why pressures in opposite directions don't cancel out effects of each other in stress-energy tensor? here is how I think: pressure can cause to flow of momentum without flow of matter. a simple example is newton's cradle. now consider pressure witch by same mechanism and intercepting of many particles. consider perfect fluid at rest. at an event in space-time flow of momentum in all directions is same and I think so the net flow of 4-momentum must be zero, but, $T$  don't give zero and I confused why is so.
 A: I think this is basically covered in Intuitive understanding of the elements in the stress-energy tensor, but maybe it's worth specifically focussing on the pressure terms in the stress-energy tensor.
If we consider our system to be made up from point particles then the diagonal terms look like:
$$ T^{ii} = \sum \gamma m (v^i)^2 \delta(x - x^i) $$
where $v^i$ is the coordinate velocity (not the four-velocity) of the point particle of mass $m$. The $\delta(x - x^i)$ term just makes the contribution to the stress-energy tensor zero everywhere apart from the position of the particle. The $\gamma m (v^i)^2$ term is basically a relativistic kinetic energy, and the kinetic energy of the particles in a gas is proportional to the pressure, which is why the diagonal terms are a pressure.
The contributions from the individual particles don't cancel because they sum $v^2$, and that is always positive regardless of the direction of $\mathbf v$. But what I want to do here is offer a simple intuitive reason why the diagonal terms can't cancel out.
Suppose we consider some ball of gas and the gravitational field it generates. If we want to increase the pressure we need to increase the particle kinetic energies, and that means increasing the temperature. So we have to add energy (as heat) to the system. But in the rest frame of the ball of gas the mass and energy are related by Einstein's famous equation:
$$ E = mc^2 $$
So if we have added energy we must have increased the mass, and that will have increased the gravitational field. So the increased velocities of the particles cannot cancel out.
