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The box balance identity checked on an explicit stress-energy field

Example

In an inertial chart $x$ on affine Minkowski space (Inertial frames and inertial charts of the affine Minkowski model, Oriented time-oriented affine Minkowski space (the spacetime primitives)) define the smooth symmetric type- $(2, 0)$ field by $T^{00} = x^{0}$ , $T^{01} = T^{10} = x^{1}$ , $T^{11} = x^{0}$ , with every other component zero. This example computes its force density and checks the finite-box balance identity of Finite-box stress balance and the flux form of conservation on an explicit box with a nonzero net side flux.

Facts & Assumptions

Given: The inertial chart with constant metric $η = diag (- 1, 1, 1, 1)$ (Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space), the coordinate partial derivatives $\partial_{a}$ (Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ), the field $T$ above, and real numbers $t_{1} < t_{2}$ , $x_{1} < x_{2}$ , $c_{2} < d_{2}$ , $c_{3} < d_{3}$ ; set $A : = (d_{2} - c_{2}) (d_{3} - c_{3}) > 0$ and $B : = [t_{1}, t_{2}] \times [x_{1}, x_{2}] \times [c_{2}, d_{2}] \times [c_{3}, d_{3}]$ .

[F1]

The box identity of the finite-flux theorem: for a $C^{1}$ symmetric field $T$ on a neighbourhood of $B$ , $\int_{B} f^{b} d^{4} x = \sum_{a} (\int_{F_{a}^{+}} T^{a b} d^{3} x - \int_{F_{a}^{-}} T^{a b} d^{3} x)$ with $f^{b} = \partial_{a} T^{a b}$ and $F_{a}^{\pm}$ the faces $x^{a} = d_{a}, c_{a}$ of $B$ . (Finite-box stress balance and the flux form of conservation,(a), Stress-energy tensor, force density and local balance on flat spacetime)

[F2]

Coordinate derivatives: $\partial_{0} x^{0} = 1$ , $\partial_{1} x^{1} = 1$ , $\partial_{0} x^{1} = \partial_{1} x^{0} = 0$ , and partial derivatives of a function of one coordinate vanish in the other coordinates, as one-variable derivatives of the coordinate restrictions. (Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ )

[F3]

Riemann integrals of functions of a single coordinate over a box factor as the product of the one-dimensional integral and the transverse volume; constant integrals equal the constant times the volume. (Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections, The lower and upper Darboux integrals over a nondegenerate rectangle in $R^{m}$ )

Verification

technique · direct

1.1F2F3given

Force density. By [F2], $f^{0} = \partial_{0} T^{00} + \partial_{1} T^{10} = 1 + 1 = 2$ , $f^{1} = \partial_{0} T^{01} + \partial_{1} T^{11} = 0 + 0 = 0$ and $f^{2} = f^{3} = 0$ ; hence $\int_{B} f^{0} d^{4} x = 2 vol (B) = 2 A (t_{2} - t_{1}) (x_{2} - x_{1})$ by [F3], and the $b = 1, 2, 3$ bulk integrals vanish.

2.1F1F3step 1.1

The $b = 0$ identity. The temporal faces contribute $\int_{F_{0}^{+}} T^{00} - \int_{F_{0}^{-}} T^{00} = A \int_{x_{1}}^{x_{2}} x^{0} ∣_{t_{2}} d x^{1} - A \int_{x_{1}}^{x_{2}} x^{0} ∣_{t_{1}} d x^{1} = A (t_{2} - t_{1}) (x_{2} - x_{1})$ ; the faces with $a = 1$ contribute $\int_{F_{1}^{+}} T^{10} - \int_{F_{1}^{-}} T^{10} = A \int_{t_{1}}^{t_{2}} x^{1} ∣_{x_{2}} d x^{0} - A \int_{t_{1}}^{t_{2}} x^{1} ∣_{x_{1}} d x^{0} = A (t_{2} - t_{1}) (x_{2} - x_{1})$ ; the transverse faces vanish because $T^{2 b} = T^{3 b} = 0$ . The sum of face differences is $2 A (t_{2} - t_{1}) (x_{2} - x_{1})$ , equal to the bulk integral of step 1.1, verifying [F1] for $b = 0$ .

2.2F1step 1.1

The $b = 1$ identity and the remaining components. For $b = 1$ , the temporal faces contribute $\pm A \int_{x_{1}}^{x_{2}} x^{1} d x^{1}$ with opposite signs, which cancel, and the faces with $a = 1$ contribute $\pm A \int_{t_{1}}^{t_{2}} x^{0} d x^{0}$ , which also cancel; the transverse faces vanish. Both sides of [F1] vanish, and the same is immediate for $b = 2, 3$ because all relevant components are zero.

3.1step 1.1step 2.1step 2.2∎

Numerical specialization: for transverse area $A = 6$ , temporal length $t_{2} - t_{1} = 2$ and axis-one length $x_{2} - x_{1} = 3$ , the bulk integral is $2 \cdot 6 \cdot 2 \cdot 3 = 72$ and the two nonzero face differences are each $6 \cdot 2 \cdot 3 = 36$ , whose sum is $72$ . The example verifies the finite-box identity on a field with nonzero force density and nonzero net side flux; it asserts no matter model, no dynamics and no empirical fact.

The box balance identity checked on an explicit stress-energy field

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels