### k-calculus and Special Relativity

I've been familiar with Special Relativity since childhood - I think I was able to give Einstein's old trains and light rays argument for time dilation when I was barely out of elementary school. (Unfortunately my brain only went downhill from there...) So I was pretty surprised when I started reading Bondi's introduction to Relativity for the layman, Relativity and Common Sense, published in the fifties, and found that I was learning something new.

At one point in the book, Bondi shows that different observers, who appear to have synchronised watches, end up having watches showing different times. Amazingly he seems to do all of this in what looks like a Newtonian framework. It took me several rereadings of the paragraphs to eventually realise where it was that he'd sneaked in the relativity assumption. I felt like I'd just seen one of those puzzles where you rearrange a square as a rectangle which appears to have a lower area. But as a result of this, he makes Special Relativity seem like the most natural thing in the world. And the really nice thing is that in his presentation he doesn't need to talk about watches that use light pulses (because the sceptic might say that time dilation only applies to watches that use laser pulses) but instead his argument applies to any kind of watch.

Anyway, the key idea is this: suppose A and B have zero relative velocity. If A signals B at one minute intervals then B will receive the signals at one minute intervals. But if A and B move apart with constant velocity, B will receive A's signals less frequently simply because the signals have to cover an increasing distance as B moves from A. This much is trivial and applies to any non-instantaneous signal in a Newtonian or Einsteinian framework.

Now suppose B receives signals every k minutes. The key step is this: according to relativity, k can only be a function of the relative velocity of A and B, not a function of the absolute velocities of A and B. Using this assumption, and some other far less radical ones that few people would have problems with, the rest of Special Relativity follows. But you'll have to read the book for the details (or google for k-calculus).

It's all very elementary stuff. But this k-factor argument seems like it would be much more convincing to the layman than Einstein's original thought experiment. In fact, I did a bit of googling and it seems that the k-factor argument has been presented in a few different places under the name "k-calculus".

Bondi also stresses that watches are like odometers. They measure something that is personal to the path that you have taken rather than some absolute observer independent quantity. This is something that is hard to get from other elementary accounts I have read. Other introductions present time dilation as some kind of influence that somehow makes your clocks go 'wrong'. Just the name 'time dilation' gives the impression that what you measure is somehow dilated from the 'correct' time, a completely erroneous view. Just as there is no 'distance' dilation for driving north-east and then north-west to get to a place due north.

Anyway, if you have some doubting Thomas relative (no pun intended) who thinks the whole Relativity thing is bogus, but they do have a bit of numerical aptitude, this might be the book to get them. You don't even need to compute a square root to compute the discrepancy between the watches in Bondi's first example. All in all, an excellent book!

Anyway, my reason for reading this book is that I really don't like the way some physicists talk about Relativity. So I thought I'd read a few elementary accounts to get an idea of what trends there are in explaining the subject.

(Sadly, Bondi passed away last year. And maybe I've finally got over my resentment of Bondi for giving me low marks for the cosmology questions in exams at Cambridge one year...)

At one point in the book, Bondi shows that different observers, who appear to have synchronised watches, end up having watches showing different times. Amazingly he seems to do all of this in what looks like a Newtonian framework. It took me several rereadings of the paragraphs to eventually realise where it was that he'd sneaked in the relativity assumption. I felt like I'd just seen one of those puzzles where you rearrange a square as a rectangle which appears to have a lower area. But as a result of this, he makes Special Relativity seem like the most natural thing in the world. And the really nice thing is that in his presentation he doesn't need to talk about watches that use light pulses (because the sceptic might say that time dilation only applies to watches that use laser pulses) but instead his argument applies to any kind of watch.

Anyway, the key idea is this: suppose A and B have zero relative velocity. If A signals B at one minute intervals then B will receive the signals at one minute intervals. But if A and B move apart with constant velocity, B will receive A's signals less frequently simply because the signals have to cover an increasing distance as B moves from A. This much is trivial and applies to any non-instantaneous signal in a Newtonian or Einsteinian framework.

Now suppose B receives signals every k minutes. The key step is this: according to relativity, k can only be a function of the relative velocity of A and B, not a function of the absolute velocities of A and B. Using this assumption, and some other far less radical ones that few people would have problems with, the rest of Special Relativity follows. But you'll have to read the book for the details (or google for k-calculus).

It's all very elementary stuff. But this k-factor argument seems like it would be much more convincing to the layman than Einstein's original thought experiment. In fact, I did a bit of googling and it seems that the k-factor argument has been presented in a few different places under the name "k-calculus".

Bondi also stresses that watches are like odometers. They measure something that is personal to the path that you have taken rather than some absolute observer independent quantity. This is something that is hard to get from other elementary accounts I have read. Other introductions present time dilation as some kind of influence that somehow makes your clocks go 'wrong'. Just the name 'time dilation' gives the impression that what you measure is somehow dilated from the 'correct' time, a completely erroneous view. Just as there is no 'distance' dilation for driving north-east and then north-west to get to a place due north.

Anyway, if you have some doubting Thomas relative (no pun intended) who thinks the whole Relativity thing is bogus, but they do have a bit of numerical aptitude, this might be the book to get them. You don't even need to compute a square root to compute the discrepancy between the watches in Bondi's first example. All in all, an excellent book!

Anyway, my reason for reading this book is that I really don't like the way some physicists talk about Relativity. So I thought I'd read a few elementary accounts to get an idea of what trends there are in explaining the subject.

(Sadly, Bondi passed away last year. And maybe I've finally got over my resentment of Bondi for giving me low marks for the cosmology questions in exams at Cambridge one year...)

## 1 Comments:

Sir,

I have a question. From reading some books using this k calculus approach, what I gather is that the constancy of the speed of light is not a postulate, but something that inevitably results if u take c=1 and use light signals for measuring distance and time coordinates. However, letâ€™s say, i use sound wave instead of light to define distance and to measure time. In that way a unit of distance will be a sound-second ( rather than light second). Now if I define the speed of sound as s=1, then I can carry out the same k calculus and in the end, after deriving the formula for velocity composition, it will turn out that the speed of sound is constant irrespective of the observers relative motion! Likewise anything other than light I use in the same way, same result will turn out!

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