Matter Coupling and Stress-Energy

Matter tells spacetime how to curve through a stress–energy tensor, but that slogan becomes predictive only after the tensor is tied to a matter model. This chapter follows the chain from covariant matter actions to metric response, then from $\nabla_\mu T^{\mu\nu}=0$ to particle, dust, and fluid motion. Scalar and electromagnetic fields provide explicit variations and contrasting traces; curvature-coupled scalars show why the split between “geometry” and “matter” can be useful without being unique. Detailed equations of state, dissipative hydrodynamics, spin–torsion theories, kinetic closure, and compact-star microphysics lie outside the chapter.

Helpful background. Events, Intervals, and Minkowski Geometry repairs four-vector and observer-energy skills; Tensor Fields and Natural Operations repairs tensor and integration language; Classical Fields and Their Actions reviews field variation and gauge symmetry; Flat-Background Hilbert Stress-Energy from Metric Variation reviews metric response; Affine Connections and Covariant Derivatives reviews divergence and commutators; Continuum Kinematics and Balance Laws reviews continuum conservation and closure; Presymplectic Potentials and Currents prepares boundary-sensitive variations.

From an action to gravitational motion

The recurring definition is

$$T_{\mu\nu} =-\frac{2}{\sqrt{-g}} \frac{\delta S_{\rm m}}{\delta g^{\mu\nu}}.$$

The sign refers specifically to variation of the inverse metric. It makes $\delta S_{\rm m}=-\tfrac12\int\sqrt{-g}\,T_{\mu\nu}\delta g^{\mu\nu}$ and produces $G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi G T_{\mu\nu}$ when combined with the gravitational action. Diffeomorphism invariance then relates $\nabla_\mu T^{\mu\nu}$ to the matter equations; conservation is generally on shell rather than an extra postulate independent of the dynamics. Reall develops this action-to-identity chain in Reall 2013, §§10.1 and 10.3, pp. 115–121, PDF.

Three checks recur throughout. First, observer energy is $T_{\mu\nu}u^\mu u^\nu$ with $u^2=-1$, which catches many sign errors. Second, every conservation statement must name the closed system: matter can exchange force with another field. Third, a flat-spacetime limit cannot determine coefficients of terms that vanish with curvature.

Chapter contents

1. Minimal Metric Coupling explains the covariantization prescription and its curvature, ordering, spinor, and boundary ambiguities.
2. Gravitationally Coupled Stress-Energy from Metric Variation derives the site normalization, observer decomposition, diffeomorphism identity, and distinction from canonical Noether stress.
3. Point Particles and Dust constructs the invariant worldline distribution and splits dust conservation into continuity and geodesic flow.
4. Perfect Fluids in General Relativity derives the parallel energy equation and orthogonal relativistic Euler equation.
5. Scalar and Electromagnetic Field Sources varies both actions and compares energy flux, arbitrary-dimension traces, and Ricci sources.
6. Nonminimal Coupling and Effective Matter tracks the derivative terms generated by $F(φ)R$, effective coupling, conformal frames, and singular limits.

Reader routes

For an action-centered route, read the first two pages, proceed to the scalar and electromagnetic calculation, and finish with nonminimal coupling. This route emphasizes what is varied, which boundary assumptions are used, and why moving a term across Einstein’s equation changes interpretation but not the underlying equation.

For a continuum route, read metric stress–energy, then particles and dust, then perfect fluids. The central thread is projection: contraction along $u^\mu$ extracts energy or continuity, while projection with $h_{\mu\nu}=g_{\mu\nu}+u_\mu u_\nu$ extracts momentum balance. Readers concerned with energy conditions should then compare the scalar, Maxwell, and nonminimal examples; Kontou and Sanders state the relevant perfect-fluid combinations and classical counterexamples in Kontou–Sanders 2020, §2, pp. 6–16, PDF.

For a quick reference route, begin with the field model of interest and return to metric variation whenever a normalization or conservation question arises. Minimal coupling is a diagnostic prerequisite for deciding whether the stated action is an assumption or a consequence of a flat theory.

Distinctions to keep visible

Metric stress–energy is the source obtained by varying the metric; canonical energy–momentum is a translation current and may require improvement. A point-particle tensor is a distribution, whereas dust is a smooth single-stream approximation. A perfect fluid is isotropic in one local rest frame, not pressureless. Maxwell tracelessness is a four-dimensional classical result and does not imply vanishing Ricci tensor. Finally, an effective tensor obtained after dividing by $F(φ)$ is meaningful only where $F$ is nonzero; a frame transformation is not an extension through $F=0$.

Check your preparation

At one event in a local inertial frame, a homogeneous canonical scalar has $\dotφ\ne0$, spatial gradient zero, and potential $V$. A source-free electromagnetic field has measured fields $\mathbf E$ and $\mathbf B$. Find the total energy density and trace in four dimensions, then determine the Ricci scalar for $\Lambda=0$. Does the Maxwell field curve spacetime when its trace vanishes?

Solution

The scalar and Maxwell energy densities add:

$$\rho_{\rm total} =\frac12\dotφ^2+V +\frac12\left(|\mathbf E|^2+|\mathbf B|^2\right).$$

For the homogeneous scalar, $(\nablaφ)^2=-\dotφ^2$, so

$$T_φ=-(\nablaφ)^2-4V =\dotφ^2-4V.$$

The four-dimensional Maxwell trace is zero. Tracing Einstein’s equation gives $-R=8\pi G T$, hence

$$R=8\pi G(4V-\dotφ^2).$$

The electromagnetic field contributes nothing to this scalar trace, but it still contributes a generally nonzero $T^{({\rm EM})}_{\mu\nu}$ and therefore a nonzero Ricci tensor. Traceless matter is not the same as absent matter.

Onward connections

The Einstein–Hilbert Action and Boundary Terms chapter completes the gravitational variation that was used here. Diffeomorphisms, Symmetry, and Conservation develops the Noether identity and exact-symmetry currents. Congruences, Focusing, and Energy Conditions tests how particular stress tensors translate into curvature inequalities and where those inequalities fail.

References

• Kontou, Eleni-Alexandra, and Ko Sanders. “Energy Conditions in General Relativity and Quantum Field Theory.” Classical and Quantum Gravity 37 (2020): 193001. DOI. Open PDF.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes, 2013. PDF.