It is frequently discussed that to find solutions having some sort of supersymmetry is easier than solving Einstein’s equations of motion.

I do not understand this discussion though. Specifically, why are Einstein's equations difficult to solve? Is it because they are second order equations and the supersymmetric ones are first order? Another thing that makes me not understand this is the fact that Einstein's equations are not supersymmetric while supersymmetric ones obviously are supersymmetric, so how can we compare two equations from two different theories?

  • $\begingroup$ By mentioning "solutions having some kind of supersymmetry" precisely which theory/model's solution are you referring to? $\endgroup$
    – Bruce Lee
    Commented Dec 25, 2015 at 18:05
  • $\begingroup$ you are right in pointing out that second order coupled differential equations (Einsteins equations) are far more difficult to solve than first order ones (the spinor equations). That is definitely one of the reasons. $\endgroup$
    – Bruce Lee
    Commented Dec 25, 2015 at 18:22
  • $\begingroup$ ... give the same answers ... in the common domain ( of application ) of the 2 theories. Like did GR with newtonian mechanics, there are adjustments even if here they may appear as very efficient... It remains to know if the results are the same in other cases. $\endgroup$
    – user46925
    Commented Dec 25, 2015 at 19:49
  • $\begingroup$ I tried to modify the title , but each word being important , it would be better you do it $\endgroup$
    – user46925
    Commented Dec 25, 2015 at 19:58

1 Answer 1


The problem with the Einstein equations is that they are non-linear partial differential equations and, as you know, there are no general algorithms to solve them. Sometimes you can impose some symmetries - make an ansatz (guess) such that the Einstein equations becomes ODE and then solve them (that is how the Schwarzschild solution was found).

On the other hand, if you have a supersymmetric background, then, besides the Einstein equations, it also satisfies the Killing spinor equations (KSE), which are much easier to solve. Different theories have different KSE, but they are always simpler than Einstein's. So usually people start with Killing spinor equations, solve them to obtain some restrictions on the metric/fluxes and then check the Einstein equations.

  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$
    – David Z
    Commented Dec 28, 2015 at 12:40

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