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Based on Lorentz factor $\gamma = \frac{1}{\sqrt {1-\frac{v^2}{c^2}}}$ it is easy to see $v < c$ since otherwise $\gamma$ would be either undefined or a complex number, which is non-physical. Also, as far as I understand this equation was known before Einstein's postulates were published. My question is: why didn't Lorentz himself conclude that no object can go faster than speed of light? Or maybe he did, I do not know. I feel I am missing some contexts here.

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    $\begingroup$ This is an interesting question, but you might get better answers on hsm.stackexchange.com . Usually the way people thought about these things at the time (in this case 130 years ago!) is very hard to wrap your head around if you've had modern training. $\endgroup$ – Ben Crowell Feb 20 at 5:54
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    $\begingroup$ I'd agree with the HSM SE idea, but also consider reading the rather more involved history of the Lorentz Transformations page on Wikipedia for a broader context. $\endgroup$ – StephenG Feb 20 at 6:09
  • $\begingroup$ @BenCrowell Thanks for the suggestion! I did not know about HSM SE. Should I ask the same question there or is there any possibility of moving this question? I am new here so it seems I cannot move my own question. $\endgroup$ – Rob Feb 20 at 6:16
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    $\begingroup$ I flagged it for a moderator that you want to move the question to HSM $\endgroup$ – anna v Feb 20 at 6:48
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    $\begingroup$ @DvijMankad Oh, I just meant how you phrased the custom close reason you entered. It's best to say something like "I'm voting to close this question as off-topic because [reason which doesn't reference other sites]. It may fit on [other site]." $\endgroup$ – David Z Feb 27 at 22:37
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If I had to sum up my findings in a sound bite it would be this: Einstein was the first to derive the Lorentz transformation laws based on physical principles--namely that the speed of light is constant and the principle of relativity. The fact that Lorentz and Poincaré were not able to do this naturally leads to why they were not able to justify making any fundamental statements about the nature of space and time--namely that nothing can go faster than light.


This is seen by a careful reading of the Einstein (1905) – Special relativity section of the History of Lorentz Transformations Wikipedia article

On June 30, 1905 (published September 1905) Einstein published what is now called special relativity and gave a new derivation of the transformation, which was based only on the principle on relativity and the principle of the constancy of the speed of light. [Emphasis mine]


Furthermore, it is stated that (idem)

While Lorentz considered "local time" to be a mathematical stipulation device for explaining the Michelson-Morley experiment, Einstein showed that the coordinates given by the Lorentz transformation were in fact the inertial coordinates of relatively moving frames of reference.

My reading of this seems so indicate that that at the time of publishing, Lorentz considered the notion of "local time" (via his transformations) to be just a convenient theoretical device, but didn’t seem to have a justifiable reason for why it it should be physically true.


It looks obvious in hindsight I know, but model building is tough. So the reason, in short, seems (to me) to be this: As far as Lorentz saw it, he was able to "explain" the Michaelson-Morely experiment in a way not unlike the way that Ptolemy could explain the orbits with epicycles. Did it work? Yes, but its mechanism lacked physical motivation.

That is, he didn't have a physical reason for such a transformation to arise. Rather it was Einstein who showed that these transformation laws could be derived from a single, physical assumption--the constancy of the speed of light. This insight was the genius of Einstein.


Picking up at the end of the last blockquote, we further have that (idem)

For quantities of first order in v/c, this was also done by Poincaré in 1900; while Einstein derived the complete transformation by this method. Unlike Lorentz and Poincaré who still distinguished between real time in the aether and apparent time for moving observers, Einstein showed that the transformations concern the nature of space and time.

This implies actually that Lorentz and Poincaré were able to derive the Lorentz transformations to first order in $\beta$, but since they believed that the Aether existed they failed to be able to make the fundamental connection to space, time and the constancy of the speed of light.

The failure to make this connection means that there would have been no justifiable reason to take it physically serious. So, to Lorentz and Poincaré the Lorentz transformation laws would remain ad-hoc mathematical devices to explain the Michaelson-Morley experiment within the context of the Aether but not saying anything fundamental about space and time. This failure to conclude any fundamental laws about the nature of spacetime subsumes, by implication, making any statements such as no moving object can surpass the speed of light.


Edit: @VladimirKalitvianski has pointed me to this source, which provides the opinions of historians on the matter.

Poincaré's work in the development of special relativity is well recognised, though most historians stress that despite many similarities with Einstein's work, the two had very different research agendas and interpretations of the work.

Poincaré developed a similar physical interpretation of local time and noticed the connection to signal velocity, but contrary to Einstein he continued to use the Aether in his papers and argued that clocks at rest in the Aether show the "true" time, and moving clocks show the local time. So Poincaré tried to keep the relativity principle in accordance with classical concepts, while Einstein developed a mathematically equivalent kinematics based on the new physical concepts of the relativity of space and time.

Indeed this resource is useful, as it adds an additional dimension as to why Lorentz didn't publish any claims about a maximum signal velocity. It reads rather clearly, so I won't bother summarizing it.

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    $\begingroup$ H. Poincaré has derived the Lorentz transformations exactly. His presentation in US, his "compte rendu" (résumé) and his full article on this subject show that he followed and mastered well the physics. His words about "coup de pouce" (experimental data) and other things were motivated by the observable physics. So A. Einstein postulated what had beed already been established from physical motivation by others. $\endgroup$ – Vladimir Kalitvianski Feb 20 at 9:03
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    $\begingroup$ @VladimirKalitvianski That's extremely interesting. What year? would you mind giving me a citation so I can update my answer? I by no means claim to be an expert on this matter, rather I'm just closely reading the wiki and related sources. $\endgroup$ – InertialObserver Feb 20 at 9:07
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    $\begingroup$ en.wikipedia.org/wiki/Henri_Poincar%C3%A9 $\endgroup$ – Vladimir Kalitvianski Feb 20 at 9:11
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    $\begingroup$ Poincare always believed in the ether as an absolute rest frame, an idea which Einstein abandoned in his 1905 paper. So Einstein usually gets credit. However, the absence of an absolute rest frame is not absolute - consider the frame in which the microwave background radiation is the same color in all directions. $\endgroup$ – Paul Young Feb 20 at 18:58
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    $\begingroup$ Actually, Lorentz did have a physical reason, it was a dynamic effect based on the hypothesis of molecular forces. And Einstein considered relativity and the speed of light invariance to be phenomenological postulates, not "physical" principles, and was dissatisfied with them. So the real difference between them is the opposite to the one described: Lorentz and Poincare derived the transformations from hypothetical dynamics, whereas Einstein did from kinematic postulates. Moral: less Wikipedia, see Zahar Why Did Einstein's Programme Supersede Lorentz's?. $\endgroup$ – Conifold Feb 20 at 20:46
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Because typically if you find an expression that seems to break down at some value of $v$, you would conclude that the expression simply loses its validity for that value of $v$, not that the value isn't attainable. Presumably this was the conclusion of Lorentz and others.

The reason Einstein concluded otherwise is that special relativity gives a physical argument for "superluminal speeds are equivalent to time running backwards" -- the argument is "does a superluminal ship hit the iceberg before or after its headlight does?"

This depends on the observer, and because the headlight would melt the iceberg, the consequences of each observation are noticeably different. The only possible conclusions are "superluminal ships don't exist", "time runs backwards for superluminal observers", or "iceberg-melting headlights don't exist".

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