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For the Monge-gauge Hamiltonian of a three-dimensional string, in Monge gauge, $X^{\prime m}=(0,1,{\Phi }^{\prime })$ and $P_m=(P_0,P_1,\Pi )$. The momentum constraint gives $P_1=-{\Phi }^{\prime }\Pi$. The other Virasoro constraint then readsChoosing the positive-energy branch,Substituting into the phase-space action leaves $I=\int dtd\sigma (\Pi \dot{\Phi }-\mathcal{H})$, soFor an infinite string this is an energy-density expression; even the undeformed string has infinite total energy. One may regulate the length or subtract its constant reference energy when a finite total energy is needed. The positive branch is unambiguous at the density level.

The canonical Hamiltonian field equations areConsequently $\Pi ={\Phi }^{\prime }$ impliesThe second equation also becomes $\dot{\Pi }={\Phi }^{\prime \prime }$, consistent with differentiating the first. These are one-direction traveling profiles along the string.

For the linear profile, the spatial string is the straight line $X^2=k{X}^1+kt$. It is tilted, and its coordinate intercept moves with speed $k$. This is also a uniformly moving straight string: the physical transverse velocity of a tilted string is the velocity normal to the line. With unit tangent $(1,k)/\sqrt{1+k^2}$ and coordinate velocity $(0,k)$, the normal speed isThus $k>1$ does not imply superluminal physical motion. Fixed-$\sigma$ labels have spacelike trajectories when $k>1$, but labels on a string can slide tangentially and are not identifiable material particles. Choosing $\dot{\sigma }=-k^2/(1+k^2)$ removes the tangential velocity and gives the subluminal velocity $(-k^2,k)/(1+k^2)$.

The induced worldsheet metric supplies a coordinate-invariant check:The string worldsheet remains Lorentzian for every finite $k$, including $k=1$ and $k>1$. Its constant energy density is $1+k^2$. This linear profile is a tilted boosted straight string, rather than a localized finite-energy wave.