Is it possible to warp space with just light? I have a question, can we make space curved with only light?
I'd be happy if you could help me edit the tags.
 A: No, it is not possible. Warped spacetime is caused by physical objects with rest mass. The stress-energy tensor ${\rm{T}}_{\mu}^{\nu}$ for electromagnetc waves is traceless, i.e. it describes objects with zero rest mass. From Einstein field equation it follows that the trace of Ricci curvature, $\rm{R}\equiv \rm{R}_{\mu}^{\mu}$, is then zero:
\begin{equation}\label{einstein}
{\rm{R}}_{\mu}^{~\nu}-\frac{1}{2}{\rm{R}}_{\lambda}^{~\lambda}~\delta_{\mu}^{\nu}=\kappa~{\rm{T}}_{\mu}^{~\nu},~~~\rm{R}-2~\rm{R}=\kappa~{\rm{T}}_{\mu}^{~\mu}=0.
\end{equation}
As in the case of Schwarzschlid vacuum solution it does not mean that spacetime is flat. The light is excitation of electromagnetic field which in turn is caused by charged particles possessing rest mass (source of warping).
Addendum
After reading some comments I have to admit that my answer could be controversial or even incorrect. Most physicists would say: ‘anything’ that has nonzero stress-energy bends spacetime. This includes light. However, the stress-energy tensor is GR is not simply a source of gravity as the charges and currents are in EM theory. It describes merely matter properties (energy density, stresses and their fluxes) that affect spacetime and vice versa, it represents a kind of interface between them, see Dennis Lehmkuhl General Relativity as a Hybrid theory: The Genesis of Einstein's work on the problem of motion. Furthermore, although light has momentum and energy it has no rest mass. In contrary to matter light does not take up space and is non-localizable. Only if absorbed or emitted the light quanta show their particles properties. In-between they have no defined trajectory. An atom that emits light diminish its rest mass by $\Delta m=h\nu/c^2$. Due to Einstein field equations this process has to be accompanied by emission of gravitational wave at the same time. That is what I understand under light warping spacetime.
