Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 311 / 2 / d / ii

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- Solution ii

Solution

In an orthonormal frame,

F_{0 i} = - E_{i}

and

F_{ij} = ϵ_{ijk} B_{k}

. The electromagnetic stress-energy tensor gives

T_{00} = \frac{E ^{2} + B ^{2}}{8 π}, T_{0 i} = - \frac{( E \times B ) _{i}}{4 π} .

The sign of

T_{0 i}

follows directly from

F_{0 j} F_{i}^{j} = - E_{j} ϵ_{ijk} B_{k}

. Thus

T_{00}^{2} - \sum_{i} T_{0 i}^{2} = \frac{( E ^{2} - B ^{2} ) ^{2} + 4 ( E \cdot B ) ^{2}}{64 π ^{2}} \geq 0,

and

T_{00} \geq 0

. These formulas hold in every local Lorentz frame, so the criterion just proved establishes the dominant energy condition. Equality occurs for a nonzero null Maxwell field with

E^{2} = B^{2}

and

E \cdot B = 0

, whose energy current is null vector.

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