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Consistency conditions of the adopted field law: conservation, trace, parameter variation and the energy conditions of a perfect fluid

Physical scope

Smooth classical domain of the framework in either presentation of the adopted field law: an oriented time-oriented smooth Lorentzian four-manifold, a constant cosmological parameter, a smooth stress-energy tensor on the model's matter domain, and the field equation G_{ab}+Lambda g_{ab}=kappa T_{ab}. The conclusions are conditional consequences of the adopted field law; they assert no existence, no propagation and no empirical fact, and none of the four energy conditions is universally forced by the field equation; particular matter models may imply some, and otherwise they are additional assumptions.

Statement

Adopt one of the two presentations of the framework's field law, that is, Adopted classical gravitational action and variational field law or Adopted Einstein field equation in the direct-equation presentation, for a model with geometric data and matter sector satisfying Adopted spacetime model and classical domain of the gravitational framework and Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions. Let $(M, g, N)$ be the model's smooth time-oriented Lorentzian four-manifold with metric-selected connection $\nabla$ and curvature tensors $R^{a}_{b c d}$ , ${Ric}_{a b}$ , $S$ , $G_{a b}$ (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, The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors), let $T$ be the model's smooth symmetric stress-energy tensor field on the matter domain (Stress-energy tensor field of a model), let $Λ \in R$ be the constant parameter and $κ = 8 π G_{N} / c^{4}$ as in SI unit and coordinate conventions of the classical gravitational framework. Let $U \subseteq M$ be the model's open matter domain and suppose the field law holds on $U$ ,

$G_{a b} + Λ g_{a b} = κ T_{a b} on U .$

(a) Necessary conservation. $\nabla^{a} T_{a b} = 0$ on $U$ . Consequently a prescribed smooth symmetric $T$ with $\nabla^{a} T_{a b} \neq 0$ at some point cannot satisfy the field law there with that proposed metric; the divergence is computed using its metric-selected connection, and may change if the metric is changed.

(b) Trace. On $U$ , $g^{a b} T_{a b}$ satisfies $- S + 4 Λ = κ g^{a b} T_{a b}$ .

(c) Parameter variation. If instead $Λ : U \to R$ is a smooth function and the field law $G_{a b} + Λ g_{a b} = κ T_{a b}$ holds on $U$ with that function in place of a constant, then on $U$

$κ \nabla^{a} T_{a b} = \partial_{b} Λ .$

Consequently, at every point where $d Λ \neq 0$ , the equation requires $\nabla^{a} T_{a b} \neq 0$ ; where $d Λ = 0$ , this identity imposes conservation. A smooth $Λ$ can be locally constant but nonconstant across disconnected components of $U$ , in which case this identity does not force nonconservation.

(d) A vacuum solution. In the global vacuum example take $U = M = R^{4}$ with the identity chart and the flat metric $g_{a b} = diag (- 1, 1, 1, 1)$ (a smooth Lorentzian metric with time orientation $N = \partial_{0}$ ), the field law with $T = 0$ and $Λ = 0$ holds: $G = 0$ and $S = 0$ for the flat metric. With $T = 0$ and $Λ \neq 0$ it fails, since $G + Λ g = Λ g \neq 0$ .

(e) Energy conditions are additional assumptions. Neither adopted presentation of the field law contains a clause imposing any of the weak, null, dominant or strong energy conditions of The pointwise energy conditions WEC, NEC, DEC and SEC; none of the four is a universal consequence of that equation, as the direct-equation perfect-fluid model in the proof shows. Particular matter models may imply energy conditions; when not derived, they are additional assumptions in the sense of clause (iv) of Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions.

(f) Perfect-fluid rest-frame reading. Let $(ε, p, n)$ be a perfect-fluid model on its matter domain, with derived tensor $T = (ε + p) n^{♭} \otimes n^{♭} + p g$ (The perfect-fluid matter model: rest energy density, pressure, four-velocity and stress-energy), and at any $p \in U$ let $(e_{0}, e_{1}, e_{2}, e_{3})$ be a $g_{p}$ -orthonormal basis with $e_{0} = n_{p}$ . Then $T_{A B} = diag (ε, p, p, p)$ in that basis, and at $p$

$T satisfies WEC ⟺ ε \geq 0 and ε + p \geq 0,$

$T satisfies NEC ⟺ ε + p \geq 0, T satisfies DEC ⟺ ε \geq ∣ p ∣,$

$T satisfies SEC ⟺ ε + 3 p \geq 0 and ε + p \geq 0.$

In particular a model with $ε > 0$ and $p = - ε$ satisfies the NEC, WEC and DEC and violates the SEC; the inequality signs here are those of the strict-timelike convention of The pointwise energy conditions WEC, NEC, DEC and SEC, under which continuity extensions to the causal cone are not assumed.

Facts & Assumptions

Given: The adopted field law in one of its two presentations on the open matter domain $U$ ; a model with smooth $(M, g, N)$ , smooth symmetric $T$ on $U$ , constant $Λ$ and coupling $κ = 8 π G_{N} / c^{4}$ ; in (f) additionally a perfect-fluid model on $U$ .

[L1]

The Einstein tensor of the metric-selected connection satisfies $\nabla^{a} G_{a b} = 0$ , and $g^{a b} G_{a b} = - S$ with $G_{a b} = {Ric}_{a b} - \frac{1}{2} S g_{a b}$ (The second Bianchi identity and the vanishing divergence of the Einstein tensor, The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors).

[L2]

$\nabla g = 0$ and $\nabla g^{- 1} = 0$ ; the extension of $\nabla$ to tensor fields obeys the product rule, and for a smooth function $f$ , $\nabla^{a} (f g_{a b}) = \partial_{b} f$ (Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields, Metric compatibility, torsion freedom and the metric-selected connection class).

[L3]

The flat metric on the identity chart of $R^{4}$ has vanishing Christoffel symbols, curvature and scalar curvature; the identity chart metric is smooth, symmetric and Lorentzian of signature $(-, +, +, +)$ , and $N = \partial_{0}$ is a time orientation (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors, Tensor fields of type (r,s) on a Lorentzian four-manifold and their components).

[L4]

An observer $n_{p}$ with $g_{p} (n_{p}, n_{p}) = - 1$ has a positive definite three-dimensional rest space $n_{p}^{⊥}$ ; choosing a $g_{p}$ -orthonormal basis of $n_{p}^{⊥}$ by Gram–Schmidt and adjoining $e_{0} : = n_{p}$ gives a $g_{p}$ -orthonormal basis $(e_{0}, e_{1}, e_{2}, e_{3})$ of $T_{p} M$ (Observers, rest spaces and the spatial projection at an event, The observer rest-space decomposition and the spatial projection, Gram–Schmidt turns every finite independent list into an orthonormal list with the same successive spans).

[L5]

The four energy conditions are the pointwise inequalities of The pointwise energy conditions WEC, NEC, DEC and SEC: WEC $T (v, v) \geq 0$ for timelike $v$ ; NEC $T (v, v) \geq 0$ for null $v$ ; DEC $- ♯ (T (v, \cdot)) \in cl (C_{p}^{+})$ for future timelike $v$ ; SEC $(T - \frac{1}{2} ({tr}_{g} T) g) (v, v) \geq 0$ for timelike $v$ .

[L6]

The coordinate curvature formula of Tensoriality of the curvature, its coordinate formula and its point-normal metric formula clause (b) applies to every smooth Lorentzian metric with its metric-selected connection; contraction gives ${Ric}_{b d} = R^{a}_{b a d}$ , $S = g^{b d} {Ric}_{b d}$ and $G = Ric - \frac{1}{2} S g$ .

Proof

technique · direct; differentiate the field equation using the contracted Bianchi identity, contract traces, test the flat metric, and evaluate the quadratic inequalities of a rest-frame diagonal tensor

1.1L1L2algebra

(Conservation and trace.) Applying $\nabla^{a}$ to the field equation and using [L1] and [L2]: $κ \nabla^{a} T_{a b} = \nabla^{a} G_{a b} + Λ \nabla^{a} g_{a b} = 0$ , so $\nabla^{a} T_{a b} = 0$ since $κ > 0$ ; this is (a). Contracting the field equation with $g^{a b}$ and using [L1]: $g^{a b} (G_{a b} + Λ g_{a b}) = - S + 4 Λ = κ g^{a b} T_{a b}$ , which is (b).

1.2L1L2algebra

(Function parameter.) If $Λ$ is a smooth function and the equation holds, then by [L1] and [L2], $κ \nabla^{a} T_{a b} = \nabla^{a} (G_{a b} + Λ g_{a b}) = \partial_{b} Λ$ , which is (c).

1.3L3algebra

(Flat solution.) By [L3], the flat metric has $Γ = 0$ , hence $R = 0$ , $Ric = 0$ , $S = 0$ and $G = 0$ ; with $T = 0$ and $Λ = 0$ the field equation reads $0 = 0$ , while for $Λ \neq 0$ it would require $Λ g_{a b} = 0$ , which fails because $g$ is nondegenerate; this is (d).

1.4L4algebra

(Rest-frame components.) By [L4] choose the orthonormal basis with $e_{0} = n_{p}$ and write $T_{A B} = T (e_{A}, e_{B})$ and let $e^{A}$ be the dual covectors, $e^{A} (e_{B}) = δ^{A}_{B}$ . From $T_{a b} = (ε + p) n_{a} n_{b} + p g_{a b}$ (The perfect-fluid matter model: rest energy density, pressure, four-velocity and stress-energy) and $n = n_{p} = e_{0}$ : $T_{00} = (ε + p) (- 1)^{2} + p (- 1) = ε$ , $T_{0 i} = 0$ , $T_{i j} = p δ_{i j}$ ; so $T_{A B} = diag (ε, p, p, p)$ . For a vector $v = v^{0} e_{0} + v$ with $v = \sum_{i \geq 1} v^{i} e_{i}$ , $T (v, v) = ε (v^{0})^{2} + p ∣ v ∣^{2}$ .

2.1step 1.4L5algebra

(WEC and NEC.) A vector $v \neq 0$ is timelike exactly when $(v^{0})^{2} > ∣ v ∣^{2}$ and null exactly when $(v^{0})^{2} = ∣ v ∣^{2}$ with $v \neq 0$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model). Writing $s : = ∣ v ∣ / ∣ v^{0} ∣ \in [0, 1)$ for timelike $v$ , $T (v, v) = (v^{0})^{2} [ε + p s^{2}]$ . This is nonnegative for all such $v$ if and only if $ε \geq 0$ (take $v = 0$ ) and $ε + p \geq 0$ (let $s \to 1$ , i.e. approach the null cone); conversely these inequalities make every value nonnegative. For null $v$ , $T (v, v) = (v^{0})^{2} (ε + p)$ , so the NEC is exactly $ε + p \geq 0$ .

2.2step 1.4L5algebra

(DEC.) By [L5] and [L2], the DEC requires $w : = - ♯ (T (v, \cdot))$ to lie in the closed future cone, including zero, for every future timelike $v$ . In the orthonormal basis, $T (v, \cdot) = ε v^{0} e^{0} + p \sum_{i} v^{i} e^{i}$ and $♯$ raises with $diag (- 1, 1, 1, 1)$ , so $w = ε v^{0} e_{0} - p \sum_{i} v^{i} e_{i}$ . Since $v$ is future and $e_{0} = n_{p}$ is the future unit normal of the rest frame, $v^{0} > 0$ . By The pointwise energy conditions WEC, NEC, DEC and SEC and Classification of future and past causal vectors by a timelike field, $w$ lies in $cl (C_{p}^{+})$ exactly when $w = 0$ or $w$ is future-directed and causal; for $w = ε v^{0} e_{0} - p \sum_{i} v^{i} e_{i}$ this is exactly the pair of conditions $ε v^{0} \geq 0$ and $(ε v^{0})^{2} \geq p^{2} ∣ v ∣^{2}$ . Requiring them for all future timelike $v$ forces $ε \geq 0$ (take $v = 0$ , so that $w = ε e_{0}$ must be $0$ or future causal) and $ε^{2} \geq p^{2}$ , that is $ε \geq ∣ p ∣$ (let $∣ v ∣ / v^{0} \to 1$ ); conversely $ε \geq ∣ p ∣$ gives both conditions. This is the DEC equivalence.

3.1step 1.4step 2.1L5algebra

(SEC.) By [L5], with ${tr}_{g} T = g^{a b} T_{a b} = - ε + 3 p$ (from step 1.4), the tensor $T - \frac{1}{2} ({tr}_{g} T) g$ has rest-frame components $\frac{ε + 3 p}{2}$ in the $(0, 0)$ slot and $\frac{ε - p}{2} δ_{i j}$ in the spatial slots, so its value on $v = v^{0} e_{0} + v$ is $\frac{(v^{0})^{2}}{2} [(ε + 3 p) + (ε - p) s^{2}]$ with $s = ∣ v ∣ / ∣ v^{0} ∣ \in [0, 1)$ . This is nonnegative for every $s \in [0, 1)$ exactly when $ε + 3 p \geq 0$ (at $s = 0$ ) and $2 (ε + p) \geq 0$ (letting $s \to 1$ ), which is the SEC equivalence; for $ε > 0$ , $p = - ε$ the conditions NEC, WEC and DEC hold with the equalities $ε + p = 0$ and $ε = ∣ p ∣$ , while SEC fails since $ε + 3 p = - 2 ε < 0$ .

4.1step 2.1step 2.2step 3.1L1L2L6givenconstructalgebra∎

(Energy conditions are not universal consequences.) The absence of inequalities from the adopted law is a formulation fact; nonimplication also needs a model. On $M = R^{4}$ with coordinates $(t, x^{1}, x^{2}, x^{3})$ , choose $L > 0$ , put $a (t) = \exp ((t / L)^{2})$ and $H = a^{'} / a = 2 t / L^{2}$ , and set $g = - d t^{2} + a^{2} \sum_{i} (d x^{i})^{2}$ , $N = n = \partial_{t}$ , $Λ = 0$ . This metric is smooth Lorentzian and $n$ is future unit timelike. Its only nonzero Christoffel symbols are $Γ^{0}_{i j} = a a^{'} δ_{i j}$ and $Γ^{i}_{0 j} = Γ^{i}_{j 0} = H δ^{i}_{j}$ . Substitution into the coordinate curvature formula of Tensoriality of the curvature, its coordinate formula and its point-normal metric formula and contraction give ${Ric}_{00} = - 3 (H^{'} + H^{2})$ , ${Ric}_{0 i} = 0$ , ${Ric}_{i j} = a^{2} (H^{'} + 3 H^{2}) δ_{i j}$ , hence $S = 6 H^{'} + 12 H^{2}$ , $G_{00} = 3 H^{2}$ , $G_{i j} = - a^{2} (2 H^{'} + 3 H^{2}) δ_{i j}$ . Choose the prescribed perfect fluid $ε = 3 H^{2} / κ$ , $p = - (2 H^{'} + 3 H^{2}) / κ$ and $T = (ε + p) n^{♭} \otimes n^{♭} + p g$ . Then $G = κ T$ , and its fluid equations hold: acceleration is zero, $ϑ = 3 H$ , spatial pressure gradient is zero, and $D ε + (ε + p) ϑ = 6 H H^{'} / κ - 6 H H^{'} / κ = 0$ . Thus it is an admissible direct-equation model, with no energy condition additionally adopted. At $t = 0$ , $ε = 0$ and $p = - 4 / (κ L^{2}) < 0$ ; steps 2.1–3.1 show that WEC, NEC, DEC and SEC all fail there. This proves (e) without inferring nonimplication merely from the wording of a postulate. In length coordinates $L$ has dimension $m$ , $H$ dimension $m^{- 1}$ , and $ε, p$ have the required energy-density units; the construction reports no observation.

Remarks

What is not proved. The theorem does not assert that any matter model satisfies any energy condition, that solutions of the field equation exist for prescribed data, or that a given solution propagates or is unique; those are separate mathematical problems.
Empirical status. The energy conditions and the field law are adopted assumptions; experimental tests compare predictions of a fully specified model with reported observations and do not prove or refute the postulate deductively.
No empirical premises. No measured input enters this item, so no empirical uncertainty is inherited by its conclusions.

Consistency conditions of the adopted field law: conservation, trace, parameter variation and the energy conditions of a perfect fluid

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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