- #36
haushofer
Science Advisor
- 2,973
- 1,524
Yes. but Einstein didn't regard it as a "gauge condition"; as far as I understand, Einstein regarded them for some time confusingly as part of the field equations. Only later did he realize his mistake.vanhees71 said:Can't one impose this as one gauge condition? I've somehow in mind that Einstein preferred this choice of a gauge condition for the choice of the coordinates, but I've not the time to look for the source. It even may be in the here discussed paper.
The crucial identity here is
[tex]
\Gamma^a_{ab} = \partial_b \Bigl( \log{\sqrt{|g|}} \Bigr)
[/tex]
In the unimodular "gauge" one thus has
[tex]
\Gamma^a_{ab} = 0
[/tex]
and this simplifies your calculation quite a bit. E.g., the Ricci tensor becomes[tex]
R_{ab} = \partial_c \Gamma^c_{ab} - \Gamma^c_{ad}\Gamma^d_{cb} \ \ \ (unimodular \ gauge!)
[/tex]
In the weak field limit, the quadratic term can even be neglected. I guess it's not a far reach to see
[tex]
R_{00} \approx \partial_c \Gamma^c_{00} \ \,,
[/tex]
.i.e. a Poisson-like equation for the Einstein field equations appearing. The Newtonian limit is then obtained by the condition of static metrical components and slowly moving particles.
See also "How Einstein found his field equations" of Renn and Janssen in Physics Today. His confusion regarding the meaning of coordinate "restrictions" also led Einstein to his infamous "hole argument".
Last edited: