Einstein tensor

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The Einstein tensor is a mathematical entity expressing the curvature of spacetime in the Einstein field equations, which describes gravitation according to the theory of general relativity. It is also sometimes called the trace-reversed Ricci tensor.

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[edit] Definition

In physics and differential geometry, the Einstein tensor \mathbf{G} is a tensor of rank 2 defined over Riemannian manifolds. In index-free notation it is defined as

\mathbf{G}=\mathbf{R}-\frac{1}{2}\mathbf{g}R,

where \mathbf{R} is the Ricci tensor, \mathbf{g} is the metric tensor and R is the scalar curvature. In component form, the previous equation reads as

G_{\mu\nu} = R_{\mu\nu} - {1\over2} g_{\mu\nu}R.

[edit] Explicit form

The Ricci tensor depends only on the metric tensor, so the Einstein tensor can be defined directly with just the metric tensor. This expression is highly complex (in fact, so complex that it is practically never quoted in textbooks). The complexity of this expression can be shown using the formula for the Ricci tensor in terms of Christoffel symbols:

\begin{align} G_{\alpha\beta} &= R_{\alpha\beta} - \frac{1}{2} g_{\alpha\beta} R \\ &= R_{\alpha\beta} - \frac{1}{2} g_{\alpha\beta} g^{\gamma\zeta} R_{\gamma\zeta} \\ &= (\delta^\gamma_\alpha \delta^\zeta_\beta - \frac{1}{2} g_{\alpha\beta}g^{\gamma\zeta}) R_{\gamma\zeta} \\ &= (\delta^\gamma_\alpha \delta^\zeta_\beta - \frac{1}{2} g_{\alpha\beta}g^{\gamma\zeta})(\Gamma^\epsilon_{\gamma\zeta,\epsilon} - \Gamma^\epsilon_{\gamma\epsilon,\zeta} + \Gamma^\epsilon_{\epsilon\sigma} \Gamma^\sigma_{\gamma\zeta} - \Gamma^\epsilon_{\zeta\sigma} \Gamma^\sigma_{\epsilon\gamma}), \end{align}

where \delta^\alpha_\beta is the Kronecker tensor and the Christoffel symbol \Gamma^\alpha_{\beta\gamma} is defined as

\Gamma^\alpha_{\beta\gamma} = \frac{1}{2} g^{\alpha\epsilon}(g_{\beta\epsilon,\gamma} + g_{\gamma\epsilon,\beta} - g_{\beta\gamma,\epsilon}).

Before cancellations, this formula results in 2 \times (6+6+9+9) = 60 individual terms. Cancellations bring this number down somewhat.

In the special case of a locally inertial reference frame near a point, the first derivatives of the metric tensor vanish and the component form of the Einstein tensor is considerably simplified:

\begin{align}G_{\alpha\beta} & = g^{\gamma\mu}\bigl[ g_{\gamma[\beta,\mu]\alpha} + g_{\alpha[\mu,\beta]\gamma} - \frac{1}{2} g_{\alpha\beta} g^{\epsilon\sigma} (g_{\epsilon[\mu,\sigma]\gamma} + g_{\gamma[\sigma,\mu]\epsilon})\bigr] \\ & = g^{\gamma\mu} (\delta^\epsilon_\alpha \delta^\sigma_\beta - \frac{1}{2} g^{\epsilon\sigma}g_{\alpha\beta})(g_{\epsilon[\mu,\sigma]\gamma} + g_{\gamma[\sigma,\mu]\epsilon}),\end{align}

where square brackets conventionally denote antisymmetrization over bracketed indices, i.e.

g_{\alpha[\beta,\gamma]\epsilon} \, = \frac{1}{2} (g_{\alpha\beta,\gamma\epsilon} - g_{\alpha\gamma,\beta\epsilon}).

[edit] Trace

The trace of the Einstein tensor can be computed by contracting the equation in the definition with the metric tensor gμν. In D dimensions (of arbitrary signature):

\begin{align}g^{\mu\nu}G_{\mu\nu} &= g^{\mu\nu}R_{\mu\nu} - {1\over2} g^{\mu\nu}g_{\mu\nu}R \\ G &= R - {1\over2} (DR) \\ G &= {{2-D}\over2}R\end{align}

The special case of 4 dimensions in physics (3 space, 1 time) gives the trace of the Einstein tensor turns as the negative of the trace of the Ricci tensor. It is for this reason that the Einstein tensor is also referred to as the trace-reversed Ricci tensor.

[edit] Use in general relativity

The Einstein tensor allows a compact expression of the Einstein field equations:

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

Using geometrized units, this simplifies to

G_{\mu\nu} = 8 \pi \, T_{\mu\nu}.

From the explicit form of the Einstein tensor above, it can be seen that the Einstein tensor is a nonlinear function of the metric tensor, but it is linear in second partial derivatives of the metric. As a symmetric 2nd rank tensor, the Einstein tensor has 10 independent components in a 4-dimensional space. It follows that the Einstein field equations are a set of 10 quasilinear second-order partial differential equations for the metric tensor.

The Bianchi identities can also be easily expressed with the aid of the Einstein tensor:

\nabla_{\mu} G^{\mu\nu} = 0.

The Bianchi identities automatically ensure the conservation of the stress-energy tensor in curved spacetimes:

\nabla_{\mu} T^{\mu\nu} = 0.

The geometric significance of the Einstein tensor is highlighted by this identity. In coordinate frames respecting the gauge condition

\Gamma^{\rho}_{\mu\nu} G^{\mu\nu} = 0

an exact conservation law for the stress tensor density can be stated:

\partial_{\mu}(\sqrt{g} T^{\mu\nu}) = 0.

The Einstein tensor plays the role of distinguishing these frames.

[edit] See also

[edit] References

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