The Bakamjian-Thomas relativistic quark model provides a Poincaré representation of bound states with a fixed number of constituents and, in the heavy quark limit, form factors of currents satisfy covariance and Isgur-Wise scaling. We compute the light cone distribution amplitudes (LCDA) of $B$ mesons ${\phi }_{\pm }^B(\omega )$ as well as the shape function $S(\omega )$, that enters in the decay $B\to X_s\gamma$, that are also covariant in this class of models. The LCDA and the shape function are related through the quark model wave function. The former satisfy, in the limit of vanishing constituent light quark mass, the integral relation given by QCD in the valence sector of Fock space. Using a Gaussian wave function, the obtained $S(\omega )$ is identical to the so-called roman shape function. From the parameters for the latter that fit the $B\to X_s\gamma$ spectrum we predict the behavior of ${\phi }_{\pm }^B(\omega )$. We discuss the important role played by the constituent light quark mass. In particular, although ${\phi }_{-}^B(0)\ne 0$ for vanishing light quark mass, a nonvanishing mass implies the unfamiliar result ${\phi }_{-}^B(0)=0$. Moreover, we incorporate the short distance behavior of QCD to ${\phi }_{+}^B(\omega )$, which has sizeable effects at large $\omega$. We obtain the values for the parameters $\overline{\Lambda }\cong 0.35\mathrm{GeV}$ and ${\lambda }_B^{-1}\cong 1.43{\mathrm{GeV}}^{-1}$. We compare with other theoretical approaches and illustrate the great variety of models found in the literature for the functions ${\phi }_{\pm }^B(\omega )$; hence the necessity of imposing further constraints as in the present paper. We briefly review also the different phenomena that are sensitive to the LCDA. The value that we find for ${\lambda }_B^{-1}$ fulfills the upper bound recently measured by BABAR.

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