# Derivation of Komar formulae from ADM formalism?

Is there a way to derive the Komar formulae from the ADM formulae? Both the formulae give the same answer for mass and angular momentum, so I was wondering if one can be derived from another.

On an unrelated note, if I know all the killing vectors, can I use it to uniquely determine the metric?

• if I know all the killing vectors, can I use it to uniquely determine the metric ?. I don't think this is possible. The Killing fields are defining the local symetries of the metric, but you could have two different metrics with the same symetries (for example : isotropy around a center and static spacetime). If the spacetime metric is maximally symetric, then I guess you could define the metric explicitly, but even then I'm not sure.
– Cham
Mar 13, 2019 at 12:46
• It's generally not a good idea to ask two unrelated questions as one question on SE. I'd suggest you edit out the second one.
– user4552
Mar 13, 2019 at 14:28
• Even in the case where the spacetime is both asymptotically flat and stationary, it wasn't proved until 15 years after Komar that the Komar and ADM masses are equal (discussed in arxiv.org/abs/1003.5015 ). So it seems unlikely to me that it's easy to get one from the other, even in that special case.
– user4552
Mar 13, 2019 at 14:48

The Komar and ADM formulae apply to different classes of spacetime. In particular, the ADM formulae apply only asymptotically flat spacetimes whereas the Komar formulae apply to any spacetime with Killing vectors. Moreover, the ADM formulae can only be applied to a sphere at "infinity", while the Komar integrals can be integrated over any surface.

There are, of course, spacetimes, such as Schwarzschild and Kerr, that are both asymptotically flat and have Killing vectors. In these cases both give the same answer. However, there are also spacetimes that have Killing vectors that are not asymptotically flat, and the other way around. Consequently, one would not necessarily expect to be able to derive one from the other.

In particular, I see no way that the Komar integrals could be derived from the ADM ones. The other way around, I believe you can obtain the ADM integrals by using the fact that asymptotic flatness implies that the spacetime has asymptotic Killing fields. (Actually, I'm not so sure about the last part)

• Just for completeness, could you show both integrals ?
– Cham
Mar 13, 2019 at 12:42
• @mmeent So, how does one derive the Komar formulae ? Can it be derived through the Hamiltonian formalism ? Mar 14, 2019 at 5:38