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Prescribed constant observer fields

Physical scope

An exact source-free constant electromagnetic two-form on a prescribed flat Minkowski background, with its electric and magnetic components in one observer tetrad. It verifies only the Maxwell subsystem and makes no claim that the nonzero field and flat metric solve the coupled Einstein equation or represent an observation.

Example

Let $E_{0} \in R$ and $B_{0} \in R$ be constant parameters with units $V m^{- 1}$ and tesla, respectively. On $M = R^{4}$ use the flat Minkowski metric $η = diag (- 1, 1, 1, 1)$ , the time orientation in which $\partial_{0}$ is future-directed, and the orientation for which $d x^{0} \land d x^{1} \land d x^{2} \land d x^{3}$ is positive. Use coordinates $x^{0} = c t, x^{1}, x^{2}, x^{3}$ in metres and the future unit observer $n = \partial_{0}$ . Set

$F : = - \frac{E_{0}}{c} d x^{0} \land d x^{1} + B_{0} d x^{2} \land d x^{3}, j : = 0.$

The covariant components are constant, with $F_{01} = - E_{0} / c$ , $F_{23} = B_{0}$ , and all other independent components zero. The observer electric and magnetic covectors have spatial components $(E_{0}, 0, 0)$ and $(B_{0}, 0, 0)$ , respectively. This is a prescribed-background example of the source-free Maxwell subsystem. It does not assert that the pair $(η, F)$ solves the coupled Einstein equation.

Facts & Assumptions

Given: The chosen flat oriented Minkowski four-manifold, its future unit observer $n = \partial_{0}$ , the constant parameters $E_{0}, B_{0}$ , the smooth two-form $F = - E_{0} d x^{0} \land d x^{1} / c + B_{0} d x^{2} \land d x^{3}$ , the zero current, and the adopted smooth curved-vacuum Maxwell postulate specialized to this flat metric.

[F1]

In an oriented orthonormal frame with signature $(-, +, +, +)$ , a smooth observer has unit timelike vector $n$ , and the observer fields are $E_{a} = c F_{a b} n^{b}$ and $B_{a} = - (⋆_{η} F)_{a b} n^{b}$ ; the local component convention is $F_{0 i} = - E_{i} / c$ , $F_{i j} = ϵ_{i j k} B_{k}$ . (Einstein–Maxwell primitive quantities and SI conventions, Observer electric/magnetic reconstruction and invariants)

[F2]

The exterior derivative of a smooth form has the coordinate formula, and the flat metric's Levi-Civita connection has zero Christoffel symbols in these coordinates. (Lorentzian exterior calculus and Hodge star, Oriented Lorentzian fields, observers and coordinate conventions)

[F3]

In the positively oriented orthonormal coframe, $⋆_{η} (d x^{0} \land d x^{1}) = - d x^{2} \land d x^{3}$ and $⋆_{η} (d x^{2} \land d x^{3}) = d x^{0} \land d x^{1}$ . (Lorentzian exterior calculus and Hodge star)

[A1]

The smooth vacuum model postulates $d F = 0$ and $\nabla_{a} F^{a b} = - μ_{0} j^{b}$ for a smooth field and source. (Smooth curved vacuum Maxwell constitutive adoption)

Verification

1.1givenF1F3algebra

The vector $n = \partial_{0}$ has $η (n, n) = - 1$ and is future-directed, so it is an observer. From $F_{10} = E_{0} / c$ , $F_{20} = F_{30} = 0$ , the definition in [F1] gives $E = (E_{0}, 0, 0)$ . By [F3], $⋆_{η} F = (E_{0} / c) d x^{2} \land d x^{3} + B_{0} d x^{0} \land d x^{1}$ , so $- (⋆_{η} F)_{i 0}$ has components $(B_{0}, 0, 0)$ ; hence $B = (B_{0}, 0, 0)$ .

2.1givenA1F2step 1.1algebra∎

Both coefficients are constant and each coordinate one-form is closed, so the coordinate exterior derivative formula in [F2] gives $d F = 0$ . Every component $F^{a b}$ is constant, and the flat connection vanishes; therefore $\nabla_{a} F^{a b} = \partial_{a} F^{a b} = 0 = - μ_{0} j^{b}$ for each $b$ , since $j = 0$ . Both adopted Maxwell equations in [A1] hold everywhere on this prescribed background.

Prescribed constant observer fields

Example

Facts & Assumptions

Verification

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