= Solution
The <Pound-Rebka experiment> measures a small <gravitational redshift> by comparing nuclear emission and resonant absorption at different heights in Earth's field. The $14.4\,\mathrm{keV}$ transition of iron-57 supplies the gamma-ray frequency standard. The <Mössbauer effect> permits recoil-free emission and absorption by nuclei bound in solids, giving a sufficiently narrow resonance. A source and absorber separated vertically by about $22.6\,\mathrm m$ have a predicted fractional frequency difference of order $2.5\times10^{-15}$. A calibrated <Doppler effect> from slow source motion measures the shift through its effect on resonant absorption. Interchanging the upper and lower arrangements reverses the gravitational contribution and helps separate it from intrinsic source-absorber offsets; temperature-dependent frequency shifts must also be controlled. The observed change agrees with the weak-field prediction.
The prediction follows either from an accelerating laboratory and the <equivalence principle>, or directly from a static metric. With $x^0=ct$, write $g_{00}=N^2\simeq1+2\Phi/c^2$. The time-translation <Killing vector field> gives a conserved photon energy $E_\xi=p_a\xi^a$, while a static observer has <four-velocity> $u^a=\xi^a/N$ and measures $h\nu=E_\xi/N$. Thus
$$
\boxed{\frac{\nu_{\rm received}}{\nu_{\rm emitted}}
=\frac{N_{\rm emitted}}{N_{\rm received}}
\simeq1+\frac{\Phi_{\rm emitted}-\Phi_{\rm received}}{c^2}.}
$$
For an upward ray over height $H$, $\Delta\nu/\nu\simeq-gH/c^2$; downward propagation has the opposite sign. The equivalent compensating Doppler speed has magnitude $gH/c$, about $7.4\times10^{-7}\,\mathrm{m\,s^{-1}}$ for this baseline. Identical clocks at rest have $d\tau=N\,dt$, so a clock at higher potential runs faster relative to the common static coordinate time. This does not mean that the nuclear transition changes its locally measured frequency with height: the comparison involves transporting a photon between different local proper-time standards.
The <weak equivalence principle> asserts universal free fall for test bodies. The <Einstein equivalence principle> adds <local Lorentz invariance> and <local position invariance> for nongravitational experiments in a freely falling laboratory. The redshift comparison tests the clock aspect of local position invariance and supports a metric description of gravity. The <strong equivalence principle> extends these requirements to self-gravitating bodies and local gravitational experiments, including the contribution of gravitational binding energy. Their outcomes should be independent of the external location and motion of a freely falling laboratory when tidal effects are negligible. A nuclear redshift experiment is therefore not a test of the whole strong equivalence principle, and its success does not uniquely establish the gravitational field equations.
The geometrical realization uses the <Levi-Civita connection> of a spacetime metric. At any event, <normal coordinates> make $g_{ab}=\eta_{ab}$ and the <Christoffel symbols> vanish. Freely falling test particles follow <geodesics>; local nongravitational equations take their special-relativistic form at that event. The <Riemann curvature tensor> need not vanish, so finite laboratories can measure tidal <geodesic deviation>. The strong principle concerns removing a uniform external gravitational field locally, not removing curvature throughout an extended region.
For a <perfect fluid>, local rest-frame isotropy and the absence of heat flux or viscosity give $T^{\hat a\hat b}=\operatorname{diag}(\rho,p,p,p)$, where $\rho$ is rest-frame energy density. With the PDF's signature $(+---)$, $u^au_a=1$ and $h^{ab}=g^{ab}-u^au^b$ projects spatially. Its covariant form is
$$
\boxed{T^{ab}=\rho u^au^b-ph^{ab}=(\rho+p)u^au^b-pg^{ab}.}
$$
The pressure sign follows from the negative spatial metric; it gives positive diagonal stresses in the rest frame. Local <stress-energy conservation>, $\nabla_aT^{ab}=0$, produces the two projections
$$
u^a\nabla_a\rho+(\rho+p)\nabla_au^a=0,\qquad
(\rho+p)u^a\nabla_au^b=(g^{ba}-u^bu^a)\nabla_ap.
$$
The first is the energy equation, including pressure work; the second is the <relativistic Euler equation>. Its rest-frame spatial pressure force is $-\nabla p$.
Further assumptions are needed to derive the <Einstein field equations>. Take a local generally covariant metric equation, linear in curvature and of second differential order, and demand compatibility with <stress-energy conservation>. The available rank-two tensors are $R_{ab}$, $Rg_{ab}$ and a constant multiple of $g_{ab}$. Defining $R_{ab}=R^c{}_{acb}$, the <contracted Bianchi identity> gives $\nabla^aR_{ab}=\tfrac12\nabla_bR$. Hence the combination $R_{ab}+\alpha Rg_{ab}$ is identically divergence-free only when $\alpha=-\tfrac12$. This determines the <Einstein tensor> $G_{ab}=R_{ab}-Rg_{ab}/2$; a <cosmological constant> term remains allowed.
The coupling sign must be matched to the actual curvature convention. Here $R^a{}_{bcd}$ represents $[\nabla_d,\nabla_c]$, and a static weak field has $R_{00}\simeq-\nabla^2\Phi/c^2$. With negligible pressure and cosmological term, a trial equation $G_{ab}=C T_{ab}$ implies $R_{00}\simeq C\rho_{\rm mass}c^2/2$. Comparing with the <Poisson equation> $\nabla^2\Phi=4\pi G\rho_{\rm mass}$ fixes $C=-8\pi G/c^4$. Thus the <Einstein equation in the sequential-derivative curvature convention> is
$$
\boxed{G_{ab}+\Lambda g_{ab}=-\frac{8\pi G}{c^4}\left[(\rho+p)u_au_b-pg_{ab}\right].}
$$
Equivalently, $R_{ab}=\Lambda g_{ab}-(8\pi G/c^4)(T_{ab}-Tg_{ab}/2)$, with $T=\rho-3p$. Pressure therefore gravitates as well as supplying a material force. The cosmological constant is not fixed by the local Newtonian matching.
Setting $c=1$ for the action normalization, the same metric equation follows from the <Einstein-Hilbert action> $S_g=(2\kappa)^{-1}\int\sqrt{-g}(R-2\Lambda)\,d^4x$, where $\kappa=8\pi G$, together with a matter action obeying $\delta S_m=\tfrac12\int\sqrt{-g}T_{ab}\delta g^{ab}\,d^4x$. After the metric boundary term is removed or supplied separately, the bulk variation is proportional to $G_{ab}+\Lambda g_{ab}+\kappa T_{ab}$. This also makes the sign convention explicit. General covariance, the field equation and the <contracted Bianchi identity> are mutually consistent with the fluid conservation laws. The experiment motivates the local metric description; the curvature and conservation assumptions, plus the Newtonian limit, supply the additional route to the field equation.
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