Am I interpreting Einstein correctly?

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bcrowell
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In section 8 of "On the electrodynamics of moving bodies," http://www.fourmilab.ch/etexts/einstein/specrel/www/ , there is a discussion of the fact that the energy of an electromagnetic wave scales by the same Doppler-shift factor as its frequency when you change frames of reference. If this hadn't been true in classical E&M, then there would have been no way for the quantization relation E=hf to be valid in all frames of reference.

What caused me endless confusion was the equation [itex](x-lct)^2+(y-mct)^2+(z-nct)^2=R^2[/itex]. I thought Einstein was talking about an expanding spherical wavefront, and I couldn't make much sense out of anything after that. Finally I decided that he must have been talking about a plane wave, and the sphere is just the shape of a region he's arbitrarily chosen out of the plane wave. Have I got this right? He does say "moving" ("bewegten") with the speed of light, not "expanding" with the speed of light.

If I've got this right, then is the following commentary wrong?
Neuenschwander, http://www.spsnational.org/radiations/2006/ecp_s06.pdf [Broken]
He refers to a "pulse emitted isotropically in all directions." On a more minor note, it seems like the subscripts in eq (23) are a mistake.

I was thinking of contacting Neuenschwander to point out the mistakes, but I wanted to make sure I had it right first.

This may also be helpful: Redzic and Strnad, "Einstein's light complex," http://fizika.phy.hr/fizika_a/av04/a13p113.pdf

Thanks!

-Ben
 
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  • #2
atyy
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I could not understand that part either.

With respect to E=hf, I believe the two things to be distinguished are the energy density and the energy. The energy density is some term in the stress tensor, and I believe it doesn't transform correctly to be E. The energy E is the thing associated to a wave packet (not a plane wave) that is wavy enough to be almost a plane wave, but finite enough in extent to be integrated over so that one gets its energy.
 
  • #3
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I don't think he is talking about any particular shape of a wave; I think he is just talking about a region which expands spherically at c and therefore there is no energy from the wave outside of this region regardless of the geometry of the wave itself. However, it is rather confusingly written and I could be misinterpreting.
 
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I don't think he is talking about any particular shape of a wave; I think he is just talking about a region which expands spherically at c and therefore there is no energy from the wave outside of this region regardless of the geometry of the wave itself. However, it is rather confusingly written and I could be misinterpreting.
That was what I thought at first. If it's not a plane wave, then l, m, and n would have to be different at different points in space, but there's no hint of that. Also, note that he treats R as a constant with respect to time, and that wouldn't make sense if the wave was expanding spherically.

I could not understand that part either.

With respect to E=hf, I believe the two things to be distinguished are the energy density and the energy. The energy density is some term in the stress tensor, and I believe it doesn't transform correctly to be E. The energy E is the thing associated to a wave packet (not a plane wave) that is wavy enough to be almost a plane wave, but finite enough in extent to be integrated over so that one gets its energy.
I suspect that there is probably a much simpler way of deriving this result if you have more of the apparatus of relativity available, such as the stress-energy tensor. Tensor notation certainly allow easier ways of deriving the Doppler shift than the technique Einstein used.
 
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In section 8 of "On the electrodynamics of moving bodies," http://www.fourmilab.ch/etexts/einstein/specrel/www/ , ....

What caused me endless confusion was the equation [itex](x-lct)^2+(y-mct)^2+(z-nct)^2=R^2[/itex].
Well, you have sparked my interest.

It seems to me that Einstein is talking about an arbitrary ray of an expanding spherical wavefront. What I don't understand, is why he would use the same direction cosines in system K that were defind in system k, given that frame rotation would require said 2 rays to not be the same ray? However, it is true that no energy passes thru said spherical surface, because said surface is defined by the expanding EM itself as it goes. I presume x,y,z is the spatial cooridinate of the POE, yes? I never looked at this section closely before, so now would be a good time.

GrayGhost
 
  • #6
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It seems to me that Einstein is talking about an arbitrary ray of an expanding spherical wavefront. What I don't understand, is why he would use the same direction cosines in system K that were defind in system k, given that frame rotation would require said 2 rays to not be the same ray? However, it is true that no energy passes thru said spherical surface, because said surface is defined by the expanding EM itself as it goes. I presume x,y,z is the spatial cooridinate of the POE, yes? I never looked at this section closely before, so now would be a good time.
This was how I first interpreted it, but I don't think that's what it really means. I think it's a plane wave, not an expanding spherical wavefront. The sphere could just as easily have been a rectangular box. If it was an expanding spherical wavefront, then the direction cosines would have to vary from point to point in space, and R would have to vary as ct -- but that isn't the case. I don't think x,y,z is the point of emission; I think he's simply giving the equation of the moving spherical surface in terms of x, y, and z. The reason R and the direction cosines also don't transform when you change from k to K is that they're simply constants that were used to define the world-tube of the traveling spherical surface.
 
  • #7
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This was how I first interpreted it, but I don't think that's what it really means. I think it's a plane wave, not an expanding spherical wavefront. The sphere could just as easily have been a rectangular box.
Well, I see where you are trying to come from there. However, all indications are (to me) that he's discussing a single arbitrary ray of an expanding spherical wavefront, and all rays may be independently considered as they all build the spherical EM surface at any time t. He does begin section 8 speaking of a volume defined in part by the value pi, which suggests a sphere. Just before stating the eqn, he states ...

If l, m, n are the direction-cosines of the wave-normals of the light in the stationary system, no energy passes through the surface elements of a spherical surface moving with the velocity of light: ... (x-lct)2+(y-mct)2+(z-nct)2 = R2.​

IMO, he's defining the spherical surface AS the leading edge of the expanding spherical EM wavefront itself, and hence the EM energy can ever pass thru it (because the surface moves with the wavefront at all times).

If it was an expanding spherical wavefront, then the direction cosines would have to vary from point to point in space, and R would have to vary as ct -- but that isn't the case.
Indeed, l m n may each vary for each unique arbitrary ray considered, however the direction cosines should never change for any single arbitrary ray of the expanding EM wavefront. Consider "any other" arbitrary ray and indeed the values of l, m, n all change, but that's fine. Doing so does not change the location of the expanding EM spherical surface in spacetime, given a specific arbitrary time t is considered. It changes only the single ray considered, which of course has a different path thru spacetime and thus a different set of l,m,n, yet it remains at length R=ct. The eqn is that of a spherical surface about a POE in system K, far as I can tell.

Far as R goes, R "should" change as t increases. However, the range R pertains always to any single arbitrary ray considered over time t. For any given time t0+t, any arbitrary ray of the expanding EM wavefront has a pathlength of length R = ct. Hence, R also applies to any other arbitrary ray of the expanding EM wavefront, and therefore applies to the expanding spherical surface (of the EM wavefront, and wrt the POE) on the whole ... for any time t.

I don't think x,y,z is the point of emission; I think he's simply giving the equation of the moving spherical surface in terms of x, y, and z.
I think both are true. It seems to me that "the spherical surface eqn" is built relative to a POE defined in system K at an arbitrary coordinate x,y,z which may be taken as origin for simplicity.

The reason R and the direction cosines also don't transform when you change from k to K is that they're simply constants that were used to define the world-tube of the traveling spherical surface.
Well, the direction cosines "are components of" the coordinates defining the ray's end point in x,y,z at time t, and so they are (in essence) transformed when the ray's endpoint (an x,y,z coordinate at time t) is transformed into terms of X,Y,Z.

His first equation defines a spherical surface in stationary system K for any arbitrary time t (ie t held constant), and the direction cosines are defined wrt that system (K) ...

(x-lct)2+(y-mct)2+(z-nct)2 = R2

He then represents this eqn in terms of system k variables, where g = gamma ...

(gX-gX*lv/c)2+(Y-gY*mv/c)2+(Z-gZ*nv/c)2 = R2

Which may be shown to be the formula for an ellipsoid as follows ...

(X(g-g*lv/c))2+(Y(1-g*mv/c))2+(Z(1-g*nv/c))2 = R2

X2((g-g*lv/c)/R)2+Y2((1-g*mv/c)/R)2+Z2((1-g*nv/c)/R)2 = 1​

X2/(R/(g-g*lv/c))2+Y2/(R/(1-g*mv/c))2+Z2/(R/(1-g*nv/c))2 = 1​

X2/a2+Y2/b2+Z2/c2 = 1​
... the formula for an ellipsoid.

So as stated by Einstein ... The spherical surface—viewed in the moving system—is an ellipsoidal surface ... .

In my prior post, I stated that it didn't makes sense that he could use the same direction cosines in the transformed eqn. I was mistaken wrt that. He can do so, and he should, and he did. So he represents the same lightsphere of system K in terms of variables of the moving system k, for a specific value t considered. The relation between sphere (in K) and ellipsoid (in k) is the same no matter what arbitrary moment in time is considered, and so Einstein tactically considers Tau=0 =t for the simplest case. That allows him to attain the ellipsoid eqn in its simplest form.

Yes?

GrayGhost
 
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  • #8
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The short version ...

In section 8 of "On the electrodynamics of moving bodies," http://www.fourmilab.ch/etexts/einstein/specrel/www/ , there is a discussion of the fact that the energy of an electromagnetic wave scales by the same Doppler-shift factor as its frequency when you change frames of reference. If this hadn't been true in classical E&M, then there would have been no way for the quantization relation E=hf to be valid in all frames of reference.
I would think it did scale by the doppler-shift factor in classical E&M, however said factor did not include relativistic redshift component. no?

What caused me endless confusion was the equation [itex](x-lct)^2+(y-mct)^2+(z-nct)^2=R^2[/itex]. I thought Einstein was talking about an expanding spherical wavefront, and I couldn't make much sense out of anything after that. Finally I decided that he must have been talking about a plane wave, and the sphere is just the shape of a region he's arbitrarily chosen out of the plane wave. Have I got this right? He does say "moving" ("bewegten") with the speed of light, not "expanding" with the speed of light.

If I've got this right, then is the following commentary wrong?

Neuenschwander, http://www.spsnational.org/radiations/2006/ecp_s06.pdf [Broken]
He refers to a "pulse emitted isotropically in all directions." On a more minor note, it seems like the subscripts in eq (23) are a mistake.

I was thinking of contacting Neuenschwander to point out the mistakes, but I wanted to make sure I had it right first.
I think Neuenschwander has it right. I believe Einstein was addressing an expanding sphertical EM wavefront, defining the region of space that encompasses it in system K, and then showing that said spherical-surface-of-space in system K (at an arbitrary instant t) is an ellipsoidal-surface-of-space in system k mapped across a duration of time Tau.

GrayGhost
 
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  • #9
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[..] I believe Einstein was addressing an expanding sphertical EM wavefront, defining the region of space that encompasses it in system K, and then showing that said spherical-surface-of-space in system K (at an arbitrary instant t) is an ellipsoidal-surface-of-space in system k mapped across a duration of time Tau.

GrayGhost
I agree: it's a continuation of section 7 in which Einstein examined light rays in all directions coming from a star (although he did not mention "star").
In section 8 he explicitly defines it as a "spherical surface moving with the velocity of light" which "permanently encloses the same light complex"; I see no reason to think that that was a mistake.
Did I miss something?

Harald
 
  • #10
bcrowell
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I agree: it's a continuation of section 7 in which Einstein examined light rays in all directions coming from a star (although he did not mention "star").
This is an excellent point about considering the context, and yes, it clearly is a continuation of section 7 -- but the equations in section 7 describe a plane wave.

Well, it looks like I'm in a position where I've solicited other people's opinions on this, have received a unanimous response, and am convinced that the unanimous response is wrong :-) But I am grateful to those who posted, since it gave me a chance to see arguments to the contrary -- which I then decided didn't convince me. There's a big difference between having a chance to weigh arguments against one's position and not having a chance to do so, since one can't anticipate what they would have been.
 
  • #11
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Well, it looks like I'm in a position where I've solicited other people's opinions on this, have received a unanimous response, and am convinced that the unanimous response is wrong :-)
No longer unanimous. I agree with you. It's a spherical surface of constant radius R moving with the same vector velocity as a continuous planar light wave. Under those conditions there is no flow of energy across the surface.

The equation of an expanding spherical wavefront would be of the form

[tex](x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2 = c^2(t-t_0)^2[/tex]​

WARNING: earlier today I tried to open the Neuenschwander ref quoted earlier on my employer's computer and received a virus alert with the cryptic reason "McAfeeGW: Heuristic.BehavesLike.Exploit.PDF.CodeExec.FFE". In view of that, I haven't tried to open it in any other way.
 
  • #12
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No longer unanimous. I agree with you. It's a spherical surface of constant radius R moving with the same vector velocity as a continuous planar light wave. Under those conditions there is no flow of energy across the surface.

The equation of an expanding spherical wavefront would be of the form

(x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2 = c^2(t-t_0)^2​

WARNING: earlier today I tried to open the Neuenschwander ref quoted earlier on my employer's computer and received a virus alert with the cryptic reason "McAfeeGW: Heuristic.BehavesLike.Exploit.PDF.CodeExec.FFE". In view of that, I haven't tried to open it in any other way.
Hmmm. Never fails, someone's always got another POV. Did Einstein draft the equation backwards? Well, how's about this one then ...

We have ...

(x-lct)2+(y-mct)2+(z-nct)2 = R2

where R = ct, and where t may be taken as t-t0.

Let's assume we consider the expanding EM spherical surface at some arbitrary time t.
Point x,y,z is an arbitrary point on the EM surface at time t.
The POE is then ...

x0,y0,z0 = lct,mct,nct​

In which case, it remains the formula for a spherical surface. Yes? I mean, for any point of the EM surface, the direction cosines l,m,n will always be such that the POE is right where it should be.

PS ... My Norton 360 antivirus gave no virus alert on the Neuenschwander related site. Not that Norton is perfect :)

GrayGhost
 
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  • #13
DrGreg
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We have ...

(x-lct)2+(y-mct)2+(z-nct)2 = R2

where R = ct, and where t may be taken as t-t0
But that would be a sphere with one point (0,0,0) at rest in the frame and the diametrically opposite point (2lct, 2mct, 2nct) moving at twice the speed of light. That doesn't seem to match Einstein's verbal description.

In my description of a spherical wavefront, all the subscripted quantities are constants. They must be, because of the isotropy of light speed.
 
  • #14
atyy
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I'm still not sure what Einstein was saying, but bcrowell's and Dr Greg's interpretation are closer to what I think (as I stated in #2) is conceptually needed to get E=hf (wave packet, close enough to plane wave to define frequency, but limited enough in extent to be enclosed).
 
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But that would be a sphere with one point (0,0,0) at rest in the frame and the diametrically opposite point (2lct, 2mct, 2nct) moving at twice the speed of light. That doesn't seem to match Einstein's verbal description.
Well, I'm not sure what to make of this. It sounds no different from saying ... let's consider the point 2ct, then ask ... how fast would light have to travel to get there in time t? Answer ... 2c. Well, of course. However, neither the question or answer have anything to do with the model at hand.

In my description of a spherical wavefront, all the subscripted quantities are constants.
OK. This makes sense. The POE cannot contain time t because POEs cannot move over time. Indeed, it would seem that Einstein's equation is not the standard equation of a spherical surface. It's not even the standard equation for the distance between 2 points. My opinion, is that Einstein assumes the POE to be x,y,z = 0,0,0. Then his equation does model the surface of a sphere. It also would then model the distance between 2 points, the ends of a ray over time t within an expanding EM spherical wavefront. As we all know, Einstein's math was not advertised as good, even per himself. He also likes to simplifiy everything as much as possible (eg Tau=0, x' =0, etc). I have little doubt that he was envisioning an expanding EM wavefront here. There is no conflict in assuming such, far as I can see.

Interesting!

GrayGhost
 
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I suppose it could have been drafted this way ...

(x-(x-lct))2+(y-(y-mct))2+(z-(z-nct))2 = R2

which is ...

(x-x0)2+(y-y0)2+(z-z0)2 = R2

where R=ct, where x,y,z is an arbitrary point on the EM surface at arbitrary time t, and where the direction cosines l,m,n change for each point considered, always guaranteeing the POE (x0,y0,z0) remains fixed in space.

Yet, Einstein's eqn was simpler, and everyone likes simple.

GrayGhost
 
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  • #17
DrGreg
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Indeed, it would seem that Einstein's equation is not the standard equation of a spherical surface.
Of course it is. I can't see why you are having so much difficulty with this. [itex](x-lct)^2+(y-mct)^2+(z-nct)^2=R^2[/itex] is the standard equation for a sphere with centre [itex](lct, mct, nct)[/itex] (therefore moving at the speed of light) and radius R. The only thing in question is whether R is constant or varying over time. I believe it to be constant: that's the only option if every point of the surface is moving at the speed of light.
 
  • #18
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There is a surprising number of different opinions about the meaning of that equation...
However we know the solution, from which it should be possible to work our way back:

- He appears to do a Lorentz Transformation at t'=0.
From that I conclude that both the light complex and the surface exist at t=0.

- The equation [itex](x-lct)^2+(y-mct)^2+(z-nct)^2=R^2[/itex] simplifies at t=0 to x2 + y2 + z2 = R2.
That is the equation of the surface of a sphere: x, y, z are the coordinates of all the points on its surface.
Apparently Einstein chose the point of light emission at the origin for this problem.

- For t>0, the expanding surface is conform the full equation.
That makes sense to me. :smile:

However, Neuschwander still appears to have made a mistake in this part, for he writes:
"the (x, y, z) coordinates of a point on the spherical surface of this light pulse are given by (lct,mct,nct)" (I harmonized it to the letters we use here, following the 1905 translation).
But then every term would be zero, and instead of R2 we get 0 as sum (right?!) - that is not what Einstein has, and zero surface at t=0 is not useful for the transformation.

Regards,
Harald

PS. if you smell a rat... it could be a mouse :biggrin:
 
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  • #19
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Of course it is. I can't see why you are having so much difficulty with this. [itex](x-lct)^2+(y-mct)^2+(z-nct)^2=R^2[/itex] is the standard equation for a sphere with centre [itex](lct, mct, nct)[/itex] (therefore moving at the speed of light) and radius R. The only thing in question is whether R is constant or varying over time. I believe it to be constant: that's the only option if every point of the surface is moving at the speed of light.
OK ... the standard form of a spherical surface is ...

(x-x0)2+(y-y0)2+(z-z0)2 = R2

The subscripted quantities are constant (as you said), given they are the POE and the POE never moves in space per inertial observers.

Einstein's eqn ...

(x-lct)2+(y-mct)2+(z-nct)2 = R2

However according to the standard form of the spherical eqn, the point x,y,z would be an arbitrary point on the spherical surface at time t, with x0,y0,z0 (which is lct,mct,nct in Einstein's eqn) being the POE. However if the POE is defined by t, then it moves thru space over time ... which in technical terms "is a no no" :)

So it's rather clear to me that Einstein assumes x,y,z = 0,0,0 (not x0,y0,z0) as the unchanging POE and x-lct,y-mct,z-nct being a point upon the EM spherical surface (wrt the zero origin) at time t.

That said, your prior post gave me the impression you disagreed that Einstein's eqn represented a sphere, but now you are saying you do believe it does represent a sphere. I've always thought it represented a sphere, however I never scrutinized it up against the standard form of a spherical suface eqn.

So, it sounds like you agree that the surface is a spherical one. Do you still disagree that the EM complex is an expanding spherical wavefront, or not? (I do)

GrayGhost
 
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  • #20
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He appears to do a Lorentz Transformation at t'=0.
From that I conclude that both the light complex and the surface exist at t=0.
Kinda, but not really. It means only that the relation-of-proportionailty alwasy exists between sphere in K and ellipsoid in k, even if a zero dimensional point in time is considered.

In section 3, Einstein took x' to be zero too, which means the separation between POE (emitter) and POR (reflector) are zero. Similarly, he took t'=0 here, and for the same reason. The proportional relationship between the sphere and ellipsoid do not depend on time, and so the ellipsoid's form may be considered even at time zero. Technically, it would not yet exist, however considering the ellipsoid at time t=t'=0 allows components of the ellipsoid related to time to drop out, leaving only components of space and motion. That puts the eqn into its simplest form, an equation of proportionality that relates the sphere in K to an ellispoid in k.

The equation [itex](x-lct)^2+(y-mct)^2+(z-nct)^2=R^2[/itex] simplifies at t=0 to x2 + y2 + z2 = R2.
That is the equation of the surface of a sphere: x, y, z are the coordinates of all the points on its surface. Apparently Einstein chose the point of light emission at the origin for this problem.
True, however you are assuming here that the POE occurs at the origin x,y,z=0,0,0 and in which case the sphere has a radius of zero. However, the standard form of a spherical surface does not restrict us to the origin. Any point in the system may be considered the sphere's center, and the equation still works. I mean, try considering Einstein's eqn if the POE were not at x,y,z=0,0,0 and then you'll see the problem. I agree though, that Einstein assumed the origin at x,y,z = 0,0,0 ... and although he did not use a general form of the eqn for a sphere, he is indeed representing a sphere.

For t>0, the expanding surface is conform the full equation.
That makes sense to me.
Me too, assuming the POE is x,y,z = 0,0,0. However again, this is not the standard formula for a spherical surface. In the standard formula, the point x,y,z is an arbitrary point on the surface of the sphere, and x0,y0,z0 is the center of the sphere. In Einstein's setup, the sphere's center is a POE, and should not move per any inertial POV. Does anyone care that Einstein didn't state his eqn consistent with the general form of a sphere? It doesn't matter to me much, but we would not be discussing it here if everyone felt like I did. It's because of the fact that his eqn was not of "the standard/general spherical form", that folks are suggesting Einstein's EM complex was not necessarily spherical.

However, Neuschwander still appears to have made a mistake in this part, for he writes:
"the (x, y, z) coordinates of a point on the spherical surface of this light pulse are given by (lct,mct,nct)" (I harmonized it to the letters we use here, following the 1905 translation).
But then every term would be zero, and instead of R2 we get 0 as sum (right?!) - that is not what Einstein has, and zero surface at t=0 is not useful for the transformation.
I disagree Harry. While it is true that R2=0 at t=t'=0, this does not matter for Einstein's purpose at hand here. He simply wants to show that the EM spherical surface that exists in system K at an instant t, must be recorded as an ellispoidal surface in system k at corresponding time t'. Had Einstein not taken t'=0, it would change nothing, because "all more time does is makes the sphere and related ellipsoid larger", it does not change the precise proportionality between ellipsoid of k and sphere of K. Taking t'=0 only allows time to drop from the eqn, thereby simplifying it. It's then an eqn of proportionality between sphere and ellipsoid as opposed to a relation of both "proportionality and size".

Again, in section 3 Einstein similarly considered x' infinitesimally small, or x'=0. Well, this means there is no time at all for the ray to complete the roundtrip interval, because the emitter/reflector separation = 0. That allowed x' to drop from the eqn, thereby simplifying it. The reason he could do so, is because the linear relationship between Tau and x,y,z,t remains true at all times and for all durations, even if a 0d point in time is considered. Consider y=mx+b. We may consider x at a single point, yet the slope is still m. One might draft y=mx+b for 1<x<5. Yet, if we know that the relation holds "for all values of x" (x<1 and x>5), then we can consider the point x=1 alone (or even x=0), and the slope is still m. Even if the sphere in K has a radius of zero, it must transform to an ellipsoid in system k. Einstein did not care, because he was interested only in establishing "the form" of the ellipsoid, not its size. The form mathematically exists even when the size is zero.

GrayGhost
 
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Yes. That section of Einstein's paper is discussed in

http://www.mathpages.com/home/kmath354/kmath354.htm
Samshorn,

Interesting. I'll have to think on that further before accepting that that's what Einstein's intent was. I've just begun looking at section 8 closely for the first time here, and I had tried to imagine the same thing as presented in your hyperlink reference there, but it just didn't make sense to me to view the situ as such. It seemed to me that a spherical surface moving at speed c in system K along +x, would be a zero length ellipsoid in system k with no volume. However, I now see the mistake there. The speed c spherical surface is not a material body, but rather a region of space. So a speed c spherical region-of-space in system K is a speed c ellipsoidal region-of-space in system k. That being the case ...

here's a question ...

What does R then represent? Is it the radius of the comoving sphere? If not, is it ever stated what said comoving sphere's radius is? Or, is R the range of the comoving sphere's center from the system origin? Or, is R the radius of an expanding EM wavefront (wrt the system origin) that only appears (after enough time) to be a plane-wave?

GrayGhost
 
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What does R then represent? Is it the radius of the comoving sphere?
Yes.
 
  • #24
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Yes.
Well, I'll be. Looks like Bcrowell and DrGreg have it right. Thanx for the hyperlink reference Samshorn.

So, there is no POE at the origin, necessarilly. The center of the sphere may be considered the center of the speed c light complex (or something thereof), the light complex having passed thru said origin prior (or emitted from there). The plane wave's location is ct from said system origin at time t. R is a fixed arbitrary radius of the co-moving spherical region-of-space (which does not expand), and we don't care much what the fixed value of R is, so long as the EM complex is always contained therein.

One point though ... Einstein's early thought experiment as to what a beam of light would look like if you could ride it, does not seem (to me) to be the same thing as considering the energy within a spherical region of space comoving with an EM complex at c.

GrayGhost
 
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Finally I decided that he must have been talking about a plane wave, and the sphere is just the shape of a region he's arbitrarily chosen out of the plane wave. Have I got this right?
At first I thought I was wrong, and later realized I was merely mistaken :)

It would seem you have it right Ben. It does make sense, given how Einstein drafted the equation.

GrayGhost
 

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