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

The dominant energy condition says that, for every future timelike vector $u^a$, the energy current $J^a=-T^a{}_b{u}^b$ is future causal or zero. Choose normal coordinates whose future unit time vector at $p$ is $u^a=(1,0,0,0)$. With metric signature $(-,+,+,+)$ and symmetric stress-energy tensor,A vector with these components is future causal or zero exactly whenThis proves necessity. Conversely, every future unit timelike vector can be made the time axis of an orthonormal frame by a proper orthochronous Lorentz transformation, and that frame can be extended to normal coordinates at $p$. Requiring the displayed inequality in every such chart therefore gives the dominant energy condition for every future timelike vector; positive rescaling handles nonunit vectors. Future null vectors follow by continuity if they are included in the definition. The condition must hold in every local Lorentz frame; one chart alone is insufficient.

In an orthonormal frame, $F_{0i}=-E_i$ and $F_{ij}={ϵ}_{ijk}{B}_k$. The electromagnetic stress-energy tensor givesThe sign of $T_{0i}$ follows directly from $F_{0j}{F}_i{}^j=-E_j{ϵ}_{ijk}{B}_k$. Thusand $T_{00}\ge 0$. 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 $\mathbf{E}\cdot \mathbf{B}=0$, whose energy current is null vector.

For the P(X, phi) scalar field theory specialized to $F(X)$, vary the inverse metric tensor while holding $\varphi$ fixed. With $q_a={\partial }_a\varphi$,Using $T_{ab}=-2(-g{)}^{-1/2}\delta S/\delta g^{ab}$ yieldsIn any local Lorentz frame, set $A=F^{\prime }(X)\ge 0$, $Y=X{F}^{\prime }(X)-F(X)\ge 0$, $s={\sum }_i{q}_i^2$, and $X=(q_0^2-s)/2$. ThenA direct calculation givesHence $\rho \ge \sqrt{T_{0i}{T}_{0i}}$ in every local Lorentz frame, proving the dominant energy condition. This proof covers timelike, null, spacelike, and zero $\nabla \varphi$ alike; it does not assume $X>0$ or use a perfect fluid in general relativity rest frame. The hypotheses need only hold at every attained value of $X$; the stated all-$X$ assumptions are sufficient. For example, $F=X-V_0$, $V_0\ge 0$, satisfies them.