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In units $c=1$, an isentropic perfect fluid in general relativity has sound speed $c_s^2=dp/d\rho$. Writing $n=p^{\prime }(ϵ)$ gives $d\rho =ϵdn$ and $c_s^2=n/(ϵp^{\prime \prime }(ϵ))$. Linearized particle number conservation in a relativistic fluid and the relativistic Euler equation then yield acoustic waves with this speed. Positive $c_s^2$ is needed for ordinary acoustic stability; a causal fluid model additionally requires $c_s^2\le 1$.

The canonical vorticity two-form is ${\omega }_{ab}={\nabla }_a(ϵu_b)-{\nabla }_b(ϵu_a)$, with specific enthalpy per particle $ϵ$. Potential flow $ϵu_a={\nabla }_a\chi$ has $\omega =0$ for a torsion-free connection. Contracting this identity with $u^a$ yields $ϵu^a{\nabla }_a{u}_b={\nabla }_bϵ-u_b{u}^a{\nabla }_aϵ$, the isentropic relativistic Euler equation in signature $(+---)$. This canonical condition does not mean the unweighted four-velocity one-form is closed.

For metric signature $(+---)$, a stationary perfect fluid follows a timelike Killing vector field $K$ with four-velocity $U=K/N$. Its Killing-flow four-acceleration is $U^b{\nabla }_b{U}_a=-{\nabla }_a\log N$. The relativistic Euler equation and stationarity of the pressure give the displayed hydrostatic equilibrium equation. This uses stationarity along $K$ and does not require hypersurface orthogonality.

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 $2.5\times 10^{-15}$. 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 $g_{00}=N^2\simeq 1+2\Phi /c^2$. 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 $h\nu =E_{\xi }/N$. ThusFor 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 $7.4\times 10^{-7}\mathrm{m}{\mathrm{s}}^{-1}$ for this baseline. Identical clocks at rest have $d\tau =Ndt$, 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 $T^{\hat{a}\hat{b}}=\mathrm{diag}(\rho ,p,p,p)$, where $\rho$ is rest-frame energy density. With the PDF's signature $(+---)$, $u^a{u}_a=1$ and $h^{ab}=g^{ab}-u^a{u}^b$ projects spatially. Its covariant form isThe pressure sign follows from the negative spatial metric; it gives positive diagonal stresses in the rest frame. Local stress-energy conservation, ${\nabla }_a{T}^{ab}=0$, produces the two projectionsThe 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 $R_{ab}$, $R{g}_{ab}$ and a constant multiple of $g_{ab}$. Defining $R_{ab}=R^c{}_{acb}$, the contracted Bianchi identity gives ${\nabla }^a{R}_{ab}=\frac{1}{2}{\nabla }_{b}R$. Hence the combination $R_{ab}+\alpha R{g}_{ab}$ is identically divergence-free only when $\alpha =-\frac{1}{2}$. This determines the Einstein tensor $G_{ab}=R_{ab}-R{g}_{ab}/2$; 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 $R_{00}\simeq C{\rho }_{\mathrm{mass}}{c}^2/2$. Comparing with the Poisson equation ${\nabla }^2\Phi =4\pi G{\rho }_{\mathrm{mass}}$ fixes $C=-8\pi G/c^4$. Thus the Einstein equation in the sequential-derivative curvature convention isEquivalently, $R_{ab}=\Lambda g_{ab}-(8\pi G/c^4)(T_{ab}-T{g}_{ab}/2)$, 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 $S_g=(2\kappa {)}^{-1}\int \sqrt{-g}(R-2\Lambda )d^{4}x$, where $\kappa =8\pi G$, together with a matter action obeying $\delta S_m=\frac{1}{2}\int \sqrt{-g}{T}_{ab}\delta g^{ab}{d}^{4}x$. 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.

Let $Q=g^{ab}{\chi }_{,a}{\chi }_{,b}$ and $\mathcal{L}=p(ϵ)+n(Q-{ϵ}^2)/(2ϵ)$. Vary $n,ϵ,\chi$ independently, taking compactly supported variations. The multiplier variation is $\partial \mathcal{L}/\partial n=(Q-{ϵ}^2)/(2ϵ)=0$. With $u_a={\chi }_{,a}/ϵ$, it gives the unit four-velocity condition $u^a{u}_a=1$. The enthalpy variation isUsing $Q={ϵ}^2$ yields $p^{\prime }(ϵ)=n$. Finally,The integration by parts boundary term vanishes. Thus the three matter equations areThe last is particle number conservation in a relativistic fluid. Off shell $n,ϵ$ are independent variational fields; on shell the thermodynamic relation ties them to the specified function $p(ϵ)$.

For the Hilbert stress-energy tensor, keep these matter fields fixed while varying the inverse metric. The volume variation is $\delta \sqrt{-g}=-\frac{1}{2}\sqrt{-g}{g}_{ab}\delta g^{ab}$, and $\delta Q={\chi }_{,a}{\chi }_{,b}\delta g^{ab}$. ThereforeOn shell, $\mathcal{L}=p$ and $nϵ=\rho +p$. With signature $(+---)$,For variation of the covariant metric the same definition reads ${\delta }_{g}S=-\frac{1}{2}\int \sqrt{-g}{T}^{ab}\delta g_{ab}{d}^{4}x$. In a local rest frame the tensor has entries $\mathrm{diag}(\rho ,p,p,p)$, which checks both the energy density and pressure sign.

Define a derivative along the flow by $\dot{F}=u^a{\nabla }_{a}F$, and set $\theta ={\nabla }_a{u}^a$, $a^b=u^a{\nabla }_a{u}^b$. The number equation immediately gives $\dot{n}+n\theta =0$. The specific enthalpy per particle relation and $dp=ndϵ$ giveConsequently the energy equation follows from the number equation: $\dot{\rho }=ϵ\dot{n}=-(\rho +p)\theta$.

The velocity equation follows directly from the potential, rather than requiring an independent assumption about the fluid force. Since $ϵu_a={\nabla }_a\chi$ and the connection has zero torsion, the canonical vorticity of a relativistic fluid vanishes:Contract with $u^a$ and use $u^a{u}_a=1$ and $u^a{\nabla }_b{u}_a=0$. This gives $ϵa_b={\nabla }_bϵ-u_b\dot{ϵ}$. Multiplication by $n$ and use of ${\nabla }_{b}p=n{\nabla }_bϵ$ produces the relativistic Euler equation. The evolution system is thereforeIt is equivalent to stress-energy conservation together with number conservation, subject to the potential-flow restriction. The action describes an isentropic barotropic sector of a perfect fluid in general relativity; it does not provide general vortical or entropy-carrying fluid motions. The constraint on canonical vorticity is stronger than the Euler equation alone.

For the weak static metric, use $c=1$. To the retained order, $\sqrt{-g}=1+\varphi$, $u^0=1-\varphi +O({\varphi }^2,|\mathbf{v}{|}^2)$ and $u^i=v^i$. The correction to $u^0$ is required by normalization, even though its zeroth-order value is one. The relevant Christoffel symbols arewith no spatial connection at this order. The cancellation $\sqrt{-g}{u}^0=1+O({\varphi }^2,|\mathbf{v}{|}^2)$ makes the leading number equation particularly simple:The corresponding leading energy transport equation is ${\partial }_t\rho +\mathbf{v}\cdot \nabla \rho +(\rho +p)\nabla \cdot \mathbf{v}=0$. The spatial Euler equation contains the gravitational acceleration $-\nabla \varphi$ and pressure force $-\nabla p/(\rho +p)$. In a strict first-order expansion, convective velocity acceleration and products of the small quantities are discarded.

To exhibit a fully linear system, perturb a homogeneous rest state: $n=n_0+\delta n$, $\rho ={\rho }_0+\delta \rho$, $p=p_0+\delta p$, and $ϵ={ϵ}_0+\delta ϵ$, with the perturbations, $\mathbf{v}$ and $\varphi$ all first order. Let $w_0={\rho }_0+p_0=n_0{ϵ}_0$. The weak-field slow-motion equations for a relativistic fluid areThe equation of state gives $\delta \rho ={ϵ}_0\delta n$ and $\delta p=n_0\delta ϵ$. Where $p^{\prime \prime }({ϵ}_0)\ne 0$, it also gives the barotropic relativistic sound speedWithout this invertibility, the primitive conservation equations remain valid and this sound-speed parametrization must not be used.

The potential furnishes a further check on the pressure and gravity signs. Write $\chi ={ϵ}_{0}t+\delta \chi$. Its time and spatial components implyTheir compatibility reproduces the linear Euler equation. Pressure gradients oppose gravitational acceleration, and hydrostatic balance is $\nabla \delta p=-w_0\nabla \varphi$. Combining the energy and Euler equations gives the forced acoustic equationFor positive $c_s^2$, disturbances propagate as sound waves; the prescribed potential supplies a static forcing. A nonrelativistic equation of state gives $p\ll \rho \simeq mn$, $ϵ\simeq m$, and reduces the inertia $\rho +p$ to mass density, recovering the ordinary linearized continuity and Euler equations. Slow velocity alone does not justify dropping $p$ from the inertia of a relativistically hot fluid. The given metric supplies an external gravitational potential; the matter action by itself supplies no Poisson equation for it.

Substitute the Killing-flow four-acceleration and stationary pressure into the relativistic Euler equation. It givesFor the equation of state of blackbody radiation, $p=\rho /3$, this becomes $d\rho =-4\rho d\log N$. The barotropic hydrostatic first integral therefore gives, on each connected nonvacuum fluid region,This is the stationary radiation density law. Combining it with the Stefan–Boltzmann law $\rho \propto T^4$ also gives the Tolman–Ehrenfest relation $TN=\mathrm{constant}$.

At a regular finite free surface, a smooth Killing vector field has finite norm. Hence $C/N^4$ cannot approach zero there. More globally, its norm has a finite upper bound on a compact regular fluid closure, which bounds the density away from zero whenever $C>0$. If the timelike norm instead tends to zero, the density diverges. Thus a radiation fluid has no regular free surface: a finite radiation-filled cavity needs a material wall supporting nonzero pressure, rather than an unsupported radiation-vacuum interface.

Use metric signature $(+---)$ throughout this solution. Stress-energy conservation for the perfect fluid is ${\nabla }_b{T}^{ab}=0$. Expanding it and using metric compatibility givesThe normalization of the four-velocity implies $U_a{U}^b{\nabla }_b{U}^a=0$. Contracting with $U_a$ therefore gives the relativistic perfect-fluid energy equation,while subtracting the component parallel to $U$ gives the relativistic Euler equation,Equivalently, apply the orthogonal projection ${\delta }_a{}^c-U_a{U}^c$ to the conservation equation. This form makes clear that a pressure gradient orthogonal to the four-velocity supplies the fluid's four-acceleration.

In cosmic time, all three Friedmann-Lemaître-Robertson-Walker metrics can be writtenwhereThe closed spatial slice is a three-sphere of radius $a(t)$, soThe complete flat and open spatial slices are noncompact and have

For a homogeneous fluid in the Spatially flat FLRW metric, the equations in part (a) reduce to the cosmological perfect-fluid continuity equationwhile homogeneity makes the spatial relativistic Euler equation automatic. The Einstein field equations give the flat Friedmann equation and Friedmann acceleration equation,equivalently $\ddot{a}/a=-4\pi G(\rho +3P)/3$.

Contract ${\nabla }_a{T}^{ab}=0$ for the perfect fluid in general relativity with $u_b$. Using $u_b{u}^b=-1$ and $u_b{\nabla }_a{u}^b=0$ gives the relativistic perfect-fluid energy equationthe local first law of thermodynamics. Project instead with $g^a{}_b+u^a{u}_b$ to obtain the relativistic Euler equation