# Reconstructing Minkowski spacetime from its Killing-algebra

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.

Questions:

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).

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.