Special relativity and moving pressure

In our August 28, 2024 blog post we were confused about if the horizontal Maxwell stress tensor pressure in the sum field E + E' "moves" with the field E', along with the moving rod.

What does a "moving pressure" mean in special relativity?

In the Lorentz transformation of the stress-energy tensor, a positive pressure to the direction of the movement shows up as a positive energy density and as positive momentum. For negative pressure, the signs are flipped.

A moving pressure is something where the action and the reaction are simultaneous in a moving frame

                positive pressure

                       <---------->

                M ●                 ● M

                       >----------<

               negative pressure

 

                             ---> v

If the system is static, but moving, then positive and negative pressures cancel each other out, and we can ignore pressures if we are concerned with the entire system.

If we remove the negative pressure, what happens?

The positive pressure shows up as momentum to the right in the stress-energy tensor, and also as energy. Does this make sense?

Simultaneousness is different in the laboratory frame relative to the moving frame. Let there be a short positive "pressure pulse" in the moving frame. An observer in the laboratory frame sees the mass M on the left to accelerate to the left before he sees the mass M on the right to accelerate. The momentum and the energy in the pressure is "buffered", and will eventually be seen in M on the right.

We conclude that in this case, momentum and energy contained in pressure makes a lot of sense.

The force field between the two M is "moving" along with them. The action and the reaction are simultaneous in the moving frame.

A static charge Q and a moving charge Q

          ●                   ● ---> v  

          Q                  Q

Does the electric field in this case "buffer" momentum, if looked at in the laboratory frame? Yes, of course. In the frame moving at the velocity v / 2 to the right there is no buffering, because of symmetry. In other frames, there is buffering.

Momentum conservation would be broken if we would not assign some momentum to the field.

          ● ---> v            ●

          Q                     Q

In the configuration above, we, similarly, see that the field does not buffer any momentum in a frame which moves at the speed v / 2 to the right.

But the field does buffer momentum to the right if looked at in the laboratory frame.

In this specific case, the "pressure" between the various Q is "moving" to the right at the speed v / 2.

Adding Poincaré stresses

        =========================   rigid rod

           ●        ● ---> v        -F <---- ●  ----> F

           Q       Q                              Q

                                                 

Let us imagine that the leftmost and the rightmost Q are attached to a rigid rod. The rod provides the Poincaré stresses for them.

If we want to keep the those two Q exactly static, we have to imitate the forces F exerted by the middle Q, but flip the sign of the force.

We could hypothetically create such forces by adding a new interaction, such that the middle Q pulls the other two Q. The moving pressure for the new interaction is the same as for the electric repulsion, but the sign flipped. In this case we can claim that the negative Poincaré stress pressure exactly cancels out the positive pressure.

Conjecture. The Poincaré stresses which are required to cancel out the pressure from an electric field, are "moving" pressures, and their contribution cancels out the momentum and the energy contribution of the pressure of the electric field.

Conclusions

We now understand what is the role of a "moving" pressure in special relativity. It has to contribute to the momentum and the energy in the stress-energy tensor, to preserve conservation of momentum and energy. Otherwise, the difference of simultaneousness in moving frames would break conservation laws.