 Solution

Because the condition is local and covariant, choose normal coordinates at the point and a Lorentz frame in which the timelike observer is $X^{\mu }={\delta }_0^{\mu }$. Rescaling a timelike vector by a positive factor does not affect the sign, so it suffices to use this unit vector. With signature $(-,+,\dots ,+)$,The measured energy density is thereforeThe value of $\varphi$, its time derivative, and its spatial first derivatives can be varied independently at a point. Nonnegativity for all such data is therefore equivalent to nonnegativity of all three coefficients:Thus the most general constraints areThese conditions are also sufficient in every timelike frame, because any timelike vector can be brought to the chosen rest frame. This is the weak energy condition for a quadratic scalar stress-energy ansatz.

The conserved values from part (a),satisfy these inequalities for real $m$, as required.