The role of angular momentum in general relativity: heuristic and covariant interpretations

TL;DR

This work shows that in General Relativity angular momentum and black-hole rotation do not act solely as a centrifugal barrier; through spin–energy and frame-dragging couplings, especially in the Kerr spacetime, angular momentum can enhance or reduce gravitational attraction relative to Schwarzschild. The authors combine a heuristic effective-potential analysis for Schwarzschild and Kerr with a covariant 1+3 Raychaudhuri framework to connect local dynamics (expansion and shear) to integrated infall times, identifying parameter regimes where rotation shortens or lengthens collapse times. The key finding is that Kerr rotation reshapes the focusing of timelike geodesic congruences by modulating shear, which can dominate over changes in expansion to govern the net infall time, a true geometric effect of relativistic kinematics. These results provide a principled relativistic benchmark for understanding infall, accretion, and early structure growth in environments where low angular momentum and high spin are relevant, while also clarifying the limitations of Newtonian intuition in strong-field regimes.

Abstract

We examine the role of angular momentum in general relativity from both heuristic and fully covariant perspectives, with the aim of clarifying conceptual ambiguities that arise when Newtonian intuition is extrapolated into the relativistic regime. Focusing on free--fall dynamics in the Schwarzschild and Kerr spacetimes in the test--particle limit, we employ an effective--potential heuristic approach to isolate the roles of the specific energy $E$, specific angular momentum $L$, and black--hole spin $a$. Within this framework, we identify well--defined regions of parameter space in which the Kerr spacetime leads to stronger or weaker local radial infall than the Schwarzschild case at the same radius. By analysing the kinematics of infalling geodesic congruences, we show how these local regimes combine along complete trajectories to either enhance or reduce gravitational focusing. We then interpret these results within a covariant 1+3 description of general relativity, in terms of the expansion, shear and Raychaudhuri evolution of timelike congruences. We demonstrate that black--hole rotation systematically modifies the shear of infalling irrotational flows, even when the magnitude of the local expansion is reduced, and that this shear modulation governs the overall rate of focusing. Our work complements previous studies of relativistic infall by providing a unified energetic and geometric interpretation of how angular momentum and rotation can strengthen or weaken gravitational collapse relative to the non--rotating case.

Figures (7)

• Figure 1: Ratio between the Newtonian infall time and the Schwarzschild proper infall time for trajectories from $r_0$ to $r_f=6M$, as a function of angular momentum $L$. The increasing deviation from unity quantifies the relativistic reduction of the integrated proper fall time relative to Newtonian expectations.
• Figure 2: Maps of the Kerr spin-induced deviation of the effective potential from Schwarzschild, $V_{\rm \Delta}(r;\lambda)/L^2$, shown as a function of the dimensionless radius $r/M$ (horizontal axis) and the parameter $\lambda = a/L$ (vertical axis), for different values of the energy $E$. Blue regions correspond to a regime that is more attractive than Schwarzschild, while green regions correspond to a more repulsive regime. Note the sharp transition between co-rotating and counter-rotating motion in the limiting case $\lambda=0$, where the regime switches from repulsive to attractive. At high energies, Kerr rotation yields a more attractive regime than Schwarzschild over essentially the entire parameter space.
• Figure 3: Profiles of $\tau$ as a function of the spin parameter $a$ for prograde (UP) and retrograde (DOWN) infall down to a fixed radius $r_f=3M$. Curves are shown for several values of the specific energy $E$ and for two representative angular momenta, $L=0.3$ and $L=1$. For low and moderate energies, $\tau$ increase markedly with $a$ for prograde orbits, while the trend is the opposite for retrograde trajectories. At higher energies, the overall infall time is shorter and becomes nearly independent of the black--hole spin.
• Figure 4: Ratio between Kerr and Schwarzschild values of $\tau$ for particles falling from $r_0=10M$ to $r_f=3M$ in prograde (UP) and retrograde families (DOWN).
• Figure 5: Parametric maps of Kerr--minus--Schwarzschild variation of the effective potential in the $(r/M,\lambda=a/L)$ plane for four representative combinations of energy $E$ and angular momentum $L$. Green (blue) regions correspond to locally repulsive (attractive) contributions of the barrier sector to the radial effective potential. Dotted black curves indicate representative infall tracks from $r_0=10M$ toward smaller radii. The event horizon $r_+$ is shown as a solid red curve, while the co-rotating and counter--rotating ISCOs are shown as dashed red curves. Reference fixed radii are marked by vertical dashed white lines. The figure highlights that retrograde orbits can, at low energies, cross attractive regimes not present in the prograde case. However, the attractive regime dominates strongly the parameter space at high energies.