Re: Division by zero

On 2025-02-03 07:56:53 +0000, Thomas Heger said:

Am Sonntag000002, 02.02.2025 um 10:30 schrieb Mikko:
 

Hi NG
 I'm actually not really certain, but found an error in Einstein's 'On the electrodynamics of moving bodies' which is quite serious.
  See page six, roughly in the middle:
 There we find an equation, which says this:
 ∂τ/∂y= 0

 Do you mean on page 899 (9th page of the article) in §3?
The operation is not division but a partial derivative.

 You should answer this question. It is not useful to talk without telling
what you are talking about.
 

I'm referring to the English translation, which can be found here
 https://www.fourmilab.ch/etexts/einstein/specrel/www/
 The English pdf version has other page numbers than the original article.
 But in a way, these original page numbers are also possible as reference.
 But unfortunately I have here only the English version (the German I have on a different computer).
 So I have to tell you the page from the English version or make the meant part available to you by other means.
 So, § 3 was meant and roughly the middle, which can be found on page 6 of the English pdf version.
 And you are absolutely right, that a partial derivative was meant.
 The problem was: of which function was a partial derivative meant?

He obviously means the function needed to determine τ. It does not matter
whether he means the function from x, y, z, t or x', y, z, t as ∂/∂y is
the same in both cases.

Einstein didn't define the used variables and simply assumed, the reader would know anyhow, what he had in mind.

Variables are clearly defined. For example, x, y, z, and t are defined as
the coordinates of the system K.

But that wasn't particularly easy, because Einstein used the symbol τ for three different types of objects.
 a) the time values of clocks in system k were named τ
 b) a function τ was derived, which should serve as coordinate transformation between system K and system k

Although modern mathematicians don't consider that correct, it is common
to use the same symbol for a quantity and for a function that computes
that quantity. It is obvious from the context which is meant: function
name is used with arguments, the quantity name without.

c) this function take (kind of) four-vectors of K as input and spits out four-vectors in k as output, while these output vectors were also called τ.

Nowhere in the article is any vector called τ.

This was rather nasty, because it could lead to several errors, if you try to interpret Einstein's intentions.

A careless reader may get a wrong idea but the target audience could
understand it.

And I have actually fallen in one of these traps, because I had regarded τ as time-value, while actually the function τ of case b) was meant.

That function is a time-valued function.
You are not in the target audience of the article. Nobody still alive is.
Therefore your comprehension problems are not an indication of a defect
in the article.
The article is incomplete. It only presents some core ideas. In later
articles Einstein filled gaps in the reasoning and extended to other
problems.
--
Mikko