Conservation Laws and the Continuity Equation

A conservation law is a promise about bookkeeping: some quantity is never created or destroyed in place, only shuffled from here to there. In pre-relativistic physics that promise is written locally as the continuity equation $\partial_t\rho + \nabla\!\cdot\!\mathbf j = 0$ — the rate at which stuff drains out of a small region equals the flux leaking through its walls. This is far stronger than global bookkeeping. It forbids matter from quietly vanishing in Delhi and rematerializing in Tokyo; every change must be accounted for by a continuous flow across every intervening surface. Special relativity then does something remarkable: it welds the separate Newtonian laws for energy and for the three components of momentum into a single geometric object. Energy density, energy flux, momentum density and stress become the ten components of one symmetric tensor, the stress-energy tensor $T^{\mu\nu}$ , and all ten continuity equations collapse to the one manifestly covariant statement $\partial_\mu T^{\mu\nu}=0$ . Carrying that law into curved spacetime, the equivalence principle demands the minimal-coupling swap $\partial_\mu\to\nabla_\mu$ , giving $\nabla_\mu T^{\mu\nu}=0$ . This is not an extra axiom bolted onto general relativity — it is forced by the field equations themselves, because the Einstein tensor is divergence-free as a pure geometric identity, $\nabla_\mu G^{\mu\nu}=0$ . Yet a deep subtlety lurks. In a general curved spacetime this local law does not integrate to a globally conserved total energy: the extra connection terms block the divergence theorem, and matter freely trades energy and momentum with a gravitational field that has no local energy density of its own. Genuine conserved charges survive only where the spacetime has a continuous symmetry — a Killing vector — the geometric face of Noether's theorem that this article builds toward.

Key takeaways

The continuity equation $\partial_\mu T^{\mu\nu}=0$ expresses local conservation of energy and momentum: the rate of change of these quantities in any region equals the flux through its boundary — the relativistic upgrade of $\partial_t\rho+\nabla\!\cdot\!\mathbf j=0$ .
The stress-energy tensor $T^{\mu\nu}$ is symmetric and holds every density and flux of energy–momentum; its vanishing four-divergence bundles four scalar laws (one energy, three momentum) into one covariant equation.
In curved spacetime the partial derivative promotes to a covariant one, $\nabla_\mu T^{\mu\nu}=0$ — the form demanded by the equivalence principle and already built into Einstein's equations.
Crucially this is only local: it does not in general give a conserved total energy, because $\nabla_\mu$ carries Christoffel terms that defeat Gauss's theorem unless a Killing vector exists.
A Killing vector $\xi^\nu$ manufactures a genuinely conserved current $J^\mu=T^{\mu\nu}\xi_\nu$ with $\nabla_\mu J^\mu=0$ , recovering true conserved charges from spacetime symmetries.
Key idea: $\nabla_\mu T^{\mu\nu}=0$ is local energy–momentum conservation, forced automatically by the Bianchi identity through Einstein's equations; genuine global conservation requires a Killing symmetry, which turns the tensor law into an ordinary conserved current $J^\mu=T^{\mu\nu}\xi_\nu$ .

The core insight

$\nabla_\mu T^{\mu\nu}=0$ is local energy–momentum conservation, automatically enforced by the contracted Bianchi identity through Einstein's equations; true global conservation requires a Killing symmetry, which converts the tensor law into an ordinary conserved current $J^\mu=T^{\mu\nu}\xi_\nu$ .

Where this lives today

Covariant conservation is the quiet engine under a surprising amount of modern gravitational physics. In cosmology it is the equation that turns a list of ingredients into a history: the $a^{-3}$ , $a^{-4}$ and constant dilution laws set the sequence of radiation-, matter- and dark-energy domination that the Planck satellite reconstructs from the microwave background, and the same law makes the accelerating expansion measured by supernova surveys a statement that some component has $w\approx-1$ . In compact-object astrophysics the spatial component of $\nabla_\mu T^{\mu\nu}=0$ for a perfect fluid is the Tolman–Oppenheimer–Volkoff equation, whose relativistic inertia $(\rho+P/c^2)$ — pressure gravitating against itself — is what caps the neutron-star mass near $2\,M_\odot$ , a limit now pinned down by NICER's X-ray timing and by the neutron-star merger GW170817. And the law's most famous failure is itself a discovery: because $\nabla_\mu T^{\mu\nu}=0$ does not localize gravitational energy, the energy radiated by an inspiralling binary can only be defined asymptotically, through the Bondi mass at null infinity — and it is precisely that slowly decreasing mass, translated into a chirping waveform, that LIGO and Virgo now detect several times a month. The subtlety this article dwells on is not a technicality; it is the reason gravitational-wave astronomy needs the machinery it does.

The physical picture

Picture energy and momentum as a fluid painted across spacetime. The continuity equation is the simplest possible statement about that fluid: whatever flows out through the walls of a small box exactly accounts for the drop in what the box holds. Nothing appears or disappears in the bulk — every change is bookkept by a flux across a boundary. The four-divergence $\partial_\mu T^{\mu\nu}$ measures the net leakage rate of the $\nu$ -component of four-momentum at each event, and demanding that it vanish is precisely the no-creation condition. Integrate the energy density $T^{00}$ over a fixed volume and the only way its total can change is for energy to stream through the enclosing surface; seal the surface and the content is frozen.

Figure

A fixed control volume V holds energy content P⁰ = ∫ T⁰⁰ d³x. The local law ∂ᵤTᵘ⁰ = 0 says the content can change only through the boundary flux Tⁱ⁰: dP⁰/dt = −∮ Tⁱ⁰ dSᵢ. With balanced walls — inflow 8.0 = outflow 8.0 — the net is zero and the energy inside is conserved

The single equation $\partial_\mu T^{\mu\nu}=0$ is really four laws bundled together by Lorentz covariance. A boost mixes energy density into momentum density, so insisting the law hold in every inertial frame welds energy conservation and momentum conservation into one object — which is exactly why $T^{\mu\nu}$ must be a tensor and not a loose collection of scalars. Carry the box into a gravitational field and its walls tilt: the extra Christoffel terms in $\nabla_\mu T^{\mu\nu}$ represent four-momentum being handed back and forth between matter and the gravitational field, like a ball trading potential for kinetic energy as it rolls downhill. Because gravitational energy has no local density, only the total is conserved — and even that needs a symmetry to define. Where the conservation law bites hardest and most cleanly is cosmology: apply it to the smooth fluid filling the universe and it dictates exactly how each ingredient dilutes as space expands.

Figure

The conservation law ρ̇ + 3(ȧ/a)(ρ + P/c²) = 0 fixes how each cosmic component dilutes with the scale factor a: matter (w = 0) thins as ρ ∝ a⁻³, radiation (w = 1/3) as ρ ∝ a⁻⁴ from the extra redshift, and a cosmological constant (w = −1) stays exactly constant. The slopes on this log–log plot are the exponents themselves

The core equations

The pre-relativistic ancestor of everything below is the ordinary continuity equation. It is the mathematical form of "what leaves must have flowed out": the local time-rate of a density plus the divergence of its current vanishes.

\partial_t\rho + \nabla\!\cdot\!\mathbf j = 0 \qquad (1)

where: $\rho$ — density of the conserved quantity (mass, charge, particle number); $\mathbf j$ — current density (flux) of that quantity; $\partial_t$ — partial time derivative at a fixed spatial point. Reading: integrate over a region and Gauss's theorem turns the divergence into a surface flux, so the content changes only through the boundary. Special relativity packages energy and momentum into the stress-energy tensor and replaces the one scalar law by a single tensor law. The four-gradient of $T^{\mu\nu}$ vanishes, one equation for each value of the free index $\nu$ .

\partial_\mu T^{\mu\nu} = 0 \qquad (2)

where: $T^{\mu\nu}$ — stress-energy tensor ( $T^{00}$ energy density, $T^{0i}$ energy flux and momentum density, $T^{ij}$ stress); $\partial_\mu = (c^{-1}\partial_t,\nabla)$ — four-gradient; $\nu$ — free index labelling the four conserved components of four-momentum. Reading: $\nu=0$ is energy conservation, $\nu=i$ is conservation of the three momentum components. Passing to curved spacetime, the equivalence principle replaces the partial derivative by the covariant derivative built from the metric's Christoffel symbols. This is the master equation of the whole topic.

\nabla_\mu T^{\mu\nu} = 0 \qquad (3)

where: $\nabla_\mu$ — covariant (metric-compatible, torsion-free) derivative; $T^{\mu\nu}=T^{\nu\mu}$ — symmetric stress-energy tensor. Reading: expanding the covariant divergence gives $\nabla_\mu T^{\mu\nu}=\tfrac{1}{\sqrt{-g}}\partial_\mu(\sqrt{-g}\,T^{\mu\nu})+\Gamma^\nu_{\mu\lambda}T^{\mu\lambda}$ , and the residual $\Gamma$ -term is the obstruction to a global divergence theorem. Most matter of interest is a perfect fluid — isotropic in its own rest frame, with no shear or heat flux. Its stress-energy is built from the four-velocity, the density and the pressure alone.

T^{\mu\nu} = \left(\rho_0 + \frac{P}{c^2}\right)u^\mu u^\nu + P\,g^{\mu\nu} \qquad (4)

where: $\rho_0 c^2$ — rest-frame energy density; $P$ — isotropic pressure in the fluid rest frame; $u^\mu$ — fluid four-velocity, $u^\mu u_\mu=-c^2$ ; $g^{\mu\nu}$ — inverse metric. Reading: the inertia of the fluid is $(\rho_0+P/c^2)$ , not $\rho_0$ — pressure itself gravitates, the hinge of relativistic astrophysics. When the spacetime has a continuous symmetry, encoded by a Killing vector, the tensor law promotes to an ordinary conserved current — a true conservation law with a divergence theorem behind it.

J^\mu = T^{\mu\nu}\,\xi_\nu,\qquad \nabla_\mu J^\mu = 0 \qquad (5)

where: $\xi_\nu$ — Killing vector satisfying $\nabla_{(\mu}\xi_{\nu)}=0$ ; $J^\mu$ — conserved four-current; the associated charge is $Q=\int_\Sigma J^\mu n_\mu\sqrt{h}\,\mathrm d^3x$ . Reading: a timelike $\xi$ gives conserved energy, a rotational $\xi$ gives conserved angular momentum. Finally, in general relativity conservation is not assumed — it is a theorem. The Einstein tensor is divergence-free by geometry, so any source it equals must be too.

\nabla_\mu G^{\mu\nu} = 0,\qquad G^{\mu\nu}=\frac{8\pi G}{c^4}\,T^{\mu\nu}\ \Rightarrow\ \nabla_\mu T^{\mu\nu}=0 \qquad (6)

where: $G^{\mu\nu}=R^{\mu\nu}-\tfrac12 R\,g^{\mu\nu}$ — Einstein tensor; $R^{\mu\nu}$ — Ricci tensor; $R$ — Ricci scalar; $8\pi G/c^4$ — Einstein coupling. Reading: covariant conservation is a consistency condition on the field equations, not an independent postulate.

Derivation AFrom the local law to the global conservation of four-momentum

The goal is to show precisely when the differential statement $\partial_\mu T^{\mu\nu}=0$ upgrades to the familiar global claim that total energy and momentum are constant in time. We work in flat spacetime and inertial coordinates, where the covariant derivative collapses to a partial one and Christoffel symbols vanish, define the total four-momentum as a volume integral of the density, differentiate it, and use Gauss's theorem to convert the interior change into a boundary flux. The final step exposes the one physical assumption — no flux at infinity — that makes an isolated system's four-momentum conserved, and it foreshadows exactly what curved spacetime will spoil.

Step 1. Define the total four-momentum in a spatial volume $V$ at fixed time, $P^\nu(t)\equiv \tfrac1c\int_V T^{0\nu}\,\mathrm d^3x$ . Why: $T^{0\nu}$ is the density of the $\nu$ -component of four-momentum, so integrating it over space is the natural "how much is in here right now" quantity whose constancy we want to test.

Step 2. Differentiate in time and use $\partial_0=c^{-1}\partial_t$ , giving $\dfrac{\mathrm dP^\nu}{\mathrm dt}=\displaystyle\int_V \partial_0 T^{0\nu}\,\mathrm d^3x$ . Why: only the integrand depends on time (the volume is fixed), so the time derivative slides inside the integral and lands on the density.

Step 3. Apply the local law. Splitting $\partial_\mu T^{\mu\nu}=0$ into time and space parts gives $\partial_0 T^{0\nu}=-\partial_i T^{i\nu}$ , so $\dfrac{\mathrm dP^\nu}{\mathrm dt}=-\displaystyle\int_V \partial_i T^{i\nu}\,\mathrm d^3x$ . Why: the continuity equation is exactly the statement that the local rate of change of the density is minus the divergence of the flux — that is what we substitute.

Step 4. Convert to a surface integral by the divergence theorem, $\dfrac{\mathrm dP^\nu}{\mathrm dt}=-\oint_{\partial V} T^{i\nu}\,\mathrm dS_i$ . If the stresses fall off fast enough that there is no flux through the boundary at infinity, then $\dfrac{\mathrm dP^\nu}{\mathrm dt}=0$ and $P^\nu$ is conserved. Why: this is the crux — a genuine conservation law needs both the local divergence law and a boundary condition, and in curved spacetime the extra $\Gamma$ -term blocks Step 3, so this chain simply cannot be run.

Go deeper · Derivation B · 1 minWhy a Killing symmetry rescues a genuine conserved charge

Derivation BWhy a Killing symmetry rescues a genuine conserved charge

The goal is to recover an honest conservation law in curved spacetime, where the bare tensor equation $\nabla_\mu T^{\mu\nu}=0$ refuses to integrate to a conserved total. The trick is to contract the stress-energy tensor with a Killing vector — a vector field along which the metric does not change — to build a genuine four-current $J^\mu=T^{\mu\nu}\xi_\nu$ whose ordinary covariant divergence vanishes, so that the density $J^0$ obeys a true divergence theorem and defines a conserved charge. This is the geometric incarnation of Noether's theorem: a continuous symmetry of spacetime yields a conserved quantity.

Step 1. Take the covariant divergence of the current and apply the Leibniz rule, $\nabla_\mu J^\mu=(\nabla_\mu T^{\mu\nu})\xi_\nu + T^{\mu\nu}\nabla_\mu\xi_\nu$ . Why: the current is a product, so its divergence splits into a piece that differentiates the tensor and a piece that differentiates the symmetry vector — each of which we can now kill separately.

Step 2. The first term vanishes identically by covariant conservation, $\nabla_\mu T^{\mu\nu}=0$ . Why: this is precisely the master equation (3); it is doing the heavy lifting, leaving only the derivative of the Killing vector to worry about.

Step 3. In the second term, symmetry of $T^{\mu\nu}$ lets only the symmetric part of $\nabla_\mu\xi_\nu$ survive, $T^{\mu\nu}\nabla_\mu\xi_\nu=T^{\mu\nu}\nabla_{(\mu}\xi_{\nu)}=0$ , which vanishes by the Killing equation $\nabla_{(\mu}\xi_{\nu)}=0$ . Why: contracting a symmetric tensor with any object projects out only the symmetric part of that object, and the Killing condition is exactly the statement that this symmetric part is zero.

Step 4. Hence $\nabla_\mu J^\mu=0$ , which is an ordinary divergence $\tfrac{1}{\sqrt{-g}}\partial_\mu(\sqrt{-g}\,J^\mu)=0$ , so $Q=\int_\Sigma J^\mu n_\mu\sqrt{h}\,\mathrm d^3x$ is conserved. Why: for a vector — unlike a rank-2 tensor — the covariant divergence collapses to a genuine partial-derivative divergence, which is what Gauss's theorem needs, restoring the conserved charge.

Go deeper · Derivation C · 1 minThe Bianchi identity forces conservation for free

Derivation CThe Bianchi identity forces conservation for free

The goal is to show that covariant conservation of the stress-energy tensor is not an extra assumption in general relativity but an automatic consequence of the geometry, so that any consistent matter source is conserved whether or not we ask it to be. We start from the differential Bianchi identity — a pure statement about the Riemann tensor of any metric connection — contract it twice to expose the divergence-free Einstein tensor, and then feed in Einstein's field equations to transfer that property directly onto the matter.

Step 1. Write the differential Bianchi identity $\nabla_{[\lambda}R_{\mu\nu]\rho\sigma}=0$ , satisfied by the Riemann tensor of any metric-compatible torsion-free connection. Why: this is a geometric identity, true for every metric with no dynamics assumed, so anything we derive from it is a constraint that gravity cannot escape.

Step 2. Contract on two index pairs to obtain the once-contracted form $\nabla_\mu R^\mu{}_\nu=\tfrac12\nabla_\nu R$ . Why: contracting the full identity collapses the four-index Riemann tensor down to the Ricci tensor and scalar, the pieces that actually appear in the field equations.

Step 3. Rearrange into the Einstein tensor: $\nabla_\mu\!\left(R^{\mu\nu}-\tfrac12 R\,g^{\mu\nu}\right)=\nabla_\mu G^{\mu\nu}=0$ . Why: the particular combination $R^{\mu\nu}-\tfrac12 Rg^{\mu\nu}$ is engineered so that the contracted Bianchi identity makes it identically divergence-free — this is the whole reason Einstein put it on the left of his equation.

Step 4. Apply the field equations $G^{\mu\nu}=8\pi G/c^4\,T^{\mu\nu}$ to get $\nabla_\mu T^{\mu\nu}=\dfrac{c^4}{8\pi G}\nabla_\mu G^{\mu\nu}=0$ . Why: because the source is proportional to the Einstein tensor, matter inherits the geometric divergence-free property — gravity self-consistently conserves energy–momentum, and a non-conserved source would make the field equations inconsistent.

Limiting cases

The one master equation specializes cleanly across every regime that matters:

Case	Result	Meaning
Flat-space / SR limit	$\nabla_\mu T^{\mu\nu}=0\ \longrightarrow\ \partial_\mu T^{\mu\nu}=0$	Christoffel symbols vanish in inertial coordinates; Gauss's theorem applies and total four-momentum $P^\nu=\int T^{0\nu}\,\mathrm d^3x$ is exactly conserved.
Newtonian limit, $\nu=0$	$\partial_t\rho+\nabla\!\cdot\!(\rho\mathbf v)=0$	with $T^{00}\to\rho c^2$ and $T^{0i}\to\rho c\,v^i$ , the time component reduces to classical mass continuity.
Newtonian limit, $\nu=i$	$\partial_t(\rho v^i)+\partial_j(\rho v^iv^j+P\delta^{ij})=-\rho\,\partial^i\Phi$	the spatial components reproduce the Euler equation, the Christoffel terms acting as the gravitational force $-\rho\nabla\Phi$ .
Killing symmetry present	$\nabla_\mu(T^{\mu\nu}\xi_\nu)=0$	a Killing vector converts the tensor law into an ordinary conserved current; $Q=\int_\Sigma T^{\mu\nu}\xi_\nu n_\mu\sqrt{h}\,\mathrm d^3x$ is a genuine charge (energy if $\xi$ is timelike).
FLRW cosmology	$\dot\rho+3\tfrac{\dot a}{a}(\rho+P/c^2)=0$	the homogeneous fluid law; equations of state $w=P/\rho c^2$ give $\rho\propto a^{-3(1+w)}$ .

Common trap∇ᵤTᵘᵛ = 0 means total energy is conserved in any curved spacetime

It does not. In a general curved spacetime the covariant divergence carries an irreducible Christoffel term, $\nabla_\mu T^{\mu\nu}=\tfrac{1}{\sqrt{-g}}\partial_\mu(\sqrt{-g}\,T^{\mu\nu})+\Gamma^\nu_{\mu\lambda}T^{\mu\lambda}$ , and that residual $\Gamma^\nu_{\mu\lambda}T^{\mu\lambda}$ is not a divergence of anything — so the volume integral of $T^{0\nu}$ generally drifts in time and Gauss's theorem gives no conserved charge. Physically, matter is exchanging energy and momentum with the gravitational field, which has no local, coordinate-independent energy density to add to the ledger. A conserved total exists only when a Killing vector is present, and even the "energy of gravitational waves" must be defined through symmetry-based, non-local constructions (Bondi/ADM mass, the Isaacson average). Expanding cosmology is the everyday example: the energy density of radiation genuinely falls as $a^{-4}$ , and the missing energy is not hiding anywhere — it has simply been redshifted away, with no global energy conservation law to forbid it.

Worked examples

1 · Matter–radiation equality from the dilution laws (fully numeric)

Today the universe holds matter at $\Omega_{m,0}=0.31$ of the critical density and radiation (photons plus neutrinos) at $\Omega_{r,0}=9.1\times10^{-5}$ . Using only the conservation-law scalings, find the scale factor and redshift at which the two densities were equal.

Step 1. The conservation law gives the dilution exponents directly: matter ( $P=0$ ) thins as $\rho_m=\rho_{m,0}\,a^{-3}$ , radiation ( $P=\rho c^2/3$ ) as $\rho_r=\rho_{r,0}\,a^{-4}$ , with $a_0=1$ today. The extra power for radiation is the redshift of each photon's energy on top of the volume dilution.

Step 2. Form the ratio and set it to one: $\dfrac{\rho_r}{\rho_m}=\dfrac{\Omega_{r,0}}{\Omega_{m,0}}\,a^{-1}=1$ at equality, so $a_{\rm eq}=\dfrac{\Omega_{r,0}}{\Omega_{m,0}}=\dfrac{9.1\times10^{-5}}{0.31}=\mathbf{2.94\times10^{-4}}$ .

Step 3. Convert to redshift: $z_{\rm eq}=a_{\rm eq}^{-1}-1=3.41\times10^{3}-1\approx\mathbf{3400}$ . Below this redshift the universe is matter-dominated (structure can grow); above it, radiation dominates and pressure resists collapse. The two exponents alone — $-3$ versus $-4$ — pin equality at $z_{\rm eq}\approx3400$ , in accord with the Planck value.

2 · The CMB was hotter in exactly the way conservation predicts (fully numeric)

The cosmic microwave background today has temperature $T_0=2.725$ K. Using the radiation dilution law, find the photon temperature at recombination ( $z\approx1100$ ) and the factor by which the radiation energy density has since dropped.

Step 1. Radiation energy density scales two ways at once: thermodynamically $\rho_r\propto T^4$ , and by conservation $\rho_r\propto a^{-4}$ . Equate the powers of $a$ : $T^4\propto a^{-4}$ , so $T\propto a^{-1}=(1+z)$ . Temperature simply tracks the inverse scale factor.

Step 2. At recombination $1+z=1101$ , so $T_{\rm rec}=T_0(1+z)=2.725\times1101=\mathbf{3.0\times10^{3}\ K}$ — about $3000$ K, the temperature at which hydrogen recombines and the universe turns transparent. This is a prediction, not an input: it follows purely from $\rho_r\propto a^{-4}$ .

Step 3. The energy density has fallen by $(1+z)^4=(1101)^4=\mathbf{1.5\times10^{12}}$ , while the photon number density (which dilutes only by volume) dropped by $(1+z)^3=1.3\times10^{9}$ . The extra factor of $1101$ between them is the energy each photon lost to redshift — the very energy that a curved-spacetime conservation law does not book anywhere. Radiation energy density is down by $1.5\times10^{12}$ since $z=1100$ , purely from the $a^{-4}$ law.

3 · Deriving the FLRW fluid equation from ∇ᵤTᵘᵛ = 0 (worked derivation)

Show that a homogeneous perfect fluid in the flat FLRW metric $\mathrm ds^2=-c^2\mathrm dt^2+a(t)^2\,\delta_{ij}\mathrm dx^i\mathrm dx^j$ obeys $\dot\rho+3(\dot a/a)(\rho+P/c^2)=0$ , and read off the three scalings.

Step 1. Use coordinate time $x^0=t$ . The comoving fluid is at rest with unit-normalized four-velocity $u^\mu=(1,0,0,0)$ (since $g_{00}=-c^2$ forces $u^\mu u_\mu=-c^2$ ), and homogeneity kills all spatial gradients, so only the $\nu=0$ component of $\nabla_\mu T^{\mu\nu}=0$ survives: $\partial_0 T^{00}+\Gamma^\mu_{\mu 0}T^{00}+\Gamma^0_{ij}T^{ij}=0$ . The relevant Christoffel symbols are $\Gamma^i_{0j}=(\dot a/a)\delta^i_j$ and $\Gamma^0_{ij}=(a\dot a/c^2)\delta_{ij}$ , giving the trace $\Gamma^\mu_{\mu 0}=3\dot a/a$ .

Step 2. Read off the components in this frame: $T^{00}=(\rho+P/c^2)u^0u^0+Pg^{00}=\rho$ (the pressure terms cancel via $g^{00}=-1/c^2$ ), and the spatial stress $T^{ij}=Pg^{ij}=P a^{-2}\delta^{ij}$ . The last connection term is then $\Gamma^0_{ij}T^{ij}=(a\dot a/c^2)\delta_{ij}\,Pa^{-2}\delta^{ij}=3(\dot a/a)(P/c^2)$ .

Step 3. Assemble the three pieces: $\dot\rho+3(\dot a/a)\rho+3(\dot a/a)(P/c^2)=0$ , i.e. $\boxed{\dot\rho+3\dfrac{\dot a}{a}\left(\rho+\dfrac{P}{c^2}\right)=0}$ , with $\rho$ the rest-mass density. With the equation of state $P=w\rho c^2$ this integrates to $\rho\propto a^{-3(1+w)}$ : dust ( $w=0$ ) gives $\rho\propto a^{-3}$ , radiation ( $w=1/3$ ) gives $\rho\propto a^{-4}$ , and a cosmological constant ( $w=-1$ ) gives $\rho=$ const. The single component $\nabla_\mu T^{\mu 0}=0$ reproduces every dilution law used in Examples 1 and 2.

Physics of the portrait

Conservation laws in relativity are the sharpest statement of a truth older than physics — that symmetry is bookkeeping. No one made that truth into a theorem more decisively than the mathematician who was summoned to Göttingen precisely because energy conservation in Einstein's new theory had stopped making sense.

Origins · 1 minEmmy Noether and how this was discovered

Emmy Noether

1882–1935 · the mathematician whose theorems tie every continuous symmetry to a conserved current and explain why energy conservation in general relativity is subtle

Emmy Noether was invited to Göttingen in 1915 by David Hilbert and Felix Klein at the exact moment general relativity was born — and for a very specific reason. Einstein's new field equations seemed to threaten the sacred law of energy conservation; the "energy" of the gravitational field would not sit still, and Hilbert suspected something deep was going on. Noether settled it completely in her 1918 paper Invariante Variationsprobleme, proving two theorems of extraordinary reach. Her first theorem shows that every continuous global symmetry of an action yields a locally conserved current — time-translation invariance gives energy conservation, spatial translations give momentum, rotations give angular momentum. Her second theorem, aimed straight at general relativity, shows that a local (gauge) symmetry like general coordinate invariance produces identities among the field equations rather than ordinary conservation laws — which is exactly why $\nabla_\mu T^{\mu\nu}=0$ is a local statement that fails to give a global conserved energy without a Killing symmetry. The Killing-vector current $J^\mu=T^{\mu\nu}\xi_\nu$ derived in this article is her first theorem wearing geometric clothes: a symmetry of the metric, not of a flat background, still delivers a genuine conserved charge. Noether worked for years unpaid, lecturing under Hilbert's name because the faculty refused to appoint a woman; Einstein called her the most significant creative mathematical genius produced since the higher education of women began. Driven out by the Nazis in 1933, she emigrated to Bryn Mawr and died two years later, in 1935, at 53. Her theorem is now the organizing principle of all of theoretical physics.

History

1757

— Euler writes the continuity equation ∂ρ/∂t + ∇·(ρv) = 0 for fluid flow

1902

— Bianchi publishes the differential identity later shown to force ∇·G = 0

1915

— Einstein and Hilbert give the field equations, tying G^{μν} to T^{μν}

1918

— Emmy Noether's theorems connect every continuous symmetry to a conserved current

1939

— Tolman, Oppenheimer and Volkoff apply ∇_μT^{μν}=0 to stellar hydrostatic equilibrium

Going further

Texts. Misner, Thorne & Wheeler, Gravitation (1973), §5.7–5.11 and §15.3 (stress-energy and automatic conservation) · Wald, General Relativity (1984), §4.3 and §11.2, Appendix E (Killing charges) · Carroll, Spacetime and Geometry (2004), §1.9, §4.2, §5.4 · Hartle, Gravity (2003), §22.4–22.5 · Schutz, A First Course in General Relativity (2009), §4.4–4.10 and §7.4. Key identities to keep in mind: the Christoffel trace satisfies $\Gamma^\mu_{\mu\lambda}=\partial_\lambda\ln\sqrt{-g}$ , which underlies the divergence-density form of $\nabla_\mu T^{\mu\nu}$ ; the FLRW fluid equation follows from the $\nu=0$ component and integrates to $\rho\propto a^{-3(1+w)}$ ; the Killing current satisfies $\nabla_\mu J^\mu=0$ as shown above; matter–radiation equality works out to $z_{\rm eq}=\Omega_{m,0}/\Omega_{r,0}-1\approx3400$ and the recombination temperature to $T_{\rm rec}=T_0(1+z)\approx3000$ K. Signature and sign conventions follow MTW/Carroll ( $-+++$ ).

Builds on. Four-Vectors · Minkowski Metric and Signature · Minkowski Spacetime · Lorentz Transformations · Proper Time. Leads to. Einstein Field Equations · Perfect Fluid Stress-Energy · Noether's Theorem and Killing Vectors · ADM Energy and Asymptotic Charges.

Ask the tutor: Ask me to walk through the FLRW fluid equation, the Killing-vector charge for a Schwarzschild orbit, or exactly why gravitational-wave energy has to be defined at infinity.

Why is the stress-energy tensor symmetric?

Symmetry $T^{\mu\nu}=T^{\nu\mu}$ is not an aesthetic choice — it encodes conservation of angular momentum and follows from the way $T^{\mu\nu}$ is defined as the response of the matter action to the metric, $T^{\mu\nu}\propto \delta S_{\rm matter}/\delta g_{\mu\nu}$ , since the metric is itself symmetric. The mixed components carry the physical content: $T^{0i}$ is simultaneously the energy flux and the momentum density, and the equality of those two readings is exactly the off-diagonal symmetry. In the Killing-current derivation the symmetry is indispensable — it is what lets you replace $\nabla_\mu\xi_\nu$ by its symmetric part $\nabla_{(\mu}\xi_{\nu)}$ , which the Killing equation then annihilates. Without symmetry, spacetime symmetries would not hand you conserved charges.

What is the difference between a local and a global conservation law?

A local conservation law is the differential statement $\nabla_\mu T^{\mu\nu}=0$ : at every event, the rate of change of a density is balanced by the divergence of a flux, so nothing is created or destroyed pointwise. A global conservation law is the integral claim that a total charge $Q=\int T^{0\nu}\,\mathrm d^3x$ is constant in time. In flat spacetime the local law implies the global one, because the divergence theorem converts the interior change into a boundary flux that vanishes for an isolated system. In curved spacetime that bridge collapses: the covariant divergence carries extra Christoffel terms that are not themselves a divergence, so the local law can hold everywhere while the global total drifts — energy leaking to or from the gravitational field.

Is energy really not conserved in an expanding universe?

In the ordinary global sense, correct — and it is not a paradox but a feature. The radiation filling the universe dilutes as $\rho\propto a^{-4}$ , one power steeper than the $a^{-3}$ you would get from volume alone, because every photon also redshifts. That lost energy does not reappear anywhere: there is no conserved total to satisfy. What is always true is the local law $\nabla_\mu T^{\mu\nu}=0$ , which in FLRW reads $\dot\rho+3(\dot a/a)(\rho+P/c^2)=0$ — a perfectly valid bookkeeping of exchange between the fluid and the expanding geometry. Because FLRW has no timelike Killing vector (the metric depends on time), Noether's theorem simply does not promise a conserved energy, and none exists.

How does pressure end up gravitating in this framework?

Through the combination $(\rho+P/c^2)$ that multiplies $u^\mu u^\nu$ in the perfect-fluid stress-energy. Because the source of gravity in Einstein's equations is the full tensor $T^{\mu\nu}$ — not just the energy density $T^{00}$ — the pressure terms feed into the geometry too. When you take the spatial part of $\nabla_\mu T^{\mu\nu}=0$ for a static star, the inertia resisting collapse is $(\rho+P/c^2)$ rather than $\rho$ , so pressure adds to the effective weight it is meant to support. This is why a sufficiently dense star cannot be saved by piling on more pressure: past a point, the pressure gravitates faster than it supports, and collapse becomes unavoidable. It is the physical origin of the maximum neutron-star mass.

Does the electromagnetic charge continuity equation fit into this picture?

Yes, as the simplest special case. Electric charge is carried by a four-current $J^\mu_{\rm em}=(c\rho_{\rm e},\mathbf j)$ , and its conservation is the vector continuity equation $\partial_\mu J^\mu_{\rm em}=0$ , or $\nabla_\mu J^\mu_{\rm em}=0$ in curved spacetime. Because $J^\mu$ is a vector, not a rank-2 tensor, its covariant divergence collapses to an honest divergence $\tfrac{1}{\sqrt{-g}}\partial_\mu(\sqrt{-g}\,J^\mu)=0$ with no obstructing Christoffel term — so total charge really is globally conserved even in a general curved spacetime, unlike energy. This is exactly why the Killing trick works: contracting $T^{\mu\nu}$ with $\xi_\nu$ demotes the two-index tensor to a one-index current, buying back the clean divergence theorem that charge enjoys automatically.

Where does Noether's theorem actually enter general relativity?

At two levels. Her first theorem — global symmetry gives a conserved current — is realized geometrically by Killing vectors: a symmetry of the metric yields the conserved current $J^\mu=T^{\mu\nu}\xi_\nu$ and hence conserved energy or angular momentum for test particles, the constants of motion that make black-hole orbits solvable. Her second theorem — local (gauge) symmetry gives identities, not conservation laws — is why general relativity, whose gauge symmetry is invariance under arbitrary coordinate changes, produces the contracted Bianchi identity $\nabla_\mu G^{\mu\nu}=0$ instead of an ordinary energy law. That identity is what forces $\nabla_\mu T^{\mu\nu}=0$ for free. So both the automatic conservation and the failure of global energy conservation trace back to Noether's two theorems.