The history of SR is actually quite enlightening. Actually, I find all physics history quite enlightening. Einstein appreciated physics history. He co-authored a book called The Evolution of Physics with Leopold Infield. Despite the frustrations relativity causes me, I find Einstein's attitude toward physics to be a very beautiful one.

 

[zoobyshoe]

Then he postulates that natural motions of bodies in a framework always have total speed of c through all 4 dimensions.

Wow! This is interesting.

A lightlike object travels only through the first 3; the rate of motion along s is 0.

s, you said, is the space-time metric. What does it mean about the lightlike object that it has no travel in the dimension of the spacetime metric?

 

[zoobyshoe]

SR predicts that when the observer goes to the track and measures the distance between the marks, he will measure a distance that is equal to L=L₀/γ. According to SR then, the length of the rod moving at speed v is really less than the length of the rod at speed 0.

This won't work, then, because if we have a guy sitting on the rod while it is in motion he will look down at the track and see it is shorter than when at rest. He will expect the flares to leave marks more than a meter apart when he checks afterward in the track rest frame.

 

[Doc Al]

when the relativity bug bites...

SR predicts that when the observer goes to the track and measures the distance between the marks, he will measure a distance that is equal to L=L0/γ. According to SR then, the length of the rod moving at speed v is really less than the length of the rod at speed 0.

This won't work, then, because if we have a guy sitting on the rod while it is in motion he will look down at the track and see it is shorter than when at rest. He will expect the flares to leave marks more than a meter apart when he checks afterward in the track rest frame.

You have forgotten that from the point of view of the observer on the rod, those marks are not made at the same time.

Well, zoobyshoe, it seems like the relativity bug has bitten you again, even though you threw in the towel in this post: https://www.physicsforums.com/showpost.php?p=255677&postcount=112

Have you gotten any of the books I recommended? (https://www.physicsforums.com/showpost.php?p=255747&postcount=114)

 

[quantumdude]

This won't work, then, because if we have a guy sitting on the rod while it is in motion he will look down at the track and see it is shorter than when at rest. He will expect the flares to leave marks more than a meter apart when he checks afterward in the track rest frame.

You have forgotten that from the point of view of the observer on the rod, those marks are not made at the same time.

Right. To the guy on the rod, the measurement is not a length measurement, because the two flares do not ignite simultaneously.

 

[quantumdude]

I think you understand what I'm talking about: the fact the speed of light doesn't behave according to the addition and subtraction of velocities was[/color] a great surprise to everyone. People have expectations and are surprised if a measurement doesn't support what they expected.

Yes, and the key word is "was[/color]". After a hundred years, peoples' expectations have changed. Now it seems strange to physicists for the speed of light not to be Lorentz invariant.

Tom: It's not an impossibility unless one insists on holding to the notion that velocities should be combined by simple addition and subtraction.

zoobyshoe: That is right, but you are trivializing what an enormous step it was for Einstein to decide if that is what was called for or if the postulate or the principle of relativity should be abandoned.

I'm not trivializing it, but I also do not allow it to stunt my growth. You could make the above remark about any of the theories of modern physics. But at some point, acceptance has to take hold, and we have to get on with the business of doing physics.

Yes I know, but I used it first, so I have dibs on the meaning.

OK, fine: The SOL postulate by itself is not enough to explain what we observe. But as I keep remarking, we all know that.

Again, you are trivializing the fact that to accommodate it Einstein had to theorize that time and space were not absolute. That's interesting. That's exiting. That's controversial. It's not peanuts.

I agree that it's interesting, but it's also old news.

And in fact, the more you get used to SR, the more strange Galilean relativity seems. One can be totally baffled by SR thinking, "How can the photon have the same speed in every frame? That shouldn't be! How can the Lorentz transform be right?"

Once you get past it, the other point of view seems nonsensical, and one will be wondering, "How can inertial frames be distinguished without making reference to an external point. That shouldn't be! How can the Galilean transform be right?"

Tom: You keep posting all this history of the development of SR, but it is really not necessary.

zoobyshoe: I posted it to answer your question "What are you talking about?"

I just needed to know what you meant by making the SOL postulate "good".

 

[zoobyshoe]

Now let the observer ignite the flares simultaneously, in his frame (The reason for simultaneous ignition is that it is the only way you could correctly say that the distance between the marks is equal to the length of the rod).

Tom,

How does the rail observer have everything arranged so that the flares are seen by him to ignite simultaneously in his frame?

 

[quantumdude]

Tom,

How does the rail observer have everything arranged so that the flares are seen by him to ignite simultaneously in his frame?

It makes no difference to the thought experiment what the actual mechanism is, but...

He could have identical wires connected to the flares, and send an electrical signal from a common source to ignite the flares simultaneously.

 

[zoobyshoe]

He could have identical wires connected to the flares, and send an electrical signal from a common source to ignite the flares simultaneously.

Yes, but if he's interested in visually confirming that the flares are simultaneous in his rest frame, he has to be positioned equidistant from the flares when they go off. The light must have a path from the front flare to him that is equal in distance from the rear flare to him. Only by knowing that these distances are equal when he causes the emission (and we can stipulate he has the ability to hit the button at the right time to make it so) will he know that the flares were simultaneous because the light from both will reach him simultaneously, no?

 

[quantumdude]

Yes, but if he's interested in visually confirming that the flares are simultaneous in his rest frame, he has to be positioned equidistant from the flares when they go off. The light must have a path from the front flare to him that is equal in distance from the rear flare to him.

OK, fine let him be equidistant from the flares.

Only by knowing that these distances are equal when he causes the emission (and we can stipulate he has the ability to hit the button at the right time to make it so) will he know that the flares were simultaneous because the light from both will reach him simultaneously, no?

No. He doesn't have to actually see the light pulses at the same time. In fact, he needn't see them at all. He can know that the pulses ignited simultaneously with photodetectors that are connected to clocks. The photodetectors need not even be equidistant from the simultaneous flash. All that needs to be known is the position of each photodetector, the position of the flash, and the time at which the button was hit.

 

[zoobyshoe]

OK, fine let him be equidistant from the flares.



No. He doesn't have to actually see the light pulses at the same time. In fact, he needn't see them at all. He can know that the pulses ignited simultaneously with photodetectors that are connected to clocks. The photodetectors need not even be equidistant from the simultaneous flash. All that needs to be known is the position of each photodetector, the position of the flash, and the time at which the button was hit.

Ok, I'm confused about in which reference frame you want the flares to be simultaneous. It sounds like you want them to be simultaneous in the rods reference frame.

edit: You mean the detector/clocks are in the track reference frame with him?

 

[quantumdude]

Ok, I'm confused about in which reference frame you want the flares to be simultaneous. It sounds like you want them to be simultaneous in the rods reference frame.

No, the flares should ignite simultaneously in the frame of the track (the same frame as the guy who is watching the rod speed by). That's the only way his measurement could be called a "length" measurement.

edit: You mean the detector/clocks are in the track reference frame with him?

Yes.

 

[zoobyshoe]

No, the flares should ignite simultaneously in the frame of the track (the same frame as the guy who is watching the rod speed by). That's the only way his measurement could be called a "length" measurement.

Good.

The clock/detectors are more realistic, but for simplicity I'd rather stick with him being equidistant from the flares when they go off.

It is no problem to arrange it so the midpoint of the rod is directly in front of him when the flares ignite. This way the light has a path of equal length from each flare to him. If both beams of light reach him at the same instant, he knows the emission was simultaneous. Which it is.

The guy on the rod could be anywhere on the rod, but if we put him exactly at the midpoint of the rod something very interesting results: he too, receives the beams of light simultaneously.

That seems ridiculous because he is in motion and should, intuitively reasoning, encounter the forward beam before the rear beam. However, that would mean you are adding and subtracting velocities, which you can't do with light. The speed of light is the same to all observers in all inertial frames. He can't encounter the forward beam at c plus his own velocity. He must regard himself as being at rest with the light approaching him at c, and the end of the rod approaching him at the velocity of the rod over the track.

The beam from the rear cannot approach him at less than c just because he is moving away from it. He must regard himself as at rest with the light approaching him at c and the back end of the rod moving away from him at the velocity of the rod over the track.

He is at the exact centerpoint of the rod, both beams of light have the same distance to travel to reach him, both will do it at the same velocity, c, both will reach him simultaneously.

Looking down, he will see a contracted rail and will expect the flare marks to measure greater than a meter apart when he goes back to measure them later in the rail rest frame.

 

[quantumdude]

The guy on the rod could be anywhere on the rod, but if we put him exactly at the midpoint of the rod something very interesting results: he too, receives the beams of light simultaneously.

No, he doesn't, because the flares don't ignite simultaneously in his frame.

 

[zoobyshoe]

No, he doesn't, because the flares don't ignite simultaneously in his frame.

Let the time t when the midpoint of the rod is directly in front of the track observer be the point of synchrony between any clock that observer may have and any clock the rod observer may have such that t=o, the time of emission, is the point of synchrony of clocks for both observers.

 

[quantumdude]

Let the time t when the midpoint of the rod is directly in front of the track observer be the point of synchrony between any clock that observer may have and any clock the rod observer may have such that t=o, the time of emission, is the point of synchrony of clocks for both observers.

It doesn't matter when you zero both clocks. The flares ignite simultaneously in exactly one frame, and no other. If the flares were at the same location, then their ignition would be simultaneous in all frames. But that isn't what is happening here--these flares are spatially separated.

Let Event 1 be the rear flare igniting and Event 2 be the front flare igniting.

For observer S (the guy watching the rod zip by):
Δx=x₂-x₁=L (the length of the rod according to him)
Δt=t₂-t₁=0 (the flares ignite simultaneously according to him)

For observer S' (the guy standing on the rod):
Δt'=t₂'-t₁'=γ(Δt-vΔx/c²)
Δt'=γ(0-vL/c²)
Δt'=-γvL/c²

See? Δt' is negative (not zero). This means that, according to the guy on the rod, event 1 (the rear flare igniting) occurs later than event 2 (the front flare igniting). If the two flares ignite at different times, and the observer is at their midpoint, then there is no way that he is going to receive both pulses simultaneously.

 

[zoobyshoe]

Tom,

I'm having question mark troubles again. I'm going to type in an example of what your equations look like, including question marks:

?t'=t₂ ' -1₁'=?(?t-v?x/c²)

Does this show up on your screen with 4 question marks in the equation? If so, that's what it looks like to me. If not my browser lacks some ability to properly translate whatever it is you are typing in. Latex seems to work fine on my browser.

 

[quantumdude]

?t'=t₂ ' -1₁'=?(?t-v?x/c²

Does this show up on your screen with 4 question marks in the equation?

Yes, I see 4 question marks. I haven't learned how to use LaTeX yet ( ), but the equation should look like this:

Delta(t')=t₂'-t₁'=(gamma)(Delta(t)-vDelta(x)/c²)

It's the Lorentz transformation for a time interval. It's the "interval" version of equation 1d on this page:

http://www.physics.nyu.edu/courses/V85.0020/node45.html