Zero Angular Momentum Observer Frame

• Zero Angular Momentum Observer (ZAMO) frames are locally nonrotating reference systems in stationary, axisymmetric spacetimes that cancel frame dragging by ensuring zero angular momentum.
• They are defined by the condition gₜφ uᵗ + gφφ uᵖ = 0, yielding a specific local angular velocity that mathematically cancels inertial effects.
• ZAMOs provide crucial insights into black hole physics, particularly in the Kerr metric, while differing from globally inertial astronomical reference frames.

A Zero Angular Momentum Observer (ZAMO) frame is a class of locally nonrotating reference frames defined in any stationary, axisymmetric spacetime (i.e., a manifold with two commuting Killing vector fields, $\partial_t$ and $\partial_\varphi$). ZAMOs are observers whose worldlines maintain zero angular momentum about the symmetry axis, achieved by circulating with a specific $\varphi$-velocity that exactly cancels the gravitomagnetic effects of inertial-frame dragging. While ZAMOs provide a natural framework for local physics around rotating bodies such as Kerr black holes, their rotation is only well defined with respect to local geometry, and they should not be conflated with globally inertial or astronomically meaningful (star-fixed) reference frames (Costa et al., 8 Oct 2025, Braeck, 2023).

1. Definition and Characterization of ZAMOs

In a general stationary, axisymmetric metric

$$ds^2 = g_{tt}(r,\theta)dt^2 + 2g_{t\varphi}(r,\theta)dtd\varphi + g_{\varphi\varphi}(r,\theta)d\varphi^2 + \cdots,$$

the angular momentum $L$ of an observer with four-velocity $u^\mu$ is

$$L = g_{t\varphi}\,u^t + g_{\varphi\varphi}\,u^\varphi.$$

A ZAMO is defined by the algebraic condition $L = 0$, explicitly,

$$g_{t\varphi} u^t + g_{\varphi\varphi} u^\varphi = 0.$$

This leads to a unique local angular velocity

$$\omega(r,\theta) \equiv \frac{u^\varphi}{u^t} = -\frac{g_{t\varphi}}{g_{\varphi\varphi}}.$$

The normalization $u^\mu u_\mu = -1$ fixes the $t$-component:

$$u^t = \bigl[-(g_{tt} + 2\omega g_{t\varphi} + \omega^2 g_{\varphi\varphi})\bigr]^{-1/2}.$$

ZAMOs thus circulate with respect to the $t$-coordinate so as to locally cancel $u_\varphi$, nullifying specific angular momentum (Costa et al., 8 Oct 2025, Braeck, 2023).

2. Orthonormal Tetrad and 3+1 Decomposition

Given the ZAMO four-velocity $u^\mu$, one constructs an adapted orthonormal tetrad:

• $$e_{(0)} = u^\mu \partial_\mu = u^t (\partial_t + \omega \partial_\varphi)$$
• $e_{(r)} = \frac{1}{\sqrt{g_{rr}}} \partial_r$
• $$e_{(\theta)} = \frac{1}{\sqrt{g_{\theta\theta}}} \partial_\theta$$
• $$e_{(\varphi)} = \frac{1}{\sqrt{g_{\varphi\varphi}}} \left(\partial_\varphi + \frac{g_{t\varphi}}{g_{\varphi\varphi}} \partial_t\right)$$

This tetrad is orthonormal with respect to the metric and ensures $g(e_{(0)}, e_{(\varphi)}) = 0$ by construction.

In the 3+1 formalism, the metric is decomposed as

$$ds^2 = -N^2 dt^2 + h_{ij}(dx^i + N^i dt)(dx^j + N^j dt),$$

with lapse $N = 1/\sqrt{-g^{tt}}$ and shift $N_i = g_{ti}$. ZAMOs coincide with Eulerian (normal) observers:

$u^\mu = \frac{1}{N}(1, -N^i).$

The local rotation $\omega = -g_{t\varphi}/g_{\varphi\varphi}$ cancels the metric's frame dragging (Costa et al., 8 Oct 2025).

3. ZAMOs in Kerr Spacetime

In Boyer–Lindquist coordinates of the Kerr metric: $$\begin{aligned} & g_{tt} = -\left(1 - \frac{2Mr}{\Sigma}\right), \qquad g_{t\varphi} = -\frac{2Mar\sin^2\theta}{\Sigma}, \ & g_{\varphi\varphi} = \left(r^2 + a^2 + \frac{2Ma^2 r\sin^2\theta}{\Sigma}\right)\sin^2\theta, \qquad \Sigma = r^2 + a^2\cos^2\theta. \end{aligned}$$ The ZAMO angular velocity is

$$\omega(r, \theta) = -\frac{g_{t\varphi}}{g_{\varphi\varphi}}.$$

In the equatorial plane:

$\omega(r) = \frac{2Ma}{r^3 + r a^2 + 2Ma^2}.$

This vanishes only at spatial infinity in the Boyer–Lindquist frame (Costa et al., 8 Oct 2025, Braeck, 2023).

4. Frame Dragging, Sagnac Effect, and Local Nonrotation

The coordinate mixing $g_{t\varphi} \neq 0$ reflects the dragging of inertial frames (Lense–Thirring effect). A ZAMO's vanishing angular momentum ensures the absence of local Sagnac effect; light sent around a closed azimuthal loop by a ZAMO experiences no time delay since the numerator in the Sagnac formula,

$g_{t\varphi} + g_{\varphi\varphi}\omega,$

vanishes identically for the ZAMO (Braeck, 2023).

Locally, ZAMOs represent a standard of nonrotation: at each spacetime point, their vorticity is zero, and they measure no local inertial rotation. However, $\omega(r,\theta)$ generally varies with position, leading to a nonzero shear $\sigma_{\alpha\beta}$ in the congruence of ZAMO worldlines; neighbors at rest in the ZAMO frame do not remain at fixed directions relative to one another, except in special cases where $\omega$ is constant (Costa et al., 8 Oct 2025).

5. Relativity of Rotation and the Thin Shell Paradigm

A salient result shown with the rotating thin-shell model (e.g., Brill–Cohen shell) is that the Einstein equations constrain only differences of ZAMO rotation rates:

$\Omega(r) - \Omega(r_Q),$

not absolute values. The transformation

$$\varphi \to \varphi - [\Omega(r_Q)-\widetilde{\Omega}_Q] t$$

effectively shifts the entire rotation profile by a constant, reflecting the coordinate freedom to set any reference ZAMO as "nonrotating." Thus, only relative rotation is physically meaningful; absolute nonrotation for ZAMOs, even at infinity, is a coordinate convention rather than an invariant property (Braeck, 2023).

6. Astronomical Reference Frames versus ZAMOs

A crucial distinction arises between ZAMOs and astronomically meaningful reference frames. While ZAMOs are well-suited for defining local energy or analyzing process slices (e.g., Penrose process near black holes), they are not shearfree and are not anchored to distant inertial objects. Shearfree, asymptotically vorticity-free congruences are required for the construction of reference frames "locked" to distant stars, as in IAU conventions. Misidentification of ZAMOs with globally nonrotating observers leads to physical misinterpretation—examples include the erroneous conclusion of "no hole rotation" at the horizon or spurious flat galactic rotation curves in rigidly rotating models (Costa et al., 8 Oct 2025).

7. Summary Table: Defining Features of ZAMO Frames

Feature	Local ZAMO Frame	Astronomically Anchored Frame
Zero angular momentum	Yes ($u_\varphi = 0$)	No (unless congruence is also shearfree)
Shearing congruence	Yes, generically ($\omega$ varies)	No (shearfree by construction)
Asymptotic star fixation	No	Yes
Sagnac effect observed	No	Depends on congruence

ZAMO frames are rigorously defined by setting $u_\varphi = 0$, with explicit formulas derived from the metric components. Their use in local diagnostics of frame-dragging is central to black-hole physics, though caution must be exercised when interpreting them as candidates for global inertial reference systems in general relativity (Costa et al., 8 Oct 2025, Braeck, 2023).