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:

$$\frac{\mbox{d}\theta}{\mbox{d}\tau}=-\left(\theta^2+\sigma^2\right)-R_{\mu\nu}v^{\mu}v^\nu$$

With $\theta^2$ and $\sigma^2$ are appropriately defined as $\theta^2=\left(\frac{\theta_\mu^\mu}{3}\right)^2$ and $\sigma^2=\sigma_{\mu\nu}\sigma^{\mu\nu}$ and also, under the assumption that the worldline of the centre of energy is completely time-like. Just for the time being, forget about $\sigma^2$.i.e. say that the object is not under any shearat all. $$\frac{\mbox{d}\theta}{\mbox{d}\tau}=-\theta^2-R_{\mu\nu}v^\mu v^{\nu}$$

Now, for there to be any anti-gravitational effect, $R_{\mu\nu}v^\mu v^{\nu}$ must be positive. Whether or not the worldlines go forward in time, or backwards, $v^\mu v^\nu$ must be negative since it is completely time-like, and thus, $R_{\mu\nu}$ must be negative. Now apply the Einstein Field Equation: $$G_{\mu\nu}=\kappa T_{\mu\nu}$$ $$R_{\mu\nu}=\kappa\left({T_{\mu\nu}-\frac{T}{2}g_{\mu\nu}}\right)$$

Thus, the anti-gravity condition imposes the following constraint: $$T_{\mu\nu}<\frac{T}{2}g_{\mu\nu}$$

Taking the trace of the inequation with respect to the Minkowski metric tensor (for fun, or maybe not really), letting $g=g_\mu^\mu$: $$T<\frac{T}{2}g$$ $$g>2$$

A negative time-time component of the SEM tensor is necessary, which isn't permitted in GR.