In this recent article, the authors write

Consequently, it is a desideratum to teach general relativity in a way that is based on elementary mathematics only. This objective, already stated by Einstein in 1916 (Einstein 1916), has been pursued in many different ways both in the development of teaching materials and in popular science publications

Where exactly, in his 1916 paper, does Einstein state the goal of simplifying the mathematics of general relativity? An English translation would be great, but the German would suffice.

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    $\begingroup$ I suspect there's little consensus as to what constitutes "elementary math." Does it include Geometry? Calculus? Abstract algebra? Topology? ODEs? $\endgroup$ Jan 15 '19 at 13:35

Not sure about "elementary" but, at least, he tried to reduce it to "the simplest and most intelligible form". He assumes the level of a graduating secondary school student, and warns of the "patience and force of will" required to get through.

The linked article refers to the German 1916 publication of Über die spezielle und allgemeine Relativitätstheorie, 1920 reprint is available on the Internet Archive. The first English translation, by Lawson, Relativity: the Special and the General Theory, first appeared also in 1920, and is available on ibiblio.org. Here is Einstein's preface in full:

"The present book is intended, as far as possible, to give an exact insight into the theory of Relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus of theoretical physics. The work presumes a standard of education corresponding to that of a university matriculation examination, and, despite the shortness of the book, a fair amount of patience and force of will on the part of the reader. The author has spared himself no pains in his endeavour to present the main ideas in the simplest and most intelligible form, and on the whole, in the sequence and connection in which they actually originated. In the interest of clearness, it appeared to me inevitable that I should repeat myself frequently, without paying the slightest attention to the elegance of the presentation. I adhered scrupulously to the precept of that brilliant theoretical physicist, L. Boltzmann, according to whom matters of elegance ought to be left to the tailor and to the cobbler. I make no pretence of having withheld from the reader difficulties which are inherent to the subject. On the other hand, I have purposely treated the empirical physical foundations of the theory in a “step-motherly” fashion, so that readers unfamiliar with physics may not feel like the wanderer who was unable to see the forest for trees. May the book bring someone a few happy hours of suggestive thought!"


Only elementary mathematics is required to formulate and understand Special Relativity as a glance of Einsteins 1905 paper On the Electrodyamics of moving bodies shows, where he put forward the main ideas involved in SR.

This is not the case for GR, as Einstein himself was forced to admit. He learned tensorial methods for working with general manifolds from his longtime colleague, Marcel Grossmann and this was the main mathematical technology he used to formulate GR; methods that were different from his rival, Hilbert, who used a variational approach - aka the Einstein-Hilbert action.

However, it is possible to develop some of the ideas of GR in an elementary way as Robert Geroch did in his small but delightful book, Relativity from A to B, the book that I first learnt about GR during my last year of high school; basically he uses the fact the Gausian curvature is a product of principal curvatures and this allows him to get to the main ideas of GR without developing the apparatus of tensor fields over manifolds, an apparatus that I might add, whose pedagogical development still has much room for improvement.

  • $\begingroup$ Only elementary mathematics is required to formulate and understand Special Relativity as a glance of Einsteins 1905 paper On the Electrodyamics of moving bodies shows, where he put forward the main ideas involved in SR. I'm not sure what you mean by "elementary" here. The initial sections of the paper are written using elementary mathematics, but certainly not the later sections, which consist of a lot of vector calculus written in clumsy old-fashioned notation. $\endgroup$
    – user466
    Nov 13 '19 at 6:40
  • $\begingroup$ @Ben Crowell: If you compare Einsteins papers with many modern papers in physics using concepts from differential geometry and differential topology you'll see what I mean. $\endgroup$ Nov 13 '19 at 22:52

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