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I've seen derivations of Lorentz transformation but are they really derivations or are we just teasing out the formula using some special cases and then assuming it to be valid for all the cases.

Is this really a derivation as in the mathematical sense or are Lorentz transformations a law in themselves apart from conservation of the spacetime metric.

By Lorentz transformation, I mean the relation between space and time 'intervals' for two inertial observers.

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    $\begingroup$ Lorentz transformation is a transformation of coordinates of an event between two inertial frames, not "relation between space and time 'intervals'". It is not a physical law per se anymore than rotation or spherical inversion of coordinates is a physical law. On the other hand, the idea that coordinate systems can be synchronized in such a way that the Lorentz transformation of measured coordinates of an event accurately predicts what coordinates observer in other inertial frame will have measured is an accepted law of physics. $\endgroup$ – Ján Lalinský Dec 6 '15 at 15:37
  • $\begingroup$ @JánLalinský The problem I face is that they are a bit different than ordinary rotations because of the negative sign in the metric, which ultimately means, you can use any coordinates of your choice but while calculating the metric, you've to ultimately go back to time and space differences. This going back to time and space seperately is what seperates them from ordinary rotations where this wasn't necessary, as space and time are measureable physical quantities. $\endgroup$ – Viesr Dec 6 '15 at 18:24
  • $\begingroup$ @Ján Lalinský What is that accepted law of physics called? can you give any reference? What I found in A.P French's book is a mention of some transformational law by Hermann Bondi in the introductory section of "Relativity according to Galileo and Newton" $\endgroup$ – Coward May 15 '16 at 11:35
  • $\begingroup$ @Coward, the law I meant is "measured coordinates with respect to two different inertial frames are related through the Lorentz transformations" and it follows from the postulates of the special theory of relativity. I do not know if there is a special name for it, it is only indirectly inferred from the experiments that studied validity of assumptions and predictions of the theory. $\endgroup$ – Ján Lalinský May 15 '16 at 12:59
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This is dangerously close to being a matter of opinion, but for what it's worth I would regard the Lorentz transformations as a special case rather than anything fundamental.

The fundamental principle in special relativity is the invariance of the proper time, which is presumably what you mean by conservation of the spacetime metric. The Lorentz transformations give you the coordinate transformations for the special case of relative motion at constant velocity. However in any system where the velocity is changing the Lorentz transformations don't apply.

In the past this has lead to the common claim that special relativity doesn't apply to accelerating systems. This is of course nonsense as the invariance of the proper time can be used to describe motion in a circle or accelerated motion in a straight line, or indeed any arbitrarily accelerated motion.

My point is that the Lorentz transformations describe only a subset of the motions that can be described using special relativity, and therefore they can't be a fundamental law.

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    $\begingroup$ Yes, but still. You use lorentz transformations in differential form. $\endgroup$ – Viesr Dec 6 '15 at 8:46
  • $\begingroup$ @ViEsr: why would you do that instead of using the metric directly? The reason we teach the Lorentz tranformations is because they are simple compared to a proper treatment using differential geometry. If you're going to use the differential form it would be better to just learn differential geometry and do it properly. $\endgroup$ – John Rennie Dec 6 '15 at 8:47
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    $\begingroup$ One doesn't need any differential geometry in SR. Also, how will you calculate space and time intervals seperately between two independent events(that may be even space-like), given the spacetime interval between those two events. $\endgroup$ – Viesr Dec 6 '15 at 9:00
  • $\begingroup$ @ViEsr: One doesn't need any differential geometry in SR, unless of course one wants to really understand what's going on. $\endgroup$ – John Rennie Dec 6 '15 at 9:10
  • $\begingroup$ Alright, I understand it finally. Thank you :P. The basic thing I found it, you need a space and time 'breaking' of the space-time interval first in a reference frame to be able to do anything using lorentz transformation at all. We use frames(lorentzian/inertial) that are naturally perceptive to us as humans to express the metric in that frame. But those frames can't be fundamental. But I still stick with 'you don't need differential geometry in SR'. You can do with basic four vectors etc. which aren't truly the essence of differential geometry. $\endgroup$ – Viesr Dec 7 '15 at 18:32
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I believe you are thinking of the Lorentz velocity boost (and rotation) transformations as the 4x4 matrices which leave invariant the metric diag(1,-1,-1,-1) (ie: they leave invariant the proper time). In this sense, perhaps they seem secondary to the SR principle of invariance of the proper time.

However, these 4x4 matrices are just one representation (called the fundamental rep) of the Lorentz Group O(1,3) or its covering group SL(4,C). There are many other representations by matrices of different dimensions. EVERY particle or object in the universe transforms under rotations and boosts as some dimensional representation of the Lorentz group. You are probably most familiar with this for rotations (subgroup of the Lorentz group). Here each different dimensional representation is labelled by the spin of the particle being rotated. For example, a spin j=1/2 particle has (2j+1)=2 states and is rotated by 2x2 matrices. There is nothing more fundamental in physics than this. Some purely abstract math SL(2,C), can be put in a correspondence with how every particle in the universe must behave! This isn't even an equation. There was no formula to tease out. For example, there was no equation to pull out of a hat by setting curvature equal to stress-energy density because it gave Newton's law in the limit.

Whereas understanding how every object in the universe transforms under rotations and boosts (a major part of quantum mechanics) is a fundamental big deal, the Lorentz group may be just the tip of the group iceberg. Rotations and boosts are not the only continuous transformations that can be done to an object. We also can do strain, space translations, and time translations. The Poincare group extends the Lorentz group by adding abelian translations. If a different group were used in which translations did not commute with each other, perhaps it would be more interesting. Raising and lowering operators would be available and much like non-commuting rotations provided the quantization of angular momentum, non-commuting translations would provide the quantization of mass. Yes, Lorentz transformations are very fundamental for physics.

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