Derivation of Lorentz Transformation

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by Jim</i>
<br />The need is to make 1+1=1 in SR is it not? The transformation is from 1+1=2 to 1+1=1 how else can you do this?
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The point is that the addition of velocities is not defined as far as the speed of light is concerned. For usual objects we can add two velocities v1 and v2 to v1+v2, but not if the speed of light is involved i.e. you can not write v1+c as it contradicts the invariance of c. Einstein simply ignored this and tried to do the velocity addition regardless. This then forced him to re-scale the time and space units, which, as shown on my pages regarding the Speed of Light and Lorentz Transformation respectively is both physically and mathematically inconsistent.
It is obvious that the only mathematically and physically consistent way to have a speed of light independent of any motion of source or observer is by dropping the usual notion of 'speed' and assume that the travel time of the light signal depends only on the distance between both at the moment the signal is emitted (with any velocities not entering the problem at all).

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The distance between two objects sending out light will change by the time the light signal arrives. If two light emiters are moving away from each other a half the speed of light then by the time signal gets to the receiver the distance is greater-isn't it? What if the two emiters are moving away at nearly the speed of light? The signal would never arrive would it? It seems to me this is a problem with an expanding universe which is moving away from every object in the universe at the speed of light.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by Jim</i>
<br />The distance between two objects sending out light will change by the time the light signal arrives. If two light emiters are moving away from each other a half the speed of light then by the time signal gets to the receiver the distance is greater-isn't it?<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">
The distance will change whilst the light signal is travelling but this does not affect its arrival time.
Consider the source at a distance x from you and moving away from you with velocity v. Now let's first assume that the source sends out a particle (rather than a light signal) towards you with velocity u (in its frame). The velocity of the particle with regard to you is then u-v and it will reach you after time t=x/(u-v) (i.e. the velocity v does affect the arrival time). Now this principle of vectorial velocity addition can not be applied to a light signal and its velocity is also u with regard to yourself (invariance of the speed of light), i.e. it will reach you after time t=x/u (or rather t=x/c using the standard notation for the speed of light).

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Thomas,

Things like this are theory dependent. Your analysis is correct under the assumptions of Special relativity (velocity is relative to the observer), but not under the assumptions of Lorentzian relativity (velocity is relative to the local gravitational potential field). This means that c is constant for ALL observers using SR, but constant only for observers in the same gravity well using LR.

To date, all of the experiments we have done support both theories ...

LB

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by Larry Burford</i>
<br />Thomas,

Things like this are theory dependent. Your analysis is correct under the assumptions of Special relativity (velocity is relative to the observer), but not under the assumptions of Lorentzian relativity (velocity is relative to the local gravitational potential field). This means that c is constant for ALL observers using SR, but constant only for observers in the same gravity well using LR.

To date, all of the experiments we have done support both theories ...

LB
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Of course my assumption above was that c is strictly constant for all observers. This actually contradicts some experiments (e.g. those related to the Sagnac effect, and also the aberration of starlight), but I think this could be due to the light propagating in the earth's magnetic field (which thus constitutes an absolute reference frame) (I personally think this is a more likely explanation than gravity given the electromagnetic nature of light).
Still, what I mentioned in my previous post should be the only consistent way of interpreting the invariance of c (although this might only apply in a vacuum and for sufficiently weak magnetic fields).

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[Thomas] "This actually contradicts some experiments (e.g. those related to the Sagnac effect, and also the aberration of starlight), ..."

SR has been tweaked to account for these and several other "exceptions", so the originally claimed falsification of SR by the discovery of these phenomena has been retracted. See the articles at the Cosmology / Gravity tabs on the home page of this Website for details.

LR accounts for them without need of tweaking.


[Thomas] " ... but I think this could be due to the light propagating in the earth's magnetic field (which thus constitutes an absolute reference frame) ... "

Earth's magnetic field, like Earth's electrical and gravitational fields, is a local phenomenon and therefore cannot be be used as absolute frame. The proper term when using any of them as a frame of reference is "locally prefered frame". Move to another planet, and you move to a different, and totally independent, locally preferred frame of reference.

[Thomas] "Still, what I mentioned in my previous post should be the only consistent way of interpreting the invariance of c ... "

Both theories generate the same predictions for all experiments we have conducted to date, making them identical in the consistency department. (All of these experiments have been conducted in the same gravity well, making the two theories functionally equivalent as far as we are able to tell.)

One of these days we will expend the resources to conduct SOL experiments in another gravity well, and then things might start changing (but there is probably room here for another tweak to SR/GR). The research community seems to have decided that the only other way to distinguish between SR and LR is FTL propagation of energy or matter. SR says, proves in fact, this is flat impossible (so no tweak can be applied); LR says go as fast as you want, there is no speed limit.

We haven't done an experiment yet to test this [*], but there are observations of natural phenomena that can be interpreted as ftl propagation. If these theory dependent interpretations are correct SR has been falsified in favor of LR. GR doesn't care. It would just inherit the starting assumtions of LR instead of SR and keep on keeping on. Except for no longer being paranoid about ftl, GR would be unchanged.

[Thomas] " ... (although this might only apply in a vacuum and for sufficiently weak magnetic fields)."

According to General relativity (which inherits its maximum speed limit and constancy of light speed for all observes from SR) the speed of light in a vacuum is a function of gravitational potential. The speed of light measured at two different altitudes (both above Earth's atmosphere on a very tall tower) will be different. Electric and magnetic fields do not seem to have any (detectable) impact, as judged by years of measurements of GPS signals looking for such things. (But stand by - newer and better GPS satellites, along with newer and better experiments, are in the planning stages.)

LB

[*] At least not one that has been independently replicated, and thus has wide creditibility within the research community.