# Geometric derivation of SR?

derz
Has anyone derived the Lorentz-transforms by using 'simple' geometrics? If so, could I get a link for the paper please. I tried to google for one but couldn't find.

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Homework Helper
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derz said:
Has anyone derived the Lorentz-transforms by using 'simple' geometrics? If so, could I get a link for the paper please. I tried to google for one but couldn't find.

Can you be more specific as to what you mean by 'simple' geometrics? Who is the target audience? and what is their background?

Do you want a geometric construction? formulation via analytic geometry? or by trigonometry? vector methods? group methods? operational construction using light-rays?

derz
robphy said:
Can you be more specific as to what you mean by 'simple' geometrics? Who is the target audience? and what is their background?

Do you want a geometric construction? formulation via analytic geometry? or by trigonometry? vector methods? group methods? operational construction using light-rays?

Target audience: everyone who wants to learn about SR or find a simpler way to derive it

I'd appreciate papers with geometric construction and papers that formulate SR with analytic geometry.

Sorry about not being specific in my first post, I'm not a native english speaker and wasn't sure what to look for

Staff Emeritus
derz said:
Target audience: everyone who wants to learn about SR or find a simpler way to derive it

I'd appreciate papers with geometric construction and papers that formulate SR with analytic geometry.

Sorry about not being specific in my first post, I'm not a native english speaker and wasn't sure what to look for

Bondi's "Relativity and common sense" would be my recommendation, or some other book using an equivalent approach. This approach is sometimes known as k-calculus, but it actually involves only high school algebra.

I may be biased, because Bondi's book was one of the first books I read.

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You might enjoy watching Ch 42 of the Mechanical Universe
http://www.learner.org/resources/series42.html

Here are some recent references that I am aware of that
emphasize the geometry of SR [with less emphasis on the algebra (i.e., explicit use of the Lorentz Transformation formula)] :

Liebscher's "The Geometry of Time". http://www.aip.de/~lie/Vorlesung/Vorlesung.RuG.html has some webpages [mainly in German] on aspects of this work.

Brill and Jacobson's preprint "Spacetime and Euclidean Geometry" http://arxiv.org/abs/gr-qc?0407022

Mermin's preprint "From Einstein's 1905 Postulates to the Geometry of Flat Space-Time" http://www.arxiv.org/abs/gr-qc/0411069. Search for his AJP papers on teaching relativity by searching for "Mermin" at http://scitation.aip.org/ajp/ [Broken] . Here are some of his course notes http://www.lassp.cornell.edu/~cew2/P209/P209_home.html [Broken]. Some info on http://people.ccmr.cornell.edu/~mermin/homepage/ndm.html.

Here's my preprint at http://arxiv.org/abs/physics/0505134 and here is a website with animations http://www.phy.syr.edu/courses/modules/LIGHTCONE/LightClock/ [Broken] .

Of course, look at the references in each to find even more references.

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pervect said:
Bondi's "Relativity and common sense" would be my recommendation, or some other book using an equivalent approach. This approach is sometimes known as k-calculus, but it actually involves only high school algebra.

I may be biased, because Bondi's book was one of the first books I read.
K calculus predated Bondi, Milne's "Kinematic Relativity" depended on it.
You will also find it in a more modern book, "Introducing Einstein's Relativity" Ray D'Inverno, Chapter 2.

Garth

drunkenfool

It Finally Dawned On Him

CompSci
Perhaps the attached derivation of the Lorentz transformations is what derz had in mind. After reading this thread, I realized it is probably original. I'm curious if the mapping of the wavefronts from a stationary and moving source to one point by the constants k and 1/k implies anything about the uniqueness of the Lorentz transformations.

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JM
Dear derz, The Lorentz transforms are part of the physics of electro-magnetic-dynamics. For clear understanding, a proper derivation should combine the physics, geometry and mathematics in logical connection. I have searched for such a derivation and not found one. Have you found one you like among the suggestions given in this forum? If so, which one?
The best one seems to be Einstein's original derivation in his 1905 paper " On the Electrodynamics of Moving Bodies". It's available in the book " The Principle of Relativity", Dover Publishers. It's not easy to follow but it relates better to the physics , and requires only early college math, e.g. calculus, differential equations, and determinants.

JM
Dear derz, ( I must have hit the wrong key, so to continue the above)
In a search for better understanding, I have looked for relations between the physics and the math/geometry. For example, the second term in the time equation of Lorentz represents a shift of the origin of the time axis of the moving coordinates, as required to satisfy the Postulate of Constant Light Speed. For more discussion go to the site 'searchforsr.com.
I hope this helps.
JM

cesiumfrog
derz said:
Has anyone derived the Lorentz-transforms by using 'simple' geometrics?

Not sure whether it's exactly how you meant, but I found http://www.mathpages.com/rr/s1-07/1-07.htm" [Broken] very interesting..

Minkowski:
Such a premonition would have been an extraordinary triumph for pure mathematics [...] though it now can display only staircase wit

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Daverz
I think the OP meant "graphical" rather than necessarily geometrical. The book Flat and Curved Spacetimes (ISBN 0198506562 ) also uses k-calculus. From the Amazon description:

The present book explains special relativity and the basics of general relativity from a geometric viewpoint. Space-time geometry is emphasised throughout, and provides the basis of understanding of the special relativity effects of time dilation, length contraction, and the relativity of simultaneity. Bondi's K-calculus is introduced as a simple means of calculating the magnitudes of these effects, and leads to a derivation of the Lorentz transformation as a way of unifying these results. The invariant interval of flat space-time is generalised to that of curved space-times, and leads to an understanding of the basic properties of simple cosmological models and of the collapse of a star to form a black hole. The appendices enable the advanced student to master the application of four-tensors to the relativistic study of energy and momentum, and of electromagnetism. In addition, this new edition contains up-to-date information on black holes, gravitational collapse, and cosmology.

An excellent book that starts with the geometry (i.e. the metric) as a given, then works out the Lorentz transformations is Spacetime Physics (though my experience is only with the old red paperback edition).

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