General Relativity and Celestial Rotation

Author: Suhang Zhang, Luoyang, Henan

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Abstract

General relativity describes gravity through spacetime curvature, and its core equations—the Einstein field equations and the geodesic equation—successfully explain the orbital motion of celestial bodies. However, with respect to celestial rotation, general relativity's treatment is structurally incomplete: rotation is an "input" in the theory (Kerr parameter, angular momentum), rather than an "output" of curvature; orbital motion and rotation are described by two independent sets of equations, lacking a unified framework; and the dynamical origin of rotation—namely, how curvature determines angular momentum—has not been systematically clarified. This paper reviews the existing ways in which general relativity handles rotation, points out what remains unfinished, and explains that the direction of "curvature determines rotation" is an area that general relativity has not yet fully explored.

Keywords: general relativity; rotation; angular momentum; geodesic; curvature gradient

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I. Introduction

The core of general relativity is "matter tells spacetime how to curve, spacetime tells matter how to move." The first half is borne by the Einstein field equations:

G_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}

The second half is borne by the geodesic equation:

\frac{d^2x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau} = 0

This framework successfully explains phenomena such as the perihelion precession of Mercury, light deflection, gravitational redshift, and gravitational waves. But on the question of celestial rotation, general relativity's treatment is not complete.

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II. How General Relativity Handles Rotation

General relativity does involve rotation, but in the manner of "from rotation to spacetime," not "from spacetime to rotation."

1. Kerr metric

Given the angular momentum $J$ and mass $M$ of a rotating celestial body, the Kerr parameter is defined:

a = \frac{J}{M}

The Kerr metric describes the spacetime around this rotating body. This is "given angular momentum → compute spacetime."

2. Lense–Thirring effect

A rotating mass drags the surrounding spacetime, producing a frame-dragging angular velocity:

\omega_{LT} \sim \frac{GJ}{c^2 r^3}

This is "rotating mass → spacetime dragging," still "from rotation to spacetime."

3. Motion of spinning particles

The spin vector $S^\mu$ is parallel transported along the geodesic:

\frac{DS^\mu}{d\tau} = 0

This is "spin moving in curved spacetime," but this is an independent equation, standing alongside the geodesic equation, not deriving spin from curvature.

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III. Where It Is Incomplete

1. The direction is reversed

· General relativity: angular momentum $J$ → spacetime (Kerr)

· Not treated: curvature → angular momentum

· Rotation is an input, not an output

2. Orbital motion and rotation are two sets of equations

· Orbital motion: geodesic equation (free particle)

· Rotation: Kerr metric (spacetime property) or spin precession equation

· The two are not unified into a single framework

3. The dynamical origin of rotation is not clarified

· The geodesic equation assumes a spinless particle

· The motion of a spinning particle requires additional assumptions (spin equation)

· How curvature determines angular momentum has not been systematically answered

4. There is no mechanism for "curvature → angular momentum"

· General relativity does not give a relation such as "curvature gradient drives rotation"

· There is no equation of the form $\omega \propto \nabla R$

· This is the part that remains unexplored

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IV. Why It Is "Incomplete"

The basic object of general relativity is the metric $g_{\mu\nu}$:

· The metric describes spacetime geometry

· Orbital motion is a direct consequence of the metric (geodesic)

· Rotation is a property of matter (angular momentum)

In general relativity, rotation is not a direct consequence of geometry; it is an input of matter. Therefore the logical directions of rotation and orbital motion differ:

· Orbital motion: geometry → motion (direct)

· Rotation: matter → geometry (Kerr), or matter → motion (spin precession)

This asymmetry in logical direction is the root of the "incompleteness."

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V. The Unfinished Direction

To "make it complete," one needs to supplement the line "curvature → angular momentum":

· Orbital motion: curvature → geodesic (already present)

· Rotation: curvature gradient → angular velocity (to be supplemented)

· Both share the same source (curvature), with different mechanisms

A possible form:

\omega \propto \nabla R

where $\omega$ is the spin angular velocity and $\nabla R$ is the curvature gradient. If this relation holds, it would incorporate rotation into the framework of "curvature determines motion."

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VI. Conclusion

General relativity successfully explains celestial orbital motion, but its treatment of celestial rotation is incomplete: rotation is an input rather than an output; orbital motion and rotation are two sets of equations; and how curvature determines angular momentum has not been systematically clarified. The line "curvature → angular momentum" is an area that general relativity has not yet fully explored.

This is not to say that general relativity is wrong—it is self-consistent within its own domain. Rather, it is to say: on the question of rotation, it has only done half the job.

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References

Omitted

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