Lately I have been pondering Einstein’s theory of Special Relativity (SR). This has long seemed a fertile area to employ ideas of rational trigonometry, as the associated geometry, called sometimes Lorentzian geometry, rests on a symmetric bilinear form, and rational trig is all about setting up the machinery to study geometry starting from such a form. Quadrance and spread, the basic two measurements between points and lines, are simple rational functions of the dot product between vectors.
Perhaps surprisingly, I have slowly come to realize that SR actually can be derived not only from Einstein’s two basic postulates (that the laws of physics are the same in any two inertial frames, and that the speed of light is constant independent of the inertial frame) but rather from simple Newtonian mechanics, once we let go of the idea of an inertial frame and replace it with the simpler, more fundamental idea of an inertial observer. We replace a grid of equally spaced observers armed with coordinated clocks with just a single observer, armed with a single clock, and with a particular method of propogating signals, be it light, sound, water waves, or something else.
The whole story can be well described using the world of bats, who employ sonar echolocation to do their hunting at night. Turns out that many of the mathematical aspects of SR are already apparent in this humble setting. Sound, not light, is the basis of measurements. It is all rather surprising to me, and really only involves some elementary first year linear algebra.
I will be giving a talk about this subject in a few weeks here at UNSW: here are the details in case any reader is in the area and would like to come along.
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Speaker: A/Prof Norman Wildberger (UNSW)
Title: Bats, echolocation, and a Newtonian view of Einstein’s Special Relativity
When: 12:00 Tuesday, 24 June 2014
Where: RC-4082, Red Centre, UNSW, (Kensington campus, Sydney)
Abstract: Einstein’s 1905 Special Relativity (SR) is a foundational theory of 20th century physics. While perhaps unintuitive and certainly surprising initially, it has a beauty and elegance which connects to a rich and interesting variant of Euclidean geometry. In this talk we present a simple but novel introduction to SR and the associated geometry, showing that the mathematical framework actually resides already in Newtonian mechanics, and could possibly have been discovered any time after 1700 if physicists had asked themselves the question: how would two (mathematically inclined) bats compare time and position measurements??
The unique abilities of bats to hunt their prey using (sonor) echolocation is one of the more remarkable aspects of the world of mammals. We will show that by adopting a `bat-centric’ point of view, and thinking about sound–not light!–as the source of physical measurement information, many of the standard pillars of SR, including Lorentz transformations, length contraction, time dilation, Einstein’s interval, and the twin paradox arise simply and naturally. Mathematically only some first year linear algebra is required. Holy Albert, Batman!
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If there is any interest, we can have a Q&A session afterwards. Hope to see some of you!
njwildberger: tangential thoughts 
Prof Wildberger, Thank you for making so many great videos and making them available on YouTube. I am especially enjoying the History of Mathematics series.
Hi Catherine, glad to hear you are enjoying the videos.
Any chance the upcoming lecture might some day get onto YouTube, for those of us who can’t attend?
You are a fucking nut!!
Hi Jim, Yes I will hopefully be posting it in the Math Seminars series.
Hi Norman.
I appreciate your work. Something that is very inspiring to me now, generally, is that there is already, in the ancient world, a level of mathematics sophisticated enough for a suprising level of engineering. So I was considering how one might, educate themselves though the work of those who came before.
Here are some example of what I think could be important original works that may very much be worth the effort of study:
https://archive.org/details/treatiseonconics00apolrich
https://www.math.nyu.edu/~crorres/Archimedes/Books/ArchimedesInternet.html
http://www.mpiwg-berlin.mpg.de/Galileo_Prototype/DHTML/DPARI.HTM
https://archive.org/details/methodmeditation00descuoft
On the subject of studying math from the roots of its development, is that it is essential to be able to have proficiency with working with the conic (or conics?). It seems this might be enough of a mathematical framework to get a basic grasp of engineering in the ancient world.
Another thing I’ve been considering is the application of logic to algebraic equalities, or using logic to explore algebra. It started with the consideration that if I took, a variable, an operator, and another variable, as say:
a [operator] b
and had that equal to:
a [perhaps some other operator] b
that i could create a systematic enumeration of equalities, like:
a + b = a + b
a – b = a + b
a * b = a + b
…
a + b = a – b
…
etc.
So I was able to do some work to widde down this set to what I considered unique equalities. One of the interesting ones was a*b = a+b when a = b/(b-1)
So I had this idea that there might be an enumeration scheme to go though all possible algerbraic equalities. Consider that any equality can be written as [group of terms] = 0.
I came up with a partial solution, thanks to the help of some other people too, but its not where I want it, because really, im only interested in enumerating through EVERY UNIQUE ALGEBRAIC IDENTITY. This is a very intriging problem to me.
Anyway, thanks again for the work you’ve shared. I dont think that I would have as clear a vision of the landscape of math if it wasnt for giving serious effort to your considerations.
You touch on an interesting line of thought. How do we systematically understand the world of algebraic identities?? In my investigations of geometry, there are many times when remarkable identities seem to come miraculously to the rescue—but one can’t help feeling sometimes that these are just random meetings with deeper patterns.
I’ve been thinking about alternative formalism of number theory that uses only relational logical operators and derives notion of equality from ‘not more not less’, instead of axiomatically from Axiom of Identity in the Peano way. Maybe this approach could offer some insights also to your line of interest.
What no heavy machinery needed?
Wow – that is nice, when can you have a video of this?
LPC
I have now posted that lecture on my YouTube channel (user: njwildberger).
The Physicist Brian Keating in one of his podcast episodes featuring Stephan Wolfram mentioned his late father’s paper in the 70s that dealt with a similar or essentially same idea of how lorentz invariance can be understood through sonar realms as processed by bats. Have you ever come across this paper?
Sounds interesting, I do not know of this paper. Do you have a link for it?