Assume the following situation is given:

We know that there is a 4 dimensional pseudo-Riemannian manifold $(M,g)$ (for now, of unknown signature) and there are 10 Killing vector fields, indexed as $K_\mu$ and $K_{\mu\nu}$ (latter is skew in $\mu\nu$) satisfying the following commutation relations: $$ [K_\mu,K_\nu]=0 \\ [K_\kappa,K_{\mu\nu}]=\eta_{\kappa\mu}K_\nu-\eta_{\kappa\nu}K_\mu \\ [K_{\mu\nu},K_{\kappa\lambda}]=\eta_{\mu\lambda}K_{\nu\kappa}+\eta_{\nu\kappa}K_{\mu\lambda}-\eta_{\nu\lambda}K_{\mu\kappa}-\eta_{\mu\kappa}K_{\nu\lambda}. $$

We wish to construct a metric whose Killing-algebra is given by these.

Approach 1:

Realize that since $[K_\mu,K_\nu]=0$, we may postulate a local chart whose coordinate frame is $\{K_\mu\}$ with $K_\mu=\partial_\mu$. The Killing condition would then force the metric components to be constant. It would not fix the signature, but I would hope the signature can be recovered from the rest of the relations.

Problem: We cannot know if the Killing vectors $K_\mu$ are pointwise linearly independent or not. Clearly the 10 KVs in a 4 dimensional space cannot be all pointwise independent. If these happen to be not independent pointwise then this reasoning fails.

Approach 2:

Define $ B_a=K_{a0} $ and $R_a=-\frac{1}{2}\epsilon_{abc}K_{bc}$ (latin indices go 1,2,3) as the boost and rotation generators.

The rotation generators assume the relation $$ [R_a,R_b]=\epsilon_{cab}R_c. $$ Recognize that this is the Lie algebra of $SO(3)$, so we essentially have a group of isometries whose orbits are 2-spheres. This gives us a metric ansatz $$ A(t,r)dt^2+B(t,r)dr^2+2C(t,r)dtdr+D(t,r)(d\vartheta^2+\sin^2\vartheta d\varphi^2), $$ with the rotation generators being the Killing vectors of the 2-sphere, and we may work from here.

Problem: This may be my general ignorance, but does it actually follow from here on that the functions $A,B,C,D$ are independent of $\vartheta$ and $\varphi$? Does it actually follow from this argument that the orbit 2-spheres are orthogonal to $\partial_t$ and $\partial_r$?

Also, I'm a bit confused, because the boost algebra satisfies $[B_a,B_b]=-\epsilon_{cab}B_c$, but if we define the generators with opposite sign, then the structure constants agree with that of the rotation algebra. So I could have done this with the boost algebra too. Does this not cause a problem?

Approach 3: Recognize that this space is maximally symmetric. Then one can basically specify a unique metric ansatz and work from there.

Problem: Let's just say I need this for a specific purpose, and I want to avoid this approach for this purpose.


1) Would any of my approaches work (aside from Approach 3)? If so, how to resolve the problems?

2) What is the most systematic way of recovering a metric from its Killing-algebra? I mean, the one that requires the least "divine inspiration" (as in foreknowledge of what we're gonna get)?

3) Can the signature actually be recovered from this, or do I need to postulate it beforehand?

If the answer would be too long, I'd also appreciate any sources that deal in detail with using Killing vectors/symmetries to construct spacetimes (I know that the book "Exact Solutions of Einstein's Field Equations" does that a lot, but that does pretty too high-level pretty too quickly imo).


1 Answer 1


Roughly speaking: you know that your manifold is maximally symmetric, so up to isometry it's either Minkowski, de Sitter or anti de Sitter. The isometry group of these are the Poincaré group, $O(n,1)$ and $O(n-1,2)$, so they can be distinguished.

However local information is not enough to reconstruct the spacetime since quotients have the same local symmetry group. See for example this question.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.