Momentum in special relativity I have been trying to understand absorption of a photon by a massive object, specifically the transfer of momentum and kinetic energy between the two.
Starting from energy-momentum relation  $E^2 = p^2c^2 + m_0^2c^4,$ or rather, for simplicity, setting $c = 1$:
$$E^2 = p^2 + m_0^2,$$
I can't understand how both the energy and momentum can be conserved in case of the full absorption of the photon. If we add the energies of a photon and massive object, then the resulting momentum is clearly not sum of the two momenta. For example, let's set the photon's momentum to be $p_1 = 1$ then photon's energy $E_1 = 1$, and let's set the massive object's momentum $p_2 = 0$ and object's energy to be $E_2 = m_0 = 1$.
By conservation of energy I would expect the final energy to continue to be $E = E_1 + E_2 = 2$.  However, if after the absorption the object has momentum $p_2=0+p_1= 1$, then it has to have energy of $E_2 = \sqrt{p^2 + m_0^2} = \sqrt{2}$, which is not $2$ as I would have expected.
What am I missing?
 A: Congratulations! You have found a very interesting result: a free particle (like an electron, for example) cannot completely absorb a photon, since energy and momentum cannot be simultaneously conserved in such an interaction.
You can see this (as you have) by explicitly calculating the four-momentum before and after the interaction, as you have. Another way is to consider Compton Scattering, which is the general case of the interaction of an electron with a photon. It can be shown that the change in wavelength of the photon will be:
$$\Delta \lambda =  \frac{h}{m_e c}\left( 1 - \cos{\theta}\right) = \lambda_c \left(1-\cos\theta\right),$$
where $\theta$ is the angle between the outgoing and incoming photon, and $\lambda_c$ is called the Compton wavelength of the electron. (It's a nice exercise, you should do it.) Clearly, $\Delta \lambda$ is a bounded quantity, with a maximum when $\theta = \pi$, or when $\Delta \lambda = 2\lambda_c$. In other words, you can never make the photon completely "disappear". (Interestingly, since the Compton wavelength varies inversely as the mass of the particle, the bound becomes tighter for more massive particles.)

It turns out, however, that if the electron isn't free (but is bound, for example, to at atom) then such an absorption can indeed occur. There's a very nice description of it in this answer here. Of course your may ask to same question for such a "composite" particle: from the outside it might "look" as though the atom has "absorbed" the photon.
The reason for this is that atoms (and other such composite particles) have internal structure: they have internal states with different energies, and absorbing a photon changes the atom's internal state. The reason that you can't model an atom as you have in your analysis above is that you haven't taken into account these "internal" degrees of freedom.
