In this post, I would like to study some basic knowledge of the \( \nabla \) symbol, as well as an important formula related with riemann tensor.
I. Connection
Definition: A connection \( \nabla \) on a smooth manifold \( (M, \mathcal{O}) \) is a map that takes a pair consisting of a vector field \( X \) and a \( (p, q) \)-tensor field \( T \) and sends them to a \( (p, q) \)-tensor field \( \nabla_X T \), satisfying:
\( \nabla_X f = X f \), \(\forall f \in C^\infty(M) \)
\( \nabla_X (T + S) = \nabla_X T + \nabla_X S \)
\( \nabla_X (T(\omega, Y)) = (\nabla_X T)(\omega, Y) + T(\nabla_X \omega, Y) + T(\omega, \nabla_X Y) \)
(For \( (1, 1) \)-tensor field \( T \), but analogously for any \( (p, q) \)-tensor field \( T \)).
(“Leibniz”)\( \nabla_{fX + Z} T = f \nabla_X T + \nabla_Z T \),
\( f \in C^\infty(M) \).
To deduce the covariant derivative of a vector field \( Y \) using the given definition of a connection \( \nabla \), we proceed as follows:
Step 1: Expand \( Y \) in a Local Basis Let \( \{ \partial_i \} \) be a coordinate basis for vector fields and \( Y = Y^i \partial_i \), where \( Y^i \) are smooth functions.
Step 2: Apply the Connection Linearity in \( X \) For \( X = X^i \partial_i \), use condition 4 (linearity in \( X \)):
\[ \nabla_X Y = X^i \nabla_{\partial_i} Y. \]Step 3: Expand \( \nabla_{\partial_i} Y \) Using the Leibniz Rule For \( Y = Y^j \partial_j \), apply condition 2 (linearity) and the Leibniz rule (condition 3) for the product \( Y^j \partial_j \):
\[ \nabla_{\partial_i} Y = \nabla_{\partial_i} (Y^j \partial_j) = (\nabla_{\partial_i} Y^j) \partial_j + Y^j \nabla_{\partial_i} \partial_j. \]By condition 1, \( \nabla_{\partial_i} Y^j = \partial_i Y^j \). Define the connection coefficients \( \Gamma^k_{ij} \) via:
\[ \nabla_{\partial_i} \partial_j = \Gamma^k_{ij} \partial_k. \]Thus:
\[ \nabla_{\partial_i} Y = (\partial_i Y^j) \partial_j + Y^j \Gamma^k_{ij} \partial_k = (\partial_i Y^k + Y^j \Gamma^k_{ij}) \partial_k. \]Step 4: Combine Results Substitute back into \( \nabla_X Y \):
\[ \nabla_X Y = X^i (\partial_i Y^k + Y^j \Gamma^k_{ij}) \partial_k. \]This gives the covariant derivative of \( Y \) in coordinates:
\[ \nabla_X Y = \left( X^i \partial_i Y^k + X^i Y^j \Gamma^k_{ij} \right) \partial_k. \]Final Expression: The covariant derivative of a vector field \( Y = Y^i \partial_i \) with respect to \( X = X^j \partial_j \) is:
\[ \boxed{ \nabla_X Y = \left( X^j \partial_j Y^i + X^j Y^k \Gamma^i_{jk} \right) \partial_i }. \]II. Geodesic Deviation Equation
The Geodesic Deviation Equation, also known as the Jacobi equation, describes how nearby geodesics in a curved spacetime deviate from each other. This equation is crucial in general relativity as it quantifies tidal forces due to spacetime curvature.
Mathematical Formulation
The equation is:
\[ \frac{D^2 \xi^\mu}{D \tau^2} + R^\mu_{\ \nu\rho\sigma} U^\nu \xi^\rho U^\sigma = 0 \]where:
- \( \xi^\mu \) is the separation vector between two neighboring geodesics.
- \( U^\mu = \frac{dx^\mu}{d\tau} \) is the tangent vector to the geodesic (4-velocity in relativity).
- \( \frac{D}{D\tau} \) is the covariant derivative along the geodesic.
- \( R^\mu_{\ \nu\rho\sigma} \) is the Riemann curvature tensor.
- \( \tau \) is the proper time along the geodesic.
Interpretation
- The first term \( \frac{D^2 \xi^\mu}{D \tau^2} \) represents the acceleration of separation between geodesics.
- The second term \( R^\mu_{\ \nu\rho\sigma} U^\nu \xi^\rho U^\sigma \) describes how spacetime curvature (encoded in the Riemann tensor) influences geodesic separation.
This equation is fundamental in understanding gravitational tidal effects, as it shows how small objects in free fall (such as in the presence of a massive body) experience relative acceleration due to the curvature of spacetime.
The geodesic deviation equation is derived by analyzing the relative acceleration between two neighboring geodesics in a curved spacetime. Here’s a step-by-step deduction:
1. Setup: Family of Geodesics
Consider a one-parameter family of geodesics \( x^\mu(\tau, s) \), where:
- \( \tau \) is the proper time along each geodesic.
- \( s \) labels neighboring geodesics.
The tangent vector to the geodesics is:
\[ U^\mu = \frac{\partial x^\mu}{\partial \tau}, \]and the separation vector between infinitesimally close geodesics is:
\[ \xi^\mu = \frac{\partial x^\mu}{\partial s}. \]2. Commutator Relation
Since \( U \) and \( \xi \) are coordinate basis vectors, their Lie bracket vanishes:
\[ [U, \xi] = 0 \implies \nabla_U \xi^\mu = \nabla_\xi U^\mu. \]This ensures the separation vector \( \xi^\mu \) is Lie-transported along \( U \).
3. First Covariant Derivative
The first derivative of \( \xi^\mu \) along the geodesic is:
\[ \frac{D \xi^\mu}{D \tau} = \nabla_U \xi^\mu = \nabla_\xi U^\mu. \]4. Second Covariant Derivative
The relative acceleration is the second covariant derivative:
\[ \frac{D^2 \xi^\mu}{D \tau^2} = \nabla_U \left( \nabla_U \xi^\mu \right) = \nabla_U \left( \nabla_\xi U^\mu \right). \]5. Commutator of Covariant Derivatives
Using the commutator identity for covariant derivatives:
\[ [\nabla_U, \nabla_\xi] U^\mu = \nabla_U \nabla_\xi U^\mu - \nabla_\xi \nabla_U U^\mu. \]The second term vanishes because \( \nabla_U U^\mu = 0 \) (geodesic equation). Thus:
\[ \nabla_U \nabla_\xi U^\mu = [\nabla_U, \nabla_\xi] U^\mu. \]6. Riemann Tensor Relation
The commutator of covariant derivatives introduces the Riemann curvature tensor:
\[ [\nabla_U, \nabla_\xi] U^\mu = R^\mu_{\ \nu\rho\sigma} U^\nu \xi^\rho U^\sigma. \]Here, \( R^\mu_{\ \nu\rho\sigma} \) encodes the spacetime curvature.
7. Final Equation
Combining the results:
\[ \frac{D^2 \xi^\mu}{D \tau^2} = R^\mu_{\ \nu\rho\sigma} U^\nu \xi^\rho U^\sigma. \]Rearranging gives the geodesic deviation equation:
\[ \boxed{\frac{D^2 \xi^\mu}{D \tau^2} + R^\mu_{\ \nu\rho\sigma} U^\nu \xi^\rho U^\sigma = 0.} \]Last modified on 2025-02-07