Republication of: On the Newtonian limit of Einstein’s theory of gravitation

Abstract

This paper has been selected by the Editors of General Relativity and Gravitation for re-publication in the Golden Oldies series because it introduced Frame Theory–a theoretical framework enabling a unified treatment of General Relativity and classical Newtonian gravitation. This in particular has applications whenever a Newtonian model is used to study a gravitating system, for example in cosmology. The accompanying editorial note by Thomas Buchert and Thomas Mädler discusses Frame Theory’s value and its impact on later developments. The original is in German and not in a readily available journal: it is presented here in English. This is the fifth Golden Oldie of which Jürgen Ehlers was a co-author or author (not to mention the several for which he contributed an editorial note): it appears in what would have been his 90th year.

Translator’s note

The copy for the translation was taken from a book owned by Ehlers. The book contained his German handwritten notes, which are translated alongside the translation of the typeset text, by permission of Frau Anita Ehlers. The original used for this translation can now be found in the Ehlers archive in the Albert Einstein Institute, Golm, Germany. The translator has added some notation which is not present in the German original:

• A word in double brackets, e.g. [[the field]], was included for the sake of easier reading.

• A phrase in double curly brackets with the label h.w., e.g. {{h.w: counter examples?}}, indicates a translation of a hand-written note in the German original.

• A phrase in double angle brackets with the label c.t.o, e.g. $\langle \langle$c.t.o: $V^aW_a<0 \rightarrow V^aW_a>0$ $\rangle \rangle$, indicates a correction of a typo in the German original.

• An asterisk $^*$ at the end of a sentence indicates that the respective sentence (and often also the one before) has been restructured, compared with the original, because the English of a more literal translation would have been hard to read.

• The typeset text had an inconsistent notation on the signature of a metric, e.g. $(+++,0)$, $(+++-)$ and $(0,0,0,+)$, which are corrected to a consistent comma separation between the items, e.g. $(+,+,+,0)$.

• With regard to the index positioning of the connection $\varGamma ^a_{bc}$, we follow the MTW convention [35].

• In Sect. 2 rule A, the German original uses the notation $U^a_\cdot :=h^{ab}U_b$. In this translation, we use a bullet point . instead of a dot, e.g. $U^a_{\mathbf{. }} :=h^{ab}U_b$, for easier readability.

• From Sect. 3 on: the phrase “families of Einstein’s solutions” is the literal translation of “Einsteinsche Lösungsscharen”, which is understood as “families of solutions of Einstein’s equations”. The literal translation is often used to be in tune with the German original.