Talk:Special Relativity/Mathematical transformations

This needs tidying up, in particular 4 vectors should be introduced after transformations. RobinH 16:08, 24 July 2006 (UTC)

The derivation of proper acceleration as a function of coordinate acceleration needs work. Look at the sentence:

"The proper acceleration, $$\alpha \!$$ is defined as the acceleration of an object in its rest frame. It is the instantaneous change in velocity for an observer for whom $$u^'=0 \!$$ and $$\alpha = du^'/dt^' \!$$. "

If $$u^'=0 \!$$ it follows immediately that $$ du^'/dt^'=0 !!! \!$$.Moriconne (talk) 04:00, 2 August 2008 (UTC)


 * Yes, the section is not good at all. In fact the whole page needs a re-write, being a series of jottings in its present form.  (Some of the jottings are excellent starting material.) Acceleration should have its own page and be properly explained. I have put a warning on the page to say "Under development"! RobinH (talk) 09:21, 4 August 2008 (UTC)


 * The rest of the page is excellent, I would only take out the section "Acceleration transformation". It isn't even about the transformation, it is about the relationship between proper and coordinate acceleration. Moriconne (talk) 14:33, 6 August 2008 (UTC)


 * OK. RobinH (talk) 11:11, 7 August 2008 (UTC)

Doppler Effect vs Time Dilatation ?
Is there any conceptual difference between the Doppler experiment and the experimental pattern described under “Time Dilatation” in the Special Relativity theory? In both cases a frequency generator is in inertial motion in respect to an observer. In both cases one computes the perception of the frequency by the distant observer… But contrary to the Doppler Effect where a red-shift or a blue-shift in frequency may be observed, Special Relativity only concludes to time dilatation in all cases. This does not look very consistent.

This immediately triggers a second question: is it legitimate for Special Relativity to invoke the Lorentz Transform in this particular context? If I understand correctly this transformation applies to the gaps in position and time between two events in a given inertial frame, and it delivers the corresponding gaps between the same events in another inertial frame. However, Special Relativity uses the Lorentz Transform to compute gaps between two events related to the observer’s life on the basis of gaps between two events relevant to the clock’s history. Moreover both pairs of events belong to the same "frame"; here I mean the same "convention" for representing the relative motion of the clock and the observer.Sugdub (talk) 21:55, 14 December 2009 (UTC)


 * It is a shame that the section on the Doppler Effect is not more complete in this book. The relativistic doppler effect has components due to time dilation and movement towards or away from a source. There is a purely relativistic "transverse doppler effect" due to time dilation and I only wish I had time to improve the text. RobinH (talk) 10:33, 15 December 2009 (UTC)

Well, your answer seems to confirm my doubts. The discussion about Time Dilatation in the framework of Special Relativity leads to a formula fairly different from the conclusions of the Relativistic Doppler Effect. In particular it is unable to properly predict the sign of the shift in frequency. So my initial questions remain unanswered: Sugdub (talk) 21:31, 15 December 2009 (UTC)
 * unless both experimental patterns are conceptually different (in which way?), the Time Dilatation theorem under Special Relativity contradicts the Relativistic Doppler Effect;
 * there is a doubt about the way the Lorentz Transform is invoked when deriving the Time Dilatation theorem.

Demonstration of the Lorentz transform
"Einstein's assumption that the speed of light is a constant can now be introduced so that x = ct and also x' = ct'."

One must keep in mind that in this case x and t represent the difference in coordinates between two events which respectively mark the emission and the reception of a light ray. However the above formulas are injected into equations where x and t represent the coordinates of an unspecified event in the space-time.

So the reasoning is invalid. On the one hand it mixes equations where x (resp. t) represents different quantities (coordinates vs delta-coordinates). On the other hand it claims being conclusive for any event in the space-time whereas it relies on a property which validity is restricted to light-related events.

One could reach the same conclusion without entering the details of the demonstration: since the law about the speed of light does not constrain the coordinates of light-related events and neither the coordinates of any other event, it is in principle unable to contribute to the determination of a transformation of coordinates of space-time events, neither the Lorentz's transform nor any other one. Sugdub (talk) 21:28, 27 January 2010 (UTC)


 * Your critique of using the "speed of light" as a foundation for relativity was spotted within about 3 years of Einstein's first publication on SR in 1905. It was rapidly understood that electrodynamics alone could not be used as the basis for such a generalised theory.  This is why the modern approach to Special Relativity does not actually depend on Einstein's original postulates.


 * In the modern approach the postulate that is to be tested is that the universe is a four dimensional manifold with a metric given by:


 * $$ ds^2 = dx^2 + dy^2 + dz^2 - cdt^2$$


 * In this type of metric the spacetime interval ds is constant for all observers.


 * According to this geometrical approach the constant "c" is simply a term that converts measurements in seconds to measurements in metres - in the same way as in the formula (centimetres = c times inches) where c ~ 2.5 centimetres per inch. So x = ct and c ~300,000,000 metres per second.


 * The derivation of the constancy of the speed of light from the metric is fully explained in the text. Please take a look at the section: Special_Relativity/Spacetime. The fact that the constancy of the speed of light is a consequence of the properties of the metric means that it can indeed be applied to sets of coordinates.  After all, what is the metric of the manifold but the relationship between the coordinates of events? The constant "c" is a geometrical entity like "pi" that arises in the Minkowski metric, not some arbitrary property of photons.


 * To be pedantic the line: "Einstein's assumption that the speed of light is a constant can now be introduced so that x = ct and also x' = ct'." should really read "Given that in a 4D manifold of this type c is a constant for all observers, no matter what their state of relative motion, x = ct and also x' = ct'."


 * I am wholly against the use of Einstein's original electrodynamic approach in the teaching of Special Relativity. It always results in students separating "light" from "geometry" and wondering what causes light to go slower or speed up! It is much better to use the geometrical approach to relativity where it can be seen that a universally constant velocity for all observers is a consequence of the metric.  The electrodynamic approach was only current between 1905 and 1908 but almost all popular physics books confuse their readers with it. In contrast nearly all advanced texts start with the bald statement "assume that the universe is a four dimensional spacetime with the metric:  $$ ds^2 = dx^2 + dy^2 + dz^2 - cdt^2$$"RobinH (talk) 10:30, 4 February 2010 (UTC)

I must admit being sceptical about your answer. I certainly can agree that soon after the inception of Einstein’s Special Relativity theory physicists recognised the need to ground it on a less peculiar feature than the law of propagation of light (at least by enlarging this law to other interactions). Certainly the initial version of the theory looked suboptimal. But I can’t imagine that physicists have consciously and deliberately masked, in all their presentations and academic works until now, the fact that it is logically flawed.

You might have misinterpreted my critic: it is logically impossible to derive a transformation applicable to any vector (x, ct) in the space-time manifold from a law which applies only to those vectors for which x = ct (eg because they actually relate to a light-ray). However, the Special Relativity theory by Einstein claims the opposite, as can be seen from the “demonstration” of the Lorentz transformation.

The change you propose to the sentence I highlighted does not make it. The issue is not about changing the rationale for injecting x = ct, it is actually about the need to address any vector, and especially those for which x is NOT equal to ct. For those, there is no way to decide how the transformation should behave.

We could also debate on the drawbacks of the “modern presentation” of the theory but this was not my point.Sugdub (talk) 18:35, 11 February 2010 (UTC)


 * At some stage all of my advanced textbooks on Relativity make the blatant statement that it will be assumed that the universe has the metric: $$ ds^2 = dx^2 + dy^2 + dz^2 - cdt^2$$ ie: that $$ ds^2 $$ is an invariant for all observers in a pseudo-Riemannian geometrical manifold. On the basis of this assumption it is then indeed possible to deal with four vectors that have a mixed spatio-temporal direction. The rest of Relativity is a set of derivations that suggest experiments and observations that test the assumption that the metric is a true description of events. This is how science works: scientific theories use the minimal set of assumptions and are tested using relations that are predicted using a mathematical formalism. It is scandalous that so many popular physics books and elementary textbooks take the electrodynamic approach. RobinH (talk) 10:32, 21 July 2010 (UTC)

No event horizon in Minkowski spacetime
The section "Accelerated Frames and Event Horizons" which describes Rindler acceleration uses the term "event horizon" which is incorrect. Yes, the line $t = r$, for $r \ge 0$ forms a sort of "horizon" between "events" in a general sense. However it is not an event horizon which has a technical meaning, as the boundary of events which can send a signal to future null infinity. Minkowski spacetime has no event horizons, nor most other types of horizon. For the present context, it is probably best called the "Rindler horizon". You might call it a particle horizon for the accelerating observers, maybe. Also it is a Killing horizon I think, because the boost Killing vector field is null there. Colin MacLaurin (discuss • contribs) 09:34, 30 May 2019 (UTC)