Why don't we denote presence of matter and energy in space-time (as we do for presence of electric charge) by a 4-vector field?
Is there any information in the stress-energy tensor that is not in 4-vector field of current of energy and matter? As I think there is a current of energy and matter about us very similar to current of charges, and we denote this by 4-vector field $J$.
I hope there is explanation as this example: if someone ask me why we don't denote the presence of charge about us just by scalar field of charge density, then I say charge density of course denote the presence of charge but it neglects current of charges and there are physical phenomenons that arise not from only charges but also from their motion.
Best Answer
Because energy is not Lorentz-invariant, whereas electric charge is. Therefore, when we compute how charge density and its flux transform to assert that charge density together with its flux is a four vector, we only need to account for the fact that different observers see the partitions of spacetime used to calculate the density and its flux differently, they do not see the charges themselves differently. The Lorentz covariance of the 4-current is equivalent to an assertion that electric charge is a Lorentz-invariant scalar.
In contrast, energy is not a Lorentz covariant scalar. It is united with momentum in the 4-momentum. So when we calculate the density of this entity, we need to account for the fact that both the underlying entity and the spacetime partitioning used to calculate a density change for different observers. We are computing the density and flux of a rank-1 tensor, not of a scalar.
This is one reason why Gravitoelectromagnetism doesn't work, although it gives pretty accurate calculations in many relativistic cases. It assumes, like Maxwell's equations, that the gravity field source is a rank 1 tensor - the mass/energy density and its flux - but this is flawed because mass / energy is not Lorentz invariant. Gravitoelectromagnetism gives the wrong values of gravitational radiation - the Larmor formula and its relativistic equivalent foretell radiation quantities that are far too big (falsified by observations of the Hulse-Taylor binary star, for example).
I think I've answered this one. Essentially the rank 2 stress energy tensor accounts for the Lorentz transformation of both the underlying quantities whose density is being calculated as well as the Lorentz transformation of the spacetime partitions used to calculate the density: a rank 1 four vector can only encode the Lorentz transformation of the latter.