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The background spatial metric is $g_{ij}=a^2{\delta }_{ij}$. For the large spatial diffeomorphismhomogeneity and isotropy imply that the purely spatial background Christoffel symbols vanish. HenceThe shift is constant, transverse and traceless, and is therefore the zero-momentum adiabatic tensor mode.

A scalar transforms by its Lie derivative:Fourier transformation and integration by parts in momentum space giveThe term proportional to ${\omega }_i{}^i$ vanishes because the deformation is traceless.

For coordinates $x^{\alpha }$, the Christoffel symbols of the Levi-Civita connection arewhere $g^{\alpha \delta }$ is the inverse metric.

For the two-dimensional hyperbolic metric in polar coordinates,Only $g_{\varphi \varphi }$ has a nonzero derivative, namelyIt follows that the nonzero connection coefficients are

An affinely parameterized geodesic $x^{\mu }(\lambda )$ satisfiesFor the conformally flat metric $g_{\mu \nu }={\phi }^2{\eta }_{\mu \nu }$, its Christoffel symbols areLet $u^{\mu }=d{x}^{\mu }/d\lambda$. Along a null geodesic, ${\eta }_{\alpha \beta }{u}^{\alpha }{u}^{\beta }=0$, so the equation reduces toThe acceleration is parallel to the tangent. A change of parameter removes it, leaving a straight null line of the Minkowski metric. Thus conformal rescaling changes the affine parameter but not the unparametrized light ray.

Consequently Nordstrom theory of gravitation predictsfor light passing a star. This contradicts the observed nonzero deflection and the successful general-relativistic value $4GM/(b{c}^2)$.

For local coordinates $x^1,\dots ,x^n$, a curve is an affinely parametrized geodesic exactly when it satisfies the geodesic equationwhere the Christoffel symbols of the Levi-Civita connection are

Write ${\dot{x}}^{\mu }=d{x}^{\mu }/ds$ and $L=\sqrt{-g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }}$. Varying $T=\int Lds$ with fixed endpoints and then choosing proper time as parameter, for which $L=1$, gives the Euler-Lagrange equationExpanding the derivative, raising the free index with the inverse metric, and using the symmetry of ${\dot{x}}^{\alpha }{\dot{x}}^{\beta }$ gives the geodesic equationThese are the Christoffel symbols of the Levi-Civita connection.

Put $h_{ab}=\delta g_{ab}$. Varying $g^{ac}{g}_{cb}={\delta }_b^a$ givesThe determinant identity $\delta \log |\det g|=g^{ab}{h}_{ab}$ givesFinally, varying the formula for the Christoffel symbols and rewriting partial derivatives covariantly gives the tensorThese are the basic metric variation identities.

Using the stated Christoffel symbols in the definition of the Riemann curvature tensor, the two nontrivial contractions areHenceand the vacuum Einstein field equations reduce toThis is the diagonal Rosen coordinates for a plane gravitational wave equation.

For the plus-polarized wave, setComparison of the transverse metric components gives, to linear order,orThereforeThe Vacuum Einstein equations are consequently satisfied at linear order. A real gravitational wave is obtained by taking the real part of the complex plane-wave notation.

Affinely parametrized geodesics are the critical curves of the geodesic LagrangianFor $w=1+r^2$ and $w^{\prime }=2r$, the nonzero Christoffel symbols, up to symmetry in the lower indices, areThey follow either from the Euler-Lagrange equations or directly from the Levi-Civita connection formula.

Divide the time component of the geodesic equation by $(P^0{)}^2$ and use $d/d\lambda =P^{0}d/d\eta$. To first order, $P^i/P^0={\hat{p}}^i(1+\delta N-\hat{\mathbf{p}}\cdot \nabla \psi )$. Substitution of the supplied Christoffel symbols makes the background $2\mathcal{H}$ terms cancel the derivative of $a^{-2}$. The derivatives of the lapse and shift also combine, leavingThus the comoving photon energy is conserved in the unperturbed spacetime and changes through the gradient of the scalar gravitational source.

A unit-speed surface curve $\alpha$ is a geodesic when its acceleration is normal to the surface, equivalently when its covariant acceleration vanishes. In coordinates $u^1,u^2$, the Christoffel symbols of the Levi-Civita connection areThe coordinate functions of a geodesic satisfy the geodesic equation

For a spatial index $j$, the background Christoffel symbols giveTogether with the $3H\delta T^0{}_j$ trace term, the first two contributions combine to $2H\delta T^0{}_j$. Varying the Levi-Civita connection and contracting it with ${\bar{T}}^{\mu }{}_{\nu }=\mathrm{diag}(-\bar{\rho },\bar{P},\bar{P},\bar{P})$ givesand the surviving combination reduces toConsequently the spatial momentum equation is

Connections ${\nabla }^1,{\nabla }^2$ on $T{M}_1,T{M}_2$ give a connection onby taking the direct sum of their pullback connections. In product coordinates its Christoffel symbols have the two factor blocks and zero mixed blocks. On arbitrary fields, the ordinary derivatives of their coefficients are still taken in both factors. On fields lifted separately from the factors it satisfies ${\nabla }_{Y_1+Y_2}(X_1+X_2)={\nabla }_{Y_1}^1{X}_1+{\nabla }_{Y_2}^2{X}_2$.

Gaussian curvature depends only on the first fundamental form and is therefore preserved by local isometries. For a parametrized surface with metric coefficients $g_{ij}$, decompose its second derivatives by the Gauss formulaTaking tangent inner products determines the lowered Christoffel symbols solely from the metric:Comparing $X_{ijk}$ and $X_{ikj}$ gives the Gauss equationConsequentlyand the final expression uses only $g_{ij}$ and its first two derivatives.