Why does four-momentum have to be tangent to the worldine? In relativity, we define momentum for a particle as a vector that is always tangent to the particle's worldline. I am wondering what is the exact reason this is necessary.
For a massive particle of mass $m$ and four-velocity $u^{\mu}$, the four-momentum is $p^{\mu} = mu^{\mu}$. What prohibits us from taking mass to be a tensor, so that $p^{\mu} = m^{\mu}_{\nu}u^{\nu}$? For massless particles, we can consider the deBroglie relation $p^{\mu} = \hbar k^{\mu}$. Again, what prohibits us from having $p^{\mu} = T^{\mu}_{\nu} k^{\nu}$ for some $T^{\mu}_{\nu}$?
The idea is that all collisions would occur in such a way that the four-momentum vector is always conserved. 


*

*Is there some catastrophic consequence that immediately rules out this possibility or is this simply a matter of experiment?

*As far as I can tell, it seems this idea is not compatible with Maxwell's equations, if there is any interaction with the $E$- and $B$-fields. We can assume force is the change in momentum per unit time, so it also has the same weird properties as momentum in this case. If we say $m_{\nu}^{\mu}$ rotates vectors 90 degrees spatially about the $z$-axis and if we say the particle has a charge $q$, then placing such a particle in the $xy$-plane in a radial field $\vec{E} = (q/4\pi\epsilon_{0}r^{2})\hat{r}$ would cause an initial acceleration to some direction perpendicular to $\hat{r}$, and that would violate conservation of energy at the initial moment, because the particle gains kinetic energy without getting closer or further away from the center of the field at that moment. Is there a way to look at Maxwell's equations that makes this violation of energy conservation apparent? 

*The above bullet point applies to particles that have mass. Can something similar be invoked in the case of massless particles with $p^{\mu} = T^{\mu}_{\nu} k^{\nu}$?
An answer to any one of the three bullet points would be appreciated. 
 A: The way I see it is as follows. 
By definition in relativity the mass is the measure of inertia (and equals to gravitational mass). Scalar mass simply means that the motion in 3-D space in whatever direction is the same. In other words, this guarantees invariance of motion in the 3-D space with respect to rotations of the space (SO(3) group). Thus, with respect to the Noether's theorem, the scalar mass will guarantee conservation of angular momentum. 
But the answer to your first question is "no", moreover, in condensed matter theory if we study, say, electron motion in crystal, its mass can be (and usually does) considered as a tensor as a consequence of its motion in a periodic electromagnetic field of the crystallin structure. 
Therefore, the answer to your second question is more "no" than "yes" -- mass tensor in condensed matter is one of the most common things to meet, and it is in agreement with electrodynamics. In the arguments to the second question, the field (as well as the energy) in which the charge is invariant to SO(3) group rotations, thus if we violate this, it will definately cause violation of energy conservation.
As for the last question -- by definition the momentum is a derivative of the Lagrangian with respect to the velocity (${\bf p} = \partial L / \partial {\dot {\bf q}}$); the Noether's theorem require invariance of $L$ with respect to SO(3) to have the angular momentum conserved, if we do not requiere this, I think an appropriate $L$ (in an appropriate metrics) can be constructed.
Using this considerations (especially Noether's theorem) you can easily thinkout the 4-vector momentum in relativity as how physicist usually does, or you can mathematically construct the action of a free particle in Einstein-Minkowski 4-space and build its mechanics as usual.
A: We want velocity vectors to be timelike or null (because otherwise we have problems with causality), and momentum vectors also to be timelike or null (because otherwise the vacuum is unstable with respect to production of pairs of particles whose momenta sum to zero).
This implies that the mass tensor must preserve the light cone. The linear transformations that preserve the light cone are (up to a constant factor) boosts and rotations, so our particle must have some built-in degree of freedom $b$ that describes its boost, or one that specifies its rotation $r$, or, in general, both. 
These $b$ and $r$ are either dynamical properties of the particle or fixed, generic properties of all particles. If they're fixed, then this is of no importance, because it just means that we're using a different frame of reference to describe velocity vectors than for momentum vectors. If they're dynamical, then for a system of particles, interactions will cause the $b$ and $r$ to change, and this will in general result in nonconservation of momentum.
A: If a massive "point" particle has a 4-momentum of the form, $p^\mu=m^\mu{}_\nu v^\nu$, then that particle has preferred directions suggested by the eigenvectors of $m^\mu{}_\nu$. So, this could be ruled out for particles that show no anisotropy.
A: Spinning particles do not always have their 4-velocity  parallel to their 4-momentum. This can be seen by considering their canonical energy momentum tensor  $T^{\mu\nu} = v^\mu p^\mu$. This tensor is not necessarily symmetric. Instead 
$$
T^{\mu \nu}-T^{\nu\mu}= \partial_\lambda S^{\lambda\mu\nu}
$$
where $S^{\lambda\mu\nu}$ is the  instrinsic spin current. So, when the spin is changing we have
$
v^\mu p^\nu\ne v^\nu p^\mu$, and the $p^\mu$ and $v^\mu$ vectors are not parallel.  
This effect is seen in numerical computations of the mergers of spinning black holes  ("bobbing") and one  can make mental  models of extended objects like a relativistic spinning hockey puck in which the origin of this effect can be understood, but it is also true for pointlike elementary particles. A general discussion can be found by reading up on the Matthisson-Papapetrou-Dixon equations which describe the effect of spin on the trajectories of particles in a gravitational field. Particles with instrinsic spin do not move along geodesics.    
