# Carter Constant with a Cosmological Constant

The Carter constant for the Kerr Newman metric

$$\rm C = p_{\theta}^{2} + \cos^{2}\theta \ \Bigg[ a^2 \ (m^2 - E^2) + \left(\frac{L_z}{\sin\theta} \right)^{2} \Bigg]$$

with (in $$[+---]$$ signature)

$${{\rm E = -p_t}=g_{\rm tt} \ {\rm \dot{t}}+g_{\rm t \phi} \ {\rm \dot{\phi}} + \rm q \ A_{t}}$$

$${\rm L_z = p_{\phi}}=-g_{\rm \phi \phi} \ {\rm \dot{\phi}}-g_{\rm t \phi} \ {\rm \dot{t}} - \rm q \ A_{\phi}$$

$${\rm p_{\theta}}=-g_{\rm \theta\theta} \ {\rm \dot{\theta} \ } { \color{#aaaaaa}{ - \ g_{\rm t \theta} \ {\rm \dot{t}} - \rm q \ A_{\theta}}}$$

(with the terms that are 0 grayed out) does not work in the Kerr Newman (Anti) De Sitter metric. In some references on the subject I've seen it mentioned though, but I've never seen its explicit form in any of them. Does it even exist in this case?

I suspect if so, there must be some additional terms dependend on $$\Lambda$$ and $$\theta$$, since even on an orbit at constant $$\rm r$$ (but variable $$\theta$$) the form that works fine with $$\Lambda=0$$ gives a sine-like function with $$\Lambda \neq 0$$ (with an orbit like this where the other constants of motion $$\rm E$$ and $$\rm L_z$$ work just fine and stay constant as they should).

Since Carter seems to have come up with his constant with a lucky guess while no one really knows why it even works, I'm not sure how to derive it by first principles, and before I try to land a lucky guess myself, maybe somebody has already figured it out or knows a reference where it is not only mentioned, but also given (for the KNdS/KNAdS metric).

It doesn't really matter since I use the second order equations of motion anyway, and the two constants of motion $$\rm E$$ and $$\rm L_z$$ are sufficient to monitor the numerical stability, but for the sake of completeness I'm still interested if there is a version of the Carter constant that works with $$\Lambda$$.

• In order to work for electric/magnetically charged particles. The Killing tensor $K_{ab}$ also needs to satisfy $K_{a(b}F_{c)}^d$ and $K_{a(b}{\star}F_{c)}^d$. The paper does not check whether that is true. It is however, true in the Kerr-Newman limit, so it's worth checking. Commented Sep 13, 2023 at 10:04