Instead of going into more recent developments, which others, such as Dilaton already have, I will discuss the general relativistic perspective on this. So, start with the Raychaudhuri Equation:
dθdτ=−(θ2+σ2)−Rμνvμvν
With θ2 and σ2 are appropriately defined as θ2=(θμμ3)2 and σ2=σμνσμν and also, under the assumption that the worldline of the centre of energy is completely time-like. Just for the time being, forget about σ2.i.e. say that the object is not under any shearat all. dθdτ=−θ2−Rμνvμvν
Now, for there to be any anti-gravitational effect, Rμνvμvν must be positive. Whether or not the worldlines go forward in time, or backwards, vμvν must be negative since it is completely time-like, and thus, Rμν must be negative. Now apply the Einstein Field Equation: Gμν=κTμν
Rμν=κ(Tμν−T2gμν)
Thus, the anti-gravity condition imposes the following constraint: Tμν<T2gμν
Taking the trace of the inequation with respect to the Minkowski metric tensor (for fun, or maybe not really), letting g=gμμ: T<T2g
g>2
A negative time-time component of the SEM tensor is necessary, which isn't permitted in GR.