Einstein described the introduction of the cosmological constant in his field equations of general theory of relativity as his Biggest Mistake.
But that was a mistake: His truly Biggest Mistake was the introduction of his special theory of relativity in 1905, which preceded the general theory by ten years.
Special relativity concerns the representations of "events" in two space-time coordinate systems (x,t) and (x',t'), where an "event" (of some nature) is recorded by specific coordinates such as (0,0).
The key example used by Einstein was the setting of train traveling along an embankment with a velocity $v<1$, with the x-axis fixed to the embankment and the x'-axis fixed to the train. The x-axis and x'-axis use the same spatial scale set by identical meter sticks. Events would then be recorded by an observer X on the embankment using the x-axis and a clock showing t-time, and by an observer X' in the train using an x'-axis fixed to the train together with an identical clock traveling with the train showing t'-time.
The windows of the train are supposed to be shut, so that X cannot look inside the train, and X' cannot look outside the train. X and X' thus use identical space-axes and identical clocks. X and X' agree on a common speed of light = 1 as measured in their respective system with identical meter sticks and clocks.
For a third outside observer Y (Galileo) the full situation would be clear and simple: The connection between the coordinates would be x' = x - vt and t' = t, expressing that the train with its x'-axis translates with respect to the x-axis of the embankment with velocity v and that identical clocks are being used.
To understand that X and X' can agree on the speed of light, although the x'-axis moves with respect to the x'axis, compare with the speed of sound. X and X' will agree on the speed of sound, because the air inside the train travels along with train and so X' will measure the same speed of sound as X measures in the still air fixed to the embankment. Moreover, if X' opens a window and measures the speed of sound in the still air of the bank, X' would get a value modified by the train velocity, and X' would also experience a Doppler effect of sound propagation in the still air of the bank. All in accordance with the Galilean transformation x' = x -vt, and t' = t.
But Einstein was not happy with a Galilean transformation between coordinates, because a medium of propagation of light (like air for propagation of sound), a unique "luminiferous aether ", could not be identified (in the Michelson-Morley experiment). Einstein then came up with the radical idea of not connecting any form of medium to light propagation and then replaced the Galilean transformation, based on an embankment-medium and a different train-medium, by the Lorentz transformation supposedly being free of any medium:
- $x^\prime =\gamma (x-vt)$, $t^\prime = \gamma (t-vx)$, $\gamma =\frac{1}{\sqrt(1-v^2)}$,
- $x =\gamma (x^\prime+vt^\prime )$, $t = \gamma (t^\prime+vx^\prime )$.
where $\vert v\vert <1$. The Lorentz transformation has the property that a $x = t$ is transformed into $x^\prime = t^\prime$, which is viewed to express the same speed of light = 1. Lorentz had introduced his transformation well before Einstein took it up, but Lorentz had been careful to note that his transformation was not to be interpreted as a transformation between physical coordinates. In particular, if t was physical time measured by a physical clock, then t' did not represent physical time measured by a physical clock.
But the young eager brave Einstein took the bold step of proclaiming, against Lorentz, that both t-time and t'-time represent physical time measured by identical clocks. In particular with x=t recording a light signal in the (x,t)-system, t' would be connected to t by
- $t' = \sqrt{\frac{1-v}{1+v}}t$
with thus $t' < t$ expressing that a moving X'-clock would be running slow compared to a still X-clock. This weird effect was termed time dilation: A moving clock would thus run slow compared to a stationary clock. Which clock was stationary and which was moving, was left in the air as an expression of Einstein's genius.
Having thrown out any form of medium for light propagation, Einstein also had to get rid of the observer Y capable of grasping the full physics of light propagation in the combined train-embankment system, and Einstein was then left with an observer X on the embankment unable to look into the train and an observer X' in the train unable to look out, as presented above.
X thus cannot make any observations in the (x',t')-system, nor can X' make any observations in the (x,t)-system. If either of them could, then the very reason to use two coordinate systems would disappear: If X' can make observations in the (x,t)-system, then X' does not need the (x',t')-system and vice versa.
Einstein was thus left with two observers X and X' locked into two different systems unable to connect in physical terms. This is where Einstein takes a leap and boldly claims that observations are to be connected by the Lorentz transformation. Einstein thus tells X' inside the train that a light signal launched by X along the embankment, which X' cannot see, has to be represented in the (x',t')-system through the Lorentz transformation coming with space contraction and time dilation.
More precisely, to support his argument Einstein made the thought experiment that both X and X' launch light signals in their respective (x,t) and (x',t')-systems at (x,t) = (0,0) and (x',t') = (0,0), which are then recorded as x=t and x'=t' respectively, which Einstein (but not Lorentz!) tells are connected by the Lorentz transformation.
X' (in the train) thus observes a light signal x' = t' travelling inside the train, and X' is told by Einstein that this is the same light signal as that launched by X following x = t along the embankment with x = t. X' is thus told by Einstein that his t'-clock is running slow as compared to the t-clock as an expression of weird time dilation of his moving clock.
But X' cannot observe the light signal traveling in the (x,t)-system (windows are shut), and Einstein's Biggest Mistake thus boils down to identifying the light signal launched by X in the (x,t)-system at (x,t) = (0,0), with the light signal launched by X' in the (x',t')-system at (x',t') = (0,0). Doing so and connecting the coordinates of the light signals by the Lorentz transformation, Einstein discovers the new physics of space contraction and time dilation.
But it is not correct to identify the two light signals, by the same reason that it would not be correct to identify two sound signals, one launched in the still air of the embankment and one in the moving air inside the train. Einstein's argument for identification is weak: Since there is no medium for light propagation, one can assume that the two signals are the same. More precisely, Einstein's identification lacks any form of rationale.
Unfortunately modern physics is built on the special theory of relativity formed from Einstein's Biggest Mistake.
Luckily, there is a different rational physics of relativity of light propagation in the spirit of Galilean physics as Many-Minds Relativity. Take a look and get enlightened (by light signals)! A short course is
- Special Relativity: There is no aether for light propagation.
- Many-Minds Relativity: There are many aethers for light propagation, one for each choice of space coordinate system.
