How does pressure create extra inertia?

UPDATE March 31, 2023: We removed most references to case 1 of the previous blog post.

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The interaction of the gravity field of a test mass, and pressure in nearby matter is a more complex interaction than gravity between two masses. This interaction presumably should add inertia to the test mass. But is the inertia a "package" as in case 1 of our previous blog post, or is it a field effect which is subject to the speed of light?

A test mass within pressurized matter

                            

                  ooooooo

                  ooo • ooo

                  ooooooo

                    •    = test mass m

                 ooo  = matter under pressure

     

The test mass stretches the radial metric around it, adding more space for the pressurized matter to occupy. This is the reason why pressure "generates" gravity. Let us denote by E the energy recovered from the pressure through the stretching of the metric.

                    |

                --- • ---   gravity field lines

                    |

           test mass m

Let us move the test mass. The field moves, and the associated energy E moves with it.

The inertia of E wants to keep the gravity field lines of m static. The field lines bend? Here we (again) encounter the self-force problem. How does the field affect the source of the field? 

Is there any inertia associated with the process where field lines bend?

There may be a significant flow of energy also in the case where the volume does not change but m moves around. The direction of the stretching changes, and the matter orders itself in a new way.

                     •  m

                     |              ooooo

                     |              ooooo   matter

                     |              ooooo

                     |

                     |

                      -----------------> 

Suppose that m moves as shown by the arrow. Energy flows from horizontal stretching to vertical stretching. What route does the energy take? Does it go to m and back to the matter?

We should perform empirical tests. If the field is an electric field and not gravity, experiments might be possible.

The energy associated with pressure probably is not "private" for each test mass

Suppose that our test mass is a symmetric sphere around the matter. The spatial metric inside the matter does not change. It would be strange if pressure would somehow affect a contracting or rotating movement of the sphere.

In this case, the "metric" seems to be different for a pointlike test mass versus a spherical test mass. In our previous blog post we suggested that the gravitational attraction between two particles is their private matter. In the case of pressure, it looks more natural that the interaction is determined by the total field produced by all the particles.

How does general relativity treat pressure?

General relativity has pressure in the diagonal of the stress-energy tensor. If we have a static system, we can calculate a static metric, and pressure affects that metric.

https://en.wikipedia.org/wiki/Geodesics_in_general_relativity

The geodesic hypothesis claims that the movement of a test mass obeys that metric. Thus, in general relativity, pressure should cause a "package" of inertia on a test mass.

However, in this blog we have presented several cases where the geodesic hypothesis fails when there are "tidal" effects. Why should the hypothesis work for pressure?

A general observation about the concept of a "field" or a metric: it only works in simple cases

How do we use the concept the "electric field" of a system C?

In certain situations we can pre-calculate the electric field E of a system C, and then we can use that field to predict how a small test charge c moves. However, if the test charge c interacts in a complex way with the system C, or there is a significant backreaction, then c will not obey the precalculated electric field F.

How does general relativity use the "metric"?

We can solve the metric of a system M from the (mathematically) beautiful Einstein equations. Then we use the geodesic hypothesis and claim that a test mass m moves obeying that metric.

Again, if the interaction of m with M is complex, or if m causes a significant backreaction in M, then it would be a miracle if the metric would predict how m moves.

We conclude that the geodesic hypothesis is bound to fail for many complex systems M.

Conclusions

The interaction of a test mass m with pressure in M might be subject to the speed of light, and the extra inertia would not be felt by a test mass m immediately. As m moves, it changes the metric inside M. Pressure inside M causes energy E to flow inside M according to the moves of m. We can even measure where the energy E is located spatially.

It would be ideal if we could empirically measure the extra inertia caused by pressure. However, we do not think it is possible with current methods. We conjecture that the inertia is not a package, and is subject to the speed of light.