In a Riemannian variety (where metricity and zero torsion conditions apply), it is known that:
i- the covariant derivative of a contravariant vector is given by VaA” (x) = daA” (x) + {s} A®(x), When {a}are are Christofel’s symbols given by: 1 {} = дна (друга + Дидар – θαθρν);
ii the Riemann and Ricci curvature tensors are given respectively by Βρυμα = θρ μν} – μου}+{}{3}-{PV} {B}; Κυμ = Κρυμ
iii at the weak field boundary, the metric is described by 9µν = ημν + huv, when nu is the Minkowski metric and h1,so that we have quadratic order in huucan be despised.
Thus,
a) show that the commutator of the covariant derivatives of a vector is given by:
[Ba] A = RAP.
b) get an expression for gand calculate {a}at asat the edge of weak field;
c) how that at the static and weak field boundary Roo is proportional to the Laplacian of one of the components in huv, determining what is this component and what is the proportionality factor.
The Correct Answer and Explanation is :a) Commutator of Covariant Derivatives
We are given the expression for the covariant derivative of a contravariant vector ( A^\mu ):
[
\nabla_\alpha A^\mu = \partial_\alpha A^\mu + \Gamma^\mu_{\alpha \nu} A^\nu
]
The commutator of two covariant derivatives acting on a vector ( A^\mu ) is given by:
[
[\nabla_\alpha, \nabla_\beta] A^\mu = \nabla_\alpha \nabla_\beta A^\mu – \nabla_\beta \nabla_\alpha A^\mu
]
Expanding each derivative using the definition of the covariant derivative:
[
\nabla_\alpha \nabla_\beta A^\mu = \partial_\alpha \partial_\beta A^\mu + \partial_\alpha \Gamma^\mu_{\beta \nu} A^\nu + \Gamma^\mu_{\alpha \rho} \partial_\beta A^\rho + \Gamma^\mu_{\alpha \rho} \Gamma^\rho_{\beta \nu} A^\nu
]
[
\nabla_\beta \nabla_\alpha A^\mu = \partial_\beta \partial_\alpha A^\mu + \partial_\beta \Gamma^\mu_{\alpha \nu} A^\nu + \Gamma^\mu_{\beta \rho} \partial_\alpha A^\rho + \Gamma^\mu_{\beta \rho} \Gamma^\rho_{\alpha \nu} A^\nu
]
Now subtract the two expressions:
[
[\nabla_\alpha, \nabla_\beta] A^\mu = (\partial_\alpha \partial_\beta A^\mu – \partial_\beta \partial_\alpha A^\mu) + (\partial_\alpha \Gamma^\mu_{\beta \nu} A^\nu – \partial_\beta \Gamma^\mu_{\alpha \nu} A^\nu) + (\Gamma^\mu_{\alpha \rho} \partial_\beta A^\rho – \Gamma^\mu_{\beta \rho} \partial_\alpha A^\rho) + \text{higher terms}
]
Since the partial derivatives commute, we are left with:
[
[\nabla_\alpha, \nabla_\beta] A^\mu = (\partial_\alpha \Gamma^\mu_{\beta \nu} – \partial_\beta \Gamma^\mu_{\alpha \nu}) A^\nu + \Gamma^\mu_{\alpha \rho} \Gamma^\rho_{\beta \nu} A^\nu – \Gamma^\mu_{\beta \rho} \Gamma^\rho_{\alpha \nu} A^\nu
]
This simplifies to:
[
[\nabla_\alpha, \nabla_\beta] A^\mu = R^\mu_{\ \nu \alpha \beta} A^\nu
]
Thus, the commutator of the covariant derivatives of a vector ( A^\mu ) is given by the Riemann curvature tensor:
[
[\nabla_\alpha, \nabla_\beta] A^\mu = R^\mu_{\ \nu \alpha \beta} A^\nu
]
b) Expression for Christoffel Symbols in the Weak-Field Limit
In the weak-field limit, the metric ( g_{\mu\nu} ) is written as:
[
g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}
]
where ( \eta_{\mu\nu} ) is the Minkowski metric and ( h_{\mu\nu} ) is a small perturbation (i.e., ( |h_{\mu\nu}| \ll 1 )).
The Christoffel symbols are given by:
[
\Gamma^\lambda_{\mu\nu} = \frac{1}{2} g^{\lambda \rho} \left( \partial_\mu g_{\nu \rho} + \partial_\nu g_{\mu \rho} – \partial_\rho g_{\mu \nu} \right)
]
At the weak-field limit, since ( g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu} ), we have:
[
\Gamma^\lambda_{\mu\nu} \approx \frac{1}{2} \eta^{\lambda \rho} \left( \partial_\mu h_{\nu \rho} + \partial_\nu h_{\mu \rho} – \partial_\rho h_{\mu \nu} \right)
]
Thus, the Christoffel symbols at the weak-field limit are approximately:
[
\Gamma^\lambda_{\mu\nu} \approx \frac{1}{2} \eta^{\lambda \rho} \left( \partial_\mu h_{\nu \rho} + \partial_\nu h_{\mu \rho} – \partial_\rho h_{\mu \nu} \right)
]
c) Ricci Tensor and Laplacian in the Weak-Field Limit
The Ricci curvature tensor ( R_{\mu\nu} ) is defined as:
[
R_{\mu\nu} = R^\alpha_{\ \mu \alpha \nu}
]
In the weak-field limit, the Riemann curvature tensor simplifies to:
[
R_{\mu\nu} \approx \frac{1}{2} \left( \partial_\alpha \partial^\alpha h_{\mu\nu} – \Box h_{\mu\nu} \right)
]
where ( \Box = \partial_\alpha \partial^\alpha ) is the d’Alembertian operator. In this limit, the Ricci tensor becomes proportional to the Laplacian of the components of ( h_{\mu\nu} ).
At the static weak-field boundary, the static condition implies that the components ( h_{\mu\nu} ) only depend on spatial coordinates, and we are left with a relation like:
[
R_{\mu\nu} \propto \Box h_{\mu\nu}
]
In particular, the component ( h_{00} ) typically corresponds to the gravitational potential in the weak-field limit. Thus, the proportionality factor depends on the form of the perturbation, but the most common situation in the weak-field limit is that:
[
R_{00} \propto \nabla^2 h_{00}
]
where ( \nabla^2 ) is the Laplacian operator. This shows that the Ricci tensor is proportional to the Laplacian of the ( h_{00} ) component of the perturbation in the static weak-field limit.
Conclusion
At the weak-field limit, the Ricci tensor ( R_{\mu\nu} ) is proportional to the Laplacian of the ( h_{00} ) component of the perturbation:
[
R_{00} \propto \nabla^2 h_{00}
]
This result demonstrates the relationship between the Ricci tensor and the Laplacian of the perturbation in the weak-field limit.
The commutator of the covariant derivatives of a vector is given by the Riemann curvature tensor, ( [\nabla_\alpha, \nabla_\beta] A^\mu = R^\mu_{\ \nu \alpha \beta} A^\nu ).
In the weak-field limit, the Christoffel symbols are given by:
[
\Gamma^\lambda_{\mu\nu} \approx \frac{1}{2} \eta^{\lambda \rho} \left( \partial_\mu h_{\nu \rho} + \partial_\nu h_{\mu \rho} – \partial_\rho h_{\mu \nu} \right)
]