Variational Definition of Stress-Energy

The variational stress-energy tensor is not a museum piece; it is the daily working definition wherever matter meets curved spacetime. In numerical relativity — the simulations that produced the waveform templates LIGO and Virgo match against merging black holes and neutron stars — the matter source is coded as the Hilbert $T_{\mu\nu}$ of a relativistic fluid or magnetofluid, and its automatic conservation $\nabla^\mu T_{\mu\nu}=0$ is what the evolution schemes enforce to stay stable. In cosmology the entire dark-energy programme is a debate about the constant term in the matter action: is the observed $w\approx-1$ (measured by DESI, Planck, and Type Ia supernovae to a few percent) a true cosmological constant, or a slowly-rolling quintessence scalar whose $T_{\mu\nu}$ only approximates $-\rho_\Lambda g_{\mu\nu}$? The variational definition is the common language in which those alternatives are even stated. In quantum field theory on curved spacetime, the renormalized $\langle T_{\mu\nu}\rangle$ — defined by varying the effective action with respect to the metric — drives Hawking radiation and the conformal (trace) anomaly, where $\langle T^\mu{}_\mu\rangle\neq0$ even for a classically traceless field because regularization smuggles in a scale. And in the effective-field-theory formulation of relativistic hydrodynamics, $T_{\mu\nu}$ is derived order by order from an action for the comoving matter variables, guaranteeing thermodynamic consistency and conservation at every order — the systematic descendant of Hilbert's single scalar functional.

Picture the matter action $S_{\mathrm m}$ not as a number but as a machine that eats a spacetime geometry and returns a single real number. Feed it flat Minkowski space and it returns one value; feed it a slightly bumped metric and it returns another. The stress-energy tensor is the sensitivity of that machine to the bump — the gradient of the action in the abstract space of all metrics. Where matter is dense and energetic, the action changes steeply as you deform the metric there: the matter "pushes back" hard against the geometry, and that push-back, component by component, is exactly the local energy density, momentum flux, and stress. Where spacetime is empty, the machine does not care how you wiggle the metric, and $T_{\mu\nu}=0$. The prefactor $-2/\sqrt{-g}$ is only a change of units — it strips off the volume-element density and fixes the sign so that ordinary matter has positive energy density — but the geometry of "action as a landscape over metrics, stress-energy as its slope" is the whole physical content.

Two further ideas sit inside this one picture, and they are what make the variational definition indispensable rather than merely elegant. The first is self-symmetrization. Because the variable you differentiate with respect to, $g^{\mu\nu}$, is a symmetric tensor, the derivative $\delta/\delta g^{\mu\nu}$ cannot help but produce something symmetric in $\mu\nu$ — the improvement terms that the canonical Noether tensor needs for spinning fields are performed for free. The second is conservation from covariance. A pure relabelling of coordinates cannot change a scalar; but a coordinate change acts on the metric as a Lie derivative, $\delta_\xi g^{\mu\nu}=-(\nabla^\mu\xi^\nu+\nabla^\nu\xi^\mu)$, and demanding that this particular deformation leaves the on-shell action unchanged is, term by term, the statement $\nabla^\mu T_{\mu\nu}=0$. Energy-momentum conservation is thus the gravitational analogue of charge conservation from gauge invariance. The second figure shows why one prescription is worth so much: a single operation, applied to three different matter Lagrangians, returns three different stress-energy tensors — and reads off their traces, which encode whether the matter has an intrinsic scale.

The definition itself is the anchor of the whole article: it says how to manufacture $T_{\mu\nu}$ from any matter Lagrangian by varying the metric. The second equality — obtained by expanding the variation of the volume element $\sqrt{-g}$ — is the working form used in every calculation, because it isolates the explicit $g^{\mu\nu}$ dependence of $\mathcal{L}_{\mathrm m}$ and adds a universal trace piece.

where: $T_{\mu\nu}$ — Hilbert (metric) stress-energy tensor; $g$ — determinant of $g_{\mu\nu}$; $\mathcal{L}_{\mathrm{m}}$ — matter Lagrangian scalar; $g^{\mu\nu}$ — inverse metric, the variational variable. Reading it: differentiate the Lagrangian with respect to the inverse metric, double it with a minus sign, and add $g_{\mu\nu}\mathcal{L}_{\mathrm m}$ for the volume-element contribution. The universal trace piece comes entirely from how the invariant volume $\sqrt{-g}\,\mathrm{d}^4x$ responds to a metric deformation. This identity, from Jacobi's formula for the derivative of a determinant, is the single most reused line in variational relativity.

where: $\delta g^{\mu\nu}$ — variation of the inverse metric; $\delta g_{\mu\nu}=-g_{\mu\alpha}g_{\nu\beta}\,\delta g^{\alpha\beta}$ — variation of the metric. The two forms differ by a sign because raising the indices on the variation flips it; this is the root of every "which sign?" convention headache. Gravity and matter meet in one functional. The total action is the Einstein-Hilbert term plus the matter term, each integrated against the same invariant volume, and the field equations are what you get by holding the whole thing stationary against $\delta g^{\mu\nu}$.

where: $R$ — Ricci scalar; $G$ — Newton's constant; $S$ — total gravity-plus-matter action. Reading it: the coefficient $1/16\pi G$ is fixed by demanding the Newtonian limit; the matter piece carries all the fields $\psi$ hidden inside $\mathcal{L}_{\mathrm m}$. Setting $\delta S/\delta g^{\mu\nu}=0$ makes the geometric variation of the first term cancel the matter variation of the second. The Einstein-Hilbert variation supplies $G_{\mu\nu}$; the matter variation supplies precisely $-\tfrac12\sqrt{-g}\,T_{\mu\nu}$, and the two balance.

where: $R_{\mu\nu}$ — Ricci tensor; $G_{\mu\nu}=R_{\mu\nu}-\tfrac12 R g_{\mu\nu}$ — Einstein tensor. Reading it: the very variation that defines $T_{\mu\nu}$ is the variation that sources the field equations, so the definition and the coupling are one and the same operation. Conservation enters not as a new equation but as a consequence of invariance. Under an infinitesimal diffeomorphism generated by $\xi^\mu$, the inverse metric shifts by minus its Lie derivative; demanding that a coordinate-scalar action be blind to this shift, on-shell, forces the covariant divergence of $T_{\mu\nu}$ to vanish.

where: $\xi^\mu$ — generator of the diffeomorphism; $\mathcal{L}_\xi$ — Lie derivative along $\xi$; $\nabla_\mu$ — metric-compatible covariant derivative. Reading it: general covariance of $S_{\mathrm m}$ is the symmetry, and $\nabla^\mu T_{\mu\nu}=0$ is its Noether identity — the exact mirror of $\nabla^\mu G_{\mu\nu}=0$. The trace of $T_{\mu\nu}$ is a physically loaded scalar: it measures whether the matter carries an intrinsic scale. It contrasts sharply with the flat-space canonical tensor, which the Hilbert construction quietly repairs.

where: $T$ — trace of stress-energy, zero for conformally invariant matter (e.g. source-free electromagnetism in four dimensions); $T_{\mu\nu}^{\mathrm{can}}$ — canonical Noether tensor from spacetime translations, generally neither symmetric nor gauge invariant; $\eta_{\mu\nu}$ — flat metric. Reading it: the Hilbert tensor is what the canonical tensor becomes once you add the Belinfante–Rosenfeld improvement, packaged automatically.

The single definition specializes to every standard matter model; each row is the output of the same $\delta/\delta g^{\mu\nu}$ machine fed a different Lagrangian.

1 · The cosmological constant from a constant Lagrangian (fully worked)

Take the simplest possible matter Lagrangian — a constant, $\mathcal{L}_{\mathrm m}=-\rho_\Lambda$, representing a uniform energy density of empty space. Find its stress-energy tensor, its equation of state, and, using the measured dark-energy density $\rho_\Lambda\approx 6\times10^{-27}\ \mathrm{kg\,m^{-3}}$, the numerical value of the cosmological constant $\Lambda$.

Step 1. Apply the working definition. A constant has no explicit $g^{\mu\nu}$ dependence, so $\partial\mathcal{L}_{\mathrm m}/\partial g^{\mu\nu}=0$ and the entire result comes from the volume-element term: $T_{\mu\nu}=-2(0)+g_{\mu\nu}(-\rho_\Lambda)=\mathbf{-\rho_\Lambda\,g_{\mu\nu}}$. Every ounce of this tensor is the $\delta\sqrt{-g}$ contribution — the term a careless calculation would drop.

Step 2. Read off the fluid content. Comparing $T_{\mu\nu}=-\rho_\Lambda g_{\mu\nu}$ with the perfect-fluid form $T_{\mu\nu}=(\rho+p)u_\mu u_\nu+p\,g_{\mu\nu}$ in the rest frame (where $u_\mu u_\nu$ contributes only to the time-time slot and $g_{\mu\nu}=\mathrm{diag}(-1,1,1,1)$): the energy density is $\rho=T_{00}/1=\rho_\Lambda$ and the pressure is $p=-\rho_\Lambda$. Hence the equation of state is $w\equiv p/\rho=\mathbf{-1}$ — the defining signature of a cosmological constant, and why vacuum energy drives accelerated expansion rather than resisting it.

Step 3. Locate $\Lambda$ in the field equations. Moving $-\rho_\Lambda g_{\mu\nu}$ to the geometry side, $G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi G\,T_{\mu\nu}^{\text{other}}$ with $\Lambda=8\pi G\,\rho_\Lambda/c^4\cdot c^2$; in the clean relation $\Lambda=8\pi G\rho_\Lambda/c^2$ (energy density) the constant and the vacuum energy are two names for one constant in the action. This is why "is $\Lambda$ geometry or matter?" has no invariant answer — both spring from the same constant term.

Step 4. Put in numbers. With $G=6.674\times10^{-11}\ \mathrm{m^3\,kg^{-1}\,s^{-2}}$, $c=3.00\times10^{8}\ \mathrm{m\,s^{-1}}$, and $\rho_\Lambda=6\times10^{-27}\ \mathrm{kg\,m^{-3}}$: $\Lambda=\dfrac{8\pi(6.674\times10^{-11})(6\times10^{-27})}{(3.00\times10^{8})^2}=\dfrac{1.006\times10^{-35}}{9.0\times10^{16}}\approx\mathbf{1.1\times10^{-52}\ m^{-2}}$. The corresponding vacuum energy density is $\rho_\Lambda c^2\approx(6\times10^{-27})(9\times10^{16})\approx\mathbf{5.4\times10^{-10}\ J\,m^{-3}}$. Both match the Planck-2018 concordance values — and both sit about $120$ orders of magnitude below the naive quantum-field-theory estimate of the vacuum energy, the cosmological-constant problem in one line.

2 · A homogeneous scalar field interpolates to the vacuum limit

A spatially uniform scalar field $\phi(t)$ in a flat cosmology has $\mathcal{L}_{\mathrm m}=-\tfrac12(\partial\phi)^2-V(\phi)$. Show that its stress-energy is that of a perfect fluid, find $\rho$, $p$, and $w$, and show that when the field rolls slowly it reproduces Example 1.

Step 1. Specialize the scalar tensor. For a homogeneous field only the time derivative survives, $\partial_\mu\phi=(\dot\phi,0,0,0)$, so $(\partial\phi)^2=g^{00}\dot\phi^2=-\dot\phi^2$. The energy density is $\rho=T_{00}=\dot\phi^2-(-1)\big(\tfrac12(-\dot\phi^2)+V\big)$; carefully, $T_{00}=\dot\phi^2-g_{00}(\tfrac12(\partial\phi)^2+V)=\dot\phi^2+(\tfrac12(-\dot\phi^2)+V)=\mathbf{\tfrac12\dot\phi^2+V}$. The pressure, from any spatial diagonal component, is $p=T_{ii}/g_{ii}=\mathbf{\tfrac12\dot\phi^2-V}$.

Step 2. Form the equation of state. $w=\dfrac{p}{\rho}=\dfrac{\tfrac12\dot\phi^2-V}{\tfrac12\dot\phi^2+V}$. Kinetic energy pushes $w$ toward $+1$ (stiff fluid); potential energy pushes it toward $-1$ (vacuum-like). A single field thus spans the entire physically interesting range $-1\le w\le +1$ depending on how fast it moves.

Step 3. Take the slow-roll limit. When the field rolls slowly, $\tfrac12\dot\phi^2\ll V$, both $\rho\to V$ and $p\to-V$, so $w\to\mathbf{-1}$ and $T_{\mu\nu}\to-V g_{\mu\nu}$ — exactly the cosmological-constant fluid of Example 1 with $\rho_\Lambda=V(\phi)$. This is the engine of cosmic inflation and of quintessence dark energy: a scalar field temporarily trapped high on its potential acts like a cosmological constant, and the variational definition is what tells us precisely how well the approximation holds — the correction is set by the size of $\tfrac12\dot\phi^2/V$.

The variational definition of stress-energy has a precise birthday and a precise author. On the same November days in 1915 that Einstein was completing the field equations by physical reasoning, a mathematician in Göttingen derived them a second way — from a single scalar action, held stationary — and in doing so fixed forever how matter is to be coupled to geometry.

Texts. Misner, Thorne & Wheeler, Gravitation, §21.2–21.3 (variational principle and the source) · Wald, General Relativity, §E.1 and §4.3 (the Hilbert tensor and its conservation) · Carroll, Spacetime and Geometry, §4.3 and Appendix E (variational derivation, worked examples) · Hartle, Gravity, Ch. 22 · Schutz, A First Course in General Relativity, Ch. 7.

Provenance. Signs and factors follow the mostly-plus convention $(-,+,+,+)$ of Wald (§E.1, §4.3) and Carroll (App. E), with cross-checks against MTW §21.3. A verifier should: (1) reproduce $\delta\sqrt{-g}=-\tfrac12\sqrt{-g}g_{\mu\nu}\delta g^{\mu\nu}$ from Jacobi's formula; (2) recompute the scalar and Maxwell stress-energy tensors and confirm the EM trace vanishes only in $d=4$; (3) verify the diffeomorphism-invariance derivation of $\nabla^\mu T_{\mu\nu}=0$ against the contracted Bianchi identity $\nabla^\mu G_{\mu\nu}=0$; (4) confirm $\Lambda=8\pi G\rho_\Lambda/c^2\approx1.1\times10^{-52}\ \mathrm{m^{-2}}$ and $\rho_\Lambda c^2\approx5.4\times10^{-10}\ \mathrm{J\,m^{-3}}$ from $\rho_\Lambda\approx6\times10^{-27}\ \mathrm{kg\,m^{-3}}$ (Planck 2018). All equations are textbook-standard; the only numerical inputs are the measured dark-energy density and fundamental constants.

Prerequisites. The Stress-Energy Tensor · The Einstein-Hilbert Action · Variational Derivation of the Field Equations · Local Conservation of Energy-Momentum. Leads to. The Einstein Field Equations · Perfect Fluid Stress-Energy · The Cosmological Constant · The Energy Conditions.

Ask the tutor: Ask me to vary any matter Lagrangian you like — a non-minimally coupled scalar, a Proca field, a viscous fluid — and watch the same $\delta/\delta g^{\mu\nu}$ machine hand back its symmetric, conserved stress-energy tensor step by step.