Einstein And Matter — Examples

Let $M={\mathbb{R}}^4$ carry the identity chart $x=(x^0,x^1,x^2,x^3)$, the flat Lorentzian metric $g=\mathrm{diag}(-1,1,1,1)$, the time orientation $N={\mathrm{\partial }}_0$ and the orientation for which the identity chart is oriented (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Oriented smooth manifolds and oriented charts); let $\mathrm{\nabla }$ be the metric-selected connection of $g$ and ${\mathrm{v}\mathrm{o}\mathrm{l}}_g$ the metric volume form (The metric volume form, current flux through 3-faces and finite-chain charges). Then:

1. The metric density and the volume form are $w=1$ and ${\mathrm{v}\mathrm{o}\mathrm{l}}_g=d{x}^0\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$ in the identity chart, and ${\mathrm{v}\mathrm{o}\mathrm{l}}_g({\mathrm{\partial }}_0,{\mathrm{\partial }}_1,{\mathrm{\partial }}_2,{\mathrm{\partial }}_3)=1$.
2. The connection is the coordinate connection, $\mathrm{\Gamma }\equiv 0$, so that ${\mathrm{\nabla }}_a{J}^a={\mathrm{\partial }}_a{J}^a$ for every smooth vector field $J$ and the divergence formula $w^{-1}{\mathrm{\partial }}_a(w{J}^a)$ reduces to the coordinate divergence.
3. For $J=x^1{\mathrm{\partial }}_1$ the divergence is ${\mathrm{\nabla }}_a{J}^a=1$, the current form is ${\iota }_J{\mathrm{v}\mathrm{o}\mathrm{l}}_g=-x^{1}d{x}^0\wedge d{x}^2\wedge d{x}^3$, and $d({\iota }_J{\mathrm{v}\mathrm{o}\mathrm{l}}_g)={\mathrm{v}\mathrm{o}\mathrm{l}}_g$, in accordance with Divergence, Stokes flux balance and the non-null face formula for a Lorentzian current clause (a).
4. Let ${\mathrm{\Delta }}^3=\{(t_1,t_2,t_3):t_i\ge 0,t_1+t_2+t_3\le 1\}$ with its coordinate orientation and coordinate volume $1\mathrm{/}6$. The affine simplex ${\sigma }_{-}(t)=(t_1,0,t_2,t_3)$ has adapted normal $-{\mathrm{\partial }}_1$, so the flux of $J={\mathrm{\partial }}_1$ through it is $-1\mathrm{/}6$. The affine simplex ${\sigma }_{+}(t)=(t_2,a,t_1,t_3)$ on the parallel plane $x^1=a$ has adapted normal $+{\mathrm{\partial }}_1$, so its flux is $+1\mathrm{/}6$. Both induced metric densities on the parameter simplex equal $1$; the signs follow from the stated parameter orientations and the ambient chart orientation.

Let $M={\mathbb{R}}^4$ carry the identity chart, the flat metric $g=\mathrm{diag}(-1,1,1,1)$, the time orientation $N={\mathrm{\partial }}_0$ and the chart orientation, with metric-selected connection $\mathrm{\nabla }$ and volume form ${\mathrm{v}\mathrm{o}\mathrm{l}}_g$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Oriented smooth manifolds and oriented charts, The metric volume form, current flux through 3-faces and finite-chain charges). Let

$J:=x^1{\mathrm{\partial }}_2$

be the smooth vector field of A smooth vector field is a smooth section of the tangent bundle. Then:

1. ${\mathrm{\nabla }}_a{J}^a={\mathrm{\partial }}_2{x}^1=0$: the current is divergence-free.
2. The current $3$-form is ${\iota }_J{\mathrm{v}\mathrm{o}\mathrm{l}}_g=x^{1}d{x}^0\wedge d{x}^1\wedge d{x}^3$, and $d({\iota }_J{\mathrm{v}\mathrm{o}\mathrm{l}}_g)=0$ identically, consistently with clause (a) of Divergence, Stokes flux balance and the non-null face formula for a Lorentzian current.
3. The pointwise flux density of $J$ through a non-null face with adapted unit normal ${\nu }^{\ast }$ is $\epsilon g(J,{\nu }^{\ast })\sqrt{\mathrm{\mid }\det (g(v_i,v_j))\mathrm{\mid }}$; for the coordinate directions this is $0\text{ for }{\nu }^{\ast }=\pm {\mathrm{\partial }}_0,\pm {\mathrm{\partial }}_1,x^1\text{ for }{\nu }^{\ast }={\mathrm{\partial }}_2,-x^1\text{ for }{\nu }^{\ast }=-{\mathrm{\partial }}_2\mathrm{.}$ Thus the divergence-free current has nonzero flux density through the faces normal to ${\mathrm{\partial }}_2$: vanishing divergence does not mean vanishing flux through every face, and the flux balance ${\mathrm{\Phi }}_J({\mathrm{\Sigma }}_2)-{\mathrm{\Phi }}_J({\mathrm{\Sigma }}_1)=-{\mathrm{\Phi }}_J(B)$ of clause (b) of Divergence, Stokes flux balance and the non-null face formula for a Lorentzian current controls the side boundary genuinely.

• Reading. The example isolates the pointwise flux density: a divergence-free current can have nonzero density through faces normal to ${\mathrm{\partial }}_2$. The flux-balance theorem concerns integrals over oriented chains and states exactly how the slice fluxes differ by the side flux; whether a whole side-chain flux vanishes depends on that chain and its orientation.

Let $M={\mathbb{R}}^4$ carry the identity chart, the flat metric $g=\mathrm{diag}(-1,1,1,1)$ and the time orientation $N={\mathrm{\partial }}_0$, with metric-selected connection $\mathrm{\nabla }$; let $u:={\mathrm{\partial }}_0$, a smooth future-directed unit timelike vector field on $M={\mathbb{R}}^4$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields, A smooth vector field is a smooth section of the tangent bundle). Then:

1. For every smooth $\rho :{\mathbb{R}}^4\to (0,\mathrm{\infty })$ with ${\mathrm{\partial }}_0\rho =0$, the pair $(\rho ,u)$ is a dust model on $M$ whose dust system holds and whose stress-energy tensor is divergence-free.
2. On the open subset $U:=\{x\in {\mathbb{R}}^4:x^0>-1\}$ the pair $(\rho {\mathrm{\mid }}_U,u{\mathrm{\mid }}_U)$ with $\rho (t,x)=1+t$ is a dust model on $U$ whose dust system fails and whose stress-energy tensor has ${\mathrm{\nabla }}^a{T}_{ab}=-{\delta }^0{}_b\mathrm{\ne }0$ on $U$.

• Physical reading. The conserved family is a constant dust congruence with time-independent, possibly spatially varying density in the flat model; the second family is not a solution of the dust system, and the field law of the framework would exclude it as a dust source. Neither statement is an empirical claim about matter.

Let $M={\mathbb{R}}^4$ carry the identity chart, the flat metric $g=\mathrm{diag}(-1,1,1,1)$ and the chart orientation, with metric-selected connection $\mathrm{\nabla }$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Metric compatibility, torsion freedom and the metric-selected connection class, Oriented smooth manifolds and oriented charts). Let $f:\mathbb{R}\to \mathbb{R}$ be a smooth function, let

$\phi (x):=f(x^0+x^1),$

and let $(\phi ,V,K)$ be the scalar-field matter model with $V\equiv 0$ and $K>0$ (The scalar-field matter model of the classical gravitational framework). Then:

1. ${\mathrm{\partial }}^c\phi {\mathrm{\partial }}_c\phi =0$: the metric dual of $d\phi$ is null wherever $d\phi \mathrm{\ne }0$ and is zero wherever $d\phi =0$.
2. ${\mathrm{\square }}_g\phi ={\mathrm{\nabla }}^a{\mathrm{\partial }}_a\phi =0$: $\phi$ satisfies the scalar equation of motion, and its stress-energy tensor is $T_{ab}=K{\mathrm{\partial }}_a\phi {\mathrm{\partial }}_b\phi$.
3. ${\mathrm{\nabla }}^a{T}_{ab}=0$: the tensor is divergence-free, and $T(v,v)=K({\mathrm{\partial }}_a\phi v^a{)}^2\ge 0$ for every $v\in T_{p}M$, with equality exactly on the kernel ${\mathrm{\partial }}_a\phi v^a=0$; this is a hyperplane where $d\phi \mathrm{\ne }0$ and the full tangent space where $d\phi =0$.

• Reading. The example is the flat-space plane-wave branch of the scalar model: the kinetic bracket vanishes identically on a null gradient, so the stress-energy tensor is a nonnegative tensor of rank at most one, with rank one wherever the gradient is nonzero. It illustrates the conservation identity of the proposition without any physical claim about the model's realization.

Let $M={\mathbb{R}}^4$ carry the identity chart, the flat metric $g=\mathrm{diag}(-1,1,1,1)$, the time orientation $N={\mathrm{\partial }}_0$ and the chart orientation, with metric-selected connection $\mathrm{\nabla }$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Metric compatibility, torsion freedom and the metric-selected connection class). Let $\xi :={\mathrm{\partial }}_0$, let $\rho :{\mathbb{R}}^4\to (0,\mathrm{\infty })$ be smooth with ${\mathrm{\partial }}_0\rho =0$, and let $u:={\mathrm{\partial }}_0$, so that $(\rho ,u)$ is a dust model on $M$ with tensor $T^{ab}=\rho u^a{u}^b$ (The dust matter model: rest density, four-velocity, stress-energy and dust system). Then:

1. $\xi ={\mathrm{\partial }}_0$ is a Killing field.
2. The current $J^a:=T^{ab}{\xi }_b$ equals $J=-\rho u$, and ${\mathrm{\nabla }}_a{J}^a=0$.
3. ${\iota }_J{\mathrm{v}\mathrm{o}\mathrm{l}}_g=-\rho d{x}^1\wedge d{x}^2\wedge d{x}^3$: the current $3$-form is (minus) the rest density carried by the constant-$x^0$ slices, and its flux through corresponding regions of two slices is equal because ${\mathrm{\partial }}_0\rho =0$.

• Signs. The minus sign in clauses 2-3 is the sign of the orientation conventions: the adapted unit normal of a constant-$x^0$ slice pointing in the $+{\mathrm{\partial }}_0$ direction is timelike, so the flux formula of the density theorem carries a factor $\epsilon =-1$. Reversing the orientation of the slice or of $M$ flips the sign of the flux but not the conservation statement.

• Reading. The current here is an energy current: the dust coefficient $\rho$ is the rest-energy density, and $J^a=T^{ab}{\xi }_b=-\rho u^a$. Under the physical normalization $U=cu$ and ${\rho }_m=\rho \mathrm{/}{c}^2$, this is $J^a=-c{\rho }_m{U}^a$, not the rest-mass current ${\rho }_m{U}^a$. The example asserts no empirical fact about matter distributions.

Statement refuted: "Let $M={\mathbb{R}}^4$ carry the flat metric $g=\mathrm{diag}(-1,1,1,1)$ and the identity chart, and let $U$ be an open subset. Let $T$ be a smooth symmetric covariant $2$-tensor field on $U$ which satisfies the weak energy condition of The pointwise energy conditions WEC, NEC, DEC and SEC: $T_p(v,v)\ge 0$ for every timelike $v\in T_{p}M$ and every $p\in U$ (Timelike, null, spacelike and causal vectors of a Lorentz vector space). Then $T$ is conserved, ${\mathrm{\nabla }}^a{T}_{ab}=0$ on $U$."

• Source scope. Carroll (4.72) and Tong section 4.5.7 supply the weak-energy-condition definition. The prescribed tensor and its nonconservation are established by the local calculation above; no source passage is cited as asserting this counterexample or the nonimplication.

• No field-equation claim. The counterexample does not assert that $T$ solves the Einstein equation with the flat metric; a model whose field law holds must additionally satisfy the conservation derived in Consistency conditions of the adopted field law: conservation, trace, parameter variation and the energy conditions of a perfect fluid, which this $T$ fails. The point is exactly that the energy condition alone does not supply that consistency.

• Sign conventions. The condition $T(v,v)\ge 0$ is evaluated with the metric convention $g=\mathrm{diag}(-1,1,1,1)$; the inequality $\mathrm{\mid }{v}^1\mathrm{\mid }<\mathrm{\mid }{v}^0\mathrm{\mid }$ used in the proof is the timelike condition of Timelike, null, spacelike and causal vectors of a Lorentz vector space for this convention.

• No empirical content. No measurement or material realization is asserted. The constants $E_{\ast }$ and $L$ set the units of the prescribed test field: every displayed tensor entry has energy-density units, and its nonzero divergence has units $\mathrm{J}{\mathrm{m}}^{-4}$. No numerical measured values are assigned (SI unit and coordinate conventions of the classical gravitational framework).

Statement refuted: "Let $M={\mathbb{R}}^4$ carry the flat metric $g=\mathrm{diag}(-1,1,1,1)$, the identity chart and the chart orientation, and let $J$ be a smooth vector field on $M$ with ${\mathrm{\nabla }}_a{J}^a=0$ on $M$ (The metric volume form, current flux through 3-faces and finite-chain charges, A smooth vector field is a smooth section of the tangent bundle). Then the flux ${\mathrm{\Phi }}_J(\sigma )$ of $J$ through every smooth singular $3$-simplex $\sigma$ of $M$ vanishes."