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Chapter 7: Riemann Curvature Tensor
Geodesic Deviation and Physical Meaning
The Riemann tensor has a direct physical manifestation: it governs the relative acceleration between nearby freely falling particles. This geodesic deviation equation is the mathematical expression of tidal forces in general relativity, and it provides the most intuitive way to understand spacetime curvature.
The Geodesic Deviation Equation
Consider a one-parameter family of geodesics \( x^\mu(\tau, s) \) where \( \tau \) is proper time along each geodesic and \( s \) labels different geodesics in the family. Define the tangent vector and the deviation vector:
$$u^\mu = \frac{\partial x^\mu}{\partial \tau} \quad \text{(4-velocity)}, \qquad \xi^\mu = \frac{\partial x^\mu}{\partial s} \quad \text{(deviation vector)}$$
The vector \( \xi^\mu \) connects corresponding points on neighboring geodesics and thus measures their separation. The tangent vector \( u^\mu \) satisfies the geodesic equation \( u^\nu \nabla_\nu u^\mu = 0 \).
A crucial identity, which follows from the equality of mixed partial derivatives and the torsion-free condition, is:
$$u^\nu \nabla_\nu \xi^\mu = \xi^\nu \nabla_\nu u^\mu$$
The deviation vector is Lie-transported along the geodesic congruence
Derivation of the Deviation Equation
The relative acceleration of neighboring geodesics is the second covariant derivative of the deviation vector along the geodesic:
$$\frac{D^2 \xi^\mu}{D\tau^2} = u^\rho \nabla_\rho (u^\sigma \nabla_\sigma \xi^\mu)$$
Using the Lie transport identity \( u^\sigma \nabla_\sigma \xi^\mu = \xi^\sigma \nabla_\sigma u^\mu \), we substitute:
$$\frac{D^2 \xi^\mu}{D\tau^2} = u^\rho \nabla_\rho (\xi^\sigma \nabla_\sigma u^\mu)$$
Expanding the right side using the Leibniz rule:
$$= (u^\rho \nabla_\rho \xi^\sigma)(\nabla_\sigma u^\mu) + \xi^\sigma (u^\rho \nabla_\rho \nabla_\sigma u^\mu)$$
The first term, using the Lie transport identity again, becomes \( (\xi^\rho \nabla_\rho u^\sigma)(\nabla_\sigma u^\mu) \). For the second term, we introduce the commutator of covariant derivatives:
$$u^\rho \nabla_\rho \nabla_\sigma u^\mu = u^\rho \nabla_\sigma \nabla_\rho u^\mu + R^\mu_{\ \nu\rho\sigma}\, u^\rho u^\nu$$
The term \( \xi^\sigma u^\rho \nabla_\sigma \nabla_\rho u^\mu \) can be rewritten as\( \xi^\sigma \nabla_\sigma(u^\rho \nabla_\rho u^\mu) - (\xi^\sigma \nabla_\sigma u^\rho)(\nabla_\rho u^\mu) \). The first part vanishes because \( u^\rho \nabla_\rho u^\mu = 0 \) (geodesic equation), and the second part exactly cancels the first term from the Leibniz expansion. We are left with:
$$\boxed{\frac{D^2 \xi^\mu}{D\tau^2} = R^\mu_{\ \nu\rho\sigma}\, u^\nu u^\rho \xi^\sigma}$$
The Geodesic Deviation Equation (Jacobi Equation)
This is one of the most important equations in general relativity. It shows that the Riemann tensor directly controls the relative acceleration of freely falling particles. In flat spacetime (\( R^\mu_{\ \nu\rho\sigma} = 0 \)), there is no relative acceleration - freely falling particles maintain constant separation, as expected.
Tidal Forces as Physical Curvature
The geodesic deviation equation is the relativistic generalization of the Newtonian tidal force equation. In Newtonian gravity, two particles separated by \( \xi^i \) falling in a gravitational potential \( \Phi \) experience a relative acceleration:
$$\frac{d^2 \xi^i}{dt^2} = -\frac{\partial^2 \Phi}{\partial x^i \partial x^j}\, \xi^j$$
Newtonian tidal acceleration
Comparing with the geodesic deviation equation in the Newtonian limit, we can identify the tidal tensor:
$$R^i_{\ 0j0} \longleftrightarrow \frac{\partial^2 \Phi}{\partial x^i \partial x^j}$$
The Riemann tensor components reduce to the Newtonian tidal tensor
This explains why tidal forces are the true signature of gravity. The equivalence principle tells us we can always find a locally inertial frame where the Christoffel symbols vanish (gravity is "eliminated"), but we can never eliminate the Riemann tensor - tidal forces persist in any frame. An astronaut in a freely falling elevator feels no gravity, but can still detect the stretching and squeezing from tidal forces.
Radial Stretching
Near a spherical mass, particles separated radially accelerate apart. The closer particle feels stronger gravity, creating a stretching tidal force along the radial direction.
Transverse Squeezing
Particles separated perpendicular to the radial direction are squeezed together because their geodesics converge toward the center of mass.
This stretching-and-squeezing pattern is the hallmark of tidal gravity. Near a black hole, these tidal forces become extreme - the process known as "spaghettification."
The Weyl Tensor
The Riemann tensor can be decomposed into a trace-free part (the Weyl tensor) and trace parts (involving the Ricci tensor and scalar). In \( n \geq 3 \) dimensions:
$$R_{\rho\sigma\mu\nu} = C_{\rho\sigma\mu\nu} + \frac{2}{n-2}\left(g_{\rho[\mu}R_{\nu]\sigma} - g_{\sigma[\mu}R_{\nu]\rho}\right) - \frac{2}{(n-1)(n-2)}\, R\, g_{\rho[\mu}g_{\nu]\sigma}$$
where \( C_{\rho\sigma\mu\nu} \) is the Weyl tensor. In 4 dimensions, this simplifies to:
$$R_{\rho\sigma\mu\nu} = C_{\rho\sigma\mu\nu} + g_{\rho[\mu}R_{\nu]\sigma} - g_{\sigma[\mu}R_{\nu]\rho} - \frac{1}{3}\, R\, g_{\rho[\mu}g_{\nu]\sigma}$$
The Weyl tensor has all the symmetries of the Riemann tensor plus being completely trace-free:
$$C^\rho_{\ \sigma\rho\nu} = 0$$
Weyl = "Free" Gravitational Field
The Weyl tensor represents the part of curvature not determined by local matter content. It encodes gravitational effects that propagate through vacuum: gravitational waves and long-range tidal forces.
Ricci = Local Matter
The Ricci part of the Riemann tensor is determined by the Einstein equations from the local stress-energy tensor. It encodes volume-changing effects of matter.
Weyl Tensor and Gravitational Radiation
In vacuum (\( R_{\mu\nu} = 0 \)), the Riemann tensor equals the Weyl tensor:
$$R_{\rho\sigma\mu\nu}\big|_{\text{vacuum}} = C_{\rho\sigma\mu\nu}$$
This means the entire curvature of the Schwarzschild spacetime is encoded in the Weyl tensor. Gravitational waves, which propagate through vacuum, are purely Weyl curvature. The Weyl tensor has 10 independent components in 4D, which encode the two polarization states of gravitational waves (plus, cross) along with their Coulomb-type and other components.
The analogy with electromagnetism is illuminating: just as the electromagnetic field tensor\( F_{\mu\nu} \) can be decomposed into electric and magnetic parts, the Weyl tensor can be decomposed into electric and magnetic parts relative to an observer with 4-velocity \( u^\mu \):
$$E_{\mu\nu} = C_{\mu\rho\nu\sigma}\, u^\rho u^\sigma \quad \text{(electric part - tidal field)}$$
$$B_{\mu\nu} = \frac{1}{2}\epsilon_{\mu\rho\alpha\beta}\, C^{\alpha\beta}_{\ \ \nu\sigma}\, u^\rho u^\sigma \quad \text{(magnetic part - frame dragging)}$$
Petrov Classification
The Petrov classification categorizes spacetimes by the algebraic structure of their Weyl tensor. The Weyl tensor can be represented as a complex \( 3 \times 3 \) symmetric traceless matrix (the Weyl spinor \( \Psi_{ABCD} \)), and the classification is determined by its eigenvalue structure. There are six algebraically distinct types:
Type I (Algebraically General)
Three distinct principal null directions. The most general case; no special algebraic symmetry. Generic gravitational fields far from isolated sources are Type I.
Type II
One double and one single principal null direction. An intermediate algebraically special type.
Type D (Degenerate)
Two double principal null directions. This is the type of the Schwarzschild and Kerr solutions. The gravitational field has a Coulomb-like structure with two preferred null directions corresponding to ingoing and outgoing principal null geodesics.
Type III
One triple principal null direction. Represents a pure longitudinal gravitational field component.
Type N (Null)
One quadruple principal null direction. This is the Petrov type of exact gravitational plane waves (pp-waves). The Weyl tensor is purely "transverse" with respect to the wave direction.
Type O (Conformally Flat)
The Weyl tensor vanishes identically. The spacetime is conformally flat. Examples: flat spacetime, de Sitter, anti-de Sitter, and all FLRW cosmological models (including our universe on large scales).
The Goldberg-Sachs Theorem
A vacuum spacetime is algebraically special (Type II, D, III, N, or O) if and only if it admits a shear-free null geodesic congruence. This deep theorem connects the algebraic properties of the Weyl tensor to the geometric properties of null geodesics, and it played a central role in the discovery of the Kerr solution.
Schwarzschild is Petrov Type D
The Schwarzschild solution is the prototypical Type D spacetime. The two double principal null directions are the ingoing and outgoing radial null directions:
$$l^\mu = \left(\frac{1}{f}, 1, 0, 0\right), \qquad n^\mu = \frac{1}{2}\left(1, -f, 0, 0\right)$$
where \( f = 1 - 2M/r \)
In the Newman-Penrose formalism, the only non-vanishing Weyl scalar for Schwarzschild is:
$$\Psi_2 = -\frac{M}{r^3}$$
The "Coulomb" component of the gravitational field
The fact that \( \Psi_2 \neq 0 \) with all other Weyl scalars vanishing is the defining characteristic of Petrov Type D. The Kerr solution, representing a rotating black hole, is also Type D with \( \Psi_2 = -M/(r - ia\cos\theta)^3 \) in Boyer-Lindquist coordinates.